MD Nastran R3 Explicit Nonlinear (SOL 700) User’s Guide

MD Nastran R3

Explicit Nonlinear (SOL 700) User’s Guide

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C o n t e n t sMD Nastran Explicit Nonlinear (SOL 700) User’s Guide

Contents

1 Introduction

MD Nastran Explicit Nonlinear (SOL 700) 16Defining the Model 16

Implicit and Explicit Methods 17Implicit Methods 17Explicit Methods 18Typical Applications 24

Principles of the Eulerian and Lagrangian Processors 24

Feature List 26

How SOL 700 Solves Explicit Problems 27Input and Output 27

This User’s Guide 28Additional Documentation for SOL 700 28

2 MD Nastran Data Files

The MD Nastran Input Data File 32Input Conventions 33Section Descriptions 33Running Existing Models with SOL 700 34

Supported SOL 700 Entries 35Supported Case Control Cards 35Supported Bulk Data Entries 37Supported Material Models in SOL 700 50Unsupported Materials in MD Patran Preference 50Supported Parameters in SOL 700 50Supported Parameters in MD Patran Preference 51

Preprocessing with MD Patran 51Generating the Bulk Data File 51Editing the Bulk Data File 52

MD Nastran Explicit Nonlinear (SOL 700) User’s Guide4

Output Requests 52Files Created by SOL 700 52

Postprocessing with MD Patran 55

3 Modeling

Coordinate Systems 58User-defined Coordinate System 58Nodal Coordinate Systems 59Element Coordinate Systems 59Material Coordinate Systems 60

Nodes 61Degrees of Freedom 61

Elements 62

Modeling in MD Patran 63Creating Geometry in MD Patran 63Creating Finite Element Meshes in MD Patran 65

Example using MD Patran 67Description of the Problem 67Solution Type 68Specifying the Solution Type 68References 68Defining the Solution Type in MD Patran 69

Input and Output Files Created During the Simulation 79Simulation on Windows Platforms 79

Postprocessing 81

Running a Batch Job 85

How to Tell When the Analysis is Done 85

How to Tell if the Analysis Ran Successfully 85Running a Parallel Job 86

5Contents

4 Special Modeling Techniques

Artificial Viscosity 90Bulk Viscosity 90

Hourglass Damping 91

Mass Scaling 95Problems Involving a Few Small Elements 95Problems Involving a Few Severely Distorted Elements 95

Time Domain NVH 97Time Domain NVH Example – Plate Subjected to a Pulse Loading 98

Prestress (Implicit to Explicit Sequential Simulation) 107Comparative Study between Nastran and MD Nastran SOL 700 Implicit

Element Formulations 107

5 Constraints and Loadings

Constraint Definition 116Single-Point Constraints 116Multi-Point Constraints 116Specifying Explicit MPCs 117Contact in SOL 700 117Contact Bodies 118Rigid Walls 119Contact Detection 119

Lagrangian Loading 120Concentrated Loads and Moments 120FORCE, FORCE2, or DAREA – Fixed-Direction Concentrated Loads 120Pressure Loads 121Initial Conditions 122

Eulerian Loading and Constraints 123Loading Definition 123Flow Boundary 123Rigid Wall 123Initial Conditions 124Detonation 129Body Forces 129Hydrostatic Preset 129Speedup for 2-D Axial Symmetric Models 129


Boundary and Loading Conditions - Theoretical Background 130Pressure Boundary Conditions 130Kinematic Boundary Conditions 132Displacement Constraints 132Prescribed Displacements, Velocities, and Accelerations 133Body Force Loads 133

6 Elements

Elements Overview 136Element Types 136

Preliminaries 140Governing Equations 141

CHEXA Solid Elements 144Volume Integration 146Hourglass Control 147Fully Integrated Brick Elements and Mid-Step Strain Evaluation 151CTETRA - Four Node Tetrahedron Element 152CPENTA - Six Node Pentahedron Element 153

CBEAM - Belytschko Beam 154Co-rotational Technique 154Belytschko Beam Element Formulation 157Calculation of Deformations 158Calculation of Internal Forces 159

CBEAM - DYSHELFORM = 1, Hughes-Liu Beam 163Geometry 163Fiber Coordinate System 167Strains and Stress Update 169

CQUADA - DYSHELLFORM = 2, Belytschko-Lin-Tsay Shell 172Co-rotational Coordinates 172Velocity-Strain Displacement Relations 174Stress Resultants and Nodal Forces 176Hourglass Control (Belytschko-Lin-Tsay) 177Hourglass Control (Englemann and Whirley) 178

CQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang Improvements 181

7Contents

CTRIA3 - DYSHELLFORM = 4, C0 Triangular Shell 184Co-rotational Coordinates 184Velocity-Strain Relations 185Stress Resultants and Nodal Forces 188

CTRIA3 - DYSHELLFORM = 3, Marchertas-Belytschko Triangular Shell (BCIZ) 190

Element Coordinates 190Displacement Interpolation 192Strain-Displacement Relations 194Nodal Force Calculations 195

CQUAD4 - DYSHELL FORM = 1, Hughes-Liu Shell 197Geometry 197Kinematics 200Fiber Coordinate System 201Lamina Coordinate System 202Strains and Stress Update 204

CQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells 207

Element Mass Matrix 210Accounting for Thickness Changes 211

Transverse Shear Treatment for Layered Shell 211

CROD, Truss Element 216

CQUAD4 - DYSHELLFORM = 9, Membrane Element 217Co-rotational Coordinates 217Velocity-Strain Displacement Relations 217Stress Resultants and Nodal Forces 218Membrane Hourglass Control 218

CELAS1D, Discrete Elements and CONM2, Masses 219Orientation Vectors 220Dynamic Magnification “Strain Rate” Effects 222Deflection Limits in Tension and Compression 223

CDAMP2D, Linear Elastic or Linear Viscous 224

Eulerian Elements 225Element Definition 225Solid Elements 225


Graded Meshes 226Requirements for Gluing Meshes 226Gluing Meshes 227Using Graded Meshes 228Visualization with MD Patran 228

7 Materials

Lagrangian Material Models for SOL 700 232Material Model 1: Elastic 238Material Model 2: Orthotropic Elastic 239Material Model 3: Elastic Plastic with Kinematic Hardening 240Material Model 5: Soil and Crushable Foam 244Material Model 6: Viscoelastic 245Material Model 7: Continuum Rubber 246Material Model 9: Null Material 246Material Model 10: Elastic-Plastic-Hydrodynamic 246Material Model 12: Isotropic Elastic-Plastic 249Material Model 13: Isotropic Elastic-Plastic with Failure 250Material Model 14: Soil and Crushable Foam With Failure 250Material Model 15: Johnson and Cook Plasticity Model 251Material Type 18: Power Law Isotropic Plasticity 252Material Type 19: Elastic Plastic Material Model with Strain Rate Dependent

Yield 253Material Type 20: Rigid 254Material Model 22: Chang-Chang Composite Failure Model 255Material Model 24: Piecewise Linear Isotropic Plasticity 256Material Model 26: Crushable Foam 257Material Model 27: Incompressible Mooney-Rivlin Rubber 261Material Model 28: Resultant Plasticity 262Material Model 29: FORCE LIMITED Resultant Formulation 263Material Model 30: Shape Memory Alloy 270Material Model 31: Slightly Compressible Rubber Model 274Material Model 32: Laminated Glass Model 275Material Model 34: Fabric 275Material Model 36: Barlat’s 3-Parameter Plasticity Model 276Material Type 37: Transversely Anisotropic Elastic-Plastic 278Material Type 38: Blatz-Ko Compressible Foam 280Material Model 39: Transversely Anisotropic Elastic-Plastic With FLD 280Material Model 53: Low Density Closed Cell Polyurethane Foam 281Material Models 54 and 55: Enhanced Composite Damage Model 282Material Model 57: Low Density Urethane Foam 284Material Type 58: Laminated Composite Fabric 286

9Contents

Material Type 62: Viscous Foam 288Material Type 63: Crushable Foam 289Material Model 64: Strain Rate Sensitive Power-Law Plasticity 290Material Model 65: Modified Zerilli/Armstrong 290Material Model 67: Nonlinear Stiffness/Viscous 3-D Discrete Beam 291Material Model 68: Nonlinear Plastic/Linear Viscous 3-D Discrete Beam 292Material Model 69: Side Impact Dummy Damper (SID Damper) 293Material Model 70: Hydraulic/Gas Damper 295Material Model 71: Cable 295Material Model 73: Low Density Viscoelastic Foam 296Material Model 74: Elastic Spring for the Discrete Beam 296Material Model 76: General Viscoelastic 297Material Model 77: Hyperviscoelastic Rubber 298Material Model 78: Soil/Concrete 300Material Model 79: Hysteretic Soil 302Material Model 80: Ramberg-Osgood Plasticity 302Material Model 81 and 82: Plasticity with Damage and Orthotropic Option 303Material Model 83: Fu-Chang’s Foam With Rate Effects 306Material Model 87: Cellular Rubber 308Material Model 89: Plasticity Polymer 310Material Model 94: Inelastic Spring Discrete Beam 310Material Model 97: General Joint Discrete Beam 310Material Model 98: Simplified Johnson Cook 311Material Model 100: Spot Weld 311Material Model 116: Composite Layup 313Material Model 119: General Nonlinear Six Degrees of Freedom Discrete

Beam 314Material Model 124: Tension-Compression Plasticity 315Material Model 126: Modified Honeycomb 315Material Model 127: Arruda-Boyce Hyperviscoelastic Rubber 318Material Model 158: Rate Sensitive Composite Fabric 319Material Model 181: Simplified Rubber Foam 319Material Model 196: General Spring Discrete Beam 322Grunseisen Equation of State (EOSGRUN) 323Tabulated Compaction Equation of State (EOSTABC) 323Tabulated Equation of State (EOSTAB) 324

Eulerian Material Models for SOL 700 325Shear Models 325Yield Models 326Equations of State 336Material Failure 339Spallation Models 340Material Viscosity 341


8 Contact Impact Algorithm

Overview 344SOL 600 Contact Capabilities not yet supported by SOL 700 344Notes on Rigid Body Modeling in SOL 700: 345SOL 600 Contact Limitations that do not apply to SOL 700 345

Penalty Methods 346

Preliminaries 346

Slave Search 347

Contact Force Calculation 351

Improvements to the Contact Searching 352

Bucket Sorting 353Bucket Sorting in Single Surface Contact 355

Accounting For the Shell Thickness 356

Initial Contact Penetrations 357

Contact Energy Calculation 358

Friction 359

9 Fluid Structure Interaction

General Coupling 362Fluid-structure Interaction 362Closed Volume 363Porosity 364

Multiple Coupling Surfaces with Multiple Euler Domains 367Coupling Surface with Failure 368Coupling Surfaces with Porous Holes 368Flow Between Domains 369Deactivation 369Output 370Using the Preference of Patran 370

Fluid- and Gas Solver for the Euler Equations 371

Modeling Fluid Filled Containers 372

Hotfilling 373

11Contents

10 Eulerian Solvers

The Standard Euler Solver 376Fluid-structure Interaction 377The Numerical Scheme 378The Time Step Criterion 380Euler with Strength 381The Multi-material Solver 382

Approximate Riemann Euler Solver 386Euler Equations of Motion 386Numerical Approach 387Entropy Fix for the Flux Difference Riemann Scheme 390Second Order Accuracy of the Scheme 390Time Integration 391

11 Airbags and Occupant Safety

Introduction 394

Airbag Definition 395Inflator Models in Airbags 397Constant Volume Tank Tests 400Porosity in Airbags 400Initial Metric Method for Airbags 403Heat Transfer in Airbags 404

Seatbelts 405Seatbelt Pretensioner 406Seatbelt Retractor 408Seatbelt Sensor 411Seatbelt Slipring 411

Occupant Dummy Models 413

Pre- and Postprocessing 414


12 System Information and Parallel Processing

Introduction 416

General Information 416Release Platforms 416MPI for MD Nastran SOL 700 417Hardware and Software Requirements 417Compatibility 418Definition 418

User Notes 419Specification of the Host file for Windows 419Specification of the Host file for UNIX and LINUX 419How to Run MD Nastran SOL 700 in Parallel 420How to Run Fluid Structure Interaction (FSI) in Parallel 421

Additional information of Different Platforms 421Windows XP and 2000 425Extra MPIch information 425

13 Examples

Crash 428Pick-up Truck Frontal Crash 428Train-barrier Impact 437

Airbags and Occupant Safety 444Simulation of Multi-compartment Airbag 444Airbag with Dummy 450

Bird Strike and Fan Blade Out (FBO) 459Bird Strike Simulation on Composite Glass Panel 459Bird Strike on Rotating Fan Blades with Prestress 466Multiple Bird-strikes on a Box Structure 479Chained Analysis - Fan Blade Out to Rotor Dynamics 490

Drop Test 510Drop Test Simulation of a Computer Package 510Wheel Drop Test 518

13Contents

Defense 524Rod Penetration 524Shaped Charge, using IG Model, Penetrating through Two Thick Plates 530Mine Blast 538Blastwave Hitting a Bunker 548

Time Domain NVH 555Chassis 555

Prestress 564Simulation of Prestress and Impact on Rotating Fan Blades 564

Smooth Particle Hydrodynamics (SPH) 576Ball Penetration using SPH Method 576

Sheet Metal Forming 582Square Cup Deep Drawing using Forming Limit Diagram 582PART 2. Square Cup Deep Drawing using Implicit Spring Back 591

Miscellaneous 596Paper Feed 596

Chapter 1 Introduction MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

1Introduction

MD Nastran Explicit Nonlinear (SOL 700) 16

Implicit and Explicit Methods 17

Principles of the Eulerian and Lagrangian Processors 24

Feature List 26

How SOL 700 Solves Explicit Problems 27

This User’s Guide 28

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideMD Nastran Explicit Nonlinear (SOL 700)

16

MD Nastran Explicit Nonlinear (SOL 700)MD Nastran Explicit Nonlinear (SOL 700) offers a powerful explicit solution to analyze dynamic events of short duration with severe geometric and material nonlinearities.

MD Nastran SOL 700 allows users to work within one common modeling environment using the same Bulk Data interface. The NVH, linear and nonlinear models can be used for explicit applications such as crash, crush, and drop test simulations. This dramatically reduces the time spent to build different models for implicit and explicit analysis and prevents you from making mistakes because of unfamiliarity between different programs.

MD Nastran SOL 700 is being developed in phases. The first phase, available in MD Nastran 2005 r3 supports the explicit structural applications.

Defining the ModelA finite element model consists of a geometric description given by the elements, their nodes, and a set of properties associated with the elements describing their attributes. These properties include material definitions, cross-section definitions in the case of structural elements like beams and shells, and other parameters for contact bodies, springs, dashpots, etc. There may also be constraints that must be included in the model - RBE elements, multi-point constraints, or equations (linear or nonlinear equations involving several of the fundamental solution variables in the model), or simple boundary conditions that are to be imposed throughout the analysis. Nonzero initial conditions, such as initial displacements and velocities, may also be specified.

The model is described and communicated to MD Nastran in the form of a text file (MD Nastran Input file). You can generate this file using a variety of preprocessor programs such as MD Patran or any text editor. It must adhere to MD Nastran conventions for the ordering and format of the model information.

User Interface

The user interface is the familiar MD Nastran card image interface comprised of executive control, case control, and bulk data cards. In most cases, they are identical to input formats defined for the MD Nastran Implicit Nonlinear module SOL 600 (and other solution sequences). The main differences involve the use of the SOL 700 materials and rigid joints (cylindrical, spherical, and revolute.) for various MD Nastran solution sequences. These materials and rigid joints may only be accessed using the SOL 700,129 executive control card. Additionally, in some specialized cases, some familiar MD Nastran cards may need small modifications and others might need to be added to specify items for defining new concepts such as moving coordinates.

Using MD Patran with SOL 700

The amount of information that needs to be conveyed in the MD Nastran Input file is extensive for even a modest size model. The amount of information and the complexity of most models makes it virtually impossible to generate the MD Nastran Input file with a text editor alone. Typically, you benefit from

17Chapter 1 IntroductionImplicit and Explicit Methods

using a preprocessor such as MD Patran. MD Patran provides a graphical user interface, an extensive line of model building tools that you can use to construct and view your model, and generate a MD Nastran Input file.

If you are using MD Patran as a preprocessor, you are required to specify an analysis code. Selecting MD Nastran Explicit Nonlinear (SOL 700) as the analysis code under the Analysis Preference menu, customizes MD Patran in five main areas:

• Material Library

• Element Library

• Loads and Boundary Conditions

• MPCs

• Analysis forms

The analysis preference also specifies that the model information be output in the MD Nastran Input File format.

Chapter 3: Modeling includes examples of forms and templates of the MD Patran Preference for SOL 700.

Implicit and Explicit MethodsA detailed theory of explicit analysis is outside the scope of this section. However, it is important to understand the basics of the solution technique, since it is critical to many aspects of using MD Nastran, SOL 700. If you are already familiar with explicit methods and how they differ from implicit methods, then you may disregard this section.

Implicit MethodsMany finite element programs use implicit methods to carry out a transient solution. Normally, they use Newmark schemes to integrate in time. If the current time step is step , a good estimate of the acceleration at the end of step will satisfy the following equation of motion:

where

= mass matrix of the structure

= damping matrix of the structure

= stiffness matrix of the structure

= vector of externally applied loads at step

= estimate of acceleration at step

n

n 1+

Ma'n 1+ Cv'n 1+ Kd'n 1++ + Fn 1+ext

=

M

C

K

Fn 1+

extn 1+

a'n 1+ n 1+

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideImplicit and Explicit Methods

18

and the prime denotes an estimated value.

The estimates of displacement and velocity are given by:

or

where is the time step and and are constants.

The terms and are predictive and are based on values already calculated.

Substituting these values in the equation of motion results in

or

The equation of motion may then be defined as

The accelerations are obtained by inverting the matrix as follows:

This is analogous to decomposing the stiffness matrix in a linear static analysis. However, the dynamics mean that mass and damping terms are also present.

Explicit MethodsThe equation of motion

= estimate of velocity at step

= estimate of displacement at step

v'n 1+ n 1+

d'n 1+ n 1+

d'n 1+ dn vnΔ t 1 2β–( )anΔ t2( ) 2 βa'n 1++⁄ Δ t

2+ +=

v'n 1+ vn 1 γ–( )anΔ t γa'n 1+ Δ t+ +=

d'n 1+ dn* βa'n 1+ Δ t

2+=

v'n 1+ vn* γa'n 1+ Δ t+=

Δ t β γ

dn* vn

*

Ma'n 1+ C v*n γa'n 1+ Δ t+( ) K d*n βa'n 1+ Δ t2

+( )+ + Fn 1+ext

=

M CγΔ t KβΔ t2

+ +[ ]a'n 1+ Fn 1+ext

Cvn*– Kdn

*–=

M*a'n 1+ Fn 1+residual

=

M*

a'n 1+ M*1–Fn 1+

residual=

Man Cvn Kdn+ + Fn

ext=


can be rewritten as

where

The acceleration can be found by inverting the mass matrix and multiplying it by the residual load vector.

In SOL 700, like any explicit finite element code, the mass matrix is lumped which results in a diagonal mass matrix.

Since is diagonal, its inversion is trivial, and the matrix equation is a set of independent equations for each degree of freedom, as follows:

The Leap-frog scheme is used to advance in time.

The position, Forces, and accelerations are defined at time level , while the velocities are defined at time level . Graphically, this can be depicted as:

The Leap-frog scheme results in a central difference approximation for the acceleration, and is second-order accurate in .

Explicit methods with a lumped mass matrix do not require matrix decompositions or matrix solutions. Instead, the loop is carried out for each time step as shown in the following diagram:

= vector of externally applied loads

= vector of internal loads (e.g., forces generated by the elements and hourglass forces)

=

= mass matrix

Man Fn

extFn

int–=

an M1–Fn

residual=

Fnext

Fnint

Fint Cvn Kdn+

M

M

ani Fniresidual

Mi⁄=

n

n 1 2⁄+

n 1– n 1 2⁄– n n 1 2⁄+ n 1+ time

d F a, , d F a, , d F a, ,v v

vn 1 2⁄+ vn 1 2⁄– an Δ tn 1 2⁄+ Δ tn 1 2⁄–+( ) 2⁄+=

dn 1+ dn vn 1 2⁄+ Δ tn 1 2⁄++=

Δ t


20

Explicit Time Step

Implicit methods can be made unconditionally stable regardless of the size of the time step. However, for explicit codes to remain stable, the time step must subdivide the shortest natural period in the mesh. This means that the time step must be less than the time taken for a stress wave to cross the smallest element in the mesh. Typically, explicit time steps are 100 to 1000 times smaller than those used with implicit codes. However, since each iteration does not involve the costly formulation and decomposition of matrices, explicit techniques are very competitive with implicit methods.

Because the smallest element in an explicit solution determines the time step, it is extremely important to avoid very small elements in the mesh.

Courant Criterion

Since it is impossible to do a complete eigenvalue analysis every cycle to calculate the timestep, an approximate method, known as the Courant Criterion, is used. This is based on the minimum time which is required for a stress wave to cross each element:

where

= Timestep

= Timestep scale factor (<1)

Grid-Point Accelerations

Grid-Point Velocities Grid-Point Displacements

Element Stain Rates

Element Stresses

Element Forces at Grid-Points

+ External Forces at Grid Points

Leap-frog Integration in Time

Element Formulation and Gradient Operator

Constitutive Model and Integration

CONTACT, Fluid-Structure Interaction, Force/Pressure boundaries

Element Formulation and Divergence Operator

tΔ SL /c=

Δ t

S


For 1-D elements, the speed of sound is defined as:

where,

When to use Explicit Analysis

The time step for implicit solutions can be much larger than is possible for explicit solutions. This makes implicit methods more attractive for transient events that occur over a long time period and are dominated by low frequency structural dynamics. Explicit solutions are better for short, transient events where the effects of stress waves are important. There is, of course, an area where either method is equally advantageous and may be used.

Explicit solutions have a greater advantage over implicit solutions if the time step of the implicit solution has to be small for some reason. This may be necessary for problems that include:

• Material nonlinearity. A high degree of material nonlinearity may require a small time step for accuracy.

• Large geometric nonlinearity. Contact and friction algorithms can introduce potential instabilities, and a small time step may be needed for accuracy and stability.

• Those analyses where the physics of the problem demands a small time step (e.g. stress wave effects as in crash, crush, and impact analyses).

• Material and geometric nonlinearity in combination with large displacements. Convergence in implicit methods becomes more difficult to achieve as the amount of nonlinearity for all types increases.

Explicit methods have increasing advantages over implicit methods as the model gets bigger. For models containing several hundred thousands of elements and including significant nonlinearity, SOL 700 may provide the most cost-effective solution even for problems dominated by low-frequency structural dynamics (see Figure 1-1).

= Smallest element dimension

= Speed of sound in the element material

= Young’s modulus

= density

L

c

c E ρ⁄=

E

ρ


22

Figure 1-1 Efficiency and Cost

Another case where explicit solutions may have advantages is in extremely large linear or nonlinear problems where implicit solvers may be limited to the number or parallel domains. There is no limit to the number of SOL 700 explicit parallel domains. Therefore, if you have more than 256 or so processors available, the explicit analysis may be faster even for linear static analyses.

SOL 700 should normally be used for structural components that may undergo large, sudden deformation, and for which the dimensions, deformed geometry, and residual stress state are of major importance. Table 1-1 summarizes the areas of overlap as well as the differences between the implicit and explicit analyses.

Table 1-1 Implicit and Explicit Technology Comparison

Material Nonlinearity Implicit Explicit

Linear isotropic elastic (metals)

Nonlinear isotropic elastic (rubber materials)

Linear orthotropic elastic (composites)

Elastic-perfectly plastic (limit analysis)

Elastoplastic, strain hardening (metals)

Viscoelastic (polymers)

Cost(CPU Time)

ImplicitExplicit

Problem Size

Cost(# of Matrix)

Implicit

Explicit

Number/Extent ofNonlinearities


Restricted orthotropic (metal-forming)

Damage accumulation and failure

Tearing and failure

Explosive detonation

Deformation Nonlinearity Implicit Explicit

Infinitesimal strains and rotations

Infinitesimal strains and finite rotations

Finite strains and rotations

Large strains (100% plus) and large rotations

(Multi) Material flow

Contact Nonlinearity Implicit Explicit

Small displacement gaps

Gaps with friction

Large displacement gaps

Contact surfaces

Single surface contact

Fluid-structure interaction

Motion Implicit Explicit

Static (infinite)

Quasi-static (noninertial)

Vibration, fundamental modes

Shock and vibration

Stress wave propagation

Shock wave propagation

High Frequency Dynamics

Detonation waves

Table 1-1 Implicit and Explicit Technology Comparison (continued)

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePrinciples of the Eulerian and Lagrangian Processors

24

Typical ApplicationsSome of the typical structural applications which are well suited for the SOL 700 explicit analysis are:

• Automotive and Aircraft crash worthiness

• Crash/Crush simulations• Drop testing• Ship Collision• Projectile penetration• Bird Strike Simulation with structural Bird• Metal forming, stamping, and deep drawing• Jet engine blade containment• Golf Club simulation• Rollover events

Principles of the Eulerian and Lagrangian Processors

MD Nastran Nonlinear Explicit (SOL 700) features two solving techniques, Lagrangian and Eulerian. The code can use either one or both and can couple the two types to produce interaction.

The Lagrangian method is the most common finite element solution technique for engineering applications.

When the Lagrangian solver is used, grid points are defined that are fixed to locations on the body being analyzed. Elements of material are created by connecting the grid points together and the collection of elements produces a mesh. As the body deforms, the grid points move with the material and the elements distort. The Lagrangian solver is, therefore, calculating the motion of elements of constant mass.

25Chapter 1 IntroductionPrinciples of the Eulerian and Lagrangian Processors

The Eulerian solver is most frequently used for analyses of fluids or materials that undergo very large deformations.

In the Eulerian solver, the grid points are fixed in space and the elements are simply partitions of the space defined by connected grid points. The Eulerian mesh is a “fixed frame of reference.” The material of a body under analysis moves through the Eulerian mesh; the mass, momentum, and energy of the material are transported from element to element. The Eulerian solver therefore calculates the motion of material through elements of constant volume.

It is important to note that the Eulerian mesh is defined in exactly the same manner as a Lagrangian mesh. General connectivity is used so the Eulerian mesh can be of an arbitrary shape and have an arbitrary numbering system. This offers considerably more flexibility than the logical rectangular meshes used in other Eulerian codes.

However, you should remember that the use of an Eulerian mesh is different from that of the Lagrangian type. The most important aspect of modeling with the Eulerian technique is that the mesh must be large enough to contain the material after deformation. A basic Eulerian mesh acts like a container and, unless specifically defined, the material cannot leave the mesh. Stress wave reflections and pressure buildup can develop from an Eulerian mesh that is too small for the analysis.

Eulerian and Lagrangian meshes can be used in the same calculation and can be coupled using a coupling surface. The surface acts as a boundary to the flow of material in the Eulerian mesh, while the stresses in the Eulerian material exerts forces on the surface causing the Lagrangian mesh to distort.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideFeature List

26

Feature ListThe complete features of MD Nastran Explicit Nonlinear (SOL 700) Phase I are:

1. MD Nastran Explicit Nonlinear (SOL 700) solves nonlinear (material, contact and/or geometric) static and transient dynamic structural finite element problems.

2. MD Nastran Explicit Nonlinear (SOL 700) supports the following elements/bodies:

• 3-noded triangular shell/membrane/plane stress/(generalized) plain strain elements.• 4-noded quadrilateral shell/membrane/plane stress/(generalized) plain strain elements.• 4-noded solid tetrahedral elements.• 6-noded solid wedge elements.• 8-noded solid hexahedral elements.• 2-noded beam element.• 2-noded bar element.• spring elements.• damper elements.• Rigid and deformable contact bodies.• Point Mass element.• RBE elements and multi-point constraint equations are supported to tie specific nodes or

degrees-of-freedom to each other. Special MPC entities are supported, (e.g. rigid links) which can be used to tie two nodes together or equate the motion of two degrees of freedoms.

3. MD Nastran Explicit Nonlinear (SOL 700) supports the following loads and boundary conditions:

• Constrained nodal displacements (zero displacements at specified degrees of freedom).• Enforced nodal velocities (nonzero velocities at specified degrees of freedom) • Forces applied to nodes • Pressures applied to element faces• Inertial body forces. Linear acceleration and rotational velocity can be applied • Contact between two bodies can be defined by selecting the contacting bodies and defining

the contact interaction properties.

4. MD Nastran Explicit Nonlinear (SOL 700) supports isotropic, orthotropic, and anisotropic material properties. Nonlinear elastic-plastic materials can be defined by specifying piecewise linear stress-strain curves, material failure can be specified.

5. Physical properties can be associated with MD Nastran Explicit Nonlinear (SOL 700) elements such as the cross-sectional properties of the beam element, the area of the beam and rod elements, the thickness of shell, plane stress, and membrane elements, spring parameters, and masses.

6. Laminated composite shell elements are supported in MD Nastran Explicit Nonlinear (SOL 700) through the PCOMP card of the materials capability. Each layer has its own material, thickness, and orientation and may represent linear or nonlinear material behavior. Failure index calculations are also supported.

7. Analysis jobs consisting of complex loading time histories are available. All loading must be applied in a single subcase. The applicable subcase for a particular analysis may be chosen from many subcases if so desired.

27Chapter 1 IntroductionHow SOL 700 Solves Explicit Problems

8. MD Nastran Explicit Nonlinear (SOL 700) jobs are submitted using text-based input decks that may be generated manually with a text editor, or by a variety of pre/post processing programs such as MD Patran. The input file is read in and a number of output files, such as the .f06, .log, f04, .dytr.dat, .dytr.d3hsp, .dytr.d3plot files are generated.

9. Results can be requested in several output formats such as .dytr.d3plot, .dytr.d3thdt, .f06 files. These files are typically read back into the pre/postprocessing programs for the purpose of evaluating the results with plots such as deformed shape plots, contour stress/strain plots, or X-Y history plots. MD Nastran native output files such as .xdb, .op2, and punch will be supported in the MD Nastran 2006r2 release.

10. Nodal displacements, velocities and accelerations, element stresses, element strains, element plastic strains, nodal reaction forces, contact interface force values, and element strain energy are output when applicable. These may be visualized with results visualization tools such as MD Patran. Composite element results are returned for individual layers of the composite, as requested.

11. Crash, impact is supported by the Phase I release. Dynamic loading described by time histories as well as static loading is also supported.

How SOL 700 Solves Explicit ProblemsSOL 700,ID solution resembles SOL 600,ID and uses the standard MD Nastran DMAP for solution ID. It is written so that there is not an effect on other solution sequences. The primary features are as follows:

1. Read MD Nastran Input File in IFP (input file processor) as in other MD Nastran solution sequences and translate executive control, case control, and bulk data to SOL 700 input formats.

2. Read SOL 700 input formats into explicit solver.

3. Run the model.

4. Write MD Nastran output result files.

Input and OutputOnce the solver is finished, the native output files are written out. You can request the type of outputs desired on the OUTR portion of the SOL 700 executive control card and by various parameters. If the OUTR options are omitted, standard outputs are available unless a request is made to delete the files. In version 2006r2, the MD Nastran native output files .xdb, op2 and punch are not supported and will be implemented in future releases. Instead d3plot, d3thdt and d3hsp are generated. They consist of displacements, velocities, accelerations, stress tensors, strain tensors and plastic strain. The geometry blocks are generated as usual using PARAM,POST,-1 or PARAM,POST,0.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideThis User’s Guide

28

The example below shows the files that are generated in the working directory:

This User’s GuideThis manual provides a complete background to SOL 700 and fully describes using SOL 700 within the MD Nastran environment. The theoretical aspects of explicit analysis methods, types, and techniques are included as well as descriptions for explicit material models.

Where appropriate, actual MD Patran forms and menus are shown so you can easily use SOL 700 from within the MD Patran environment.

Additional Documentation for SOL 700MD Nastran Reference Manual - provides supporting information that relates to MD Nastran input formats, element libraries, and loads and boundary conditions.

MD Nastran Quick Reference Guide (QRG) - contains a complete description of all the input entries for MD Nastran. Within each section, entries are organized alphabetically so they are easy to find. Each entry provides a description, formats, examples, details on options, and general remarks. Within the QRG, you will find the full descriptions for all of the SOL 700 input entries.

Input file:

filename.bdf Input deck for MD Nastran

filename.dytr.datfilename.dytr.str

Intermediate internal files generated for checking input errors

Output file:

filename.dytr.d3hsp Contains model summary, calculation process, CPU time, etc.

filename.dytr.out Contains time step summary and CPU timing information.

filename.dytr_prep.d3hspfilename.dytr_prep.out

Contains translation summary of the preparation phase.

filename.f06 Nastran native output that contains summary of model.

Binary Result file:

filename.dytr.d3plot Contains complete model, used for plotting deformed shape and stress contour.

filename.dytr.d3thdt Contains subset of the model, used for time history plots.

binout000 Contains additional time-history data.

29Chapter 1 IntroductionThis User’s Guide

The MD Patran library includes three key books which will be of assistance in running SOL 700:

• MD Patran User’s Guide - gives the essential information needed to immediately begin using MD Patran for SOL 700 projects. Understanding and using the information in this guide requires no prior experience with CAE or finite element analysis.

• MD Patran Reference Manual - gives complete descriptions of basic functions in MD Patran, geometry modeling, finite element modeling, material models, element properties, loads and boundary conditions, analysis, and results. It is the counterpart to the MD Nastran Reference Manual.

• MD Nastran Preference Guide - gives specific information that relates to using MD Patran with MD Nastran as the intended analysis code. All application forms and required input are tailored to MD Nastran.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideThis User’s Guide

30

Chapter 2: MD Nastran Data FilesMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

2MD Nastran Data Files

The MD Nastran Input Data File 32

Supported SOL 700 Entries 35

Preprocessing with MD Patran 51

Output Requests 52

Postprocessing with MD Patran 55

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideThe MD Nastran Input Data File

32

The MD Nastran Input Data FileThe MD Nastran Input File, usually given the extension .dat or .bdf, is made up of three distinct sections:

1. Executive Control - describes the problem type and size.2. Case Control - defines the load history.3. Bulk Data - gives a detailed model description.

Input data is organized in (optional) blocks. Key words identify the data for each optional block. This form of input enables you to specify only the data for the optional blocks that you need to define your problem. The various blocks of input are “optional” in the sense that many have built-in default values which are used in the absence of any explicit input from you.

A typical input file setup for the MD Nastran program is shown below.

• Executive Control Statements

Terminated by a CEND parameter

• Case Control Commands

Terminated by the BEGIN BULK option

• Bulk Data Entries

Model data starting with the BEGIN BULK option and terminated by the ENDDATA option

BulkData

CaseControl

Executive Control

Element andMaterial PropertiesFixed Displ,Etc.

Load Incrementation,Applied Loads, Applied DisplacementsEtc.

Title, Job Control,Solution Sequence,Etc.

MD

Nas

tran

Exp

licit

Non

linea

rC

ompl

ete

Inpu

t Dec

k

Con

trol

Info

rmat

ion

Mod

el D

ata

- gr

ids,

elem

ents

, etc

.

33Chapter 2: MD Nastran Data FilesThe MD Nastran Input Data File

Input ConventionsMD Nastran Explicit Nonlinear performs all data conversion internally so that the system does not abort because of data errors made by you. The program reads all input data options alphanumerically and converts them to integer, floating point, or keywords, as necessary. MD Nastran Explicit Nonlinear issues error messages and displays the illegal option image if it cannot interpret the option data field according to the specifications given in the manual. When such errors occur, the program attempts to scan the remainder of the data file and ends the run with a FATAL ERROR message. For SOL 700, certain entries generate Warning or Severe Warning messages and, in some cases, the run is terminated immediately after the message is written.

Two input format conventions can be used: fixed and free format. You can mix fixed and free format options within a file.

The syntax rules for fixed fields are as follows:

• Give floating point numbers with or without an exponent. If you give an exponent, it must be preceded by the character E or D and must be right-justified (no embedded blanks). If data is double precision, a D must be used.

The syntax rules for free fields are as follows:

• Check that each option contains the same number of data items that it would contain under standard fixed-format control. This syntax rule allows you to mix fixed-field and free-field options in the data file because the number of options you need to input any data list are the same in both cases.

• Separate data items on a option with a comma (,). The comma can be surrounded by any number of blanks. Within the data item itself, no embedded blanks can appear.

• Give keywords exactly as they are written in the manual.

• Enter data as uppercase or lowercase text.

• Limit to 8 columns per field for small field format whether using fixed-field or free-field. Large field is 16 columns; see the MD Nastran Quick Reference Guide for more details.

Section Descriptions

Executive Control

This group of entries provides overall job control for the problem and sets up initial switches to control the flow of the program through the desired analysis. This set of input must be terminated with a CEND parameter. See “Executive Control Statements” in Chapter 3 of the MD Nastran Quick Reference Guide for additional descriptions on input formats.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideThe MD Nastran Input Data File

34

Case Control

This group of options provides the loads and constraints. Case Control options also include blocks which allow changes in the initial model specifications. Case Control options can also specify print-out and postprocessing options. See “Case Control Commands” in Chapter 4 of the MD Nastran Quick Reference Guide for additional descriptions on input formats.

Bulk Data Entries

This set of data options enters the loading, geometry, and material data of the model and provides nodal point data, such as boundary conditions. This group of options must be terminated with the ENDDATA option. See “Bulk Data Entries” in Chapter 8 of the MD Nastran Quick Reference Guide for additional descriptions on input formats. Multiple BEGIN entries and superelements are not allowed.

Running Existing Models with SOL 700MD Nastran SOL 700 supports the following solution sequence:

• SOL 129 - Nonlinear Transient Dynamic Analysis

Some users may have existing models that have been developed and analyzed using MD Nastran Solution Sequences mentioned above. These models may be run through MD Nastran Explicit Nonlinear (SOL 700) by changing the SOLUTION procedure input to MD Nastran Explicit Nonlinear (SOL 700) input.

The following is an example of the change required to run existing models through SOL 700. The first line shows an existing MD Nastran SOL 129 Executive Control Statement and the second shows its revision for MD Nastran Explicit Nonlinear (SOL 700):

SOL 129SOL 700,129

By changing the Solution Sequence, there is no guarantee that the job will run. A successful run depends on the nature of the problem and the extent that the input entries are supported for a SOL 700 analysis. SOL 700 supports only the input entries that are relevant for an explicit simulation (see Section 2 for supported entries for details). All other entries that are unique to a certain Solution Sequence, are ignored and warning messages will be printed out in the output files as the user information. You are advised to check your models for accuracy and completeness before starting the simulation. In many occasions, it is wise to check the model in the preprocessor for “poor modeling” to flag out the skewed or small elements. The CPU time in the explicit analysis is directly controlled by the smallest element dimensions in the model. Small or poor elements may not abort a dynamic analysis but they certainly cause an explicit simulation to run infinitely longer.

It is recommended that proper material models with well-defined properties are prescribed in the model to accurately predict the material behavior as the model undergoes severe deformations and failure. For example, defining a suitable material failure criteria for the model, ensures that the distorted elements actually fail and are taken out of the calculation once they reach their failure limits. Otherwise, the smallest length in the distorted elements causes the time step to fall infinitely low, resulting in the premature termination of the job.

35Chapter 2: MD Nastran Data FilesSupported SOL 700 Entries

Supported SOL 700 Entries

Supported Case Control Cards1. The following summarizes the Case Control Commands for SOL 700:

(Only those which are supported or produce fatal errors are listed)

Item Available in SOL 700Patran

Support

$ Y Y

ACCELERATION Y Y

BCONTACT Y Y

BEGIN BULK Y (other BEGIN forms are not allowed) Y

DEFAULT Y

DISPLACEMENT Y Y

DLOAD Y Y

ECHO Y Y

ELFORCE (see FORCE) Y

ENDTIME Y (new)

FORCE & ELFORCE Y (automatically produced in d3plot files; no user control)

GROUNDCHECK Y (Nastran f06 only)

IC Y Y

INCLUDE Y Y

LABEL Y (Nastran f06 only)

LINE Y (Nastran f06 only)

LOAD Y (for dynamic pseudo-statics only)

LOADSET Y Y

MAXLINES Y (Nastran f06 only) Y

MPC Y

NLPARM Y (pseudo static analysis only)

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideSupported SOL 700 Entries

36

NLSTRESS Y (changed to STRESS)

PAGE Y (Nastran only)

PARAM Y (only applicable parms are used) Y

PRESSURE

SET Y Y

SET – OUTPUT(PLOT) N

SKIP Y (required if multiple subcases are present)

SPC Y Y

STRAIN Y Y

STRESS Y Y

SUBCASE Y

Note: Only one subcase can be selected for a particular SOL 700 analysis. Many subcases may be entered in the input deck, but the one to be used must be selected using the SKIP ON and SKIP OFF Case Control commands. If the SKIP ON/OFF commands are not found or are in the wrong place, the first subcase encountered is used and the others are ignored.

Y

SUBTITLE Y Y

SURFACE Y

TITLE Y Y

TSTEP Y (same as TSTEPNL) Y

TSTEPNL Y Y

VELOCITY Y Y

WEIGHTCHECK Y (Nastran only)

Item Available in SOL 700Patran

Support


Supported Bulk Data Entries

Item Available in Sol 700Fatal Error

Patran Support

ACCMETR Y

ABINFL Y (new)

AIRBAG Y Y

AXIC N Y

AXIF N Y

AXSLOT N Y

BARRIER Y (new) Y

BAROR Y N

BCBODY Y Y

BCBOX Y Y

BCGRID Y Y

BCHANGE N Y

BCMATL Y N

BCONP N

BCPROP Y Y

BCSEG Y Y

BCTABLE Y (revised) Y

BEAMOR Y N

BJOIN Y

BLSEG N Y

BSURF Y Y

CBAR Y Y

CBEAM Y Y

CBELT Y Y


38

CBEND N Y

CBUSH N Y

CBUTT Y N

CCONEAX N Y

CCRSFIL Y N

CDAMP1 Y Y

CDAMP2 Y N

CDAMP1D Y Y

CDAMP2D Y N

CELAS1 Y Y

CELAS2 Y Y

CELAS1D Y Y

CELAS2D Y Y

CFILLET Y N

CFLUID N Y

CGAP N Y

CHACAB N Y

CHACBR N Y

CHEXA Y (8 nodes only) Y

CMARKB2 Y (new)

CMARKN1 Y (new)

COMBWLD Y N

CONM2 Y Y

CONROD Y Y

CONSPOT Y


Patran Support


CORD1C Y Y

CORD1R Y Y

CORD1RX Y

CORD1S Y Y

CORD2C Y Y

CORD2R Y Y

CORD2RX Y

CORD2S Y Y

CORD3G N Y

CORD3R Y N

CORD3RX Y

COUOPT Y (new) Y

COUP1FL Y (new)

COUPINT Y (new)

COUPLE Y (new) Y

CYLINDR Y (new) Y

CPENTA Y (6 nodes only) Y

CQUAD4 Y Y

CQUAD8 Y (4 nodes only) Y

CQUADR Y Y

CQUADX N Y

CREEP N Y

CROD Y Y

CSHEAR N Y

CSHP Y (new) Y


Patran Support


40

CSPOT Y Y

CSPR Y Y

CTETRA Y (4 or 10 nodes only) Y

CTQUAD Y

CTRIA3 Y Y

CTRIA6 Y (3 nodes only) Y

CTRIAR Y Y

CTRIAX N Y

CTRIAX6 N Y

CTTRIA Y

CTUBE Y Y

CVISC Y Y

CWELD N Y

D2R0000 Y Y

D2RAUTO Y Y

D2RINER Y Y

DAMPGBL Y Y

DAMPMAS Y Y

DAMPSTF Y Y

DAREA Y

DBEXSSS Y

DETSPH Y (new)

DLOAD Y Y

DYCHANG Y

DYDELEM Y


Patran Support


DYPARAM Y

DYRELAX Y

DYRIGSW Y

DYTERMT Y

DYTIMHS Y N

ENDDATA Y Y

ENDDYNA Y

EOSGAM Y Y

EOSGRUN Y (new) Y

EOSIG Y (new) Y

EOSJW Y (new) Y

EOSMG Y (new)

EOSPOL Y Y

EOSTAB Y Y

EOSTABC Y Y

EOSTAIT Y (new) Y

FAILMPS Y (new)

FFCONTR Y (new)

FLOW Y (new) Y

FLOWDEF Y (new) Y

FLOWSPH Y (new)

FLOWT Y (new) Y

FORCE Y Y

FORCE1 N Y

FORCE2 Y N


Patran Support


42

FORCEAX N Y

GBAG Y Y

GBAGCOU Y (new)

GENEL N Y

GRAV Y Y

GRDSET Y N

GRIA Y Y

GRID Y Y

HEATLOS Y (new)

HGSUPPR Y Y

HTRCONV Y (new)

HTRRAD Y (new)

HYDSTAT Y (new) Y

INCLUDE Y N

INFLCG Y (new)

INFLFRC Y

INFLGAS Y (new)

INFLHB Y (new)

INFLTNK Y (new)

INFLTR Y (new)

INITGAS Y (new)

IPSTRAIN N Y

ISTRESS N Y

ISTRSBE Y Y

ISTRSSH Y Y


Patran Support


ISTRSSO Y Y

ISTRSTS Y

LEAKAGE Y (new) Y

LOAD Y Y

LSEQ Y Y

MAT1 Y Y

MAT2 Y Y

MAT3 Y Y

MAT8 Y Y

MATDxxx Y (see Supported Material Models in SOL 700 in this chapter)

Y

MATDERO Y Y

MATDEUL Y (new) Y

MATEP N Y

MATF Y

MATG N Y

MATHE N Y

MATHED N Y

MATHP Y Y

MATORT N Y

MATRIG Y

MATS1 Y Y

MATVE N Y

MATVORT N Y

MATVP N Y

MESH Y (new) Y


Patran Support


44

MFLUID N Y

MOMAX N Y

MOMENT Y Y

MOMENT2 Y N

MPC Y Y

MPCAX N Y

NLPARM Y (for pseudo statics) Y

NLRGAP N Y

NOLINi N Y

NTHICK N Y

PANEL N Y

PBAR Y Y

PBARL Y Y

PBCOMP N Y

PBEAM Y Y

PBEAM71 Y

PBEAMD Y

PBEAML Y

PBELTD Y

PBEND N Y

PBSPOT Y Y

PBUSH N Y

PCOMP Y Y

PCOMPA Y

PCOMPG N Y


Patran Support


PDAMP Y Y

PDAMP5 N Y

PELAS Y Y

PELAS1 Y

PELAST N Y

PERMEAB Y (new) Y

PERMGBG Y (new)

PEULER Y (new) Y

PEULER1 Y (new) Y

PGAP N Y

PHBDY N Y

PINTC N Y

PINTS N Y

PLOAD Y N

PLOAD1 N Y

PLOAD2 Y N

PLOAD4 Y Y

PLOADX1 N Y

PLPLANE Y

PLSOLID Y

PMARKER Y (new)

PMINC Y (new) Y

PORFCPL Y (new) Y

PORFGBG Y (new)

PORFLOW Y (new) Y


Patran Support


46

PORFLWT Y (new) Y

PORHOLE Y (new) Y

PORHYDS Y (new)

PRESTRS Y Y

PMASS N Y

PRESPT N Y

PROD Y Y

PSHEAR N Y

PSHELL Y Y

PSHELL1 Y Y

PSHELLD Y

PSOLID Y Y

PSOLIDD Y Y

PSPH Y (new) Y

PSPRMAT Y

PTSHELL Y

PTUBE Y Y

PVISC Y Y

RBAR Y Y

RBE1 N Y

RBE2 Y Y

RBE2A Y

RBE2D Y

RBE2F Y

RBE3 Y Y


Patran Support


RBE3D Y

RBJOINT Y N

RBJSTIFF Y N

RCONN Y

RESTART Y

RFORCE Y (CID, METHOD, continuation line not supported)

Y

RLOADi N Y

RROD N Y

RSPLINE N Y

RTRPLT N Y

SBPRET Y

SBRETR Y

SBSENSR Y

SBSLPR Y

SLOAD N Y

SEQROUT Y (new)

SHREL Y (new) Y

SHRPOL Y (new)

SPC Y Y

SPC1 Y Y

SPCADD Y Y

SPCAX N Y

SPCD Y Y

SPCD2 Y Y

SPHDEF Y (new) Y


Patran Support


48

SPHERE Y (new) Y

SPHSYM Y (new)

SUPAX N Y

SURFINI Y

SPWRS Y (new)

TABLED1 Y Y

TABLED2 Y N

TABLED3 Y N

TABLEDR Y Y

TABLES1 Y Y

TEMP N Y

TEMPD N Y

TIC Y Y

TICD Y

TIC3 Y Y

TICEL Y (new) Y

TICEUL1 Y (new) Y

TICREG Y (new) Y

TICVAL Y (new) Y

TIMNAT Y

TIMNVH Y

TIMSML Y

TLRE1 Y

TLOAD1 Y Y

TLOAD2 Y N


Patran Support


TODYNA Y

TSTEP Y (changed to TSTEPNL) Y

TSTEPNL Y Y

UGASC Y (new) Y

WALL Y Y

WALLGEO Y (new) Y

YLDHY Y (new) Y

YLDJC Y (new) Y

YLDMC Y (new) Y

YLDMSS Y (new)

YLDPOL Y (new) Y

YLDRPL Y (new) Y

YLDSG Y (new) Y

YLDTM Y (new) Y

YLDVN Y (new) Y

YLDZA Y (new) Y


Patran Support


50

Supported Material Models in SOL 700In addition to the MD Nastran material models listed in the table above, the following material models are supported in SOL 700:

MATD001, MATD2AN, MATD20M, MATD003, MATD005, MATD006, MATD007, MATD009, MATD010, MATD012, MATD013, MATD014, MATD015, MATD018, MATD019, MATD020, MATD022, MATD024, MATD026, MATD027, MATD028, MATD029, MATD030, MATD031, MATD032, MATD034, MATD040, MATD054, MATD055, MATD057, MATD058, MATD059, MATD062, MATD063, MATD064, MATD066, MATD067, MATD068, MATD069, MATD070, MATD071, MATD072, MATD72R, MATD073, MATD074, MATD076, MATD077, MATD080, MATD081, MATD083, MATD087, MATD089, MATD093, MATD094, MATD095, MATD097, MATD098, MATD099, MATD100, MATD112, MATD114, MATD119, MATD0121, MATD123, MATD126, MATD127, MATD158, MATD181, MATD196, MATDB01, MATDS01, MATDS02, MATDS03, MATDS04, MATDS05, MATDS06, MATDS07, MATDS08, MATDS13, MATDS14, MATDS15, MATDSW1, MATDSW2, MATDSW3, MATDSW4, MATDSW5

Unsupported Materials in MD Patran PreferenceAll the material models listed above are supported with the exception of the following:

MATD018, MATD034, MATD040, MATD066, MATD072, MATD071, MATD72R, MATD087, MATD089, MATD093, MATD094, MATD095, MATD097, MATD112, MATD114, MATD119, MATD0121, MATD123, MATD158

Supported Parameters in SOL 700PARAM,DYBEAMIP, PARAM,DYBLDTIM, PARAM,DYBULKL, PARAM,DYBULKQ1, PARAM,DYCMPFLG, PARAM*,DYCONSLSFAC, PARAM*,DYCONRWPNAL, PARAM*,DYCONPENOPT, PARAM*,DYCONTHKCHG, PARAM*,DYCONENMASS, PARAM*,DYCONECDT, PARAM*,DYCONIGNORE, PARAM*,DYCONORIEN, PARAM*,DYCONSKIPRWG, PARAM,DYDCOMP, PARAM,DYCOWPRD, PARAM,DYCOWPRP, PARAM, DYDEFAUL, PARAM,DYDTOUT, PARAM,DYENDTIM, PARAM*,DYENERGYHGEN, PARAM,DYENGFLG, PARAM,DYEPSFLG, PARAM,DYHRGIHQ, PARAM,DYHRGQH, PARAM,DYIEVERP, PARAM*,DYINISTEP, PARAM,DYLDKND, PARAM,DYMAXINT, PARAM*,DYMAXSTEP, PARAM*,DYMINSTEP, PARAM,DYNAMES, PARAM,DYNEIPH, PARAM,DYNEIPS, PARAM,DYNINTSL, PARAM,DYN3THDT, PARAM,DYRBE3, PARAM,DYRLTFLG, PARAM*,DYSHELLFORM, PARAM,DYSHGE, PARAM*,DYSHTHICK, PARAM,DYSIGFLG, PARAM,DYSTATIC, PARAM*,DYSTEPFCTL, PARAM,DYSTRFLG, PARAM,DYSTSSZ, PARAM*,DYTERMNDTMIN, PARAM*,DYTERMNENDMAS, PARAM*,DYTSTEPERODE,

51Chapter 2: MD Nastran Data FilesPreprocessing with MD Patran

PARAM*,DYTSTEPDT2MS, PARAM,DYXPENE, PARAM,S700NVH, PARAM,SBOLTZ, PARAM,SCALEMAS, PARAM,UGASC, PARAM,DYININT, PARAM,DYNREAL,

PARAM,COPOR, PARAM,EULBND, PARAM,EULBULKL, PARAM,EULBULKQ, PARAM,EULBULKT, PARAM, EULSTRES, PARAM,FBLEND, PARAM,FMULTI, PARAM,GRADMESH,PARAM,MICRO, PARAM,RKSCHEME, PARAM,ROHYDRO, PARAM,ROMULTI, PARAM,ROSTR, PARAM,VELCUT, PARAM,S700NVH1, DYPARAM,AXIALSYM, DYPARAM,EULERCUB, DYPARAM, EULTRAN, DYPARAM,FASTCOUP, DYPARAM,FSIDMP, DYPARAM,HYDROBOD, DYPARAM,LIMITER, DYPARAM,VELMAX

Supported Parameters in MD Patran PreferenceDYBEAMIP, DYBLDTIM, DYBULKL, DYCMPFLG, DYCONECDT, DYCONENMASS, DYCONIGNORE, DYCONPENOPT, DYCONRWPNAL, DYCONSLSFAC, DYCONSKIPRWG, DYCONTHKCHG, DYCOWPRD, DYCOWPRP, DYDCOMP, DYDTOUT, DYENERGYHGEN, DYENGFLG, DYEPSFLG, DYHRGIHQ, DYHRGQH, DYIEVERP, DYINISTEP, DYLDKND, DYMAXINT, DYMAXSTEP, DYMINSTEP, DYN3THDT, DYNEIPH, DYNEIPS, DYNINTSL, DYRLTFLG, DYSHELLFORM, DYSHGE, DYSHTHICK, DYSIGFLG, DYSTATIC, DYSTRFLG, DYSTEPFCTL, DYSTSSZ, DYTERMNENDMAS, DYTSTEPDT2MS, DYTSTEPERODE, PARAM,S700NVH1

Preprocessing with MD PatranMD Patran offers a MD Nastran interface that provides a communication link between MD Patran and MD Nastran. It provides for the generation of the MD Nastran Input file and customization of certain features in MD Patran. The interface is a fully integrated part of the MD Patran system.

Generating the Bulk Data FileSelecting MD Nastran as the analysis code preference in MD Patran ensures that sufficient and appropriate data is generated for the MD Nastran. Specifically, the MD Patran forms in these main areas are modified:

• Materials

• Element Properties

• Finite Elements/MPCs and Meshing

• Loads and Boundary Conditions

• Analysis Forms

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideOutput Requests

52

Using MD Patran, you can run a MD Nastran analysis or you may generate the MD Nastran Input File to run externally. For information on generating the MD Nastran Input file from within MD Patran, see “Analysis Form” in Chapter 3 of the MD Patran MD Nastran Preference Guide, Volume 1: Structural Analysis.

Editing the Bulk Data FileOnce the bulk data file has been generated, you can edit the file directly from MD Patran.

1. Click the Analysis Application button to bring up the Analysis Application form.

2. On the Analysis form set the Action>Object>Method combination to Analyze>Existing Deck>Full Run and click Edit Input File.

MD Patran finds the bulk data file with the current database name and displays the file for editing in a text editing window.

Output RequestsThe SOL 700 default output file is the d3plot file which is a binary file including the displacements, stresses and strain output results. The file d3plot can be postprocessed with a number of commercially available tools such as MD Patran, ETA/Femb, Hypermesh, CEI/Ensight and LS-POST.

In addition to d3plot, other output files are optionally available for output request.

Files Created by SOL 700SOL 700 creates a number of files during the analysis.

In the main sections of this manual, generic names are used when referring to a particular MD Nastran SOL 700 file. These generic reference names and the actual generic file names are given below:

FileGeneric

Reference Name SOL 700 Name

Input (ASCII) dat or bdf Filename.dat (or .bdf)

Output/d3hsp (ASCII) d3hsp Filename.dytr.d3hsp

Result Output (Binary) d3plot Filename.dytr.d3plot

Time History (Binary) d3thdt Filename.dytr.d3thdt

Result Output (Binary) ARC Filename.dytr.ARC

Result Output (Binary) binout Filename.dytr.binout

Result Output (ASCII) ssstat Filename.dytr.ssstat

53Chapter 2: MD Nastran Data FilesOutput Requests

Input File

The input file contains all the input data and must be present in order to run MD Nastran SOL 700. It is a text file with up to 80 characters on each line. The primary input file may reference one or more include files.

Output (d3hsp) File

The output (d3hsp) file is a text file suitable for printing purposes or viewing with a standard editor. It contains messages produced by explicit solver in SOL 700, which contains summary of model input, calculation status at every 100 cycles and error messages This file is normally also included in the f06 file. The quantities that are placed on this file are controlled by PARM statements.

d3plot

MD Nastran can create any number of output files containing results at times during the analysis. The d3plot files are binary files generated by explicit solver in SOL 700. They contain a complete description of the geometry and connectivity of the analysis model and the requested results. MD Patran can read d3plot files for postprocessing.

ARC

MD Nastran outputs the fluid structure interaction (FSI) results in the *ARC file. The ARC files are binary files and are consistant with Dytran output. To postporcess the FSI results by Patran, both d3plot and ARC files have to be read.

binout

The binout file is the file generated by SOL 700 and is consistent with the LS-DYNA native binout file. It includes the time history results in binary format.

Restart Dump file (Binary) d3dump Filename.dytr.d3dump

Execution Summary file (ASCII) log Filename.dytr.log

Warnings and Errors (ASCII) f06 Filename.f06

MD Nastran Result file (Binary) op2 Filename.op2

MD Nastran Result file (Binary) xdb Filename.xdb

MD Nastran Punch file (ASCII) pch Filename.pch

FileGeneric

Reference Name SOL 700 Name

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ssstat File

The sstat file is generated by SOL 700 and contains the information of the subsystem. The DYPARAM,LSDYNA,DATABASE,SSTATM and DBEXSSS bulk data entries can be activated to generate the sstat file. The information of the sstat file is useful for FBO-RD chaining analysis.

Time-History (d3thdt) Files

SOL 700 can also create any number of time-history files containing results for particular grid points and elements during the calculation. They are also binary files. Time-history files only contain results, no geometry or connectivity.

Restart Files

SOL 700 supports the three kinds of restart. These are named simple, minor (or small) and full. Simple restarts are jobs that for some reason did not finish (for example the computer went down) and involve no changes in the original Nastran input deck. They can use either the running restart file (jid.dytr.runrsf) or the d3dump file (jid.dytr.d3dumpxx where xx is a numerical value such as 00, 01, 02 ...). Minor restarts involve simple changes to the Nastran input deck such as deleting contact surfaces, changing the run time or output time intervals, switching rigid bodies to deformable or visa versa. These require a Nastran input deck with the executive control, case control and only the changes as well as the jid.dytr.d3dumpxx file. The full restart normally involves massive changes to the Nastran deck where the complete executive control, case control and bulk data are input as well as restarting with the jid.dytr.d3dumpxx file.

Error File

SOL 700 produces an f06 file containing a summary of all warnings and errors issued during reading and subsequent data processing. SOL 700 has five levels of messages below:

1. Informative messages.

An example of a Level 1 message is a message that indicates that a new processor has begun execution. These message provide job information.

2. Nonfatal warning message of something that could affect the results.

An example of a Level 2 message is one indicating that the aspect ratio is greater than 15. This may or may not be a serious problem.

3. Serious warnings.

An example of a Level 3 message is a warning about a highly distorted element.

4. Fatal error (all occurrences will be found before aborting).

An example of a Level 4 message is the warning “undefined node used in rigid element.”

5. Immediately fatal errors.

An example of a Level 5 message is “Unable to open file” message. The job is immediately aborted.

55Chapter 2: MD Nastran Data FilesPostprocessing with MD Patran

MD Nastran Native Result Files

In addition to d3plot which is native result file, SOL 700 optionally outputs MD Nastran native result files such as op2, xdb, and pch files for postprocessing in version 2006r2 and subsequent versions.

Postprocessing with MD PatranThe results from an MD Nastran Explicit Nonlinear Analysis can be read into and postprocessed by MD Patran. Typically, you will get the most complete set of results if you use the d3plot results options (see Section 14.1 “Output from the Analysis” on how to select which output files will be created), but you can also postprocess using an .xdb or .op2 formatted file.

The Results application in MD Patran provides the capabilities for creating, modifying, deleting, posting, unposting, and manipulating results visualization plots and viewing the finite element model. In addition, results can be derived, combined, scaled, interpolated, extrapolated, transformed, and averaged in a variety of ways which are controllable by you.

Control is provided for manipulating the color/range assignment and other attributes for plot tools, and for controlling and creating animations of static and transient results.

Results are selected from the database and assigned to plot tools using simple forms. Results transformations are provided to derive scalars from vectors and tensors and to derive vectors from tensors. This allows for a wide variety of visualization tools to be used with all of the available results.

If the job was created within MD Patran such that a MD Patran jobname of the same name as the MD Nastran jobname exists, you only need to use the Results tools and MD Patran imports or attaches the jobname.xxx file without you having to select it. If you did not create the job in MD Patran, you can still import the model and results and postprocess.

For more information, see MD Patran Reference Manual, Part 6: Results Postprocessing.

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Chapter 3: ModelingMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

3Modeling

Coordinate Systems 58

Nodes 61

Elements 62

Modeling in MD Patran 63

Example using MD Patran 67

Input and Output Files Created During the Simulation 79

Postprocessing 81

Running a Batch Job 85

How to Tell When the Analysis is Done 85

How to Tell if the Analysis Ran Successfully 85

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Coordinate SystemsThe basic coordinate system in MD Nastran Explicit Nonlinear is a right-handed rectangular coordinate system as shown in Figure 3-1.

Figure 3-1 Basic Coordinate System

User-defined Coordinate SystemMD Nastran provides six Bulk Data entry options for defining coordinate systems. The new coordinate system is directly or indirectly related to the basic coordinate system. The ten options are:

The CORD1R, CORD1C, and CORD1S entries define coordinate systems by referencing three defined grid points. The CORD2R, CORD2C, and CORD2S entries define coordinate systems by specifying the location of three points.

X

Y

Z

CORD1RCORD2R

Rectangular

CORD1CCORD2C

Cylindrical

CORD1SCORD2S

Spherical

CORD3RCORD1RXCORD2RXCORD3RX

59Chapter 3: ModelingCoordinate Systems

Nodal Coordinate SystemsEach grid point refers to two coordinate system. One system is used to locate the grid point (CP in field 3) and the other is used to establish the grid point’s displacement coordinate system (CD in field 7). The displacement coordinate system defines the direction of displacements, constraints and applied loads. The basic (default coordinate system) is indicated by a zero or blank in the CP and CD fields. CD and CP do not have to be the same coordinate system.

The grid point output is with respect to basic coordinate system.

Element Coordinate SystemsThe connectivity of the CQUAD4s defines the element coordinate system. It is a rectangular coordinate system and the direction of axes depends on the order of the grid points in the connectivity entry. The z-axis is normal to the shell as shown in the figure below. The x-axis is a vector from grid point1 to grid point 2. The y-axis is perpendicular to both the x-axis and the z-axis and is the direction defined by the right-hand rule.

The counter clockwise node numbering determines the positive z-direction and is the top surface of the shell element and the bottom surface is the negative z-direction.

By default, shell stresses/strains written to d3plots are in the basic coordinate system and shell stresses/strains written to elout (time history element output) are in the element local coordinate system.

The local coordinate system of CBAR/CBEAM is explained in the Remarks of PBAR and PBEAM entries respectively in the quick reference guide. By default, stresses/resultants for beams are written to d3plot and elout (time history element output) in the element coordinate system.

The output of solid elements in d3plot and elout (time history element output) is always with respect to the basic coordinate system.

G1

G4G3

G2

Xelem

Zelem

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Material Coordinate SystemsFor shell elements of anisotropic material, there are three options for defining the initial direction of the material axes, AOPT = 0, 2, and 3 (see MATD2AN, MATD022, MAT054, MATD059). During the solution, the element coordinate system rotates and transforms with the element, so the angle between the element system and material system can be assumed to remain constant. In other words, the material direction is constantly updated as the element rotates and deforms. It is, therefore, sufficient to define the material coordinate system in the undeformed geometry.

For this discussion, the material coordinate system is called the a-b-c system to be consistent with the user's manual. For shell elements, the c-axis coincides with the element normal, the a-axis is in the plane of the shell, and the b-axis is determined by the cross product, b = c x a. Actually, for warped elements, the a-axis is not exactly in the element plane, but is projected along the c-axis such that it is orthogonal to c. This projection is also done for the local element system, so a simple 2-D transformation between the two systems is possible.

For reasons discussed in the last paragraph, the three available shell element options for defining the a-b-c system boil down to defining the a-axis.

For AOPT = 0, the a-axis is assumed to be equal to the local element system x-axis.

For AOPT = 2, the a-axis is defined as the user-defined vector , projected onto the surface of the element. Note that the user-defined vector is not used at all. Also, note that the user-defined vector , is not equivalent to the a-axis of the material unless is orthogonal to the element normal.

For AOPT = 3, the a-axis is defined by the cross product of a user-defined vector with element normal; i.e., a = v x c. Given the same user-defined vector, AOPT = 3 defines a material coordinate system that is rotated exactly 90 degrees from coordinate system defined by AOPT.

The material axes as defined by the AOPT option in the material input are rotated by the THETA element (in CQUAD4) angle to get the reference direction for the element. The material axes for the element integration points are then rotated by the integration point THETA angles in PCOMP option. In summary, AOPT, THETA in CQUAD4, and the integration point THETA angles in PCOMP go into defining the material directions at each integration point.

a

d a

a

v

61Chapter 3: ModelingNodes

NodesModel geometry is defined in MD Nastran with grid points. A grid point is a point on or in the structural continuum which is used to define a finite element. A simple model may have only a handful of grid points; a complex model may have many hundreds of thousands. The structure’s grid points displace with the loaded structure. Each grid point of the structural model has six possible components of displacement: three translations (in the x-, y-, or z-directions) and three rotations (about the x-, y-, or z-axes). These components of displacement are called degrees of freedom (DOFs).

Degrees of FreedomThe degrees of freedom in MD Nastran Nonlinear are always referred to as:

MD Nastran only activates those degrees of freedom needed at a node. Thus, some of the degrees of freedom listed above may not be used at all nodes in a model, because each element type only uses those degrees of freedom which are relevant. For example, two-dimensional solid (continuum) stress/displacement elements only use degrees of freedom 1 and 2. The degrees of freedom actually used at any node are thus the envelope of those variables needed in each element that uses the node.

1 x-displacement

2 y-displacement

3 z-displacement

4 Rotation about the x-axis

5 Rotation about the y-axis

6 Rotation about the z-axis

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62

ElementsOnce the geometry (grid points) of the structural model has been established, the grid points are used to define the finite elements.

MD Nastran has an extensive library of finite elements covering a wide range of physical behavior. Some of these elements and their names are shown in figure below. The C in front of each element name stands for “connection.”

• Point Element (not a finite element, but can be included in the finite element model)

• Spring Elements (they behave like simple extensional or rotational springs)

• Line Elements (they behave like rods, bars, or beams)

• Surface Elements (they behave like membranes or thin plates)

• Solid Elements (they behave like bricks or thick plates)

CONM2 (Concentrated mass)

CELAS1, CELAS2, CELAS1D, CELAS2D

CROD, CBAR, CBEAM, CONROD

CTRIA3 CQUAD4

CHEXA CPENTA CTETRA

63Chapter 3: ModelingModeling in MD Patran

• Rigid Bar (infinitely stiff without causing numerical difficulties in the mathematical model)

Structural elements are defined on Bulk Data connection entries that identify the grid points to which the element is connected. The mnemonics for all such entries have a prefix of the letter C, followed by an indication of the type of element, such as CBAR and CROD. The order of the grid point identification defines the positive direction of the axis of a one-dimensional element and the positive surface of a plate element. The connection entries include additional orientation information when required. Some elements allow for offsets between its connecting grid points and the reference plane of the element. The coordinate systems associated with element offsets are defined in terms of the grid point coordinate systems. For most elements, each connection entry references a property definition entry. If many elements have the same properties, this system of referencing eliminates a large number of duplicate entries.

Details for each element type are described in Chapter 6: Elements.

Modeling in MD PatranIn MD Patran, geometric models are the foundation on which most finite element models are built. Geometric curves, surfaces, or solids provide the base for creating nodes, elements, and loads and boundary conditions; the geometric model also serves as the structure to which material properties, as well as element properties, may be assigned even before any mesh is actually generated.

Creating Geometry in MD PatranModel geometry may be constructed in MD Patran, accessed directly from a CAD application, or imported in specially formatted translator files. Whatever the source of the geometry, a single geometric model will be maintained throughout all geometric and finite element operations. Geometric entities, even if obtained from external files, retain their original mathematical representation without any approximations or substitutions.

Accessing the Geometry Application

In MD Patran you can create, modify, and delete points, curves, surfaces, and solids. MD Patran assigns a default color to the display of all geometric entities.

Pick the Geometry icon in the MD Patran Main Form to access the Geometry application.

RBAR

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64

The Geometry form controls all processes in the Geometry application. The top portion of the form contains three keywords, Action, Object, and Method; these remain the same throughout all activities. The rest of the entries will vary depending on the requirements posed by the specified action, object, and method.

There are hundreds of action, object, method combinations available for creating geometric entities in MD Patran. For complete descriptions on creating geometry models, see the MD Patran Reference Manual, Part 2: Geometry.

Utilizing External Geometry (CAD) Files

MD Patran can make use of geometry created in databases outside of MD Patran by either accessing geometric data directly from one of several CAD systems, or importing geometry using special files.

Geometry access, performed through the unique Direct Geometry Access (DGA) feature, does not require any translation. MD Patran accesses the original geometry and uses the geometric definitions of all entities.

On the other hand, when geometry is imported, MD Patran first evaluates the mathematical definition of entities in their originating CAD system, and then formulates the information to be appropriate for MD Patran operations.

Imported geometry comes to MD Patran via IGES, Express Neutral files, or MD Patran Neutral files.

IGES (Initial Graphic Exchange Specification) is an ANSI standard formatted file that makes it possible to exchange data among most commercial CAD systems. Express Neutral files are intermediate files created during a Unigraphics or CV CAD model access. MD Patran Neutral files are specially formatted for the purpose of providing a means of importing and exporting model data.

Action Names the operation to be performed; for example Create, Edit, or Delete.

Object Identifies the geometric entity upon which the action is performed, for example, Solid. In this case, if the Action is Create, then the command requests that a solid be created.

Method Specifies the procedure which perform the action. Taking the above example one step further, if the Method is Surface, a solid is created by one of the techniques that utilize surfaces.

65Chapter 3: ModelingModeling in MD Patran

Geometry received into the database, whether through direct access or import, is treated as if it had been built in MD Patran; meshing, load and boundary condition assignments, element and material properties definitions are all performed as if on MD Patran’s own “native” geometry.

Creating Finite Element Meshes in MD PatranFinite elements themselves are defined by both their topology (i.e., their shape) and their properties. For example, the elements used to create a mesh for a surface may be composed of quadrilaterals or triangles. Similarly, one element may be a steel plate modeling structural effects such as displacement and rotation, while another may represent an air mass in an acoustic analysis.MD Patran provides numerous ways to create a finite element mesh.

At this stage of using MD Patran, where you are creating a finite element mesh using the Finite Elements application form, elements are defined purely in terms of their topology. Other properties such as materials, thickness and behavior types are then defined for these elements in subsequent applications, and discussed in Chapter 6: Elements of this guide.

The most rudimentary method of creating a finite element mesh is to manually generate individual nodes, and then to create individual elements from previously defined nodes. Individual nodes can be either be generated from the geometry model or directly created using node creation tools that bypass the need for point definitions. A finite element model created manually supports the entire MD Patran element library and where applicable, MD Patran automatically generates midedge, midface and midbody nodes.

MD Patran contains many capabilities to help you manually create the right kind of finite element mesh for your model, and capabilities that automate the process of finite element creation. MD Patran provides the following capabilities for finite element modeling (FEM):

• Mesh seeding tools to control specific mesh densities in specific areas of your geometry.

• Several highly automated techniques for mesh generation.

• Equivalencing capabilities for joining meshes in adjacent regions.

• Tools to verify the quality and accuracy of your finite element model.

• Capabilities for direct input and editing of finite element data.

Automatic Meshing Tools

There are four basic mesh generation techniques available in MD Patran: IsoMesh, Paver Mesh, Auto TetMesh, and 2-1/2D Meshing. Selecting the right technique for a particular model must be based on geometry, model topology, analysis objectives, and engineering judgment.

Isomesh

Creates a traditional mapped mesh on regularly shaped geometry via simple subdivision. This method creates Quad and Tria elements on surfaces and brick elements on solids. The resulting mesh supports all element configurations in MD Patran.

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66

Paver

The Paver is an automated surface meshing technique that you can use with any arbitrary surface region, including trimmed surfaces, composite surfaces, and irregular surface regions. Unlike the IsoMesh approach, the Paver technique creates a mesh by first subdividing the surface boundaries into mesh points, and then operates on these boundaries to construct interior elements.

TetMesh

Arbitrary solid mesher generates tetrahedral elements within MD Patran solids defined by an arbitrary number of faces or volumes formed by collection of triangular element shells. This method is based on MSC plastering technology.

2-1/2D Mesher

Transforms a planar 2-D mesh to produce a 3-D mesh of solid elements, using sweep and extrude operations.

Accessing the Finite Element Application

All of MD Patran’s finite element modeling capabilities are available by selecting the Finite Element button on the main form.

Like the Geometry Application, the top portion of the Finite Element form contains three keywords, Action, Object, and Method; these remain the same throughout all activities. Finite Element (FE) Meshing, Node and Element Editing, Nodal Equivalencing, ID Optimization, Model Verification, FE Show, Modify and Delete, and ID Renumber, are all accessible by setting the Action/Object/Method combination on the Finite Elements form.

For complete descriptions on creating geometry models, see the MD Patran Reference Manual, Part 3: Finite Element Modeling.

Caution: In all finite element simulations, the size and shape of the elements is one aspect that controls the accuracy. In explicit simulations, small elements will also result in the need to reduce the time step or increase the mass scaling. See Chapter 4: Special Modeling Techniques.

67Chapter 3: ModelingExample using MD Patran

Example using MD PatranThe objective of the following example is to highlight the templates and forms that are used in MD Nastran SOL 700 Preference to set up the model, define the material, boundary conditions, type of output files generated and post processing.

Description of the ProblemThe figure below is an example of a tapered beam which impact a rigid wall at 10,000 inches/seconds.

The beam is 40 inches in length and is made of steel with the following properties:

Density = 0.000783 lbm/in3

Elastic Modulus = 2.1E9 psiYield Stress = 1.4E7 psiPoisson’s ratio = 0.3

The example is designed to make the user familiar with the following objectives:

1 – How to define a Contact Body and Contact Table.

2 – How to define the initial velocity for the tapered beam.

3 – How to define elastoplastic material properties using MAT24 material model.

Beam Tip = 2x2 in

Rigid Plate = 6x6 in

Beam End = 4x4 in Beam Length = 40 in

Impact Velocity = 10,000in/sec

Impact of a Tapered Beam to a Rigid Wall

Steel PropertiesDensity = 0.000783 Ibm/in3

Elastic Modulus = 21E9 psiYield Stress = 1.4E7 psiPoisson’s Ratio = 0.3

YX

Z

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideExample using MD Patran

68

Solution TypeMD Nastran can simulate many different types of structural response. In general an analysis type can be either static or dynamic. In a static analysis, loads and boundary conditions are applied to a model and the response is assumed to remain constant over time. In dynamic analysis the response changes over time. In MD Nastran, both static and dynamic analysis may simulate linear response or nonlinear response. SOL 700 incorporates the formulations and functionality to simulate nonlinear dynamic structural responses. The specific procedure MD Nastran uses is specified on the Executive Control Statement by the ID entry.

Specifying the Solution TypeMD Nastran Explicit Nonlinear (SOL 700) is designated with the following Executive Control Statement in the MD Nastran Bulk Data file, where the ID entry indicates which analysis procedure is to be run.

References• “SOL 700,ID” in the MD Nastran Quick Reference Guide. In this release, ID = 129 indicates

a nonlinear dynamic analysis.


Defining the Solution Type in MD PatranPrior to selecting a Solution Type, check to see that under Analysis Preferences the Analysis Code is set to MD Nastran, and then set the Analysis Type to Explicit Nonlinear.

Open a new data base and name it tapered_beam:

a. Open File and click New. c. Type tapered_beam under File name and click OK.

b. Select MD Nastran for Analysis Code. d. Select Explicit Nonlinear for analysis and click OK.

aa

bb

c

d

c

d


70

Defining Material Model

Define material properties using MAT24 constitutive material model:

a. Materials: e. Enter

Action: Create Density = 0.000783

Object: Isotropic (lsdyna) Elastic Modulus = 2.1E9

Method: Manual Input Poisson Ratio = 0.3

b. Material Name: steel Yield Strength = 1.4E7

c. Click Input Properties f. Click OK

d. Constitutive Model: Elastoplastic g. Click Apply

Implementation: Piecewise Linear (MAT24)

Curve Type Bilinear

c

b

a

g

c

b

a

g f

e

d

f

e

d


Defining the Initial Velocity

Define a group called “beam” and proceed with creating the beam finite element model. Post group beam and define an impact velocity of 10000 in/sec in positive Z direction for the beam.

a. Group: Post h. Click OK

b. Select beam i. Click Select Application Region

c. Click Apply and then Cancel j. Click FEM

d. Loads/Boundary Conditions k. Select Nodes: Select all beam nodes

Action: Create l. Click ADD

Object: Initial Velocity m. Click OK

Method: Nodal n. Click Apply

e. Enter

New Set Name initial_velocity

f. Click Input Data

g. Enter

Trans Veloc = <0 0 10000>

aa

c

b

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b

i

f

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d

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i

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72

Defining the Rigid Plate

Create a group called rigid_plate and proceed with constructing the finite element model of the rigid plate. Post group rigid_plate and define a constraint on the plate.

a. Group: Post g. Enter

b. Select rigid_plate Translations: <0 0 0>

c. Click Apply and then Cancel Rotations: <0 0 0>

d. Loads/Boundary Conditions h. Click OK

Action: Create i. Click Select Application Region

Object: Displacement j. Select Nodes: Select all the nodes on the plate.

Method: Nodal k. Click ADD

e. Enter l. Click OK

New Set Name fix_plate

m. Click Apply

f. Click Input Data

aa

c

b

c

b

i

e

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f

m

i

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d

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m

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Defining the Contact Between the Beam and the Rigid Plate

Post both group beam and rigid_plate and define a contact between the plate and the beam.

a. Loads/Boundary Conditions: f. Click OK

Action: Create g. Click Apply

Object: Contact h. New Set Name: beam

Type: Element Uniform i. Target Element Type: 3D

Option: Deformable Body j. Click Select Application Region

b. New Set Name: rigid_plate k. Choose beam elements and click ADD

c. Target Element Type: 2D l. Click OK

d. Click Select Application Region m. Click Apply

e. Choose plate elements and click ADD

g

d

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b

a

g

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b

a

g

d

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b

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74

Defining the Simulation End Time and Contact Table

Now, we are ready for analysis.

a. Analysis: d. Available Subcases: Default

Action: Analyze e. Available Load Cases: Default(Subcase Default is related to Load Case Default)

Object: Entire Model f. Click Subcase Parameters

Method: Full Run g. Enter

b. Job Name: tapered_beam End Time 0.005

c. Click Select Application Region h. Click Contact Table

a

b

c

a

b

c

d

e

f

d

e

f

Note: Only when “Contact Table” in Solution Parameter is clicked in MD Patran is the Bulk Data Input BCTABLE written to the input deck. Otherwise, BCONTACT - ALLBODY is written in the Case Control section of the input file.


i. Global Contact Detection: First-> Second l. Click Apply

j. Click OK m. Click Cancel

k. Click OK

l mll m


76

Defining the Time Step for Output Files

Specify the time step for the output files.

a. Analysis:

Action: Analyze

Object: Entire Model

Method: Full Run

b. Enter in Direct Text Input

Bulk Data Section: PARAM, DYDTOUT, 0.0001

c. Click OK.

aa

b

c

b

c


Running the Model

Now you are ready to write the input deck.

a. Analysis:

Action: Analyze

Object: Entire Model

Method: Full Run

b. Click Apply

b

a

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78

Examine the MD Nastran Input File

$ NASTRAN input file created by the MSC MSC.Nastran input file$ translator ( MSC.Patran 13.1.073 ) on February 14, 2005 at 15:15:28.$ Direct Text Input for Nastran System Cell Section$ Direct Text Input for File Management Section$ Explicit NonLinear AnalysisSOL 700,NLTRAN STOP=1$ Direct Text Input for Executive ControlCENDTITLE = MSC.Nastran job created on 14-Feb-05 at 15:01:47$ Direct Text Input for Global Case Control Data....$ Direct Text Input for this SubcaseBEGIN BULKPARAM POST 0PARAM LGDISP 1PARAM PRTMAXIM YESTSTEPNL 1 100 .00005 1 ADAPT 2 10....BCTABLE 1 3

SLAVE 4 0. 0. 0. 0. 0 0.0 0 0

MASTERS 4SLAVE 4 0. 0. 0. 0. 0 0.

0 0 0MASTERS 3SLAVE 3 0. 0. 0. 0. 0 0.

0 0 0MASTERS 3

$ Direct Text Input for Bulk DataPARAM, DYDTOUT, 0.0001$ Elements and Element Properties for region : steel_beamPSOLID 1 1 0$ Pset: "steel_beam" will be imported as: "psolid.1"CHEXA 1 1 1 2 5 4 10 11

$ Referenced Material Records$ Material Record : steel$ Description of Material : Date: 14-Feb-05 Time: 14:52:07MATD024 1 7.83-4 2.1+9 .3 1.4+7

-1.

$ Loads for Load Case : DefaultSPCADD 2 1$ Initial Velocities of Load Set : initial_velocityTIC 1 1 3 10000.TIC 1 2 3 10000.TIC 1 3 3 10000.TIC 1 4 3 10000.TIC 1 5 3 10000.TIC 1 6 3 10000.....TIC 1 368 3 10000.TIC 1 369 3 10000.$ Displacement Constraints of Load Set : fix_plateSPC1 1 123456 370 THRU 418

$ Deform Body Contact LBC set: contact_mid$ Deform Body Contact LBC set: plateBCBODY 3 3D DEFORM 3 0BSURF 3 161 162 163 164 165 166 167

168 169 170 171 172 173 174 175....$ Deform Body Contact LBC set: beamBCBODY 4 3D DEFORM 4 0BSURF 4 1 2 3 4 5 6 7

8 9 10 11 12 13 14 15

This section describesthe Explicit NonlinearSolution Sequence.

This TSTEPNL describes

simulation. The actual number of Time Stepsand Time Increment is determined by SOL 700 during theanalysis.

This PARAM entrydescribes the timeinterval of d3plot outputs.

This section of describesthe contact table definition,as well as the master bodyand the slave body in eachcontact relation.

This section describesthe Elastic-PlasticMaterial definition.

This section describes the initial velocity to thetapered beam and fixedboundary condition forthe plate.

This section describesthe contact bodies of the tapered beam and the plate.

the End Time of the

79Chapter 3: ModelingInput and Output Files Created During the Simulation

Input and Output Files Created During the Simulation

Simulation on Windows PlatformsThe following files are created in your working directory:

Input file:

tapered_beam.bdf Input deck for MD Nastran

tapered_beam.dytr.dattapered_beam.dytr.str Intermediate input files

Output file:

tapered_beam.dytr.d3hsp Contains model summary, calculation process, CPU time, etc.

tapered_beam.dytr.out Contains time step summary and CPU timing information

tapered_beam.dytr_prep.d3hsptapered_beam.dytr_prep.out Contains translation summary of the preparation phase

tapered_beam.f06 MD Nastran analysis output that contains model summary

Binary Result file:

tapered_beam.dytr.d3plot Contains complete model used for plotting deformed shape and stress contour.

tapered_beam.dytr.d3thdt Contains subset of the model used for time history plotting of element and grid point data.

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Examine the Output file tapered_beam.dytr.out for Errors and Debugging

1 t 0.0000E+00 dt 4.35E-07 flush i/o buffers1 t 0.0000E+00 dt 4.35E-07 write d3plot file

239 t 9.9943E-05 dt 4.11E-07 write d3plot file488 t 1.9964E-04 dt 3.86E-07 write d3plot file740 t 2.9994E-04 dt 3.86E-07 write d3plot file

::

12088 t 4.7999E-03 dt 3.85E-07 write d3plot file12341 t 4.9000E-03 dt 3.85E-07 write d3plot file12593 t 5.0000E-03 dt 4.09E-07 write d3plot file

Warning: Airbag data not written to full restart file

******** termination time reached ********

12593 t 5.0004E-03 dt 4.09E-07 write d3plot file

N o r m a l t e r m i n a t i o n::

C P U T i m i n g i n f o r m a t i o n

Processor Hostname CPU(seconds)---------------------------------------------------------------------------

# 1 RWC-DYTRAN 9.4844E+00---------------------------------------------------------------------------T o t a l s 9.4844E+00

Start time 02/14/2005 15:15:31 End time 02/14/2005 15:15:40 Elapsed time 9 seconds ( 0 hours 0 minutes 9 seconds)

N o r m a l t e r m i n a t i o n

This section (*.dytr.out) describes increment (740) and time step (3.86E-07) at this increment.

This section describes successful completion ofthe analysis.

This section describes CPU time and Elasped Time of the job.

81Chapter 3: ModelingPostprocessing

PostprocessingThe MD Nastran output the results in the binary native file d3plot and d3thdt for X-Y plots. The MD Nastran native output files *.op2 and *xdb will be supported in the future releases.

Read in the results by attaching the d3plot file where the analysis results are stored.

a. Analysis: d. Click OK

Action: Access Results e. Click Apply

Object: Attach d3plot

Method: Result Entries

b. Click Select Results File

c. Click tapered_beam_dytr.d3plot

b

a

e

b

a

e

d

c

d

c

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePostprocessing

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Let’s have a Quick Plot of the Deformed Shape of the last cycle.

a. Results: e. Select Deformation Result

Action: Create Select Displacement

Object: Quick Plot f. Click on Deform Attributes

b. Select Results Cases g. Scale Interpretation

Click ...:Time 0.005 Click True Scale

c. Select Fringe Result h. Deactivate Show Undeformed

Click Stress Components i. Click Apply

d. Position ...((NON-LAYERED))

Quantity: von Mises

e

d

c

b

a

f

e

d

c

b

a

f

h

g

i

h

g

i

83Chapter 3: ModelingPostprocessing

a. Results: f. Scale Interpretation

Action: Create Click True Scale

Object: Deformation g. Deactivate Show Undeformed

b. Select Results Cases h. Click on Animation Options

Select all cases i. Animation Graphics

c. Select Deformation Result Click 3D

Select Displacement Components j. Number of Frames 52

d. Click Animate k. Click Apply

e. Click Display Attributes

e

d

c

b

a

e

d

c

b

a

h

g

f

h

g

f

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePostprocessing

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a. Results: c. Position ...((NON-LAYERED))

Action: Create Quantity: von Mises

Object: Fringe d. Enable Animate

b. Select Fringe Result e. Click Apply

Stress Components

dc

b

a

e

dc

b

a

e

85Chapter 3: ModelingRunning a Batch Job

Running a Batch JobAfter the generation of the input file is complete, it is submitted for execution as a batch process. Once the input file has been submitted, you have no additional interaction with MD Nastran until the job is complete except that you can terminate the job prior to completion if it becomes necessary and monitor several keys files such as: .f06, .out, etc. MD Nastran is executed with a command called nastran. (Your system manager may assign a different name to the command.) The nastran command permits the specification of keywords used to request options affecting MD Nastran job execution. The format of the nastran command is:

nastran input_data_file [keyword1 = value1 keyword2 = value2 ...]

where input_data_file is the name of the file containing the input data and keywordi=valuei is one or more optional keyword assignment arguments. For example, to run an a job using the data file example1.dat, enter the following command:

nastran example1

See “The nastran Command” in MD Nastran Quick Reference Guide.

The details of submitting an MD Nastran job are specific to your computer system — contact your computer system administrator or your MD Nastran Installation and Operations Guide for further information.

How to Tell When the Analysis is DoneIf you submit the job from the MD Nastran icon (i.e., outside MD Patran), as long as the parent window the job was run from is active, the analysis is still running. If you submit the job from within MD Patran and use -stdout when you execute MD Patran, you can look in the MD Patran parent window and it will tell you when it submits the Nastran job, and also when the Nastran job is completed. Of course, you can always use the Analysis Manager. Once the job is complete look in the parent window to see what files were generated.

How to Tell if the Analysis Ran SuccessfullyLook in the working directory and you will see the typical jobname.f06, jobname.out. If these files are there, you successfully submitted the Nastran job. If you submitted a job with SOL 700, there will also be some jobname.dytr.xxx files in the subdirectory. To see if the run was successful, open jobname.dytr.out and search for errors. If you see “Normal termination” at the end of the file, it means the run was successful.

It is always a good practice to monitor the time step of the calculation. If you noticed that your job has taken a long time to be completed, check the time step and if it is too small (E-08 to E-09 seconds or smaller), you should investigate the model to determine the causes of the small time step. Post processing

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the results can give you clues as to what is happening to the model during the simulation and which elements are controlling the timestep. See Chapter 4: Special Modeling Techniques to control small timesteps.

Running a Parallel Job

When Not Connected to the Network

If you disconnect a PC system from the network and want to run a parallel job on that system, you will have to install the Microsoft Loopback Adapter. Follow these steps:

Go to Control Panel, Add/Remove Hardware.

Select the hardware task you want to perform:

Add/Troubleshoot a device

Choose a Hardware Device:

Add a new device

Do you want Microsoft Windows to search for your new hardware?

No, I want to select the hardware from a list

Select the type of hardware you want to install:

Network adapters

Select Network Adapter:

Manufacturers: Microsoft

Network Adapter: Microsoft Loopback Adapter

It will now install the loopback adapter. You will have to enable/disable the loopback adapter as you remove/connect your machine to the network.

On Windows XP System When Not a Member of a Domain

If you will be running a parallel job on a Windows XP system that is not a member of a domain, you will have to modify a registry entry.

Using regedt32, look for the following key:

HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Lsa

"forceguest" : REG_DWORD : 00000001

If you find this key, change the REG_DWORD value to 0. The name may also appear as ForceGuest.

If you do not have this registry entry, your system will function properly.

87Chapter 3: ModelingHow to Tell if the Analysis Ran Successfully

On Windows XP SP2

After you install or upgrade to Windows XP SP2, the RPC protocol does not permit anonymous requests to the RPC Endpoint Mapper but requires client requests be authenticated.

To workaround this problem, do the following:

From a command prompt, run gpedit.msc.

Select Computer Configuration,expand Administrative Templates,expand System,click Remote Procedure Call, double click RPC Endpoint Mapper Client Authentication.

Change the value to Enabled.

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Chapter 4: Special Modeling TechniquesMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

4Special Modeling Techniques

Artificial Viscosity 90

Hourglass Damping 91

Mass Scaling 95

Time Domain NVH 97

Prestress (Implicit to Explicit Sequential Simulation) 107

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Artificial ViscosityTwo types of artificial viscosity are used in SOL 700; bulk viscosity and hourglass viscosity are used to control the numerical process. The parameters for bulk viscosity are material parameters. The hourglass viscosity parameters are defined per property.

Bulk ViscosityArtificial bulk viscosity is used to control the formation of shock waves. Shock waves are the propagation of discontinuities in velocity. The simplest example of a shock wave is a “square wave.” An ideal impact between two flat surfaces generates a square wave. Materials that stiffen upon deformation can produce a shock wave from a smooth wave profile. A finite element model of a continuous body cannot numerically represent this propagating discontinuity. When a time integration scheme without algorithmic damping (such as the explicit central difference method) is used to integrate the response, severe oscillations in amplitude trail the shock front. These oscillations can be traced to the limitations imposed by the finite frequency spectrum of the finite element mesh.

To control the oscillations trailing the shock front, artificial bulk viscosity is introduced. Artificial bulk viscosity is designed to increase the pressure in the shock front as a function of the strain rate. The effect on the shock wave is to keep it smeared over approximately five elements. Reducing the coefficients in an attempt to steepen the wave front may result in undesirable oscillations trailing the shock, a condition sometimes referred to as “overshoot.”

An artificial viscosity term “ ” is added to the pressure. An artificial viscosity term can be considered as a modification to the pressure , which is replaced by:

The definition of the artificial viscosity term that modifies the pressure:

where

The following parameters can be used in the SOL 700 to define the linear and quadratic bulk viscosity coefficients:

PARAM,DYBULKL, value (Linear coefficient, Default =0.06)

PARAM,DYBULKQ1, value (Quadratic coefficient, default = 1.5)

See How to control Hourglassing in SOL 700.

= constant = 1.5

= constant = 0.06

is the characteristic element dimension and is the material speed of sound.

Q

p

p* p Q+=

Q max ρ CQ2

d2 V·

V--- V·

V---⋅ ⋅ ⋅ ⋅– 0,⎝ ⎠

⎛ ⎞ max ρ CL d cV·

V---⋅ ⋅ ⋅ ⋅– 0,⎝ ⎠

⎛ ⎞+=

CQ

CL

d c

91Chapter 4: Special Modeling TechniquesHourglass Damping

Hourglass DampingThe solid and shell elements in SOL 700 have only one integration point at the center of the element. This makes the program very efficient since each element requires relatively little processing, but it also introduces the problem of hourglassing.

For simplicity, consider the two-dimensional membrane action of a CQUAD4 element.

The element has four grid points, each with two degrees of freedom. There are, therefore, a total of eight degrees of freedom and eight modes of deformation. There are three rigid body modes, two translational modes, and one rotational mode.

With a single integration point, two direct and one shear stress are calculated at the center of the element. This means that only three modes of deformation have stiffness associated with them.

Two modes of deformation remain, that correspond to the linear stress terms. With a single integration point, these have no stiffness associated with them and are called the zero energy or hourglass modes.

When no measures are taken to stop these modes from occurring, they rapidly spread through the mesh and degrade the accuracy of the calculation (see Figure 4-1), reduce the time step, and ultimately cause the analysis to abort when the length of the side of an element becomes zero.

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Similar zero energy modes exist for the bending deformation of CQUAD4 elements, in CHEXA and CPENTA elements. CTRIA3 and CTETRA elements do not suffer from hourglassing, since no zero energy modes exist in these elements.

Figure 4-1 Deformation of a Mesh Showing Hourglassing

How to control Hourglassing in SOL 700

There are two methods available in SOL 700 to control the hourglassing. Viscous and Stiffness damping. The viscous form damps out hourglass modes and is carefully tuned so that other modes of deformation are not affected. The stiffness form applies forces to restrict the hourglass deformation by controlling the nonlinear part of the strain field that produces hourglassing. Normally the viscous forms work well, but, in some instances, are not adequate. The stiffness form is more effective but tends to make the elements overly stiff, depending on the input parameters selected.

Each of the hourglass forms has slightly different characteristics. The default model is efficient and recommended for general use.

The hourglass type can be specified using the PARAM,DYHRGIHQ. The hourglass coefficients can be specified using the PARAM options DYHRGQH, DYBULKQ1, and DYBULKQW. The hourglass type and the hourglass coefficients can be defined for each individual property using the HGSUPPR bulk data entry. Note that the value specified on the HGSUPPR overwrite the default values defined by using the PARAM options.

Careful modeling can help prevent the occurrence of hourglassing in a mesh. Try to avoid sharp concentrations of load and isolated constraints. Rather, try to spread the loading and constraint over as large an area as possible. Some examples of how to avoid hourglassing are shown in Figure 4-2.

In the majority of cases, hourglassing does not cause any problem. In those instances where it does begin to occur, adjustment of the type of hourglass control and the hourglass viscosity should allow the analysis to be completed successfully. Extreme cases of hourglassing are normally caused by coarse meshes. The only solution is to refine the mesh.

Increasing the hourglass coefficient helps prevent hourglassing. However, excessively large values can cause numerical problems. Start with the default value and only increase it if excessive hourglassing occurs.

93Chapter 4: Special Modeling TechniquesHourglass Damping

Figure 4-2 Hourglass Promotion and Avoidance

The following parameters are used in MD Nastran SOL 700 to define viscous and stiffness methods of hourglassing:

Where QH is the default hourglass coefficient. Values of QH that exceed 0.15 may cause instabilities. The recommended default applies to all options except for Belytschko-Bindeman (type 6) of hourglass control. The stiffness forms, however, can stiffen the response especially if deformations are large and therefore should be used with care. For the shell and membrane elements QH is taken as the membrane

PARAM,DYHRGIHQ

(Integer > 0, Default=1)Default hourglass viscosity type

= 1: standard (default)= 2: Flanagan-Belytschko integration= 3: Flanagan-Belytschko with exact volume integration= 4: stiffness form of type 2 (Flanagan-Belytschko)= 5: stiffness form of type 3 (Flanagan-Belytschko)= 6: Belytschko-Bindeman assumed strain co-rotational stiffness form for

solid elements only.

PARAM,DYHRGQH (Real >= 0.0)

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94

hourglass coefficient, the bending as QB, and warping as QW. These coefficients can be specified independently but in SOL 700 the default is set at QH=QB=QW which is adequate (see PARAM,DYBULKL and PARAM,DYBULKQ1 below). For type 6 solid element hourglass control, see note 4 below.

Below are some guidelines to use the different formulations for the hourglass viscosity:

1. Viscous hourglass control is recommended for problems deforming with high velocities. Stiffness control is often preferable for lower velocities, especially if the number of time steps are large. For solid elements the exact volume integration provides some advantage for highly distorted elements.

2. For automotive crash, the stiffness form of the hourglass control with a coefficient of 0.05 is preferred by many users.

3. Bulk viscosity is necessary to propagate shock waves in solid materials and therefore applies only to solid elements. Generally, the default values are okay except in problems where pressures are very high, larger values may be desirable. In low density foams, it may be necessary to reduce the viscosity values since the viscous stress can be significant. It is not advisable to reduce it by more than an order of magnitude.

4. Type 6 hourglass control is for 2-D and 3-D solid elements only. Based on elastic constants and an assumed strain field, it produces accurate coarse mesh bending results for elastic material when hourglass coefficient QH=1.0. For plasticity models with a yield stress tangent modulus that is much smaller than the elastic modulus, a smaller value of QH (0.001 to 0.1) may produce better results. For any material, keep in mind that the stiffness is based on the elastic constants, so if the material softens, a QH value smaller than 1.0 may work better.

5. In part, the computational efficiency of the Belytschko-Lin-Tsay and the under integrated Hughes-Liu shell elements are derived from their use of one-point quadrature in the plane of the element. To suppress the hourglass deformation modes that accompany one-point quadrature, hourglass viscous or stiffness based stresses are added to the physical stresses at the local element level.

PARAM,DYBULKL, value (Linear coefficient CL, Default = 0.06)

The linear bulk viscosity coefficient CL. The defaults are: CL=QB=QH. See remark 4 above.

PARAM,DYBULKQ1, value (Quadratic coefficient, default = 1.5)

PARAM,DYBULKQW,value (Hourglass coefficient for warping). The default is QW=QB

95Chapter 4: Special Modeling TechniquesMass Scaling

Mass ScalingThe explicit dynamics procedure of SOL 700 uses relatively small time steps dictated by the shortest natural period of the mesh: the analysis cost is in direct proportion to the size of the mesh. There are two types of problems where the cost effectiveness of the analysis can be increased:

• If a mesh consists of a few, very small (or stiff) elements, the smallest (or stiffest) element determines the time step for all elements of the mesh.

• If a few severely distorted elements are obtained by the analysis, the most distorted element determines the time step for all elements of the mesh. This may result in a very small stable time step.

Speedup of those problems can be achieved by using mass scaling. Mass scaling is based on adding numerical mass to an element so that its time step never becomes less than the minimum allowable time step defined by the user. One can also employ mass scaling in a selective manner by artificially increasing material density of the parts needed for mass-scaling. This manual form of mass scaling is done independently of the automatic mass scaling invoked with PARAM,DYTSTEPDT2MS. Note that mass scaling can be risky in areas where either inertia effects are relevant or contact with other parts is expected to occur.

Problems Involving a Few Small ElementsIt is common practice that meshing of real-life problems may involve some relatively small elements: elements frequently localized in transition region and meant to connect large structural parts to each other. Those elements determine the time step of the whole calculation although they might be present in the model to a very limited extent. Speedup can be realized by using mass scaling.

• Make a run for one cycle and retrieve the time step of all elements.

• By using a postprocessing program, view which elements are determining the time step and filter out the elements whose time steps exceed a user-defined minimum time step.

• See what the impact would be of specifying this new minimum time step. Select the value of minimum time step such that hardly any elements would be scaled in the area of interest (for example, as much as possible outside the impact region in a crash simulation)

Problems Involving a Few Severely Distorted ElementsThere are many applications where elements are distorted to such a high extent that a few of them determine the time step for all elements of the mesh. For example, crushing of a subfloor structure frequently involves failure modes associated with the occurrence of severely distorted elements. Modeling this kind of crushing behavior without including a failure mechanism result in a stable time step that is too small. Since those elements are often present in a relatively small region, the mass scaling method might be a good means to artificially speed up the calculation without losing the capability to model the global crushing behavior. Note that to prevent severely distorted elements, it is recommended that a proper failure mechanism be included, instead of coping with the distorted elements by making use of the mass scaling method. Some guidelines:

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• Since you do not know in advance which elements will become too distorted, you should first run the analysis as far as possible without mass scaling. You should gauge and monitor the time step of all elements.

• If the problem ends up with a too small stable time step, the analysis finishes prematurely. See which elements are so severely distorted and decide what a reasonable minimum time step might be without affecting elements in the area of interest. See the guidelines of the previous section.

• Rerun the analysis specifying mass scaling if the region of highly distorted elements is relatively small compared to the whole model.

• If there is too much mass added to the grid points of those elements, the model might show significantly different inertia effects, and subsequently, different global structural response. In order to avoid this, no more mass is added if the numerically added mass exceeds a certain percentage (PARAM, DYTERMNENDMAS).

How to define Mass Scaling in SOL 700

The following PARAM entries can be used to define mass scaling in SOL 700:

PARAM,DYTSTEPDT2MS,value (Default=0.0, time step size for mass scaled solution)

PARAM,DYSTEPFCTL,value (Default=0.9, scale factor for internally calculated time step)

PARAM,DYTERMNENDMAS,value (Default=1.0e20, Percent change in total mass for termination of calculation).

Anytime you add nonphysical mass to increase the time step in a dynamic analysis, you affect the results (think of F = ma). Sometimes the effect is insignificant and in those cases adding nonphysical mass is justifiable. Examples of such cases may include the addition of mass to just a few small elements in a noncritical area or quasi-static simulations where the velocity is low and the kinetic energy is very small relative to the peak internal energy. In the end, it's up to the judgement of the analyst to gauge the affect of mass scaling. You may have to reduce or eliminate mass scaling in a second run to gauge the sensitivity of the results to the amount of mass added.

When the value of DT2MS in PARAM,DYTSTEPDT2MS is input as a negative value, mass is added only to those elements whose time step would otherwise be less than Fact * |DT2MS |. The value of Fact is specified in PARAM,DYSTEPFCTL. By adding mass to these elements, their time step becomes equal to Fact *|DT2MS|. An infinite number of combinations of Fact and DT2MS will give the same product; i.e., time step but the added mass will be different for each of those combinations.

The trend is that the smaller the time step factor Fact, the greater the added mass. In return, stability may improve as time step factor Fact is reduced (just as in non-mass-scaled solutions). If stability is a problem with the default time step factor of 0.9, try 0.8 or 0.7. If you reduce Fact, you can increase |DT2MS| proportionally so that the product/time step is unchanged.

The difference between using a positive or negative number for DT2MS is:

Negative: Mass is added to only those elements whose time step would otherwise be less than Fact*abs(DT2MS). When mass scaling is appropriate, this is the recommended approach.

Positive: Mass is added or taken away from elements so that the time step of every element is the same. There is no clear advantage in using this method over the negative DT2MS method.

97Chapter 4: Special Modeling TechniquesTime Domain NVH

Time Domain NVHThis is a new methodology in MD Nastran SOL 700 that is used to compute the natural frequencies of a structure similar to what is done in experimental modal identification using impact or other transient testing. The user applies an impact, simulates driving a vehicle over a rough road or some other type of transient loading to the structure such that it is likely to excite all the important modes. A standard SOL 700 analysis is run to output accelerations, velocities and/or displacements at selected grids using a reasonably fine output delta time. After the nonlinear analysis finishes, a postprocessing operation is used to transform the selected outputs from the time domain to the frequency domain using Fast Fourier Transforms. Various criteria are available to select which peaks will be approximately defined as modes (eigenvectors). This approximation is reasonably accurate if the damping is small. The amplitudes of all the selected degrees of freedom for each of the selected modal frequencies are normalized and become the eigenvectors and saved on a file. A "restart" capability is available to change the criteria selections. In addition, a user may pick modes from plots made from data saved during the first run and then compute the eigenvectors associated with the chosen natural frequencies.

The advantages of Time Domain NVH include the ability to consider the material and structural nonlinearity, damping and contact interface between various components. In other words, instead of utilizing a linear implicit method, an explicit approach is under taken to first, predict the nonlinear behavior of the structure, and then compute the NVH characteristics by FFT method. The double precision version of MD Nastran Nonlinear Explicit (SOL 700) is used to ensure higher fidelity of the solution.

The disadvantage of time domain NVH technique is the excessive CPU time since an analysis may be required to run for a few seconds, considered a long runtime for an explicit simulation, before an FFT is performed to compute frequencies and mode shapes. However, this problem is becoming less significant due to the advanced Distributed Memory Parallel (DMP) technology in SOL 700 and dramatic hardware performance improvements.

Figure 4-3 Time Domain NVH of vehicle running on proving grounds (courtesy of ETA)

Time Domain Data Frequency Domain Data

FFT Process

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To define a time domain NVH simulation, refer to the new entries of TIMNVH, TIMSML, and TIMNAT in the QRG and see the chassis example in SOL 700 User Guide. A simple example is listed below.

Time Domain NVH Example – Plate Subjected to a Pulse Loading

SOL 700 entries included:SOL 700TSTEPNLPARAM,S700NVH1TIMNVHTIMNAT

Description:

This is an example of a virtual dynamic test. A plate was modeled and a impulse loading was applied at one of corner points. Time histories were obtained at six points and they were translated by Fast Fourier Transformation (FFT) method to a frequency domain to obtain modal frequencies and shapes.

Model

To build a model, a plate was constructed with 231 grid points and 200 quadratic elements. A fixed boundary condition was applied along one end. All shell elements were Belytschko-Wong-Chiang formulation. The impact loading at the corner of the plate was modeled by defining the load time history as shown in Figure 4-4. The simulation time is 1.024 seconds.


Figure 4-4 Impluse Load applied at One Corner of the Plate

Time NVH Scheme

Input

The loading was applied at node 231 by using FORCE and TABLE1 entries. Using TIMNVH, TIMNAT, and PARAM, S700NVH1, dynamic properties of the plate were obtained. There were three steps to acquire the dynamic properties as follows:

Step 1: Find modal properties using TIMNVH entry in the first trialStep 2: Check the obtained modal properties and select required natural frequenciesStep 3: Re-run with selected natural frequencies

MD Nastran SOL 700 (impluse loading)

Time history- Displacement- Velocity- Acceleration (default)

Time domain -> Frequency domain using FFT

Extract dynamic properties:Natural frequencies and Mode shapes


100

Input file timnvh.bdf

SOL 700 is a executive control card similar to SOL 600. It activates an explicit nonlinear transient analysis.

Case control section is below.

Bulk entry section starts.

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (10.24 ms) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

TIMNVH entry is for Time NVH analysis.

Detail for TIMNVH entry:

TIMNVH, 1, , , 1.0, 500., 3, 0.00005, 2,+

The range of natural frequencies to obtain is from 1.0 Hz to 500 Hz and only z-translational degrees of freedom is considered (3). The sampling rate is 0.00005 seconds. The peaking criterion is two, which means a peak is selected if the number of increasing and decreasing amplitude around a peak is over 2.

+, 0, 3, 1, 0.015, 0, 3, 13, .0030,+

Acceleration is selected for the response (0) and translational eigenvectors are only requested as ASCII format (3). Eignevalues are normalized by 1.0 (1) and 0.015 is selected as CLOSE value which means if there are two modes which distance is smaller than 0.015 Hz, it is assumed to be the same mode. ACII

SOL 700,NLTRAN path=3 stop=1

CENDTITLE = MD Nastran job created on 12-Dec-06 at 11:21:25LOADSET = 1$ Direct Text Input for Global Case Control DataSUBCASE 1

TITLE=This is a default subcase. TSTEPNL = 1 SPC = 2 DLOAD = 2 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL

BEGIN BULKTSTEPNL 1 100 .01024 1 ADAPT 2 10TIMNVH,1,,,1.0,500.,3,0.00005,2,++,0,3,1,0.015,0,3,13,.0030,++,11,21,116,126,221,231


file format of natural frequencies and eigenvalues are asked (0) and translational time histories of z-direction are requested (3). Frequency-amplitude data of z-direction are asked (13) and a peak which amplitude is less than 0.0030×the maximum amplitude is ignored (.0030)

+,11,21,116,126,221,231

The grid points 11, 21, 116, 126, 221 and 231 are only considered for Time Domain NVH analysis.

Bulk data entry that defines grid points and elements.

Bulk data entry that defines boundary conditions.

Bulk data entry that defines material properties.

MAT1 is an isotropic-elastic material card of SOL 700 bulk data entry.

Bulk data entry that defines properties for shell elements with .001 thickenss.

Bulk data entry that defines applying forces.

CQUAD4 1 1 1 2 23 22..$ Nodes of the Entire ModelGRID 1 0. 0. 0...

SPCADD 2 1$ Displacement Constraints of Load Set : fixedSPC1 1 123456 1 22 43 64 85 106 127 148 169 190 211

$ Referenced Material Records$ Material Record : steel$ Description of Material : Date: 12-Dec-06 Time: 11:09:13

PSHELL 1 1 .001 1 1

$ Loads for Load Case : DefaultTLOAD1 4 5 1LSEQ 1 5 3DLOAD 2 1. 1. 4$ Nodal Forces of Load Set : impactFORCE 3 231 0 .01 0. 0. -1.


102

Bulk data entry that defines tables.

Input file timnvh1.bdf

This input is for the refinement of the selection of modal frequencies and mode shapes.

Only different part is shown

PARAM, S700NVH is for the re-run of Time NVH analysis. Using 1 as option Time Domain NVH analysis is carried out without re-running SOL 700.

The PEAK value (original: 2) in TIMNVH card is changed to -2 to use TIMNAT card.

TIMNAT card is for the control of the natural frequency selection. In this job, 20.5, 88, 129, 285, 360 Hz are selected to get the results.

Results

There are three types of new results file from Time Domain NVH analysis.

1. mode.out: the natural frequencies and eigenvalues selected are restored.

2. ampl-freq-00000116-3.txt : amplitude-frequency output of DOF =3 at grid point 116.

3. time-hist-00000116-3.txt: time history output of DOF =3 at grid point 116.

From the ampl-freq-*** files, the frequency-amplitude plots are obtained.

$ Dynamic Load Table : hammerTABLED1 1 -10. 0. 0. 0. .001 1. .002 0. 10. 0. ENDT..ENDDATA

PARAM,S700NVH1,1TSTEPNL 1 100 .01024 1 ADAPT 2 10TIMNVH,1,,,1.0,500.,3,0.00005,-2,++,0,3,1,0.015,0,3,13,.0030,++,11,21,116,126,221,231TIMNAT,1,20.5,88.,129.,285.,360.


1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

0 100 200 300 400 500

Frequency (Hz)

Am

pli

tud

e (a

ccel

erat

ion

)

node 11

node 21

node 116

node 126

node 221

node 231

2nd mode ≈ 88.3rd mode ≈ 129.

4th mode ≈ 285.


104

In mode.out file,

Comparison of natural frequencies between SOL 103 and SOL 700 (Hz)

Comparison of mode shapes between SOL 103 and SOL 700

Mode SOL103 SOL 700 Diff(%)

1 2.0508E+01 2.0864E+01 1.74

2 8.7894E+01 8.8639E+01 0.85

3 1.2891E+02 1.2966E+02 0.58

4 2.8516E+02 2.8771E+02 0.89

5 3.6036E+02 3.6269E+02 0.65

MODES 1 18EIGV 1 2.050821E+01 11-9.13892833E-06 3.86241316E-06 3.19331321E-01 21 2.14690732E-05-1.14613790E-05 9.95118635E-01 116-9.93770531E-06 2.13863912E-06 3.30875181E-01 126 7.27358928E-06-2.94809676E-06 1.00000000E+00 221-1.23613298E-05-3.06943202E-06 3.20993777E-01 231 1.35196599E-05 6.87379272E-06 9.99026322E-01EIGV 2 8.789234E+01 11-5.69607103E-06 1.65922185E-06-6.09298304E-01 21 5.33656071E-06-7.67633319E-06-9.90453563E-01 116-3.47585317E-06 8.00428604E-07 1.97687095E-02 126 5.34868995E-06-3.96739670E-06-7.95414322E-03 221-4.07797686E-06-2.98546031E-06 5.90684541E-01 231 3.21720171E-06 7.30456344E-06 1.00000000E+00

1st Mode

Natural Frequency

Total Number of Modes

Node Number Mode Shape of x-direction Mode Shape of y-direction Mode Shape of z-direction


Mode SOL 103 SOL 700

1

2

3


106

4

5

6 points sampling

12 points sampling


107Chapter 4: Special Modeling TechniquesPrestress (Implicit to Explicit Sequential Simulation)

Prestress (Implicit to Explicit Sequential Simulation)Many applications require a prestress analysis prior to transient analysis. For example in bird strike and blade out analysis, the blades of a running jet engine have residual stresses due to high rotational velocities. SOL 700 uses the implicit solver for prestress analysis to reduce the CPU time. Prestress calculations are performed by MD Nastran Nonlinear Explicit (SOL 700) implicit double precision version where initial state of the model is written into a file that can only be used for subsequent SOL 700 transient run. This is an automated process and there is no need for dynamic relaxation to eliminate the high frequency oscillations.

The following entries may be used to define a pre-stressed state.

Caution must be exercised to use the appropriate element formulation to get consistent results during prestress analysis. Implicit and explicit element formulations might differ from each other resulting in different displacement or stress values. For this purpose, the following study is done to identify the best element formulations in MD Nastran Nonlinear Explicit (SOL 700) implicit solver that match closest to those in MD Nastran. It is therefore recommended that the users follow the guideline below for prestress simulation.

Comparative Study between Nastran and MD Nastran SOL 700 Implicit Element FormulationsModel description: Two simple models were constructed for the comparative study. The first model is a 1 x 1 square plate with a thickness of 0.01 and isotropic material. The element sizes are 0.2 x 0.2. The plate model was constructed using the standard MD Nastran CQUAD4, CTRIA3 shells and then were compared to MD Nastran Nonlinear Explicit (SOL 700) shell formulations under different loading conditions.

Similarly, the second model is a 1 x 1 x 1 solid cube constructed with Nastran solid elements CHEXA, CPENTA, and CTETRA as well as MD Nastran Nonlinear Explicit (SOL 700) different solid element formulations. The size of elements are 0.2x0.2x0.2 with isotropic material. The solid model was subjected to different loading condition to determine the consistency of the results between MD Nastran and MD Nastran Nonlinear Explicit (SOL 700).

Two sets of analyses were done to compare the results in linear and nonlinear domain. For linear analysis, only partial loading was used to remain in elastic range whereas the loading was ramped up beyond the yield point to compare the nonlinear behavior of the elements.

PRESTRS Perform prestress run to calculate an initially stressed model and write out the initial state to a file that can be used for a subsequent explicit SOL 700 run

ISTRSBE Initialize stresses and plastic strains in the Hughes-Liu beam elements

ISTRSSH Initialize stresses, history variables and the effective plastic strain for shell elements

ISTRSTS Initialize stresses, history variables and the effective plastic strain for thick shell elements.

ISTRSSO Initialize stresses and plastic strains for solid elements

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePrestress (Implicit to Explicit Sequential Simulation)

108

Model

Material: Isotropic & elastic (E=1000., v=0.3 and density=1.)

1. Plate Model:

Size: 1x1 Thickness: 0.01 Element size: 0.2x0.2

a. Boundary Condition – Fixed at all edges

b. loading condition:

case 1. Z- Translational Loading at the every grid point: 1.E-2

case 2. X-Rotational Loading at the every grid point: 1.E-2

case 3. Pressure at the top surface 1E-2


c. CTRIA

Loading and boundary conditions are the same as CQUAD4

d. CSHEAR

Not tested yet.

2. Solid Model

Cube Size: 1x1x1

a. CHEXA Element size: 0.2x0.2x0.2

Case 1 Case 2

Case 3


110

Boundary Condition: Fixed at the bottom surface.

b. loading condition

case 1. Z- Translational Loading on the every grid point at the top: 1.E-2

case 2. Pressure at the top surface 1E-2

c. CPENTA The same loading and boundary conditions as the Hexa case

d. TETRA

The same loading and boundary conditions as the Hexa case


LS-DYNA Model:

The following entries were used for the linear analysis of the shells:

*CONTROL_SHELL ,1*CONTROL_IMPLICIT_AUTO0*CONTROL_IMPLICIT_GENERAL1,0.1,,0,0,0,0*CONTROL_IMPLICIT_SOLUTION2,5,,0.01,0.01*CONTROL_IMPLICIT_SOLVER , , , ,1,100.*CONTROL_TERMINATION1.0000

Similarly for solid model:

*CONTROL_SOLID1,2*CONTROL_IMPLICIT_AUTO0*CONTROL_IMPLICIT_GENERAL1,0.1,,0,0,0,0*CONTROL_IMPLICIT_SOLUTION2,5,,0.01,0.01*CONTROL_IMPLICIT_SOLVER , , , ,1,100.

LS-DYNA Element formulations used in the study:

CQUAD4 : ELFORM= 1,2,6,10,16,18,20CTRIA: ELFORM= 1,2,6,10,16,17,18,20,21HEXA: ELFORM= 1,2,6,18TETRA: ELFORM= 1,2,4,10,16,18PENTA: ELFORM= 1,2,15,18

Results:

The results for the linear and nonlinear static analyses are shown in Table 4-1. Even though many elements showed close results (within 10% differences), several cases had a large variance of more than 10%. These are highlighted in red.


112

Table 4-1 Comparative Results Between MD Nastran and LS-DYNA Implicit Elements

Description of LS-DYNA element formulations:

SECTION_SHELL

ELFORM=1: Hughes-Liu,

ELFORM=2: Belytschko-Tsay (default),

ELFORM=6: S/R Hughes-Liu,

ELFORM=10: Belytschko-Wong-Chiang,

ELFORM=16: Fully integrated shell element,

ELFORM=17: Fully integrated DKT, triangular shell element,

ELFORM=18: Taylor 4-node quadrilateral and 3-node triangle (linear only)

ELFORM=20: Wilson 3 & 4-node DSE quadrilateral (linear only)

ELFORM=21: Fully integrated linear assumed strain C0 shell (5 DOF).

ELFORM=22: Linear shear panel element (3 DOF per node)

SECTION_SOLID

ELFORM=1: constant stress solid element,

ELFORM=2: fully integrated S/R solid. See remark 5 below,

ELFORM=3: fully integrated 8 node solid with rotational DOFs,

DYNA (linear) DYNA (nonlinear) SOL101 SOL106 %Diff (linear and SOL101) %Diff (nFORCE 1.008E+01 1.008E+01 1.040E+01 8.400E-02 3.08PRESSURE 4.031E-01 4.03E-01 4.140E-01 2.680E-02 2.63MOMENT 3.559E+00 3.559E+00 3.400E+00 Not converged 4.68

SECTION_SHELL with ELFORM=1 FORCE 1.019E+01 8.030E-02 1.040E+01 8.400E-02 2.02SECTION_SHELL with ELFORM=2 FORCE 1.019E+01 8.030E-02 1.040E+01 8.400E-02 2.02SECTION_SHELL with ELFORM=6 FORCE 1.002E+01 8.300E-02 1.040E+01 8.400E-02 3.65SECTION_SHELL with ELFORM=10 FORCE 8.620E+00 8.539E-02 1.040E+01 8.400E-02 17.12SECTION_SHELL with ELFORM=16 FORCE 1.003E+01 8.411E-02 1.040E+01 8.400E-02 3.56SECTION_SHELL with ELFORM=20 FORCE 1.084E+01 7.781E-02 1.040E+01 8.400E-02 4.23

FORCE 1.040E+01 7.369E-02 9.660E+00 7.870E-02 7.66PRESSURE 4.147E-01 2.546E-02 3.860E-01 2.680E-02 7.44MOMENT 3.557E+00 Not converged 3.470E+00 Not converged 2.51

SECTION_SHELL with ELFORM=1 FORCE 3.819E+00 7.382E-02 9.660E+00 7.870E-02 60.47SECTION_SHELL with ELFORM=2 FORCE 3.819E+00 7.382E-02 9.660E+00 7.870E-02 60.47SECTION_SHELL with ELFORM=6 FORCE 3.819E+00 7.382E-02 9.660E+00 7.870E-02 60.47SECTION_SHELL with ELFORM=10 FORCE 3.819E+00 7.382E-02 9.660E+00 7.870E-02 60.47SECTION_SHELL with ELFORM=16 FORCE 3.819E+00 7.382E-02 9.660E+00 7.870E-02 60.47SECTION_SHELL with ELFORM=17 FORCE 9.778E+00 7.397E-02 9.660E+00 7.870E-02 1.22SECTION_SHELL with ELFORM=20 FORCE 1.040E+01 7.369E-02 9.660E+00 7.870E-02 7.66SECTION_SHELL with ELFORM=21 FORCE 1.040E+01 7.369E-02 9.660E+00 7.870E-02 7.66

FORCE 6.416E-02 6.416E-02 6.500E-02 6.290E-02 1.29PRESSURE 9.952E-04 9.958E-04 9.950E-04 9.960E-04 0.02

SECTION_SOLID with ELFORM=1 FORCE 9.356E-02 9.132E-02 6.500E-02 6.290E-02 43.94SECTION_SOLID with ELFORM=2 FORCE 6.577E-02 6.734E-02 6.500E-02 6.290E-02 1.18SECTION_SOLID with ELFORM=3 FORCE 1.016E-01 1.243E-01 6.500E-02 6.290E-02 56.31


SECTION_SOLID with ELFORM=1 FORCE 6.047E-02 6.179E-02 6.050E-02 5.890E-02 0.05SECTION_SOLID with ELFORM=2 FORCE 6.047E-02 6.179E-02 6.050E-02 5.890E-02 0.05SECTION_SOLID with ELFORM=4 FORCE 4.021E-02 9.077E-02 6.050E-02 5.890E-02 33.54SECTION_SOLID with ELFORM=10 FORCE 6.047E-02 6.179E-02 6.050E-02 5.890E-02 0.05SECTION_SOLID with ELFORM=16 FORCE 6.047E-02 6.179E-02 6.050E-02 5.890E-02 0.05


SECTION_SOLID with ELFORM=1 FORCE 7.200E-02 7.191E-02 7.060E-02 6.820E-02 1.98SECTION_SOLID with ELFORM=2 FORCE 7.200E-02 7.191E-02 7.060E-02 6.820E-02 1.98SECTION_SOLID with ELFORM=15 FORCE 7.200E-02 7.191E-02 7.060E-02 6.820E-02 1.98

LS-DYNA

SOLID

CHEXA

SECTION_SOLID with ELFORM=18

CTETRA


CPENTA


LOADINGOutput

Displacement

SHELL

CQUAD4

SECTION_SHELL with ELFORM=18

CTRIA

SECTION_SHELL with ELFORM=18

Nastran


ELFORM=4: fully integrated S/R 4 node tetrahedron with rotational DOFs,

ELFORM=10: 1 point tetrahedron.

ELFORM=15: 2 point pentahedron element.

ELFORM=16: 5 point 10 noded tetrahedron

ELFORM=18: 8 point enhanced strain solid element for linear statics only,


114

Chapter 5: Constraints and LoadingsMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

5Constraints and Loadings

Constraint Definition 116

Lagrangian Loading 120

Eulerian Loading and Constraints 123

Boundary and Loading Conditions - Theoretical Background 130

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideConstraint Definition

116

Constraint DefinitionThe motion of part or all of a mesh can be prescribed by application of constraints.

Single-Point ConstraintsA single-point constraint is used to prescribe the motion of a translational or rotational degree of freedom. The constraint is effective throughout the analysis and is used to specify boundary conditions or planes of symmetry.

A single-point constraint is defined by an SPCn entry. The SPC entry defines the constraints on one grid point, while the SPC1 defines the constraints to be applied to a set of grid points. Several sets of SPC entries can be defined in the Bulk Data Section, but only those selected in the Case Control Section using the SPC = n command are incorporated in the analysis.

Single-point constraints can also be defined using the GRIDt entry. These constraints are present for the entire analysis and do not need to be selected in Case Control. This is valid only for SPC and SPC1.

Since Sol 700 is an explicit code, there is no matrix decomposition. Therefore, the problems of singular matrices that occur with some implicit codes do not exist. All, or part of the Lagrangian mesh can be entirely unconstrained and can undergo rigid body motion. SOL 700 correctly calculates the motion of the mesh. Similarly, the redundant degrees of freedom, such as the in-plane rotation of shell elements, do not need to be constrained since they do not affect the solution. The only constraints that are needed are those representing the boundary conditions of the model and those necessary for any planes of symmetry.

The following SPC types are supported in MD Nastran SOL 700.

• SPC

• SPC1

• SPCADD

• SPCD (at present, only velocity is available)

• SPCD2

Multi-Point ConstraintsMPCs are special element types which define a rigorous behavior between several specified nodes. The following MPC types which are supported for MD Nastran SOL 700:

• MPC

• RBAR

• RBE2

• RBE2A

• RBE2D

• RBE2F

• RBE3

117Chapter 5: Constraints and LoadingsConstraint Definition

• RBE3D

• RBJOINT

• REJSTIFF

• RCONN

Specifying Explicit MPCsMPC’s may be created between a dependent degree of freedom and one or more independent degrees of freedom. The dependent term consists of a node ID and a degree of freedom; an independent term consists of a coefficient, a node ID, and a degree of freedom. An unlimited number of independent terms can be specified, but only one dependent term can be specified. The constant term is not allowed in MD Nastran.

References

“MPC” in the MD Nastran Quick Reference Guide.

Contact in SOL 700The detailed theory behind the implementation of contact in SOL 700 is discussed in Chapter 8: Contact Impact Algorithm. However, here we discuss the contact methodology and definition as implemented in MD Nastran Explicit Nonlinear.

The simulation of many physical problems requires the ability to model the contact phenomena. This includes impact simulations, drop testing, component crush, crash, and manufacturing processes among others. The analysis of contact behavior is complex because of the requirement to accurately track the motion of multiple geometric bodies, and the motion due to the interaction of these bodies after contact occurs. This includes representing the friction between surfaces if required. The numerical objective is to detect the motion of the bodies, apply a constraint to avoid penetration, and apply appropriate boundary conditions to simulate the frictional behavior and heat transfer. Several procedures have been developed to treat these problems including the use of Perturbed or Augmented Lagrangian methods, penalty methods, and direct constraints. Furthermore, contact simulation has often required the use of special contact or gap elements. MD Nastran Explicit Nonlinear allows contact analysis to be performed automatically without the use of special contact elements. A robust numerical procedure to simulate these complex physical problems has been implemented in MD Nastran Explicit Nonlinear.

Contact problems can be classified as one of the following types of contact.

• Deformable-Deformable contact between two and three-dimensional deformable bodies.

• Rigid - Deformable contact between a deformable body and a rigid body, for two- or three-dimensional cases.

MPC Types Description

MPC Defines a multipoint constraint equation.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideConstraint Definition

118

Contact problems involve a variety of different geometric and kinematic situations. Some contact problems involve small relative sliding between the contacting surfaces, while others involve large sliding or penetration followed by perforation and failure. Some contact problems involve contact over large areas, while others involve contact between discrete points. The general Contact Body approach in SOL 600 is also adopted by MD Nastran Explicit Nonlinear (SOL 700) to model contact and can be used to handle most contact problem definitions.

Contact BodiesThere are two types of contact bodies in MD Nastran Explicit Nonlinear – deformable and rigid. Deformable bodies are simply a collection of finite elements as shown below.

Figure 5-1 Deformable Body

This body has three key aspects to it:

1. The elements which make up the body.

2. The nodes on the external surfaces which might contact another body or itself. These nodes are treated as potential contact nodes.

3. The edges (2-D) or faces (3-D) which describe the outer surface which a node on another body (or the same body) might contact. These edges/faces are treated as potential contact segments.

Note that a body can be multiply connected (have holes in itself). It is also possible for a body to have a self contact where the entire surface folds on to itself. A contact may include 1-D elements such as beams and rods, 2-D elements such as shells and membranes, and 3-D elements such as solids.

Each element should be in, at most, one body. The elements in a body are defined using the BCBODY option. It is not necessary to identify the grid points on the exterior surfaces as this is done automatically. The algorithm used is based on the fact that grid points on the boundary are on element edges or faces that belong to only one element. Each node on the exterior surface is treated as a potential slave grid point.

The second type of a contact body is called Rigid bodies. Rigid bodies are composed of (2-D) or (3-D) meshes and their most significant aspect is that they do not distort. Deformable bodies can contact rigid bodies and contact between rigid bodies is also allowed.

119Chapter 5: Constraints and LoadingsConstraint Definition

Rigid WallsA rigid wall is a plane through which specified slave grid points cannot penetrate. The rigid wall provides a convenient way of defining rigid targets in impact analyses.

Any number of rigid walls can be specified using WALL entries. The orientation of each wall is defined by the coordinates of a point on the wall and a vector that is perpendicular to the wall and points towards the model.

At each time step, a check is made to determine whether the slave grid points have penetrated the wall. These slave points are defined using a Case Control SET entry, and there can be any number of them. Since a check is made for every slave point at each time step, you should specify only those points as slave points that are expected to contact the wall in order to ensure the most efficient solution.

If a slave point is found to have penetrated the wall, it is moved back towards the wall so that its momentum is conserved. If the slave point subsequently moves away from the wall, it is allowed to do so.

Slave points cannot have any other constraint. They can, however, be part of other contact.

Contact DetectionDuring the incremental procedure, each potential contact node is first checked to see whether it is near a contact segment. The contact segments are either edges of other 2- D deformable bodies, faces of 3-D deformable bodies, or segments from rigid bodies. By default, each node could contact any other segment including segments on the body that it belongs to. This allows a body to contact itself. To simplify the computation, it is possible to use the BCTABLE entry to indicate that a particular body will or will not contact another body. This is often used to indicate that a body will not contact itself. During the iteration process, the motion of the node is checked to see whether it has penetrated a surface by determining whether it has crossed a segment.

For more detailed discussion on the Contact, refer to Chapter 8: Contact Impact Algorithm of this guide and see Chapter 12 in the SOL 600 User’s Guide.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideLagrangian Loading

120

Lagrangian LoadingThis section covers the different ways that the analysis model can be loaded. The facilities available are:

• Concentrated Loads and Moments

• FORCE, FORCE2, or DAREA – Fixed-Direction Concentrated Loads

• Pressure Loads

• Initial Conditions

Concentrated Loads and Moments

Concentrated loads and moments can be applied to any grid point using the DAREA, FORCE, FORCE2, entries in combination with a TLOADn entry.

The types of concentrated load that can be applied are discussed in the following section.

FORCE, FORCE2, or DAREA – Fixed-Direction Concentrated LoadsThe FORCE, FORCE2, and DAREA entries define fixed direction loads. In other words, the direction of the force is constant throughout the analysis and does not change as the structure moves.

FORCE, FORCE2, and DAREA entries have the same effect but define the loading in different ways. With the DAREA entry, you specify the grid point, the direction in the basic coordinate system in which the load acts, and the scale factor. With the FORCE entry, a concentrated force remains in the same direction for the entire problem. For FORCE2, the direction of the force follows the deformation and you define the grid point and the components of a vector giving the loading direction and the scale factor. In this case, the magnitude of the vector also acts as a scale factor, so the force in direction i is given by

.Pi ANi=

121Chapter 5: Constraints and LoadingsLagrangian Loading

On a rigid body, the concentrated load or enforced motion is specified by defining the load at the rigid body center of gravity. To do so, set the TYPE field of the TLOAD1 or TLOAD2 entries to 13 and 12, respectively. The G field in the FORCE entry references the property number of the rigid body.

Pressure LoadsPressure loads are applied to the faces of solid elements and to shell elements. Pressure loads are defined using the PLOAD or PLOAD4 entry in combination with a TLOADn entry.

TLOAD2 also references a set of PLOAD and/or PLOAD4 entries. Each entry selects the face of the element to be loaded by its grid points and defines the scale factor to be applied to the curve of pressure

versus time. The actual pressure acting on the element is given as follows:

where is the scale factor.

The direction of positive pressure is calculated according to the right-hand rule using the sequence of grid points on the PLOAD entry. For PLOAD4 entries, the pressure is inwards for solid elements and in the direction of the element normal vector for shell elements.

The RFORCE entry defines enforced motion due to a centrifugal acceleration field. This motion affects all structural elements present in the problem. The GRAV defines an enforced motion due to a gravitational acceleration field. This motion affects all Lagrangian elements.

Grid points with enforced motion cannot be:

• Attached to a rigid body.

• A slave point for a rigid wall.

To specify the motion of a rigid body, the enforced motion of the rigid-body center of gravity must be defined. To do so, set the TYPE field of the TLOAD1 and TLOAD2 entries to 12. The G field on the DAREA, FORCE entry references the property number of the rigid body.

pel

pel t( ) Ap t( )=

A

G3

G2

G4

G1

G3

G2

G1

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideLagrangian Loading

122

Initial ConditionsThe initial velocity of grid points can be defined using TIC, TIC3, and TICD entries. This allows the initial state of the model to be set prior to running the analysis. It is important to recognize the difference between initial velocities and enforced velocities. Enforced velocities specify the motion of grid points throughout the transient analysis. Initial velocities, on the other hand, specify the velocity of grid points at the beginning of the analysis. Thereafter, the velocities are determined by the calculation.

Where TIC and TIC3 set only the initial grid-point velocity, the TICD entry can be used to set the initial value of any valid grid point variable. It can also refer to a local coordinate system by including the CID1 and/or CID2 entry in the list.

123Chapter 5: Constraints and LoadingsEulerian Loading and Constraints

Eulerian Loading and Constraints

Loading DefinitionThe implementation of loading and constraints within Eulerian meshes is somewhat different than that in a Lagrangian mesh. Eulerian constraints apply to element faces within the mesh rather than to the grid points. The code allows you to set the initial conditions for material in Eulerian elements, constrain material with fixed barriers, apply gravitational body forces, apply pressure boundaries to element faces, apply flow boundaries where material enters or leaves the mesh, and couple the mesh so that the material interacts with the Lagrangian parts of the model.

If an exterior or free face of an Eulerian mesh does not have a specific boundary condition, by default, it forms a barrier through which the material cannot flow. The default can be redefined by using a FLOWDEF entry.

Flow BoundaryA flow boundary defines the physical properties of material flowing in or out of Eulerian elements and the location of the flow. The FLOW entry is referenced by a TLOAD1. The TYPE field on the TLOAD1 must be set to 4.

The FLOW entry references a set of segments, specified by BCSEG entries, through which the material flows. The subsequent fields allow you to specify the x, y, or z velocity, the pressure, and the density or specific internal energy of the flowing material. If only the pressure is defined, this gives a pressure boundary. Any variables not specified take the value in the element that the material that is flowing into or out of the flow.

Rigid WallThe BARRIER entry defines a wall that is equivalent to a Lagrangian rigid wall. This is a barrier to material transport in an Eulerian mesh. The barrier is defined by a set of faces generated from a BCSEG entry through which no material can flow.

This is the default condition for any exterior faces of the Eulerian mesh that do not have a FLOW boundary specified. However, the BARRIER entry can be used to specify rigid walls within an Eulerian mesh.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideEulerian Loading and Constraints

124

Initial ConditionsThe initial conditions of Eulerian elements can be defined using the TICEL or TICEUL1 entry or by using the option “initeul=” in the sol700.pth file. This allows the initial state of the model to be set prior to running the analysis. It is important to recognize the difference between initial conditions and enforced conditions. Enforced conditions specify the loading and constraints of material throughout the transient analysis. Initial conditions, on the other hand, specify the state of the material only at the beginning of the analysis. Thereafter, the material state is determined by the calculation.

TICEL

The TICEL entry defines transient initial conditions for elements. Any valid element variable can be given an initial value.

INITEUL

The INITEUL entry imports an Euler archive into a follow-up run and maps it onto the defined Euler elements. If the follow-up run uses a coupling surface, then in the first run, this coupling surface can be left out provided that physics in the Euler have not reached the coupling surface at the end of the first run. In the follow-up run, a coarser mesh can be used to reduce CPU time.

TICEUL1

The TICEUL1 entry defines transient initial conditions for geometrical regions in the Eulerian mesh using the TICREG entries. The TICEUL1 entry must be used together with the PEULER1 property definition.

Eulerian Mesh

BARRIER Boundary


With the TICREG entry, it is possible to generate initial conditions inside or outside multifaceted surfaces, analytical cylindrical or spherical geometry shapes and in sets of elements.

Each geometrical region (multifaceted surface, cylinder, sphere, box or set of elements) has a level number. This allows the creation of regions of arbitrary shape by allowing the regions to overlap. An element that lies in two or more geometrical regions is assigned to the region that has the highest level number.

Think of geometrical regions as shapes cut out of opaque paper. Position the region of the lowest level number on the mesh. Then, place the next higher region on top of the first and continue until all the regions are in place. When the last region is placed, you have a map indicating to which region each element in the problem is assigned.

The following figure shows how three different geometrical regions can be used to create regions of arbitrary shape. The solid line represents the boundary of the mesh. Region one (LEVEL = 1) is the large dashed rectangle. Region two (LEVEL = 2) is the long narrow rectangle. Region three (LEVEL = 3) is a circular region. The numbers on the diagram indicate how the elements in different parts of the mesh are assigned to these three regions.

If two or more regions with the same level number but different initial value sets or materials overlap, the regions are ambiguously defined resulting in an error.

Multifaceted surfaces that are used in initial value generation must be closed and form a positive volume and are defined on the SURFINI entry. The SURFINI entry is referred to from the TICREG entry and together with a TICVAL entry the initial condition for the initialization surface is defined.

1

1

1

1

22

3

3

3

LEVEL = 1

LEVEL = 2

LEVEL = 3


126

Example 1:

In the example below, a combination of two multi-faceted surfaces, a cylindrical and a spherical shape together with a block of elements (all Eulerian elements) are used to define the initial conditions in an Eulerian rectangular mesh.

The different shapes, initial value sets and levels used are shown below. All elements of the Eulerian mesh are defined as an ELEMENT shape with void and the lowest level 6 (see input file below). This means that the part of an element that doesn’t fall inside any of the shapes will be initialized as being void.7.

Plot of the material inside the Eulerian mesh after initialization.

Input:TICEUL1,101,101TICREG,1,101,SURF,6,12,204,9.TICREG,1,101,SURF,7,12,205,10.TICREG,1,101,CYLINDER,8,12,206,7.TICREG,1,101,SPHERE,9,12,207,8.TICREG,1,101,ELEM,444,,,6.SET1,444,50000,THRU,53375$TICVAL,204,,DENSITY,1000.,XVEL,50.TICVAL,205,,DENSITY,1000.,XVEL,50.


TICVAL,206,,DENSITY,1000.,XVEL,50.TICVAL,207,,DENSITY,1000.,XVEL,50.$BCPROP 116 SURFINI 6 6 Inside On On $BCPROP 117 SURFINI 7 7 Inside On On $CYLINDR,8,,1.0,.5,.5,2.0,.5,.5,++,.075SPHERE,9,,1.75,.5,.5,.15

Example 2:

In the example below a multifaceted surface is used with the OUTSIDE option. When the OUTSIDE option is used on the SURFINI entry, the parts of the Eulerian elements that fall outside the initialization surface are initialized.

Plot of the material inside the Eulerian mesh after initialization:


128

Input:TICEUL1,102,102TICREG,1,102,SURF,8,12,204,9.TICREG,2,102,,CYLINDER,7,12,206,7.TICREG,3,102,,ELEM,445,,,5.SET1,445,60000,THRU,63375$BCPROP 118 SURFINI 8 8 Outside On On $CYLINDR,7,,1.0,.20,.50,3.0,.40,.50,++,.1

Any initial condition that you define will act on the material as it is defined within the confines of the Eulerian mesh.

The user can define a radial velocity field for the material in an Eulerian domain. The definition does not apply the standard element variables, but a sequence of four definitions that completely specify the radial velocity field. You need to define the center from where the radial is to emerge (X-CENTER, Y-CENTER, and Z-CENTER), the velocity in the direction of the radial (R-VEL) and the decay coefficient (DECAY).

Assume the element center at location and the location of where the radial emerges as .

With , the velocity along the radial and the decay coefficient , the velocity components for the element (mass) can be computed:

and

The velocity components resulting from the radial field are added to the velocity components otherwise defined for the element.

Note that the dimension of RVEL changes with the value of DECAY.

Example 3:

Assume the initialization is with sphere with origin at (0,0,0) and has a radius of 2.

The DECAY coefficient is 3 and the velocity of air at the sphere boundary is 400.

The value of R-VEL is:

Input:PEULER1,100,,2ndOrder,101$TICEUL1,101,101TICREG,1,101,SPHERE,1,1,4,2.0TICREG,2,101,ELEM,2,1,5,1.0 $SPHERE,1,,0.,0.,0.,2.0

x y z, ,( ) x y z, ,( )

R· β

rx xr–

x xr–-------------------= v r R· x xr–( )β⋅ ⋅=

R VEL–400

23

--------- 50= =


TICVAL 4 DENSITY 1.1468-7SIE 3.204+8 ZVEL 20000. +CONT+CONT X-CENTER0.0 Y-CENTER0.0 Z-CENTER0.0 R-VEL 50. +CONT+CONT DECAY 3.0$SET1,2,1,THRU,10000TICVAL,5,,DENSITY,1.1468-7,SIE,3.204+8

DetonationEulerian elements that reference a JWL equation of state (EOSJWL) have to be detonated. A DETSPH entry must be present that defines a spherical detonation wave. You define the location of the detonation point, the time of detonation, and the speed of the detonation wave. The solver then calculates the time at which each explosive element detonates. Elements that do not have a JWL equation of state are unaffected.

Body ForcesIf the GRAV entry is specified, the Eulerian material also has body forces acting on the material mass. The GRAV entry defines acceleration in any direction. All Eulerian material present in the problem is affected.

Hydrostatic PresetWith PARAM, HYDSTAT, the Euler element densities are initialized in accordance to a hydrostatic pressure profile. This PARAM requires the use of the GRAV entry.

To impose matching boundary conditions FLOW,,,HYDSTAT and PORHYDS can be used. These two entries use the following boundary conditions:

• The pressure given by hydrostatic pressure profile. This is defined by the HYDSTAT entry

• The velocities are set equal to their element values.

• If fluid flows in, its density is derived from the hydrostatic pressure.

Speedup for 2-D Axial Symmetric ModelsTo simulate a 2-D axial symmetric model, a 3-D pie model can be used. To get sufficient accuracy, the angle of the pie should be smaller than five degrees but not too small. The small angle gives a mesh size in circumferential direction that is much smaller than the mesh sizes in the other directions. This results in a small time step. In principle, the mesh size in the circumferential direction can be skipped for the time step computation. But often there are small errors in the circumferential normals and the circumferential direction has to be taken into account. With DYPARAM,AXIALSYM, these normals are automatically aligned. This allows a time step that is only based on the axial and radial directions, resulting in a significant larger time step. The larger time step is automatically computed when using DYPARAM,AXIALSYM.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideBoundary and Loading Conditions - Theoretical Background

130

Boundary and Loading Conditions - Theoretical Background

Pressure Boundary ConditionsConsider pressure loadings on boundary in (5-1). To carry out the surface integration indicated by

the integral:

(5-1)

a Gaussian quadrature rule is used. To locate any point of the surface under consideration, a position vector, , is defined:

(5-2)

where:

(5-3)

and , , and are unit vectors in the , , and directions (see (5-2)).

Nodal quantities are interpolated over the four-node linear surface by the functions:

(5-4)

so that the differential surface area may be written in terms of the curvilinear coordinates as:

where is the surface Jacobian defined by:

(5-5)

in which:

(5-6)

∂b1

Ntt sd∂b1∫

γ

r f1 ξ η,( ) i1 f2 ξ η,( ) i2 f3 ξ η,( ) i3+ +=

fi ξ η,( ) φjxij

j 1=

4

∑=

i1 i2 i3 x1 x2 x3

φi14--- 1 ξξi+( ) 1 ηηi+( )=

ds

ds J dξdη=

J

J∂r∂ξ------ ∂r

∂η-------× EG F2–( )1 2⁄= =

E∂r∂ξ------ ∂r

∂ξ------⋅=

F∂r∂ξ------ ∂r

∂η-------⋅=

G∂r∂η------- ∂r

∂η-------⋅=

131Chapter 5: Constraints and LoadingsBoundary and Loading Conditions - Theoretical Background

A unit normal vector to the surface segment is given by:

(5-7)

and the global components of the traction vector can now be written:

(5-8)

where is the applied pressure at the jth node.

Figure 5-2 Parametric Representation of a Surface Segment

The surface integral for a segment is evaluated as:

(5-9)

One such integral is computed for each surface segment on which a pressure loading acts. Note that the Jacobians cancel when (5-7) and (5-6) are put into (5-9). Equation (5-9) is evaluated with one-point integration analogous to that employed in the volume integrals. The area of an element side is approximated by where .

n

n J 1– ∂r∂ξ------ ∂r

∂η-------×⎝ ⎠

⎛ ⎞=

ti ni φjpj

j 1=

4

∑=

pj

x3

x2

x1

i3

i2

i1

rξ,η1

23

4ξ

η

Ntt J1–

1

∫⎩ ⎭⎨ ⎬⎧ ⎫

ξd( ) ηd1–

1

∫

4 J J J 0 0,( )=


132

Kinematic Boundary ConditionsIn this subsection, the kinematic constraints are briefly reviewed. SOL 700 tracks reaction forces for each type of kinematic constraint and provides this information as output if requested. For the prescribed boundary conditions, the input energy is integrated and included in the external work.

Displacement ConstraintsTranslational and rotational boundary constraints are imposed either globally or locally by setting the constrained acceleration components to zero. If nodal single point constraints are employed, the constraints are imposed in a local system. The user defines the local system by specifying a vector in

the direction of the local x-axis, , and a local in-plane vector . After normalizing , the local , ,

and axes are given by:

(5-10)

(5-11)

(5-12)

A transformation matrix is constructed to transform the acceleration components to the local system:

(5-13)

and the nodal translational and rotational acceleration vectors and , for node I are transformed to

the local system:

(5-14)

(5-15)

and the constrained components are zeroed. The modified vectors are then transformed back to the global system:

(5-16)

(5-17)

ul

xl vl ul xl yl

zl

xl

ul

ul---------=

zl

xl vl×xl vl×

--------------------=

yl zl xl×=

q

q

xlt

ylt

zlt

=

aI ω· I

aIlqaI=

ω· Ilqω· I=

aI qtaIl

=

ω· I qtω· Il

=

133Chapter 5: Constraints and LoadingsBoundary and Loading Conditions - Theoretical Background

Prescribed Displacements, Velocities, and AccelerationsPrescribed displacements, velocities, and accelerations are treated in a nearly identical way to displacement constraints. After imposing the zero displacement constraints, the prescribed values are imposed as velocities at time, . The acceleration versus time curve is integrated or the displacement versus time curve is differentiated to generate the velocity versus time curve. The prescribed nodal components are then set.

Body Force LoadsBody force loads are used in many applications. For example, in structural analysis the base accelerations can be applied in the simulation of earthquake loadings, the gun firing of projectiles, and gravitational loads. The latter is often used with dynamic relaxation to initialize the internal forces before proceeding with the transient response calculation. In aircraft engine design the body forces are generated by the application of an angular velocity of the spinning structure.

For base accelerations and gravity, we can fix the base and apply the loading as part of the body force loads element by element according to (5-18).

(5-18)

where is the base acceleration and is the element (lumped) mass matrix.

tn 1 2⁄+

febody ρNtNabase υdvm∫ meabase= =

abase me


134

Chapter 6: ElementsMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

6 Elements

Elements Overview 136

Preliminaries 140

CHEXA Solid Elements 144

CBEAM - Belytschko Beam 154

CBEAM - DYSHELFORM = 1, Hughes-Liu Beam 163

CQUADA - DYSHELLFORM = 2, Belytschko-Lin-Tsay Shell 172

CQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang Improvements 181

CTRIA3 - DYSHELLFORM = 4, C0 Triangular Shell 184

CTRIA3 - DYSHELLFORM = 3, Marchertas-Belytschko Triangular Shell (BCIZ) 190

CQUAD4 - DYSHELL FORM = 1, Hughes-Liu Shell 197

CQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells 207

Transverse Shear Treatment for Layered Shell 211

CROD, Truss Element 216CQUAD4 - DYSHELLFORM = 9, Membrane Element 217

CELAS1D, Discrete Elements and CONM2, Masses 219

CDAMP2D, Linear Elastic or Linear Viscous 224

Eulerian Elements 225

Graded Meshes 226

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideElements Overview

136

Elements OverviewThe heart of a finite element program lies in its element library which allows you to model a structure for analysis. MD Nastran has a very comprehensive element library which lets you model 1-D, 2-D, or 3-D structures. This section gives some basic definitions of the element types available in MD Nastran Explicit Nonlinear. Please note, these elements differ from those used in other portions of MD Nastran.

Element TypesEach element has five definitive characteristics that determine its behavior:

• Class

• Number of Nodes

• Interpolation

• Degrees of Freedom

• Integration Method

Class

The type of geometric domain that an element represents determines the class of the element. Listed below are the classes of elements in the MD Nastran Explicit Nonlinear element library.

• Beam Elements - a 3-D bar with axial, bending, and torsional stiffness.

• Shell Elements - a curved, thin or thick structure with membrane/bending capabilities.

• Plate Elements - a flat thin structure carrying in-plane and out-of-plane loads.

• Continuum Elements - Solid elements used to model thick sections.

• Plane stress - a thin plate with in-plane stresses only. All normal and shear stresses associated with the out-of-plane direction are assumed to be zero. (All plane strain elements lie in the global x-y plane.)

• Plane strain - a region where there is no out-of-plane motion and the normal and transverse strains are zero.

• Generalized plane strain is the same as plane strain except that the normal z-strain can be a prescribed constant or function of x and y.

• Axisymmetric elements are describe in 2-D, but represent a full 3-D structure where the geometry and loading are both axisymmetric.

• 3-D solid - a solid structure with only translational degrees of freedom for each node (linear or quadratic interpolation functions).

• Truss Elements - a 3-D rod with axial stiffness only (no bending).

• Membrane Elements - a thin sheet with in-plane stiffness only (no bending resistance).

• Concentrated mass/Springs/Damper Elements -

• Rigid Constraints -

137Chapter 6: ElementsElements Overview

Number of Nodes

The number of nodes for an element define where the displacements are calculated in the analysis. Elements with only corner nodes are classified as first order elements and the calculation of displacements at locations within the element are made by linear interpolation. Elements that contain midside nodes are second order elements and quadratic interpolations are made for calculating displacements. MD Nastran SOL 700 do not support second order elements.

In MD Nastran, the number of nodes is designated at the end of the element name. For example, a CQUAD4 has 4 nodes.

Interpolation

Interpolation (shape) function is an assumed function relating the displacements at a point inside an element to the displacements at the nodes of an element. In MD Nastran, three types of shape functions are used: linear, quadratic, and cubic.

Degrees of Freedom

Degrees of freedom is the number of unknowns at a node. In the general case, there are six degrees of freedom at a node in structural analysis (three translations, three rotations). For example there are three (translations) for 3-D truss element; six (three translations, three rotations) for a 3-D beam element and only three translations for the 3-D solid elements.

Integration

Numerical integration is a method used for evaluating integrals over an element. Element quantities – such as stresses, strains, and temperatures – are calculated at each integration point of the element. Full integration (quadrature) requires, for every element, 2-D integration points for linear interpolation and 3-D points for quadratic interpolation, where scalar “d” is the number of geometric dimensions of an element (that is, d = 2 for a quad; d = 3 for a hexahedron). Reduced integration uses a lower number of integration points than necessary to integrate exactly. For example, for an 8-node quadrilateral, the number of integration points is reduced from 9 to 4 and, for a 20-node hexahedron, from 27 to 8. For some elements, an “hourglass” control method is used to insure an accurate solution.

Table 6-1 summarizes the elements available in MD Nastran SOL 700. This section describes the characteristics of the elements. For shell elements CTRIA3 and CQUAD4, shell element PARAM,DYNSHELLFORM is used to control the behavior.

Table 6-1 Elements in MD Nastran SOL 700

MD Nastran Available in SOL 700 Fatal Error

CBAR Y

CBEAM Y

CBELT Y

CBEND N Y

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideElements Overview

138

CBUSH N Y

CBUTT Y

CCONEAX N Y

CCRSFIL Y

CDAMP1 Y

CDAMP2 Y

CDAMP1D Y

CDAMP2D Y

CELAS1 Y

CELAS2 Y

CELAS1D Y

CELAS2D Y

CFILLET Y

CFLUID N Y

CGAP N Y

CHACAB N Y

CHACBR N Y

CHEXA Y (8 Nodes only)

COMBWLD Y

CONM2 Y

CONROD Y

CONSPOT Y

CPENTA Y (6 Nodes only)

CQUAD4 Y

CQUAD8 Y (4 Nodes only)

CQUADR Y

CQUADX N Y

CROD Y

CSHEAR N Y

CSPOT Y

CSPR Y



139Chapter 6: ElementsElements Overview

CTETRA Y (4 and 10 Nodes)

CTRIA3 Y

CTRIA6 Y (3 Nodes only)

CTRIA3R Y

CTRIAX N Y

CTRIAX6 N Y

CTQUAD Y

CTTRIA Y

CTUBE Y

CVISC Y

CWELD N Y



MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePreliminaries

140

PreliminariesConsider the body shown in Figure 6-1. We are interested in time-dependent deformation in which a point in b initially at in a fixed rectangular Cartesian coordinate system moves to a point

in the same coordinate system. Since a Lagrangian formulation is considered, the deformation can be expressed in terms of the convected coordinates , and time

(6-1)

At time , we have the initial conditions

(6-2)

(6-3)

where defines the initial velocities.

Figure 6-1 Notation

Xα α 1 2 3, ,=( ) xi

i 1 2 3, ,=( )

Xα t

xi xi Xα t,( )=

t 0=

xi Xα 0,( ) Xα=

x· i Xα 0,( ) Vi Xα( )=

Vi

x3 X3

x2 X2

t 0=

∂B

B0

b

n

∂b

x1X1

141Chapter 6: ElementsPreliminaries

Governing EquationsWe seek a solution to the momentum equation

(6-4)

satisfying the traction boundary conditions

(6-5)

on boundary , the displacement boundary conditions

(6-6)

on boundary , the contact discontinuity

(6-7)

along an interior boundary when . Here is the Cauchy stress, is the current density, is

the body force density, is acceleration, the comma denotes covariant differentiation, and is a unit

outward normal to a boundary element of .

Mass conservation is trivially stated

(6-8)

where is the relative volume, i.e., the determinant of the deformation gradient matrix, ,

(6-9)

and is the reference density. The energy equation

(6-10)

is integrated in time and is used for equation of state evaluations and a global energy balance. In Equation (6-10), and represent the deviatoric stresses and pressure,

(6-11)

(6-12)

respectively, is the bulk viscosity, is the Kronecker delta ( if ; otherwise ) and

is the strain rate tensor. The strain rates and bulk viscosity are discussed later.

σi j j, ρ fi+ ρx··i=

σi jnj ti t( )=

∂b1

xi Xα t,( ) Di t( )=

∂b2

σi j+ σ i j

-–( )ni 0=

∂b3 Xi+ xi

-= σi j ρ f

x·· nj

∂b

ρV ρ0=

V Fij

Fij

∂xi

∂Xj--------=

ρ0

E· Vsijε·

ij p q+( )V·–=

si j p

si j σ i j p q+( )δij+=

p13---σ ijδi j– q–

13---σkk– q–= =

q δi j δi j 1= i j= δi j 0=

ε· i j

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePreliminaries

142

We can write:

(6-13)

where satisfies all boundary conditions on , and the integrations are over the current geometry.

Application of the divergence theorem gives

(6-14)

and noting that

(6-15)

leads to the weak form of the equilibrium equations:

(6-16)

a statement of the principle of virtual work.

We superimpose a mesh of finite elements interconnected at nodal points on a reference configuration and track particles through time; i.e.,

(6-17)

where are shape (interpolation) functions of the parametric coordinates , is the number of

nodal points defining the element, and is the nodal coordinate of the jth node in the ith direction.

Summing over the n elements we may approximate with

(6-18)

and write

(6-19)

ρx··i σi j j,– ρ fi–( )δxi ν σi jnj ti–( )δxi s

σi j+ σi j

-–( )njδxi sd

∂b3

∫+

d

∂b1

∫+d

v∫

0=

δxi ∂b2

σijδxi( ) dυj,υ∫ σi jnjδxi s σi j

+ σi j-

–( )njδxi sd

∂b3

∫+d

∂b1

∫=

σi jδxi( ) σi j j, δxi–j, σijδxi j,=

δπ ρx··iδxi υ σi jδxi j, υ ρ fiδxi υ tiδxi sd

∂b1

∫–d

υ∫–d

υ∫+d

υ∫ 0= =

xi Xα t,( ) xi Xα ξ η ζ,,( ) t,( ) φj ξ η ζ,,( )xij t( )

j 1=

k

∑= =

φj ξ η ζ,,( ) k

xij

δπ

δπ δπm

m 1=

n

∑ 0= =

ρx··iΦim υ σij

mΦi j,m υ ρ fiΦi

m υ tiΦim sd

∂b1

∫–d

υm

∫–d

υm

∫+d

υm

∫⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

m 1=

n

∑ 0=

143Chapter 6: ElementsPreliminaries

where

(6-20)

In matrix notation (6-19) becomes:

(6-21)

where is an interpolation matrix, is the stress vector

(6-22)

is the strain-displacement matrix, is the nodal acceleration vector

(6-23)

is the body force load vector, and are applied traction loads.

(6-24)

Φim φ1 φ2 … φk,,,( )i

m=

ρNtNa υ Btσ υ ρNtb υ Ntt sd

∂b1

∫–d

υm

∫–d

υm

∫+d

υm

∫⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫m

m 1=

n

∑ 0=

N σ

σ t σxx σyy σzz σxy σyz σzx, , , , ,( )=

B a

x··1x··2x··3

N

ax1

ay1

ayk

azk

Na= =…

b t

b

fx

fy

fz

= t

tx

ty

tz

=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCHEXA Solid Elements

144

CHEXA Solid ElementsFor a mesh of 8-node hexahedron solid elements, (6-17) becomes:

(6-25)

The shape function is defined for the 8-node hexahedron as

(6-26)

where take on their nodal values of and is the nodal coordinate of the th node in

the th direction (see Figure 6-2).

Figure 6-2 Eight-node Solid Hexahedron Element

For a solid element, is the 3 x 24 rectangular interpolation matrix given by

(6-27)

xi Xα t,( ) xi Xα ξ η ζ,,( ) t,( ) φj ξ η ζ,,( )xij t( )

j 1=

8

∑= =

φj

φj18--- 1 ξξj+( ) 1 ηηj+( ) 1 ζζj+( )=

ξj ηj ζj,, 1 1 1±,±,±( ) xij j

i

2

3

4

8

7

5

6

1ξ

ζ

η

Node12345678

ξ-111

-1-111

-1

η-1-111

-1-111

ζ-1-1-1-11111

N

N ξ η ζ,,( )φ1 0 0 φ2 0 … 0 0

0 φ1 0 0 φ2 … φ8 0

0 0 φ1 0 0 … 0 φ8

=

145Chapter 6: ElementsCHEXA Solid Elements

is the stress vector

(6-28)

is the 6 x 24 strain-displacement matrix

(6-29)

In order to achieve a diagonal mass matrix the rows are summed giving the th diagonal term as

(6-30)

since the basis functions sum to unity.

Terms in the strain-displacement matrix are readily calculated. Note that

(6-31)

which can be rewritten as

(6-32)

σ

σ t σxx σyy σzz σxy σyz σzx, , , , ,( )=

B

B

∂∂x------ 0 0

0∂

∂y------ 0

0 0∂∂z-----

∂∂y------ ∂

∂x------ 0

0∂∂z----- ∂

∂y------

∂∂z----- 0

∂∂x------

N=

k

mkk ρφk φi

i 1=

8

∑ υd

υ∫ ρφk υd

υ∫= =

∂φi

∂ξ--------

∂φi

∂x-------- ∂x

∂ξ------

∂φi

∂y-------- ∂y

∂ξ------

∂φi

∂z-------- ∂z

∂ξ------+ +=

∂φi

∂η--------

∂φi

∂x-------- ∂x

∂η-------

∂φi

∂y-------- ∂y

∂η-------

∂φi

∂z-------- ∂z

∂η-------+ +=

∂φi

∂ζ--------

∂φi

∂x-------- ∂x

∂ζ------

∂φi

∂y-------- ∂y

∂ζ------

∂φi

∂z-------- ∂z

∂ζ------+ +=

∂φi

∂ξ--------

∂φi

∂η--------

∂φi

∂ζ--------

∂x∂ξ------ ∂y

∂ξ------ ∂z

∂ξ------

∂x∂η------- ∂y

∂η------- ∂z

∂η-------

∂x∂ζ------ ∂y

∂ζ------ ∂z

∂ζ------

∂φi

∂x--------

∂φi

∂y--------

∂φi

∂z--------

J

∂φi

∂x--------

∂φi

∂y--------

∂φi

∂z--------

= =


146

Inverting the Jacobian matrix, , we can solve for the desired terms

(6-33)

Volume IntegrationVolume integration is carried out with Gaussian quadrature. If is some function defined over the volume, and is the number of integration points, then

(6-34)

is approximated by

(6-35)

where are the weighting factors,

(6-36)

and is the determinant of the Jacobian matrix. For one-point quadrature

(6-37)

and we can write

(6-38)

Note that approximates the element volume.

Perhaps the biggest advantage to one-point integration is a substantial savings in computer time. An anti-symmetry property of the strain matrix

(6-39)

J

∂φi

∂x--------

∂φi

∂y--------

∂φi

∂z--------

J 1–

∂φi

∂ξ--------

∂φi

∂η--------

∂φi

∂ζ--------

=

g

n

g υd

υ∫ g J ξd ηd ζd

1–

1∫1–

1∫1–

1∫=

gjkl Jjkl wjwkwl

l 1=

n

∑k 1=

n

∑j 1=

n

∑

wj wk wl, ,

gjkl g ξj ηk ζl, ,( )=

J

n 1=

wi wj wk 2= = =

ξ1 η1 ζ1 0= = =

g vd∫ 8g 0 0 0,,( ) J 0 0 0,,( )=

8 J 0 0 0,,( )

∂φ1

∂xi

---------∂φ7

∂xi

---------–=∂φ3

∂xi

---------∂φ5

∂xi

---------–=

∂φ2

∂xi---------

∂φ8

∂xi---------–=

∂φ4

∂xi---------

∂φ6

∂xi---------–=


at reduces the amount of effort required to compute this matrix by more than 25 times over an 8-point integration. This cost savings extends to strain and element nodal force calculations where the number of multiplies is reduced by a factor of 16. Because only one constitutive evaluation is needed, the time spent determining stresses is reduced by a factor of eight. Operation counts for the constant stress hexahedron are given in Table 6-2. Included are counts for the Flanagan and Belytschko [1981] hexahedron and the hexahedron used by Wilkins [1974] in his integral finite difference method, which was also implemented [Hallquist 1979]..

It may be noted that 8-point integration has another disadvantage in addition to cost. Fully integrated elements used in the solution of plasticity problems and other problems where Poisson’s ratio approaches .5 lock up in the constant volume bending modes. To preclude locking, an average pressure must be used over the elements; consequently, the zero energy modes are resisted by the deviatoric stresses. If the deviatoric stresses are insignificant relative to the pressure or, even worse, if material failure causes loss of this stress state component, hourglassing will still occur, but with no means of resisting it. Sometimes, however, the cost of the fully integrated element may be justified by increased reliability and if used sparingly may actually increase the overall speed.

Hourglass ControlThe biggest disadvantage to one-point integration is the need to control the zero energy modes, which arise, called hourglassing modes. Undesirable hourglass modes tend to have periods that are typically much shorter than the periods of the structural response, and they are often observed to be oscillatory. However, hourglass modes that have periods that are comparable to the structural response periods may be a stable kinematic component of the global deformation modes and must be admissible. One way of resisting undesirable hourglassing is with a viscous damping or small elastic stiffness capable of stopping the formation of the anomalous modes but having a negligible affect on the stable global modes. Two of the early three-dimensional algorithms for controlling the hourglass modes were developed by Kosloff and Frazier [1974] and Wilkins et al. [1974].

Table 6-2 Operation Counts for Constant Stress Hexahedron*

Flanagan-Wilkins

SOL 700Belytschko

[1981] FDM

Strain displacement matrix 94 357 843

Strain rates 87 56

Force 117 195 270

Subtotal 298 708 1,113

Hourglass control 130 620 680

Total 428 1328 1,793

*Includes adds, subtracts, multiplies, and divides in major subroutines, and is independent of vectorization.

ξ η ζ 0= = =


148

Since the hourglass deformation modes are orthogonal to the strain calculations, work done by the hourglass resistance is neglected in the energy equation. This may lead to a slight loss of energy; however, hourglass control is always recommended for the under integrated solid elements. The energy dissipated by the hourglass forces reacting against the formations of the hourglass modes is tracked and reported in the output files MATSUM and GLSTAT. This is only provided if the PARM,DYENERGYGE is included.

It is easy to understand the reasons for the formation of the hourglass modes. Consider the following strain rate calculations for the 8-node solid element

(6-40)

Whenever diagonally opposite nodes have identical velocities; i.e.,

(6-41)

the strain rates are identically zero:

(6-42)

due to the asymmetries in (6-39). It is easy to prove the orthogonality of the hourglass shape vectors, which are listed in Table 6-3 and shown in Figure 6-2 with the derivatives of the shape functions:

(6-43)

The hourglass modes of an 8-node element with one integration point are shown [Flanagan and Belytschko 1981] (Figure 6-3). A total of twelve modes exist.

Table 6-3 Hourglass Base Vectors

1 1 1 1

-1 1 -1 -1

1 -1 -1

-1 -1 1 -1

1 -1 -1 -1

-1 -1 1 1

1 1 1 -1

-1 1 -1 1

ε· i j12---

∂φk

∂xi--------x· j

k ∂φk

∂xj--------x· i

k+

k 1=

8

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

x· i1 x· i

7 x· i2, x· i

8 x· i3, x· i

5 x· i4, x· i

6= = = =

ε· i j 0=

∂φk

∂xi--------Γαk

k 1=

8

∑ 0= i 1 2 3,,= α 1 2 3 4,,,=

α 1= α 2= α 3= α 4=

Γ j1

Γ j2

Γ j3

Γ j4

Γ j5

Γ j6

Γ j7

Γ j8


Figure 6-3 Hourglass Modes

The product of the base vectors with the nodal velocities

(6-44)

are nonzero if hourglass modes are present. The 12 hourglass-resisting force vectors, are

(6-45)

where

(6-46)

in which is the element volume, is the material sound speed, and is a user-defined constant

usually set to a value between .05 and .15. This is defined by DYHRGQH PARAM. The hourglass resisting forces of (6-45) are not orthogonal to rigid body rotations; however, the approach of Flanagan and Belytschko [1981] is orthogonal. This is controlled by using the DYHRGIHQ PARAM.

Material subroutines add as little as 60 operations for the bilinear elastic-plastic routine to ten times as much for multi-surface plasticity and reactive flow models. Unvectorized material models will increase that share of the cost a factor of four or more.

Γ1k Γ2k

Γ3k Γ4k

hiα x· ikΓαk

k 1=

8

∑ 0= =

fiαk

fiαk ahhiαΓαk=

ah Qhgρνe2 3⁄ c

4---=

νe c Qhg


150

Instead of resisting components of the bilinear velocity field that are orthogonal to the strain calculation, Flanagan and Belytschko resist components of the velocity field that are not part of a fully linear field. They call this field, defined below, the hourglass velocity field

(6-47)

where

(6-48)

(6-49)

Flanagan and Belytschko construct geometry-dependent hourglass shape vectors that are orthogonal to the fully linear velocity field and the rigid body field. With these vectors they resist the hourglass velocity deformations. Defining hourglass shape vectors in terms of the base vectors as

(6-50)

and setting

(6-51)

the 12 resisting force vectors become

(6-52)

where is a constant given in (6-45).

The hourglass forces given by (6-45) and (6-52) are identical if the, hexahedron element is a parallelepiped. The default hourglass control method for solid element is given by (6-45); however, we recommend the Flanagan-Belytschko approach for problems that have large rigid body rotations since the default approach is not orthogonal to rigid body rotations.

A cost comparison in Table 6-2 shows that the default hourglass viscosity requires approximately 130 adds or multiplies per hexahedron, compared to 620 and 680 for the algorithms of Flanagan-Belytschko and Wilkins.

Hourglass stabilization for the 3-D hexahedral element is available. Based on material properties and element geometry, this stiffness type stabilization is developed by an assumed strain method [Belytschko and Bindeman 1993] such that the element does not lock with nearly incompressible material. This is activated by using the PARAM,DYHRGIHG,6. When the user-defined hourglass constant is set to 1.0,

accurate coarse mesh bending stiffness is obtained for elastic material. For nonlinear material, a smaller value of is suggested and the default value is set to 0.1.

x· ikHG

x· i x· ikLIN

–=

x· ikLIN

x· i x· i j, xjk xj–( )+=

xi18--- xi

k

k 1=

8

∑= x· i18--- x· i

k

k 1=

8

∑=

γak Γak φk i, xinΓan

n 1=

8

∑–=

gia x· ikγαk

k 1=

8

∑ 0= =

fiαk ahgiαγαk=

ah

Qhg

Qhg


Fully Integrated Brick Elements and Mid-Step Strain EvaluationTo avoid locking in the fully integrated brick elements strain increments at a point in a constant pressure, solid element are defined by [see Nagtegaal, Parks, and Rice 1974]

(6-53)

where modifies the normal strains to ensure that the total volumetric strain increment at each integration point is identical

(6-54)

and is the average volumetric strain increment in the midpoint geometry

(6-55)

, and are displacement increments in the , and directions, respectively, and

(6-56)

(6-57)

(6-58)

To satisfy the condition that rigid body rotations cause zero straining, it is necessary to use the geometry at the mid-step in the evaluation of the strain increments. The explicity formulation uses the geometry at step to save operations; however, for calculations, which involve rotating parts, the mid-step geometry should be used especially if the number of revolutions is large.

Δεyy∂Δv

∂yn 1 2⁄+----------------------- φ+= Δεyz

∂Δw

∂yn 1 2⁄+----------------------- ∂Δv

∂zn 1 2⁄+----------------------+

2-----------------------------------------------------=

Δεxx∂Δu

∂xn 1 2⁄+----------------------- φ+= Δεxy

∂Δν∂xn 1 2⁄+----------------------- ∂Δu

∂yn 1 2⁄+-----------------------+

2------------------------------------------------------=

Δεzz∂Δw

∂zn 1 2⁄+---------------------- φ+= Δεzx

∂Δu

∂zn 1 2⁄+---------------------- ∂Δw

∂xn 1 2⁄+-----------------------+

2-----------------------------------------------------=

φ

φ Δεvn 1 2⁄+

∂Δu

∂xn 1 2⁄+----------------------- ∂Δν

∂yn 1 2⁄+----------------------- ∂Δw

∂zn 1 2⁄+----------------------+ +

3------------------------------------------------------------------------------------–=

Δενn 1 2⁄+

13--- ∂Δu

∂xn 1 2⁄+----------------------- ∂Δν

∂yn 1 2⁄+----------------------- ∂Δw

∂zn 1 2⁄+----------------------+ +⎝ ⎠

⎛ ⎞ vn 1 2⁄+d

vn 1 2⁄+∫

vn 1 2⁄+d

vn 1 2⁄+∫

------------------------------------------------------------------------------------------------------------------------------------------

Δu Δv, Δw x y, z

xn 1 2⁄+ xn xn 1++( )2

------------------------------=

yn 1 2⁄+ yn yn 1++( )2

------------------------------=

zn 1 2⁄+ zn zn 1++( )2

-----------------------------=

n 1+


152

Since the bulk modulus is constant in the plastic and viscoelastic material models, constant pressure solid elements result. In the thermoelastoplastic material, a constant temperature is assumed over the element. In the soil and crushable foam material, an average relative volume is computed for the element at time step , and the pressure and bulk modulus associated with this relative volume is used at each integration point. For equations of state, one pressure evaluation is done per element and is added to the deviatoric stress tensor at each integration point.

The foregoing procedure requires that the strain-displacement matrix corresponding to equation (6-53) and consistent with a constant volumetric strain, , be used in the nodal force calculations [Hughes 1980]. It is easy to show that:

(6-59)

and avoid the needless complexities of computing .

CTETRA - Four Node Tetrahedron ElementThe four node tetrahedron element with one point integration, shown in Figure 6-4, is a simple, fast, solid element that has proven to be very useful in modeling low density foams that have high compressibility. For most applications, however, this element is too stiff to give reliable results and is primarily used for transitions in meshes. The formulation follows the formulation for the one point solid element with the difference that there are no kinematic modes, so hourglass control is not needed. The basis functions are given by:

(6-60)

Figure 6-4 Four-node tetrahedron

n 1+

B

F Bn 1t+ σn 1+ vn 1+d

vn 1+∫ Bn 1t+ σn 1+ vn 1+d

vn 1+∫= =

B

N1 r s t, ,( ) r=

N2 r s t, ,( ) s=

N3 r s t, ,( ) 1 r– s– t–=

N4 r s t, ,( ) t=

r1

2

s

3

4

t


If a tetrahedron element is needed, this element should be used instead of the collapsed solid element since it is, in general, considerably more stable in addition to being much faster.

CPENTA - Six Node Pentahedron ElementThe pentahedron element with two point Gauss integration along its length, shown in Figure 6-5, is a solid element that has proven to be very useful in modeling axisymmetric structures where wedge shaped elements are used along the axis-of-revolution. The formulation follows the formulation for the one point solid element with the difference that, like the tetrahedron element, there are no kinematic modes, so hourglass control is not needed. The basis functions are given by:

(6-61)

If a pentahedron element is needed, this element should be used instead of the collapsed solid element since it is, in general, more stable and significantly faster. Selective-reduced integration is used to prevent volumetric locking; i.e., a constant pressure over the domain of the element is assumed.

Figure 6-5 Six Node Pentahedron

N1 r s t, ,( ) 12--- 1 t–( )r=

N2 r s t, ,( ) 12--- 1 t–( ) 1 r– s–( )=

N3 r s t, ,( ) 12--- 1 t+( ) 1 r– s–( )=

N4 r s t, ,( ) 12--- 1 t+( )r=

N5 r s t, ,( ) 12--- 1 t–( )s=

N6 r s t, ,( ) 12--- 1 t+( )s=

51

r

4

2

6

s

t

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCBEAM - Belytschko Beam

154

CBEAM - Belytschko BeamThe Belytschko beam element formulation [Belytschko et. al.1977] is part of a family of structural finite elements, by Belytschko and other researchers that employ a ‘co-rotational technique’ in the element formulation for treating large rotation. This section discusses the co-rotational formulation, since the formulation is most easily described for a beam element, and then describes the beam theory used to formulate the co-rotational beam element.

Co-rotational TechniqueIn any large displacement formulation, the goal is to separate the deformation displacements from the rigid body displacements, as only the deformation displacements give rise to strains and the associated generation of strain energy. This separation is usually accomplished by comparing the current configuration with a reference configuration.

The current configuration is a complete description of the deformed body in its current spatial location and orientation, giving locations of all points (nodes) comprising the body. The reference configuration can be either the initial configuration of the body (i.e., nodal locations at time zero) or the configuration of the body at some other state (time). Often, the reference configuration is chosen to be the previous configuration, say at time .

The choice of the reference configuration determines the type of deformations that will be computed: total deformations result from comparing the current configuration with the initial configuration, while incremental deformations result from comparing with the previous configuration. In most time stepping (numerical) Lagrangian formulations, incremental deformations are used because they result in significant simplifications of other algorithms, chiefly constitutive models.

A direct comparison of the current configuration with the reference configuration does not result in a determination of the deformation, but rather provides the total (or incremental) displacements. We will use the unqualified term displacements to mean either the total displacements or the incremental displacements, depending on the choice of the reference configuration as the initial or the last state. This is perhaps most obvious if the reference configuration is the initial configuration. The direct comparison of the current configuration with the reference configuration yields displacements, which contain components due to deformations and rigid body motions. The task remains of separating the deformation and rigid body displacements. The deformations are usually found by subtracting from the displacements an estimate of the rigid body displacements. Exact rigid body displacements are usually only known for trivial cases where they are prescribed a priori as part of a displacement field. The co-rotational formulations provide one such estimate of the rigid body displacements.

The co-rotational formulation uses two types of coordinate systems: one system associated with each element; i.e., element coordinates which deform with the element, and another associated with each node; i.e., body coordinates embedded in the nodes. (The term ‘body’ is used to avoid possible confusion from referring to these coordinates as ‘nodal’ coordinates. Also, in the more general formulation presented in [Belytschko et al., 1977], the nodes could optionally be attached to rigid bodies. Thus, the term ‘body coordinates’ refers to a system of coordinates in a rigid body, of which a node is a special case.) These two coordinate systems are shown in the upper portion of Figure 6-6.

tn tn 1+ Δ t–=

155Chapter 6: ElementsCBEAM - Belytschko Beam

Figure 6-6 Co-rotational Coordinate System

Y

IX

Y^

e2

b2

e1

b1

J

X^

X

YY

X

e10

b1

e2

b2

e1 J

X

J

e10

I

e20

X

b1

b2

YY^

(a) Initial Configuration

(b) Rigid Rotation Configuration

(c) Deformed Configuration


156

The element coordinate system is defined to have the local x-axis originating at node and terminating

at node ; the local y-axis and, in three dimension, the local z-axis , are constructed normal to . The element coordinate system and associated unit vector triad are updated at every time

step by the same technique used to construct the initial system; thus the unit vector deforms with the

element since it always points from node to node .

The embedded body coordinate system is initially oriented along the principal inertial axes; either the assembled nodal mass or associated rigid body inertial tensor is used in determining the inertial principal values and directions. Although the initial orientation of the body axes is arbitrary, the selection of a principal inertia coordinate system simplifies the rotational equations of motion; i.e., no inertial cross product terms are present in the rotational equations of motion. Because the body coordinates are fixed in the node (or rigid body), they rotate and translate with the node and are updated by integrating the rotational equations of motion, as will be described subsequently.

The unit vectors of the two coordinate systems define rotational transformations between the global coordinate system and each respective coordinate system. These transformations operate on vectors with global components , body coordinates components , and element coordinate

components which are defined as:

(6-62)

where are the global components of the body coordinate unit vectors. Similarly for the element

coordinate system:

(6-63)

where are the global components of the element coordinate unit vectors. The inverse

transformations are defined by the matrix transpose: i.e.,

(6-64)

(6-65)

since these are proper rotational transformations.

The following two examples illustrate how the element and body coordinate system are used to separate the deformations and rigid body displacements from the displacements.

x I

J y z x

x y z, ,( ) e1 e2 e3, ,( )

e1

I J

A Ax Ay Az, ,( )= A Ax Ay Az, ,=

A Ax Ay Az, ,=

A

Ax

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

b1x b2x b3x

b1y b2y b3y

b1z b2z b3z

Ax

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

λ[ ] A{ }= = =

bix biy biz, ,

A

Ax

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

e1x e2x e3x

e1y e2y e3y

e1z e2z e3z

Aˆ

x

Aˆ

y

Aˆ

z⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

μ[ ] Aˆ{ }= = =

eix eiy eiz, ,

A{ } λ[ ]TA{ }=

A{ } μ[ ]TA{ }=


Rigid Rotation

First, consider a rigid body rotation of the beam element about node , as shown in the center of Figure 6-6b; i.e., consider node to be a pinned connection. Because the beam does not deform during the rigid rotation, the orientation of the unit vector in the initial and rotated configuration will be the

same with respect to the body coordinates. If the body coordinate components of the initial element unit

vector were stored, they would be identical to the body coordinate components of the current element

unit vector .

Deformation Rotation

Next, consider node to be constrained against rotation; i.e., a clamped connection. Now node is moved, as shown in the lower portion of Figure 6-6, causing the beam element to deform. The updated element unit vector is constructed and its body coordinate components are compared to the body

coordinate components of the original element unit vector . Because the body coordinate system did

not rotate, as node was constrained, the original element unit vector and the current element unit vector are not co-linear. Indeed, the angle between these two unit vectors is the amount of rotational deformation at node ; i.e.,

(6-66)

Thus the co-rotational formulation separates the deformation and rigid body deformations by using:

• a coordinate system that deforms with the element; i.e., the element coordinates or

• a coordinate system that rigidly rotates with the nodes; i.e., the body coordinates.

Then, it compares the current orientation of the element coordinate system with the initial element coordinate system, using the rigidly rotated body coordinate system, to determine the deformations.

Belytschko Beam Element FormulationThe deformation displacements used in the Belytschko beam element formulation are:

(6-67)

where

= length change

= torsional deformation

= bending rotational deformations

The superscript emphasizes that these quantities are defined in the local element coordinate system,

and and are the nodes at the ends of the beam.

I

I

e1

e10

e1

I J

e1

e10

I

I

e1 e10× θle3=

dˆ T

δIJ θˆ xJI θˆ yI θˆ yJ θˆ zI θˆ zJ, , , , ,{ }=

δIJ

θˆ xJI

θˆ yI θˆ yJ θˆ zI θˆ zJ, , ,

ˆ

I J


158

The beam deformations, defined in equation (6-62), are the usual small displacement beam deformations (see, for example, [Przemieniecki 1986]). Indeed, one advantage of the co-rotational formulation is the ease with which existing small displacement element formulations can be adapted to a large displacement formulation having small deformations in the element system. Small deformation theories can be easily accommodated because the definition of the local element coordinate system is independent of rigid body rotations and hence deformation displacement can be defined directly.

Calculation of DeformationsThe elongation of the beam is calculated directly from the original nodal coordinates and the

total displacements :

(6-68)

where

(6-69)

etc. (6-70)

The deformation rotations are calculated using the body coordinate components of the original element

coordinate unit vector along the beam axis; i.e., , as outlined in the previous section. Because the body

coordinate components of initial unit vector rotate with the node, in the deformed configuration it

indicates the direction of the beam’s axis if no deformations had occurred. Thus comparing the initial unit

vector with its current orientation indicates the magnitude of deformation rotations. Forming the

vector cross product between and :

(6-71)

where

is the incremental deformation about the local axis

is the incremental deformation about the local axis

The calculation is most conveniently performed by transforming the body components of the initial element vector into the current element coordinate system:

(6-72)

XI YI ZI, ,( )

uxI uyI uzI, ,( )

δIJ1

l lo+------------ 2 XJIuxJI YJI uyJI ZJIuzJI+ +( ) uxJI

2 uyJI2 uzJI

2+ + +[ ]=

XJI XJ XI–=

uxJI uxJ uxI–=

e10

e10

e10 e1

e10 e1

e1 e10× θˆ ye2 θˆ ze3+=

θˆ y y

θˆ z z

e1x0

e1y0

e1z0

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

μ[ ]T λ[ ]e1x

0

e1y0

e1z0

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=


Substituting equation (6-72) into ((6-71))

(6-73)

Thus,

(6-74)

(6-75)

The torsional deformation rotation is calculated from the vector cross product of initial unit vectors, from

each node of the beam, that were normal to the axis of the beam, i.e., and ; note that and

could also be used. The result from this vector cross product is then projected onto the current axis of the beam; i.e.,

(6-76)

Note that the body components of and are transformed into the current element coordinate system

before performing the indicated vector products.

Calculation of Internal ForcesThere are two methods for computing the internal forces for the Belytschko beam element formulation:

• Functional forms relating the overall response of the beam- e.g., moment-curvature relations,

• Direct through-the-thickness integration of the stress.

Currently only the former method, as explained subsequently, is implemented; the direct integration method is detailed in [Belytschko et al., 1977].

Axial Force

The internal axial force is calculated from the elongation of the beam , as given by equation (6-68), and an axial stiffness:

(6-77)

where:

is the axial stiffness

is the cross sectional area of the beam

e1 e10× det

e1 e2 e3

1 0 0

e1x0 e1y

0 e1z0

e1z0 e2– e1y

0 e3+ θˆ ye2 θˆ ze3+= = =

θyˆ e1z

0–=

θzˆ e1y

0=

e2 I0 e2J

0 e3 I0 e3J

0

θˆ xJI e1 e2I0 e2J

0×( )⋅ e1det

e1 e2 e3

ex2 I0 ey2 I

0 ez2I0

ex2J0 ey2J

0 ez2J0

ey2 I0 ez2J

0 ey2J0 ez2 I

0–= = =

e2 I0 e2J

0

δ

fˆxJ Kaδ=

Ka AE l0⁄=

A


160

is Young’s Modulus

is the original length of the beam

Bending Moments

The bending moments are related to the deformation rotations by

(6-78)

(6-79)

where equation (6-78) is for bending in the plane and equation (6-79) is for bending in the plane. The bending constants are given by

(6-80)

(6-81)

(6-82)

(6-83)

(6-84)

Hence is the shear factor, the shear modulus, and is the effective area in shear.

Torsional Moment

The torsional moment is calculated from the torsional deformation rotation as

(6-85)

where

(6-86)

(6-87)

E

l0

myI

myJ⎩ ⎭⎨ ⎬⎧ ⎫ Ky

b

1 φy+---------------

4 φy+ 2 φy–

2 φy– 4 φy+

θˆ yI

θˆ yJ⎩ ⎭⎨ ⎬⎧ ⎫

=

mzI

mzJ⎩ ⎭⎨ ⎬⎧ ⎫ Kz

b

1 φz+---------------

4 φz+ 2 φz–

2 φz– 4 φz+

θˆ zI

θˆ zJ⎩ ⎭⎨ ⎬⎧ ⎫

=

x z– x y–

Kyb EIyy

l0----------=

Kzb EIzz

l0----------=

Iyy z2 yd zd∫∫=

Izz y2 yd zd∫∫=

φy

12EIyy

GAsl2-----------------= φz

12EIzz

GAsl2----------------=

φ G As

mxJ Ktθˆ xJI=

Kt GJl0

-------=

J y z yd zd∫∫=


The above forces are conjugate to the deformation displacements given previously in equation (6-67); i.e.,

(6-88)

where

(6-89)

(6-90)

The remaining internal force components are found from equilibrium:

(6-91)

Updating the Body Coordinate Unit Vectors

The body coordinate unit vectors are updated using the Newmark -Method [Newmark 1959] with , which is almost identical to the central difference method [Belytschko 1974]. In particular, the

body component unit vectors are updated using the formula

(6-92)

where the superscripts refer to the time step and the subscripts refer to the three unit vectors comprising the body coordinate triad. The time derivatives in the above equation are replaced by their equivalent forms from vector analysis:

(6-93)

(6-94)

where and are vectors of angular velocity and acceleration, respectively, obtained from the rotational equations of motion. With the above relations substituted into equation (6-92), the update formula for the unit vectors becomes

(6-95)

dˆ T δIJ θˆ xJI θˆ yI θˆ yJ θˆ zI θˆ zJ, , , , ,{ }=

dˆ{ }T

fˆ{ } W

int=

fˆT

fˆxJ mxJ myI myJ mzI mzJ, , , , ,{ }=

fˆxI f

ˆxJ–= mxI m– xJ=

fˆzI

myI myJ+

l0------------------------–= f

ˆzI f

ˆzJ–=

fˆyJ

mzI mzJ+

l0------------------------–= f

ˆyI f

ˆyJ–=

β

β 0=

bij 1+

bij

Δ tdbi

j

dt-------- Δ t2

2--------

d2bij

dt2-----------+ +=

dbij

dt-------- ω bi×=

d2bij

dt2----------- ω ω bi×( )× αi bi×( )+=

ω α

bij 1+

bij

Δ t ω bi×( ) Δ t2

2-------- ω ω bi×( ) αi bi×( )+×[ ]{ }+ +=


162

To obtain the formulation for the updated components of the unit vectors, the body coordinate system is temporarily considered to be fixed and then the dot product of equation (6-95) is formed with the unit vector to be updated. For example, to update the component of , the dot product of equation (6-95),

with , is formed with , which can be simplified to the relation

(6-96)

Similarly,

(6-97)

(6-98)

The remaining components and are found by using normality and orthogonality, where it is

assumed that the angular velocities are small during a time step so that the quadratic terms in the

update relations can be ignored. Since is a unit vector, normality provides the relation

(6-99)

Next, if it is assumed that , orthogonality yields (6-100)

(6-101)

The component is then found by enforcing normality:

(6-102)

The updated components of and are defined relative to the body coordinates at time step . To

complete the update and define the transformation matrix, equation (6-62), at time step , the updated unit vectors and are transformed to the global coordinate system, using equation (6-62) with

defined at step , and their vector cross product is used to form .

x b3

i 3= b1

bx3j 1+

b1j

by3j 1+

⋅ Δ tωyj Δ t2

2-------- ωx

jωz

jαy

j+( )+= =

by3j 1+

b2j

b3j 1+

⋅ Δ tωxj Δ t2

2-------- ωy

jωz

jαx

j+( )+= =

bx2j 1+

b1j

b2j 1+

⋅ Δ tωzj Δ t2

2-------- ωx

jωy

jαz

j+( )+= =

b3j 1+

b1j 1+

ω

b3j 1+

bz3j 1+

1 bx3j 1+

( )2

– by3j 1+

( )2

–=

bx1j 1+

1≈

bz1j 1+ bx3

j 1+by1

j 1+by3

j 1++

bz3j 1+

--------------------------------------------–=

bx1j 1+

bx1j 1+

1 by1j 1+

( )2

– bz1j 1+

( )2

–=

b1 b3 j

j 1+

b1 b3 λ[ ]

j b2

163Chapter 6: ElementsCBEAM - DYSHELFORM = 1, Hughes-Liu Beam

CBEAM - DYSHELFORM = 1, Hughes-Liu BeamThe Hughes-Liu beam element formulation, based on the shell [Hughes and Liu 1981a, 1981b] discussed later. It has several desirable qualities:

• It is incrementally objective (rigid body rotations do not generate strains), allowing for the treatment of finite strains that occur in many practical applications.

• It is simple, which usually translates into computational efficiency and robustness.

• It is compatible with the brick elements, because the element is based on a degenerated brick element formulation.

• It includes finite transverse shear strains. The added computations needed to retain this strain component, compare to those for the assumption of no transverse shear strain, are insignificant.

GeometryThe Hughes-Liu beam element is based on a degeneration of the isoparametric 8-node solid element, an approach originated by Ahmad et al., [1970]. Recall the solid element isoparametric mapping of the biunit cube

(6-103)

(6-104)

where is an arbitrary point in the element, are the parametric coordinates, are the global

nodal coordinates of node , and are the element shape functions evaluated at node , i.e.,

are evaluated at node .

In the beam geometry, determines the location along the axis of the beam and the coordinate pair defines a point on the cross section. To degenerate the 8-node brick geometry into the 2-node beam geometry, the four nodes at and at are combined into a single node with three translational and three rotational degrees of freedom. Orthogonal, inextensible nodal fibers are defined at each node for treating the rotational degrees of freedom. Figure 6-7 shows a schematic of the bi-unit cube and the beam element. The mapping of the bi-unit cube into the beam element is separated into three parts:

(6-105)

where denotes a position vector to a point on the reference axis of the beam, and and are position

vectors at point on the axis that define the fiber directions through that point. In particular,

(6-106)

(6-107)

(6-108)

x ξ η ζ,,( ) Na ξ η ζ,,( )xa=

Na ξ η ζ,,( )1 ξaξ+( ) 1 ηaη+( ) 1 ζaζ+( )

8--------------------------------------------------------------------------=

x ξ η ζ,,( ) xa

a Na a ξa ηa ζa,,

ξ η ζ,,( ) a

ξ η ζ,( )

ξ 1–= ξ 1=

x ξ η ζ,,( ) x ξ( ) X ξ η ζ,,( )+ x ξ( ) Xζ ξ ζ,( ) Xη ξ η,( )+ += =

x Xζ Xη

x

x ξ( ) Na ξ( )xa=

Xη ξ η,( ) Na ξ( )Xηa η( )=

Xζ ξ ζ,( ) Na ξ( )Xζa ζ( )=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCBEAM - DYSHELFORM = 1, Hughes-Liu Beam

164

Figure 6-7 Hughes-Liu Beam Element

With this description, arbitrary points on the reference line are interpolated by the one- dimensional shape function operating on the global position of the two beam nodes that define the reference axis, i.e., . Points off the reference axis are further interpolated by using a one-dimensional shape function

along the fiber directions; i.e., and where

(6-109)

(6-110)

ζ

ξ

η

Bi-unit Cube

xζ

ξ

η

Beam Element

zζ

ζ

η

+

Xζ+

Xζ

Xˆ

Xζ

Xˆ

ζ

zζ

-1

0

+1Top Surface

Nodal Fibers

Bottom Surface

x

N ξ( )

xa

Xηa η( ) Xζa ζ( )

Xηa η( ) zη η( )Xηa=

zη η( ) N+ η( )zηa+ N- η( )zηa

-+=


(6-111)

(6-112)

(6-113)

(6-114)

(6-115)

(6-116)

where and are “thickness functions”.

The Hughes-Liu beam formulation uses four position vectors, in addition to , to locate the reference

axis and define the initial fiber directions. Consider the two position vectors and located on the

top and bottom surfaces, respectively, at node . Then

(6-117)

(6-118)

(6-119)

(6-120)

(6-121)

(6-122)

(6-123)

(6-124)

where is the Euclidean norm. The reference surface may be located at the midsurface of the beam or offset at the outer surfaces. This capability is useful in several practical situations involving contact surfaces, connection of beam elements to solid elements, and offsetting elements such as for beam stiffeners in stiffened shells. The reference surfaces are located within the beam element by specifying

N+ η( ) 1 η+( )2

------------------=

N- η( ) 1 η–( )2

------------------=

Xζa ζ( ) zζ ζ( )Xζa=

zζ ζ( ) N+ ζ( )zζa+ N- ζ( )zζa

-+=

N+ ζ( ) 1 ζ+( )2

-----------------=

N- ζ( ) 1 ζ–( )2

-----------------=

zζ ζ( ) zη η( )

ξ

xζa+ xζa

-

a

xζa12--- 1 ζ–( )xηa

– 1 ζ+( )+[ ]xζa+=

Xζaxζa

+ xζa-–( )

xζa+ xζa

-–----------------------------=

zζa+ 1

2--- 1 ζ–( ) xζa

+ xζa-–⋅=

zζa- 1

2--- 1 ζ+( ) xζa

+ xζa-–⋅–=

xηa12--- 1 ζ–( )xηa

– 1 ζ+( )+[ ]xηa+=

Xηaxηa

+ xηa-–( )

xηa+ xηa

-–-----------------------------=

zηa+ 1

2--- 1 η–( ) xηa

+ xηa-–⋅=

zηa- 1

2--- 1 η+( ) xηa

+ xηa-–⋅–=

.


166

the value of the parameters and , (see lower portion of Figure 6-7). When these parameters take on the values or , the reference axis is located on the outer surfaces of the beam. If they are set to zero, the reference axis is at the center.

The same parametric representation used to describe the geometry of the beam elements is used to interpolate the beam element displacements; i.e., an isoparametric representation. Again, the displacements are separated into the reference axis displacements and rotations associated with the fiber directions:

(6-125)

(6-126)

(6-127)

(6-128)

(6-129)

(6-130)

where is the displacement of a generic point, is the displacement of a point on the reference surface, and is the ‘fiber displacement’ rotations. The motion of the fibers can be interpreted as either displacements or rotations as will be discussed.

Hughes and Liu introduced the notation that follows, and the associated schematic shown in Figure 6-8, to describe the current deformed configuration with respect to the reference configuration:

(6-131)

(6-132)

(6-133)

(6-134)

(6-135)

(6-136)

(6-137)

In the above relations, and in Figure 6-8, the quantities refer to the reference configuration, the quantities refer to the updated (deformed) configuration and the quantities are the displacements. The notation consistently uses a superscript bar to indicate reference surface quantities, a superscript caret

to indicate unit vector quantities, lower case letter for translational displacements, and upper case

letters for fiber displacements. Thus to update to the deformed configuration, two vector quantities are needed: the reference surface displacement and the associated nodal fiber displacement . The nodal fiber displacements are defined in the fiber coordinate system, described in the next subsection.

η ζ

1– +1

u ξ η ζ, ,( ) u ξ( ) U ξ η ζ, ,( )+ u ξ( ) Uζ ξ ζ,( ) Uη ξ η,( )+ += =

u ξ( ) Na ξ( )ua=

Uη ξ η,( ) Na ξ( )Uηa η( )=

Uζ ξ ζ,( ) Na ξ( )Uζa ζ( )=

Uηa η( ) zηa η( )Uηa=

Uζa ζ( ) zζa ζ( )Uζa=

u u

U

y y Y+=

y x u+=

ya xa ua+=

Y X U+=

Ya Xa Ua+=

Yηa Xηa Uηa+=

Yζa Xζa Uζa+=

x y

u

( )

ˆ( )

u U


Figure 6-8 Schematic of Deformed Configuration Displacements and Position Vectors

Fiber Coordinate SystemFor a beam element, the known quantities will be the displacements of the reference surface obtained from the translational equations of motion and the rotational quantities at each node obtained from the rotational equations of motion. What remains to complete the kinematics is a relation between nodal rotations and fiber displacements . The linearized relationships between the incremental components

the incremental rotations are given by

(6-138)

(6-139)

Equations (6-138) and (6-139) are used to transform the incremental fiber tip displacements to rotational increments in the equations of motion. The second-order accurate rotational update formulation due to Hughes and Winget [1980] is used to update the fiber vectors:

(6-140)

(6-141)

Y

Uu

u

X

Deformed ConfigurationReference Surface

Reference axis inundeformed geometry

Parallel Construction

x

xy

y

u

U

ΔU

ΔUη1

ΔUη2

ΔUη3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

0 Yη3 Yη2–

Yη3– 0 Yη1

Yη2 Yη1– 0

Δθ1

Δθ2

Δθ3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

hηΔθ= =

ΔUζ1

ΔUζ2

ΔUζ3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

0 Yζ3 Yζ2–

Yζ3– 0 Yζ1

Yζ2 Yζ1– 0

Δθ1

Δθ2

Δθ3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

hζΔθ= =

Yη in 1+

Rij Δθ( ) Yη jn

=

Yζ in 1+

Rij Δθ( ) Yζ in

=


168

then

(6-142)

(6-143)

where

(6-144)

(6-145)

(6-146)

Here is the Kronecker delta and is the permutation tensor.

Local Coordinate System

In addition to the above described fiber coordinate system, a local coordinate system is needed to enforce the zero normal stress conditions transverse to the axis. The orthonormal basis with two directions and

normal to the axis of the beam is constructed as follows:

(6-147)

(6-148)

From the vector cross product of these local tangents.

(6-149)

and to complete this orthonormal basis, the vector

(6-150)

is defined. This coordinate system rigidly rotates with the deformations of the element.

The transformation of vectors from the global to the local coordinate system can now be defined in terms of the basis vectors as

(6-151)

ΔUηa Yηan 1+

Yηan–=

ΔUζa Yζan 1+

Yζan–=

Rij Δθ( ) δi j

2δik ΔSik+( )ΔSjk

2D---------------------------------------------+=

ΔSij eikjΔθk=

2D 212--- Δθ1

2 Δθ22 Δθ3

2+ +( )+=

δi j eikj

e2

e3

e1y2 y1–

y2 y1–----------------------=

e2′ Yη1 Yη2+

Yη1 Yη2+------------------------------=

e3 e1 e2′×=

e2 e3 e1×=

Aˆ

Aˆ

x

Aˆ

y

Aˆ

z⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

e1x e2x e3x

e1y e2y e3y

e1z e2z e3z

TAx

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

q[ ] A{ }= = =


where are the global components of the local coordinate unit vectors, is a vector in the local

coordinates, and is the same vector in the global coordinate system.

Strains and Stress Update

Incremental Strain and Spin Tensors

The strain and spin increments are calculated from the incremental displacement gradient

(6-152)

where are the incremental displacements and are the deformed coordinates. The incremental strain

and spin tensors are defined as the symmetric and skew-symmetric parts, respectively, of :

(6-153)

(6-154)

The incremental spin tensor is used as an approximation to the rotational contribution of the

Jaumann rate of the stress tensor. Here the Jaumann rate update is approximated as

(6-155)

where the superscripts on the stress tensor refer to the updated and reference configurations. This update of the stress tensor is applied before the constitutive evaluation, and the stress and strain are stored in the global coordinate system.

Stress Update

To evaluate the constitutive relation, the stresses and strain increments are rotated from the global to the local coordinate system using the transformation defined previously in (6-151).

(6-156)

(6-157)

where the superscript indicates components in the local coordinate system. The stress is incrementally updated:

(6-158)

eix eiy eiz, , Aˆ

A

Gij

∂Δui

∂yj------------=

Δui yj

Gij

Δεi j12--- Gij Gji+( )=

Δωij12--- Gij Gji–( )=

Δωij

σi j σi jn σip

n Δωpj σ jpn Δωpi+ +=

n 1+( ) n( )

σi jln qikσknqjn=

Δεi jl qikΔεknqjn=

l

σi jln 1+

σijln

Δσi jln

12---+

+=


170

and rotated back to the global system:

(6-159)

before computing the internal force vector.

Incremental Strain-Displacement Relations

After the constitutive evaluation is completed, the fully updated stresses are rotated back to the global coordinate system. These global stresses are then used to update the internal force vector

(6-160)

where are the internal forces at node and is the strain-displacement matrix in the global

coordinate system associated with the displacements at node . The matrix relates six global strain components to eighteen incremental displacements [three translational displacements per node and the six incremental fiber tip displacements of (6-142)]. It is convenient to partition the matrix:

(6-161)

Each sub matrix is further partitioned into a portion due to strain and spin with the following sub

matrix definitions:

(6-162)

where

(6-163)

With respect to the strain-displacement relations, note that:

• The derivative of the shape functions are taken with respect to the global coordinates;

• The matrix is computed on the cross-section located at the mid-point of the axis;

• The resulting matrix is a matrix.

σi jn 1+

qkiσknln 1+

qnj=

faint

BaTσ υd∫=

faint

a Ba

a B

B

B B1 B2,[ ]=

Ba

Ba

B1 0 0 B4 0 0 B7 0 0

0 B2 0 0 B5 0 0 B8 0

0 0 B3 0 0 B6 0 0 B9

B2 B1 0 B5 B4 0 B8 B7 0

0 B3 B2 0 B6 B5 0 B9 B8

B3 0 B1 B6 0 B4 B9 0 B7

=

Bi =

Na i,∂Na

∂yi----------=

Naηa( ) i 3–( ),

∂ Nazηa( )∂yi 3–

-----------------------=

Nazζa( ) i 6–( ),

∂ Nazηa( )∂yi 6–

-----------------------=

for i = 1,2,3

for i = 4,5,6

for i = 7,8,9

B

B 6 18×


The internal force, , given by

(6-164)

is assembled into the global right hand side internal force vector. is defined as (also see equation (6-138)):

(6-165)

where is a identity matrix.

f

f ′ Ttfaint=

T

T

I 0

0 hη

0 hζ

=

I 3 3×

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUADA - DYSHELLFORM = 2, Belytschko-Lin-Tsay Shell

172

CQUADA - DYSHELLFORM = 2, Belytschko-Lin-Tsay ShellThe Belytschko-Lin-Tsay shell element ([Belytschko and Tsay 1981], [Belytschko et al., 1984a]) was implemented as a computationally efficient alternative to the Hughes-Liu shell element. For a shell element with five through thickness integration points, the Belytschko-Lin-Tsay shell elements requires 725 mathematical operations compared to 4050 operations for the under integrated Hughes-Liu element. The selectively reduced integration formulation of the explicit Hughes-Liu element requires 35,350 mathematical operations. Because of its computational efficiency, the Belytschko-Lin-Tsay shell element is usually the shell element formulation of choice. For this reason, it has become the default shell element formulation for explicit calculations.

The Belytschko-Lin-Tsay shell element is based on a combined co-rotational and velocity-strain formulation. The efficiency of the element is obtained from the mathematical simplifications that result from these two kinematical assumptions. The co-rotational portion of the formulation avoids the complexities of nonlinear mechanics by embedding a coordinate system in the element. The choice of velocity-strain or rate-of-deformation in the formulation facilitates the constitutive evaluation, since the conjugate stress is the physical Cauchy stress. We closely follow the notation of Belytschko, Lin, and Tsay in the following development.

Co-rotational CoordinatesThe midsurface of the quadrilateral shell element, or reference surface, is defined by the location of the element’s four corner nodes. An embedded element coordinate system (see Figure 6-9) that deforms with the element is defined in terms of these nodal coordinates. Then the procedure for constructing the co-rotational coordinate system begins by calculating a unit vector normal to the main diagonal of the element:

(6-166)

(6-167)

(6-168)

where the superscript caret is used to indicate the local (element) coordinate system.

e3

s3

s3----------=

s3 s312 s32

2 s332+ +=

s3 r31 r42×=

ˆ( )

173Chapter 6: ElementsCQUADA - DYSHELLFORM = 2, Belytschko-Lin-Tsay Shell

Figure 6-9 Construction of Element Coordinate System

It is desired to establish the local axis approximately along the element edge between nodes 1 and 2.

This definition is convenient for interpreting the element stresses, which are defined in the local coordinate system. The procedure for constructing this unit vector is to define a vector that is nearly

parallel to the vector .

(6-169)

(6-170)

The remaining unit vector is obtained from the vector cross product

(6-171)

If the four nodes of the element are coplanar, then the unit vectors and are tangent to the midplane

of the shell and is in the fiber direction. As the element deforms, an angle may develop between the

actual fiber direction and the unit normal . The magnitude of this angle may be characterized as

(6-172)

where is the unit vector in the fiber direction and the magnitude of depends on the magnitude of the strains. According to Belytschko et al., for most engineering applications, acceptable values of are on

the order of 10-2 and if the condition presented in equation (6-172) is met, then the difference between the rotation of the co-rotational coordinates and the material rotation should be small.

3

r31

s1

2r21

x

e3

s3

4

y

r42

1

e2

e1

x x

x y–

s1

r21

s1 r21 r21 e3⋅( ) e3–=

e1s1

s1----------=

e2 e3 e1×=

e1 e2

e3

e3

e3 f 1–⋅ δ<

f δ

δ

e


174

The global components of this co-rotational triad define a transformation matrix between the global and local element coordinate systems. This transformation operates on vectors with global components

and element coordinate components , and is defined as:

(6-173)

where are the global components of the element coordinate unit vectors. The inverse

transformation is defined by the matrix transpose; i.e.,

(6-174)

Velocity-Strain Displacement RelationsThe above small rotation condition, equation (6-172), does not restrict the magnitude of the element’s rigid body rotations. Rather, the restriction is placed on the out-of-plane deformations, and, thus, on the element strain. Consistent with this restriction on the magnitude of the strains, the velocity-strain displacement relations used in the Belytschko-Lin-Tsay shell are also restricted to small strains.

As in the Hughes-Liu shell element, the displacement of any point in the shell is partitioned into a midsurface displacement (nodal translations) and a displacement associated with rotations of the element’s fibers (nodal rotations). The Belytschko-Lin-Tsay shell element uses the Mindlin [1951] theory of plates and shells to partition the velocity of any point in the shell as:

(6-175)

where is the velocity of the mid-surface, is the angular velocity vector, and is the distance along the fiber direction (thickness) of the shell element. The corresponding co-rotational components of the velocity strain (rate of deformation) are given by

(6-176)

Substitution of equation (6-175) into the above yields the following velocity-strain relations:

(6-177)

(6-178)

(6-179)

A Ax Ay Az, ,( )= Aˆ

Aˆ

x Aˆ

y Aˆ

z, ,( )=

A{ }Ax

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

e1x e2x e3x

e1y e2y e3y

e1z e2z e3z

Ax

Aˆ

y

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

μ[ ] Aˆ{ } q[ ]T A

ˆ{ }= = = =

eix eiy eiz, ,

A{ } μ[ ]T A{ }=

v vm ze3 θ×–=

vm θ z

dˆ

i j12--- ∂υˆ i

∂ xj

-------- ∂υˆ j

∂ xi

--------+⎝ ⎠⎜ ⎟⎛ ⎞

=

dˆ

x∂ vx

m

∂ x---------- z

∂θˆ y

∂ x--------+=

dˆ

y∂υˆ y

m

∂ y---------- z

∂θˆ x

∂ y--------–=

2dˆ

xy∂υˆ x

m

∂ y---------- ∂υˆ y

m

∂ x---------- z

∂θˆ y

∂ y-------- ∂θˆ x

∂ x--------–⎝ ⎠

⎛ ⎞+ +=


(6-180)

(6-181)

The above velocity-strain relations need to be evaluated at the quadrature points within the shell. Standard bilinear nodal interpolation is used to define the mid-surface velocity, angular velocity, and the element’s coordinates (isoparametric representation). These interpolations relations are given by

(6-182)

(6-183)

(6-184)

where the subscript is summed over all the nodes of the element and the nodal velocities are obtained by differentiating the nodal coordinates with respect to time, i.e., . The bilinear

shape functions are

(6-185)

(6-186)

(6-187)

(6-188)

The velocity-strains at the center of the element (i.e., at , and ) are obtained by substitution of the above relations into the previously defined velocity-strain displacement relations, equations (6-177) through (6-181). After some algebra, this yields

(6-189)

(6-190)

(6-191)

(6-192)

(6-193)

2dˆ

yz∂υˆ z

m

∂ y---------- θˆ x–=

2dˆ

xz∂υˆ z

m

∂ x---------- θˆ y+=

νm NI ξ η,( )νI=

θm NI ξ η,( )θI=

xm NI ξ η,( )xI=

I

υI x· I=

N114--- 1 ξ–( ) 1 η–( )=

N214--- 1 ξ+( ) 1 η–( )=

N314--- 1 ξ+( ) 1 η+( )=

N414--- 1 ξ–( ) 1 η+( )=

ξ 0= η 0=

dˆ

x B1 Iυˆ

xI zB1 Iθˆ

yI+=

dˆ

y B2 Iυˆ

yI zB2Iθˆ

xI–=

2dˆ

xy B2Iυˆ

xI B1 Iυˆ

yI z B2Iθˆ

yI B1 Iθˆ

xI–( )+ +=

2dˆ

xz B1 Iυˆ

zI NIθˆ

yI+=

2dˆ

yz B2 Iυˆ

zI NIθˆ

xI–=


176

where

(6-194)

(6-195)

The shape function derivatives are also evaluated at the center of the element; i.e., at , and

.

Stress Resultants and Nodal ForcesAfter suitable constitutive evaluations using the above velocity-strains, the resulting stresses are integrated through the thickness of the shell to obtain local resultant forces and moments. The integration formula for the resultants are

(6-196)

(6-197)

where the superscript, , indicates a resultant force or moment, and the Greek subscripts emphasize the limited range of the indices for plane stress plasticity.

The above element midplane force and moment resultants are related to the local nodal forces and moments by invoking the principle of virtual power and integrating with a one-point quadrature. The relations obtained in this manner are

(6-198)

(6-199)

(6-200)

(6-201)

(6-202)

(6-203)

B1I

∂NI

∂ x---------=

B2I

∂NI

∂ y---------=

BaI ξ 0=

η 0=

fˆ

αβR

σˆ αβ zd∫=

mαβR

z σαβ zd∫–=

R

fˆxI A B1I f

ˆxxR

B2 I fˆxyR

+⎝ ⎠⎛ ⎞=

fˆyI A B2I f

ˆyyR

B1 I fˆxyR

+⎝ ⎠⎛ ⎞=

fˆzI Aκ B1I f

ˆxzR

B2 I fˆyzR

+⎝ ⎠⎛ ⎞=

mxI A B2I myyR B1I mxy

R κ4--- f

ˆyzR

–+⎝ ⎠⎛ ⎞=

myI A B1 I mxxR B2 Imxy

R κ4--- f

ˆxzR

–+⎝ ⎠⎛ ⎞–=

mzI 0=


where is the area of the element, and is the shear factor from the Mindlin theory. In the Belytschko-Lin-Tsay formulation, is used as a penalty parameter to enforce the Kirchhoff normality condition as the shell becomes thin.

The above local nodal forces and moments are then transformed to the global coordinate system using the transformation relations given previously as equation (6-173). The global nodal forces and moments are then appropriately summed over all the nodes and the global equations of motion are solved for the next increment in nodal accelerations.

Hourglass Control (Belytschko-Lin-Tsay)In part, the computational efficiency of the Belytschko-Lin-Tsay and the under integrated Hughes-Liu shell elements are derived from their use of one-point quadrature in the plane of the element. To suppress the hourglass deformation modes that accompany one-point quadrature, hourglass viscosity stresses are added to the physical stresses at the local element level. The discussion of the hourglass control that follows pertains to the Hughes-Liu and the membrane elements as well. The hourglass procedure is controlled by the DYHRGIHQ PARAM.

The hourglass control used by Belytschko et al., extends an earlier derivation by Flanagan and Belytschko [1981], (see also Kosloff and Frazier [1978], Belytschko and Tsay [1983]). The hourglass shape vector, , is defined as

(6-204)

where

(6-205)

is the basis vector that generates the deformation mode that is neglected by one-point quadrature. In equation (6-204) and the reminder of this subsection, the Greek subscripts have a range of 2; e.g.,

.

The hourglass shape vector then operates on the generalized displacements, in a manner similar to equations (6-189) through (6-193), to produce the generalized hourglass strain rates

(6-206)

(6-207)

(6-208)

where the superscripts and denote bending and membrane modes, respectively. The corresponding hourglass stress rates are then given by

A κ

κ

τI

τI hI hJ xaJ( )BaI–=

h

+1

1–

+1

1–

=

xaI x1 I x2 I,( ) xI yI,( )= =

q· αB τIθ

ˆαI=

q·3

B τIυˆ

zI=

q· αM τIυ

ˆαI=

B M


178

(6-209)

(6-210)

(6-211)

where is the shell thickness and the parameters, , and are generally assigned values between

0.01 and 0.05.

Finally, the hourglass stresses, which are updated from the stress rates in the usual way; i.e.,

(6-212)

and the hourglass resultant forces are then

(6-213)

(6-214)

(6-215)

where the superscript emphasizes that these are internal force contributions from the hourglass deformations. These hourglass forces are added directly to the previously determined local internal forces due to deformations equations (6-198) through (6-203). These force vectors are orthogonalized with respect to rigid body motion.

Hourglass Control (Englemann and Whirley)Englemann and Whirley [1991] developed an alternative hourglass control, which they implemented in the framework of the Belytschko, Lin, and Tsay shell element. We will briefly highlight their procedure here that has proven to be cost effective-only twenty percent more expensive than the default control.

In the hourglass procedure, the in-plane strain field (subscript ) is decomposed into the one point strain field plus the stabilization strain field:

(6-216)

where the stabilization strain field, which is obtained from the assumed strain fields of Pian and Sumihara [1984], is given in terms of the hourglass velocity field as

(6-217)

Q·

αB rθEt3A

192------------------BβIBβI q· α

B=

Q·

3

B rwκGt3A

12-----------------------BβIBβI q·

3

B=

Q·

αM rmEtA

8----------------BβIBβI q· α

M=

t rθ rw, rm

Qn 1+ Qn Δ tQ·

+=

mαIH τIQα

B=

f3 I

HτIQ3

B=

f αIH

τIQαM

=

H

p

ε·

p ε·

p0

ε·

ps

+=

ε·

ps

Wmq· m zWbq· b+=


Here, and play the role of stabilization strain velocity operators for membrane and bending:

(6-218)

(6-219)

where the terms , are rather complicated and the reader is referred to the reference

[Englemann and Whirley, 1991].

To obtain the transverse shear assumed strain field, the procedure given in [Bathe and Dvorkin, 1984] is used. The transverse shear strain field can again be decomposed into the one point strain field plus the stabilization field:

(6-220)

that is related to the hourglass velocities by

(6-221)

where the transverse shear stabilization strain-velocity operator is given by

(6-222)

Again, the coefficients and are defined in the reference.

In their formulation, the hourglass forces are related to the hourglass velocity field through an incremental hourglass constitutive equation derived from an additive decomposition of the stress into a “one-point stress,” plus a “stabilization stress.” The integration of the stabilization stress gives a resultant constitutive equation relating hourglass forces to hourglass velocities.

The in-plane and transverse stabilization stresses are updated according to:

(6-223)

where the tangent matrix is the product of a matrix , which is constant within the shell domain, and a scalar that is constant in the plane but may vary through the thickness.

Wm Wb

Wm

f1

pξ η,( ) f

4

pξ η,( )

f2

pξ η,( ) f

5

pξ η,( )

f3

pξ η,( ) f

6

pξ η,( )

=

Wb

f4

pξ η,( ) – f

1

pξ η,( )

f5

pξ η,( ) – f

2

pξ η,( )

f6

pξ η,( ) – f

3

pξ η,( )

=

fi

pξ η,( ) i 1 2 … 6, , ,=

ε·

s ε·

s0

ε·

ss

+=

ε·

ss

Wsq· s=

Ws

Wsf

1

s ξ η,( ) g1

s ξ– g2

s η g3

s ξ g3

s η

f2s ξ η,( ) g

4s ξ g

4s η g

2s ξ– g

1s η

=

f1

s ξ η,( ) g1s

τss n 1+, τs

s n, Δ tcsCsε·

ss+=

C

c


180

The stabilization stresses can now be used to obtain the hourglass forces:

(6-224)

Qm WmT

τps Ad zd

A∫h

2---–

h2---

∫=

Qb Wb

Tτp

s Ad zd

A∫h

2---–

h2---

∫=

Qs Ws

Tτs

s Ad zd

A∫h

2---–

h2---

∫=

181Chapter 6: ElementsCQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang Improvements

CQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang ImprovementsSince the Belytschko-Tsay element is based on a perfectly flat geometry, warpage is not considered. Although this generally poses no major difficulties and provides for an efficient element, incorrect results in the twisted beam problem, See Figure 6-10, are obtained where the nodal points of the elements used in the discretization are not coplanar. The Hughes-Liu shell element considers non-planar geometry and gives good results on the twisted beam, but is relatively expensive. The effect of neglecting warpage in typical a application cannot be predicted beforehand and may lead to less than accurate results, but the latter is only speculation and is difficult to verify in practice. Obviously, it would be better to use shells that consider warpage if the added costs are reasonable and if this unknown effect is eliminated. In this section, we briefly describe the simple and computationally inexpensive modifications necessary in the Belytschko-Tsay shell to include the warping stiffness. The improved transverse shear treatment is also described which is necessary for the element to pass the Kirchhoff patch test. Readers are directed to the references [Belytschko, Wong, and Chang 1989, 1992] for an in depth theoretical background.

Figure 6-10 The Twisted Beam Problem Fails with the Belytschko-Tsay Shell Element

In order to include warpage in the formulation it is convenient to define nodal fiber vectors as shown in Figure 6-11. The geometry is interpolated over the surface of the shell from:

(6-225)

where: and is a parametric coordinate which varies between -1 to +1.

00

5 10 15 19

4

8

12

16

20

24

2930

Belystchkno-Tsay

Hughes-LiuBelytschko-Wong-Chiang

time (ms)

Displacement-time History

Y D

ispl

acem

ent (

104 )

Twisted Beam Problem

L = 12b = 1.1t = .32twist = 90 degreesE = 29 000 000ν = .22

x xm ζp+ xI ζpI+( )NI ξ η,( )= =

ζ ζh2

------= ζ

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang Improvements

182

Figure 6-11 Nodal Fiber Vectors , and where is the Thickness

The in plane strain components are given by:

(6-226)

(6-227)

(6-228)

The coupling terms are come in through which is defined in terms of the components of the fiber

vectors as:

(6-229)

For a flat geometry the normal vectors are identical and no coupling can occur. Two methods are used by

Belytschko for computing and the reader is referred to his papers for the details. Both methods have

been tested and comparable results were obtained.

The transverse shear strain components are given as

(6-230)

(6-231)

where the nodal rotational components are defined as:

(6-232)

hp2

p3

p1

p1 p2, p3 h

dxx bxI vxI ζ bxIc

vxI bxIp·xI+( )+=

dyy byI vyI ζ byIc

vyI byIp·yI+( )+=

dxy12---bxIvyI byI vxI ζ bxI

cvyI bxIp

·yI byI

cvxI byIp

·xI+ + +( )+ +=

biIc

bxI

c

byI

c

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

py2

py4

– py3

py1

– py4

py2

– py1

py3

–

px2

px4

– px3

px1

– px4

px2

– px1

px3

–=

biI

c

γxz N– I ξ η,( )θy I=

γyz N– I ξ η,( )θx I=

θ x I enI

ex

⋅( )θnI

enK

ex

⋅( )θnK

+=

183Chapter 6: ElementsCQUAD4 DYSHELLFORM = 10, Belytschko-Wong-Chiang Improvements

(6-233)

The rotation comes from the nodal projection

(6-234)

where the subscript refers to the normal component of side as seen in Figure 6-12 and is the length of side .

Figure 6-12 Vector and Edge Definitions for Computing the Transverse Shear Strain Components

θy I enI

ey

⋅( )θnI

enK

ey

⋅( )θnK

+=

θnI

θnI 1

2--- θnI

I θnJI

+( ) 1LIJ------- υzJ υzJ–( )+=

n I LIJ

IJ

J

K

K

I

eni

eX

enk

LK

ey

y

r

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCTRIA3 - DYSHELLFORM = 4, C0 Triangular Shell

184

CTRIA3 - DYSHELLFORM = 4, C0 Triangular ShellThe shell element due to Kennedy, Belytschko, and Lin [1986] has been implemented as a computationally efficient triangular element complement to the Belytschko-Lin-Tsay quadrilateral shell element ([Belytschko and Tsay 1981], [Belytschko et al., 1984a]). For a shell element with five through-the-thickness integration points, the element requires 649 mathematical operations (the Belytschko-Lin-Tsay quadrilateral shell element requires 725 mathematical operations) compared to 1417 operations for the Marchertas-Belytschko triangular shell [Marchertas and Belytschko 1974] (referred to as the BCIZ [Bazeley, Cheung, Irons, and Zienkiewicz 1965] triangular shell element).

Triangular shell elements are offered as optional elements primarily for compatibility with local user grid generation and refinement software. Many computer aided design (CAD) and computer aided manufacturing (CAM) packages include finite element mesh generators, and most of these mesh generators use triangular elements in the discretization. Similarly, automatic mesh refinement algorithms are typically based on triangular element discretization. Also, triangular shell element formulations are not subject to zero energy modes inherent in quadrilateral element formulations.

The triangular shell element’s origins are based on the work of Belytschko et al., [Belytschko, Stolarski, and Carpenter 1984b] where the linear performance of the shell was demonstrated. Because the triangular shell element formulations parallels closely the formulation of the Belytschko-Lin-Tsay quadrilateral shell element presented in the previous section, the following discussion is limited to items related specifically to the triangular shell element.

Co-rotational CoordinatesThe mid-surface of the triangular shell element, or reference surface, is defined by the location of the element’s three nodes. An embedded element coordinate system (see Figure 6-13) that deforms with the element is defined in terms of these nodal coordinates. The procedure for constructing the co-rotational coordinate system is simpler than the corresponding procedure for the quadrilateral, because the three nodes of the triangular element are guaranteed coplanar.

Figure 6-13 Local Element Coordinate System for Shell Element

C0

2x

3

1

y

z

e3 e2

e1

C0

185Chapter 6: ElementsCTRIA3 - DYSHELLFORM = 4, C0 Triangular Shell

The local x-axis, , is directed from node 1 to 2. The element’s normal axis, , is defined by the vector cross product of a vector along with a vector constructed from node 1 to node 3. The local y-axis, ,

is defined by a unit vector cross product of with , which are the unit vectors in the directions,

respectively. As in the case of the quadrilateral element, this triad of co-rotational unit vectors defines a transformation between the global and local element coordinate systems (see equations (6-173) and (6-174)).

Velocity-Strain RelationsAs in the Belytschko-Lin-Tsay quadrilateral shell element, the displacement of any point in the shell is partitioned into a mid-surface displacement (nodal translations) and a displacement associated with rotations of the element’s fibers (nodal rotations). The Kennedy-Belytschko-Lin triangular shell element also uses the Mindlin [Mindlin 1951] theory of plates and shells to partition the velocity of any point in the shell (recall equation (6-175)):

(6-235)

where is the velocity of the mid-surface, is the angular velocity vector, and is the distance along the fiber direction (thickness) of the shell element. The corresponding co-rotational components of the velocity strain (rate of deformation) were given previously in equations (6-189) through (6-193).

Standard linear nodal interpolation is used to define the midsurface velocity, angular velocity, and the element’s coordinates (isoparametric representation). These interpolation functions are the area coordinates used in triangular element formulations. Substitution of the nodally interpolated velocity fields into the velocity-strain relations (see Belytschko et al., for details), leads to strain rate-velocity relations of the form

(6-236)

where are the velocity strains (strain rates), the elements of are derivatives of the nodal interpolation

functions, and the are the nodal velocities and angular velocities.

It is convenient to partition the velocity strains and the matrix into membrane and bending contributions. The membrane relations are given by

(6-237)

x z

x y

e3 e1 z

ν vm ze3 θ×–=

νm θ z

d B v=

d B

ν

B

dx

dy

2 dxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

M

1x2 y3----------

y3 0 y3 0 0 0

0 x3 x2 – 0 x3 – 0 x2

x3 x2 – y3 – x3 – y3 x2 0

υx1

υy1

υx2

υy2

υx3

υy3⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=


186

or

(6-238)

The bending relations are given by

(6-239)

or

(6-240)

The local element velocity strains are then obtained by combining the above two relations:

(6-241)

The remaining two transverse shear strain rates are given by

(6-242)

dˆ M BM ν=

κx

κy

2 κxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1–x2 y3

----------

0 y3 – 0 y3 0 0

x3 x2 – 0 x3 0 x2 – 0

y3 x3 x2 – y3 – x3 – 0 x2

θx1

θˆ y1

θˆ x2

θˆ y2

θˆ x3

θˆ y3⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

κM BMθˆ def=

dˆ

x

dˆ

y

2dˆ

xy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

dˆ

x

dˆ

y

2dˆ

xy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

M

z

κx

κy

2 κxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

dˆ M

z κ–=

2dˆ

xz

2dˆ

yz⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

16 x2 y3

--------------=

y3

2– y3 2 x2 x3+( ) y

32 y3 3 x2 x3–( ) 0 x2 y3

y3 x2 2 x2–( ) x2

2x3

2– y

3

2x2 x3+( )– x3 x3 2 x2–( ) 3 x2 y3– x2 2 x3 x2–( )

θx1

θˆ y1

θˆ x2

θˆ y2

θˆ x3

θˆ y3⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

def


or

(6-243)

All of the above velocity-strain relations have been simplified by using one-point quadrature.

In the above relations, the angular velocities are the deformation component of the angular velocity

obtained by subtracting the portion of the angular velocity due to rigid body rotation; i.e.,

(6-244)

The two components of the rigid body angular velocity are given by

(6-245)

(6-246)

The first of the above two relations is obtained by considering the angular velocity of the local x-axis about the local y-axis. Referring to Figure 6-14, by construction nodes 1 and 2 lie on the local x-axis and the distance between the nodes is ; i.e., the distance from node 2 to the local coordinate origin at

node 1. Thus, the difference in the nodal velocities divided by the distance between the nodes is an average measure of the rigid body rotation rate about the local y-axis.

Figure 6-14 Element Configurations with Node 3 Aligned with Node 1 (left) and Node 3 Aligned with Node 2 (right)

dˆ S BSθˆ def

=

θˆ def

θˆ

θˆ def θˆ θˆ rig–=

θˆ yrig υz1 υz2–

x2

----------------------=

θxrig υz3 υz1–( ) x2 υz2 υz1–( ) x3–

x2 y3

-------------------------------------------------------------------------=

x2 x

z

1 2

3

y

x

z

y

z

3

x21


188

The second relation is conceptually identical, but is implemented in a slightly different manner due to the arbitrary location of node 3 in the local coordinate system. Consider the two local element configurations shown in Figure 6-14. For the left-most configuration, where node 3 is the local y-axis, the rigid body rotation rate about the local x-axis is given by

(6-247)

and for the rightmost configuration the same rotation rate is given by

(6-248)

Although both of these relations yield the average rigid body rotation rate, the selection of the correct relation depends on the configuration of the element; i.e., on the location of node 3. Since every element in the mesh could have a configuration that is different in general from either of the two configurations shown in Figure 6-14, a more robust relation is needed to determine the average rigid body rotation rate about the local x-axis. In most typical grids, node 3 will be located somewhere between the two configurations shown in Figure 6-14. Thus, a linear interpolation between these two rigid body rotation rates was devised using the distance as the interpolant:

(6-249)

Substitution of equations (6-247) and (6-248) into equation (6-249) and simplifying produces the relations given previously as equation (6-246).

Stress Resultants and Nodal ForcesAfter suitable constitutive evaluation using the above velocity strains, the resulting local stresses are integrated through the thickness of the shell to obtain local resultant forces and moments. The integration formulae for the resultants are

(6-250)

(6-251)

where the superscript indicates a resultant force or moment and the Greek subscripts emphasize the limited range of the indices for plane stress plasticity.

The above element midplane force and moment resultant are related to the local nodal forces and moments by invoking the principle of virtual power and performing a one-point quadrature. The relations obtained in this manner are

θx lef t–rig υz3 υz1–

y3

----------------------=

θx right–rig υz3 υz2–

y3

----------------------=

x3

θˆ xrig θˆ x left–

rig 1x3

x2

-----–⎝ ⎠⎛ ⎞ θˆ x right–

rig x3

x2

-----⎝ ⎠⎛ ⎞+=

fˆ

αβR σαβ zd∫=

mαβR z σαβ zd∫–=

R


(6-252)

(6-253)

where is the area of the element .

The remaining nodal forces, the component of the force , are determined by successively

solving the following equilibration equations

(6-254)

(6-255)

(6-256)

which represent moment equilibrium about the local x-axis, moment equilibrium about the local y-axis, and force equilibrium in the local z-direction, respectively.

fˆx1

fˆy1

fˆx2

fˆy2

fˆx3

fˆy3

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

ABMT

fˆ

xxR

fˆ

yyR

fˆ

xyR

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

mx1

my1

mx2

my2

mx3

my3⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

ABMT

mxxR

myyR

mxyR

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

ABST f

ˆxzR

fˆ

yzR

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

+=

A 2A x2 y3=( )

z fˆz3

fˆz2

fˆz1

, ,( )

mx1 mx2 mx3 y3 fˆz3

+ + + 0=

my1 my2 my3 x3 fˆz3– x2 f

ˆz2–+ + 0=

fˆz1

fˆz2

fz3

+ + 0=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCTRIA3 - DYSHELLFORM = 3, Marchertas-Belytschko Triangular Shell (BCIZ)

190

CTRIA3 - DYSHELLFORM = 3, Marchertas-Belytschko Triangular Shell (BCIZ)The Marchertas-Belytschko [1974] triangular shell element, or the BCIZ triangular shell element, was developed in the same time period as the Belytschko beam element [Belytschko, Schwer, and Klein, 1977], see CBEAM - Belytschko Beam, forming the first generation of co-rotational structural elements developed by Belytschko and co-workers. Although the Marchertas-Belytschko shell element is relatively expensive (i.e., the triangular shell element with five through-the-thickness integration points requires 649 mathematical operations compared to 1,417 operations for the Marchertas-Belytschko triangular shell), it is maintained in SOL 700 for compatibility with earlier user models. However, as the user community moves to application of the more efficient shell element formulations, the use of the Marchertas-Belytschko triangular shell element will decrease.

As mentioned above, the Marchertas-Belytschko triangular shell has a common co-rotational formulation origin with the Belytschko beam element. The interested reader is referred to the beam element description, see Co-rotational Technique for details on the co-rotational formulation. In the next subsection a discussion of how the local element coordinate system is identical for the triangular shell and beam elements. The remaining subsections discuss the triangular element’s displacement interpolants, the strain displacement relations, and calculations of the element nodal forces.

Element CoordinatesFigure 6-15a shows the element coordinate system, originating at Node 1, for the Marchertas-Belytschko triangular shell. The element coordinate system is associated with a triad of unit vectors

the components of which form a transformation matrix between the global and local coordinate

systems for vector quantities. The nodal or body coordinate system unit vectors are defined at

each node and are used to define the rotational deformations in the element, see Co-rotational Technique.

The unit normal to the shell element is formed from the vector cross product

(6-257)

where and are unit vectors originating at Node 1 and pointing towards Nodes 2 and 3, respectively

(see Figure 6-15b).

C0

x y z, ,( )

e1 e2 e3, ,( )

b1 b2 b3, ,

e3

e3 l21 l31×=

l21 l31

191Chapter 6: ElementsCTRIA3 - DYSHELLFORM = 3, Marchertas-Belytschko Triangular Shell (BCIZ)

Figure 6-15 Construction of Local Element Coordinate System

1

2

3

x

y

b3

b2

b1

e1

e2

e3

z

(a) Element and Body Coordinates

2

1

3

x

y

z l31α

α 2⁄

β

g

l21

e3

(b) Construction of Element Coordinates


192

Next a unit vector, , see Figure 6-15b, is assumed to be in the plane of the triangular element with its origin at Node 1 and forming an angle with the element side between Nodes 1 and 2, i.e., the vector

. The direction cosines of this unit vector are represented by the symbols . Since is the unit

vector, its direction cosines will satisfy the equation:

(6-258)

Also, since and are orthogonal unit vectors, their vector dot product must satisfy the equation

(6-259)

In addition, the vector dot product of the co-planar unit vectors and satisfies the equation

(6-260)

where are the direction cosines of .

Solving this system of three simultaneous equation; i.e., equations (6-258), (6-259), and (6-260), for the direction cosines of the unit vector g yields

(6-261)

These equations provide the direction cosines for any vector in the plane of the triangular element that is oriented at an angle from the element side between Nodes 1 and 2. Thus the unit vector components of , and are obtained by setting and in (6-261), respectively. The angle

is obtained from the vector dot product of the unit vectors and ,

(6-262)

Displacement InterpolationAs with the other large displacement and small deformation co-rotational element formulations, the nodal displacements are separated into rigid body and deformation displacements,

(6-263)

where the rigid body displacements are defined by the motion of the local element coordinate system, i.e., the co-rotational coordinates, and the deformation displacement are defined with respect to the co-rotational coordinates. The deformation displacement are defined by

g

β

l21 gx gy gz, ,( ) g

gx2 g

y2 g

z2+ + 1=

g e3

e3xgx e3ygy e3zgz+ + 0=

g l21

I21xgx I21ygy I21zgz+ + βcos=

l21x + l21y + l21z, ,( ) l21

gy l21y β e3zl21x e3xl21z–( ) βsin+cos=

gx l21x β e3yl21z e3z l21y–( ) βsin+cos=

gz l21z β e3xl21y e3y l21x–( ) βsin+cos=

β

e1 e2 β α 2⁄= β π α+( ) 2⁄= α

l21 l31

αcos l21 l31⋅=

u urigid udef+=


(6-264)

where

(6-265)

are the edge elongations and

(6-266)

are the local nodal rotation with respect to the co-rotational coordinates.

The matrices and are the membrane and flexural interpolation functions, respectively. The

element’s membrane deformation is defined in terms of the edge elongations. Marchertas and Belytschko adapted this idea from Argyris et. al., [1964], where incremental displacements are used, by modifying the relations for total displacements,

(6-267)

where , etc.

The non-conforming shape functions used for interpolating the flexural deformations, were originally

derived by Bazeley, Cheung, Irons, and Zienkiewicz [1965], called the BCIZ element. Explicit

expressions for are quite tedious and are not given here. The interested reader is referred to Appendix

G in the original work of Marchertas and Belytschko [1974].

The local nodal rotations, which are interpolated by these flexural shape functions, are defined in a manner similar to those used in the Belytschko beam element. The current components of the original element normal are obtained from the relation

(6-268)

where and are the current transformations between the global coordinate system and the element

(local) and body coordinate system, respectively. The vector is the original element unit normal

expressed in the body coordinate system. The vector cross product between this current-original unit normal and the current unit normal,

(6-269)

ux

uy

---

uz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

def φxm

φym

---

φzf

δ---

θˆ⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

δ{ }T δ12δ23δ31{ }=

θ{ } θˆ 1xθˆ 1yθˆ 2xθˆ 2yθˆ 3xθˆ 3y{ }=

φxm φy

m, φzf

δi j

2 xjiuj ix yjiujiy zj iuj iz+ +( ) ujix2

ujiy2

ujiz2

+ + +

li j0

lij+------------------------------------------------------------------------------------------------------------------=

xji xj xi–=

φz

f

φz

f

e30 μTλe3

0=

μ λ

e30

e3 e30× θˆ xe1 θˆ ye2+=


194

define the local nodal rotations as

(6-270)

(6-271)

Note that at each node the corresponding transformation matrix is used in equation (6-268).

Strain-Displacement RelationsMarchertas-Belytschko impose the usual Kirchhoff assumptions that normals to the midplane of the element remain straight and normal, to obtain

(6-272)

(6-273)

(6-274)

where it is understood that all quantities refer to the local element coordinate system.

Substitution of equation (6-264) into the above strain-displacement relations yields

(6-275)

where

(6-276)

with

(6-277)

and

θˆ x e– 3y0=

θˆ y e3x0=

λ

exx

∂ux

∂x-------- z

∂2uz

∂x2-----------–=

eyy

∂uy

∂y-------- z

∂2uz

∂y2-----------–=

2exy

∂ux

∂y--------

∂uy

∂x-------- 2z

∂2uz

∂x∂y------------–+=

ε{ } Em[ ] δ{ } z Ef

[ ] θˆ{ }–=

ε{ } t εxx εyy 2εxy, ,{ }=

Em[ ]

∂φxim

∂x----------

∂φyim

∂y----------

∂φxim

∂y----------

∂φyim

∂x----------+

=


(6-278)

Nodal Force CalculationsThe local element forces and moments are found by integrating the local element stresses through the thickness of the element. The local nodal forces are given by

(6-279)

where

(6-280)

(6-281)

where the side forces and stresses are understood to all be in the local convected coordinate system.

Similarly, the local moments are given by

(6-282)

where

(6-283)

The through-the-thickness integration portions of the above local force and moment integrals are usually performed with a 3- or 5-point trapezoidal integration. A three-point in-plane integration is also used; it is, in part, this three-point in-plane integration that increases the operation count for this element over the

shell, which used one-point inplane integration with hourglass stabilization.

The remaining transverse nodal forces are obtained from element equilibrium considerations. Moment equilibrium requires

(6-284)

where is the area of the element. Next transverse force equilibrium provides

(6-285)

Ef

[ ]

∂2φzif

∂x2-------------

∂2φzif

∂y2-------------

2∂2φzi

f

∂x∂y-------------

=

f Em[ ]t σ Vd∫=

fˆT

f12 f23 f31{ }=

σT σxxσyyσxy{ }=

mT z Ef[ ]t σ vd∫–=

mT m1xm1ym2xm2ym3xm3y{ }=

C0

fˆ2z

fˆ3z⎩ ⎭

⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

12A-------

x3– y3

x2 y2–

m1x m2x m3x+ +

m1y m2y m3y+ +⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

A

fˆ1z f

ˆ2z– f

ˆ3z–=


196

The corresponding global components of the nodal forces are obtained from the following transformation

(6-286)

Finally, the local moments are transformed to the body coordinates using the relation

(6-287)

fix

fiy

fiz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fij

lij----

xij uijx+

yij uijy+

zij ui jz+⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fik

lik-----

xik uikx+

yik uiky+

zik uikz+⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆi z

e3x

e3y

e3z⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

+ +=

mix

miy

miz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

λTμmix

miy

miz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

197Chapter 6: ElementsCQUAD4 - DYSHELL FORM = 1, Hughes-Liu Shell

CQUAD4 - DYSHELL FORM = 1, Hughes-Liu ShellThe Hughes-Liu shell element formulation ([Hughes and Liu 1981a, b], [Hughes et al., 1981], [Hallquist et al., 1985]) has several desirable qualities:

• It is incrementally objective (rigid body rotations do not generate strains), allowing for the treatment of finite strains that occur in many practical applications;

• It is simple, which usually translates into computational efficiency and robustness;

• It is compatible with brick elements, because the element is based on a degenerated brick element formulation. This compatibility allows many of the efficient and effective techniques developed for brick elements to be used with this shell element;

• It includes finite transverse shear strains;

• A through-the-thickness thinning option (see [Hughes and Carnoy 1981]) is also available when needed in some shell element applications.

The remainder of this section reviews the Hughes-Liu shell element (referred to by Hughes and Liu as the U1 element) which is a four-node shell with uniformly reduced integration, and summarizes the modifications to their theory as it is implemented in SOL 700. A detailed discussion of these modifications are presented in an article by Hallquist and Benson [1986].

GeometryThe Hughes-Liu shell element is based on a degeneration of the standard 8-node brick element formulation, an approach originated by Ahmad et al. [1970]. Recall from the discussion of the solid elements the isoparametric mapping of the bi-unit cube:

(6-288)

(6-289)

where is an arbitrary point in the element, are the parametric coordinates, are the global

nodal coordinates of node , and are the element shape functions evaluated at node , i.e.,

are evaluated at node .

In the shell geometry, planes of constant will define the lamina or layers of the shell and fibers are defined by through-the-thickness lines when both and are constant (usually only defined at the nodes and thus referred to as ‘nodal fibers’). To degenerate the 8-node brick geometry into the 4-node shell geometry, the nodal pairs in the direction (through the shell thickness) are combined into a single node, for the translation degrees of freedom, and an inextensible nodal fiber for the rotational degrees of freedom. Figure 6-16 shows a schematic of the bi-unit cube and the shell element.

The mapping of the bi-unit cube into the shell element is separated into two parts

(6-290)

x ξ η ζ,,( ) Na ξ η ζ,,( )xa=

Na ξ η ζ,,( )1 ξaξ+( ) 1 ηaη+( ) 1 ζaζ+( )

8--------------------------------------------------------------------------=

x ξ η ζ, ,( ) xa

a Na a ξa ηa ζa, ,( )

ξ η ζ, ,( ) a

ζ

ξ η

ζ

x ξ η ζ,,( ) x ξ η,( ) X ξ η ζ,,( )+=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUAD4 - DYSHELL FORM = 1, Hughes-Liu Shell

198

where denotes a position vector to a point on the reference surface of the shell and is a position vector, based at point on the reference, that defines the fiber direction through that point. In particular, if we consider one of the four nodes which define the reference surface, then

(6-291)

(6-292)

Figure 6-16 Mapping of the Bi-unit Cube into the Hughes-Liu Shell Element and Nodal Fiber Nomenclature

x X

x

x ξ η,( ) Na ξ η,( )xa=

X ξ η ζ,,( ) Na ξ η,( )Xa ζ( )=

x

ζ

ξ

η

Bi-unit Cube

xζ

ξ

η

Shell Element

z

ζ

+

x+

xx

-1

0

+1Top Surface

Nodal Fiber

Bottom Surface

Reference Surface

z


With this description, arbitrary points on the reference surface are interpolated by the two-dimensional shape function operating on the global position of the four shell nodes that define the reference surfaces; i.e., . Points off the reference surface are further interpolated by using a one-dimensional

shape function along the fiber direction, i.e., , where

(6-293)

(6-294)

(6-295)

(6-296)

As shown in the lower portion of Figure 6-16, is a unit vector in the fiber direction and is a thickness function. (Thickness changes (see [Hughes and Carnoy 1981]) are accounted for by explicitly adjusting the fiber lengths at the completion of a time step based on the amount of straining in the fiber direction. Updates of the fiber lengths always lag one time step behind other kinematical quantities.)

The reference surface may be located at the mid-surface of the shell or at either of the shell’s outer surfaces. This capability is useful in several practical situations involving contact surfaces, connection of shell elements to solid elements, and offsetting elements such as stiffeners in stiffened shells. The

reference surface is located within the shell element by specifying the value of the parameter (see lower

portion of Figure 6-16). When , the reference surface is located at the bottom, middle, and top surface of the shell, respectively.

The Hughes-Liu formulation uses two position vectors, in addition to , to locate the reference surface

and define the initial fiber direction. The two position vectors and are located on the top and bottom

surfaces, respectively, at node . From these data the following are obtained:

(6-297)

(6-298)

(6-299)

(6-300)

(6-301)

where is the Euclidean norm.

x

N ξ η,( )

xa

Xa ζ( )

Xa ζ( ) za ζ( )Xa=

za ζ( ) N+ ζ( )za+ N- ζ( )za

-+=

N+ ζ( ) 1 ζ+( )2

-----------------=

N- ζ( ) 1 ζ–( )2

-----------------=

Xa z ζ( )

ζ

ζ 1 0 +1, ,–=

ζ

xa+ xa

-

a

xa12--- 1 ζ–( )xa

- 1 ζ+( )+[ ]xa+=

Xaxa

+ xa-–( )

ha-----------------------=

za+ 1

2--- 1 ζ–( )ha=

za- 1

2---– 1 ζ+( )ha=

ha xa+ xa

-–=

.


200

KinematicsThe same parametric representation used to describe the geometry of the shell element (i.e., reference surface and fiber vector interpolation) are used to interpolate the shell element displacement; i.e., an isoparametric representation. Again, the displacements are separated into the reference surface displacements and rotations associated with the fiber direction:

(6-302)

(6-303)

(6-304)

(6-305)

where is the displacement of a generic point; is the displacement of a point on the reference surface, and is the ‘fiber displacement’ rotations; the motion of the fibers can be interpreted as either displacements or rotations as will be discussed.

Hughes and Liu introduce the notation that follows, and the associated schematic shown in Figure 6-17, to describe the current deformed configuration with respect to the reference configuration:

(6-306)

(6-307)

(6-308)

(6-309)

(6-310)

(6-311)

Figure 6-17 Schematic of Deformed Configuration Displacements and Position Vectors

u ξ η ζ,,( ) u ξ η,( ) U ξ η ζ,,( )+=

u ξ η,( ) Na ξ η,( )ua=

U ξ η ζ,,( ) Na ξ η,( )Ua ζ( )=

Ua ζ( ) za ζ( )Uˆ

a=

u u

U

y y Y+=

y x u+=

ya xa ua+=

Y X U+=

Ya Xa Ua+=

Ya Xa Ua+=

Y

Uu

u

X

Deformed ConfigurationReference Surface

Initial Configuration Reference Surface

Parallel Construction

x

x

y

y


In the above relations, and in Figure 6-17, the quantities refer to the reference configuration, the quantities refer to the updated (deformed) configuration and the quantities are the displacements. The notation consistently uses a superscript bar to indicate reference surface quantities, a superscript caret

to indicate unit vector quantities, lower case letters for translational displacements, and upper case letters indicating fiber displacements. To update to the deformed configuration, two vector quantities are

needed: the reference surface displacement and the associated nodal fiber displacement . The nodal fiber displacements are defined in the fiber coordinate system, described in the next subsection.

Fiber Coordinate SystemFor a shell element with four nodes, the known quantities will be the displacements of the reference surface obtained from the translational equations of motion and some rotational quantities at each node obtained from the rotational equations of motion. To complete the kinematics, we now need a relation between nodal rotations and fiber displacements .

At each node a unique local Cartesian coordinate system is constructed that is used as the reference frame for the rotation increments. The relation presented by Hughes and Liu for the nodal fiber displacements (rotations) is an incremental relation; i.e., it relates the current configuration to the last state, not to the

initial configuration. Figure 6-18 shows two triads of unit vectors: comprising the

orthonormal fiber basis in the reference configuration (where the fiber unit vector is now ) and

indicating the incrementally updated current configuration of the fiber vectors. The reference

triad is updated by applying the incremental rotations, and , obtained from the rotational

equations of motion, to the fiber vectors ( and ) as shown in Figure 6-18. The linearized relationship

between the components of in the fiber system, , and the incremental rotations is

given by

(6-312)

Although the above Hughes-Liu relation for updating the fiber vector enables a reduction in the number of nodal degrees of freedom from six to five, it is not implemented in SOL 700 because it is not applicable to beam elements.

x y

u

( )

ˆ( )

u U

u

U

b1f

b2f

b3f

, ,( )

Y b3f

=

b1 b2 b3,,( )

Δθ1 Δθ2

b1f

b2f

ΔU ΔU1

fΔU2

fΔU3

f, ,

ΔU1

f

ΔU2

f

ΔU3

f

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

1 – 0

0 1–

0 0

Δθ1

Δθ2⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=


202

Figure 6-18 Incremental Update of Fiber Vectors Using Hughes-Liu Incremental Rotations

In SOL 700, three rotational increments are used, defined with reference to the global coordinate axes:

(6-313)

Equation (6-313) is adequate for updating the stiffness matrix, but for finite rotations the error is significant. A more accurate second-order technique is used in SOL 700 for updating the unit fiber vectors:

(6-314)

where

(6-315)

(6-316)

(6-317)

Here, is the Kronecker delta and is the permutation tensor. This rotational update is often referred

to as the Hughes-Winget formula [Hughes and Winget 1980]. An exact rotational update using Euler angles or Euler parameters could easily be substituted in equation (6-314), but it is doubtful that the extra effort would be justified.

Lamina Coordinate SystemIn addition to the above described fiber coordinate system, a local lamina coordinate system is needed to enforce the zero normal stress condition; i.e., plane stress. Lamina are layers through the thickness of the shell that correspond to the locations and associated thicknesses of the through-the-thickness shell

Fiber

b3f

Yˆ=b3

b2Δθ1

b2f

b1f

b1

Δθ2

ΔU1

ΔU2

ΔU3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

0 Y3 Y2–

Y3 – 0 Y1

Y2 Y1 – 0

Δθ1

Δθ2

Δθ3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

Yi

n 1+Rij Δθ( ) Yj

n=

Rij Δθ( ) δi j12---

2δik ΔSik+( )ΔSjk

D---------------------------------------------+=

ΔSij eikjΔθk=

2D 212--- Δθ1

2 Δθ22 Δθ3

2+ +( )+=

δi j eijk


integration points; the analogy is that of lamina in a fibrous composite material. The orthonormal lamina basis (Figure 6-19), with one direction normal to the lamina of the shell, is constructed at every

integration point in the shell.

Figure 6-19 Schematic of Lamina Coordinate Unit Vectors

The lamina basis is constructed by forming two unit vectors locally tangent to the lamina:

(6-318)

(6-319)

where, as before, is the position vector in the current configuration. The normal to the lamina at the integration point is constructed from the vector cross product of these local tangents:

(6-320)

To complete this orthonormal lamina basis, the vector

(6-321)

is defined, because , although tangent to both the lamina and lines of constant , may not be normal

to and . The lamina coordinate system rotates rigidly with the element.

The transformation of vectors from the global to lamina coordinate system can now be defined in terms of the lamina basis vectors as

(6-322)

e3

e3

e2

e1

ηη = constant

ξ = c

onst

ant

ξ

e1

y ,ξ

y ,ξ

------------=

e2′ y ,η

y ,η-------------=

y

e3 e1 e2′×=

e2 e3 e1×=

e2 ξ

e1 e3

A

Aˆ

x

Aˆ

y

Aˆ

z⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

e1x e2x e3x

e1y e2y e3y

e1z e2z e3z

TAx

Ay

Az⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

q[ ] A{ }= = =


204

where are the global components of the lamina coordinate unit vectors; is a vector in the

lamina coordinates, and is the same vector in the global coordinate system.

Strains and Stress Update

Incremental Strain and Spin Tensors

The strain and spin increments are calculated from the incremental displacement gradient

(6-323)

where are the incremental displacements and are the deformed coordinates. The incremental

strain and spin tensors are defined as the symmetric and skew-symmetric parts, respectively, of :

(6-324)

(6-325)

The incremental spin tensor is used as an approximation to the rotational contribution of the

Jaumann rate of the stress tensor; SOL 700 implicit uses the more accurate Hughes-Winget transformation matrix equation (6-314) with the incremental spin tensor for the rotational update. The Jaumann rate update is approximated as:

(6-326)

where the superscripts on the stress refer to the updated ( ) and reference ( ) configurations. The Jaumann rate update of the stress tensor is applied in the global configuration before the constitutive evaluation is performed. In the Hughes-Liu shell the stresses and history variables are stored in the global coordinate system.

Stress Update

To evaluate the constitutive relation, the stresses and strain increments are rotated from the global to the lamina coordinate system using the transformation defined previously in equation (6-322).

(6-327)

(6-328)

where the superscript indicates components in the lamina (local) coordinate system.

eix eiy eiz, , Aˆ

A

Gij

∂Δui

∂yj------------=

Δui yj

Gij

Δεi j12--- Gij Gji+( )=

Δωij12--- Gij Gji–( )=

Δωij

σi jn 1+ σi j

n σipn Δωpj σ jp

n Δωpi+ +=

n 1+ n

σi jln 1+

qikσknn 1+ qjn=

Δεi jln 1 2⁄+

qikΔεknn 1 2⁄+ qjn=

l


The stress is updated incrementally:

(6-329)

and rotated back to the global system:

(6-330)

before computing the internal force vector.

Incremental Strain-Displacement Relations

The global stresses are now used to update the internal force vector

(6-331)

where are the internal forces at node , is the strain-displacement matrix in the lamina

coordinate system associated with the displacements at node , and is the transformation matrix

relating the global and lamina components of the strain-displacement matrix. Because the matrix relates six strain components to twenty-four displacements (six degrees of freedom at four nodes), it is convenient to partition the matrix into four groups of six:

(6-332)

Each submatrix is further partitioned into a portion due to strain and spin:

(6-333)

with the following submatrix definitions:

(6-334)

(6-335)

σi jln 1+

σijln 1+

Δσijln 1 2⁄+

+=

σijn 1+ qkiσkn

ln 1+

qnj=

f aint

TaT

BaTσ υd∫=

f aint

a Ba

a Ta

B

B

B B1 B2 B3 B4, , ,[ ]=

Ba

BaBa

ε

Baω

=

Baε

B1 0 0 B4 0 0

0 B2 0 0 B5 0

B2 B1 0 B5 B4 0

0 B3 B2 0 B6 B5

B3 0 B1 B6 0 B4

=

Baω

B2 B– 1 0 B5 B– 4 0

0 B3 B– 2 0 B6 B– 5

B– 3 0 B1 B– 6 0 B4

=


206

where

(6-336)

Notes on strain-displacement relations:

• The derivatives of the shape functions are taken with respect to the lamina coordinate system,

e.g., . The superscript bar indicates the ’s are evaluated at the center of the lamina

. The strain-displacement matrix uses the ‘B-Bar’ approach advocated by Hughes

[1980]. In the SOL 700 implementations, this entails replacing certain rows of the matrix and the strain increments with their counterparts evaluated at the center of the element. In particular, the strain-displacement matrix is modified to produce constant shear and spin increments throughout the lamina.

• The resulting -matrix is a 8 x 24 matrix. Although there are six strain and three rotations increments, the -matrix has been modified to account for the fact that will be zero in the

integration of equation (6-331).

Bi

Na i,

∂Na

∂yil

---------- =

Naza( ) ,i 3–

∂ Naza( )

∂yi 3–l

--------------------=

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

for i 1 2 3, ,=

for i 4 5 6, ,=

yl qy= B0 0 ζ, ,( ) B( )

B

B

B σ33

207Chapter 6: ElementsCQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells

CQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu ShellsIt is well known that one-point integration results in zero energy modes that must be resisted. The four-node under integrated shell with six degrees of freedom per node has nine zero energy modes, six rigid body modes, and four unconstrained drilling degrees of freedom. Deformations in the zero energy modes are always troublesome but usually not a serious problem except in regions where boundary conditions such as point loads are active. In areas where the zero energy modes are a problem, it is highly desirable to provide the option of using the original formulation of Hughes-Liu with selectively reduced integration.

The major disadvantages of full integration are two-fold:

• nearly four times as much data must be stored;

• the operation count increases three- to fourfold. The level 3 loop is added as shown in Figure 6-20.

However, these disadvantages can be more than offset by the increased reliability and accuracy.

Figure 6-20 Selectively Reduced Integration Rule Results in Four Inplane Points Being Used

There are two version of the Hughes-Liu shell with selectively reduced integration. The first closely follows the intent of the original paper, and therefore no assumptions are made to reduce costs, which are outlined in operation counts in Table 6-4. This is activated by using DYSHELLFORM = 6. These operation counts can be compared with those in Table 6-5 for the Hughes-Liu shell with uniformly reduced integration. The second formulation, which reduces the number of operation by more than a factor of two, is referred to as the co-rotational Hughes-Liu shell. This is activated by using DYSHELLFORM = 7. This shell is considerably cheaper due to the following simplifications:

• Strains rates are not centered. The strain displacement matrix is only computed at time

and not at time .

η

ξ

n 1+

n 1 2⁄+

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells

208

• The stresses are stored in the local shell system following the Belytschko-Tsay shell. The transformations of the stresses between the local and global coordinate systems are thus avoided.

• The Jaumann rate rotation is not performed, thereby avoiding even more computations. This does not necessarily preclude the use of the shell in large deformations.

To study the effects of these simplifying assumptions, we can compare results with those obtained with the full Hughes-Liu shell. Thus far, we have been able to get comparable results.

Figure 6-21 An Inner Loop, Level 3, is added for the Hughes-Liu Shell with Selectively Reduced Integration

Table 6-4 Operation Counts for the Hughes-Liu Shell with Selectively Reduced Integration

LEVEL L1 - Once per element

Midstep translation geometry, etc. 204

Midstep calculation of 318

LEVEL L2 - For each integration point through thickness (NT points)

Strain increment at 316

Hughes-Winget rotation matrix 33

Square root of Hughes-Winget matrix 47

LEVEL L1 - Do over each element group, gather data, midstep geometry calculations

LEVEL 2 - For each thickness integration point center of element calculations for selective reduced integration

LEVEL 3 - Do over four Gauss points stress update and force contributions

LEVEL 2 - Completion

LEVEL L1 - Completion

Y

0 0 ζ, ,( )

209Chapter 6: ElementsCQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells

Rotate strain increments into lamina coordinates 66

Calculate rows 3 through 8 of B matrix 919

LEVEL L3 - For each integration point in lamina

Rotate stress to configuration 75

Incremental displacement gradient matrix 370

Rotate stress to lamina system 75

Rotate strain increments to lamina system 55

Constitutive model model dependent

Rotate stress back to global system 69

Rotate stress to configuration 75

Calculate rows 1 and 2 of B matrix 358

Stresses in lamina system 75

Stress divergence 245

TOTAL 522 +NT {1381 +4 * 1397}

Table 6-5 Operation Counts for the SOL 700 Implementation of the Uniformly Reduced Hughes-Liu Shell

LEVEL L1 - Once per element

Calculate displacement increments 24

Element areas for time step 53

Calculate 238

LEVEL L2 and L3 - Integration point through thickness (NT points)

Incremental displacement gradient matrix 284

Jaumann rotation for stress 33

Rotate stress into lamina coordinates 75

Rotate stain increments into lamina coordinates 81

Constitutive model model dependent

Rotate stress to global coordinates 69

Stress divergence 125

Table 6-4 Operation Counts for the Hughes-Liu Shell with Selectively Reduced Integration (continued)

n 1 2⁄+

n 1+

n 1+

Y

n 1+

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUAD4, DYSHELLFORM = 6, 7, Fully Integrated Hughes-Liu Shells

210

Element Mass MatrixHughes, Liu, and Levit [Hughes et al., 1981] describe the procedure used to form the shell element mass matrix in problems involving explicit transient dynamics. Their procedure, which scales the rotary mass terms, is used for all shell elements in SOL 700 including those formulated by Belytschko and his co-workers. This scaling permits large critical time step sizes without loss of stability.

The consistent mass matrix is defined by

(6-337)

but cannot be used effectively in explicit calculations where matrix inversions are not feasible. In SOL 700 only three and four-node shell elements are used with linear interpolation functions; consequently, we compute the translational masses from the consistent mass matrix by row summing, leading to the following mass at element node :

(6-338)

The rotational masses are computed by scaling the translational mass at the node by the factor :

(6-339)

where

(6-340)

(6-341)

(6-342)

(6-343)

(6-344)

and and are the volume and the thickness of the element, respectively.

LEVEL L1 - Cleanup

Finish stress divergence 60

Hourglass control 356

TOTAL 731 +NT * 667

Table 6-5 Operation Counts for the SOL 700 Implementation of the Uniformly Reduced Hughes-Liu Shell (continued)

M ρNtN υmd

υm

∫=

a

Mdispaρφa υd

v∫=

α

Mrota∞Mdispa

=

∞ max ∞1 ∞2,{ }=

∞1 za⟨ ⟩ 2 112------ za[ ]2+=

∞2V8h------=

za⟨ ⟩za

+ za-+( )

2----------------------=

za[ ] za+ za

-–=

V h

211Chapter 6: ElementsTransverse Shear Treatment for Layered Shell

Accounting for Thickness ChangesHughes and Carnoy [1981] describe the procedure used to update the shell thickness due to large membrane stretching. Their procedure with any necessary modifications is used across all shell element types in SOL 700. One key to updating the thickness is an accurate calculation of the normal strain component . This strain component is easily obtained for elastic materials but can require an iterative

algorithm for nonlinear material behavior. In SOL 700, we, therefore, default to an iterative plasticity update to accurately determine .

Hughes and Carnoy integrate the strain tensor through the thickness of the shell in order to determine a mean value :

(6-345)

and then project it to determine the straining in the fiber direction:

(6-346)

Using the interpolation functions through the integration points the strains in the fiber directions are extrapolated to the nodal points if 2 x 2 selectively reduced integration is employed. The nodal fiber lengths can now be updated:

(6-347)

Transverse Shear Treatment for Layered ShellThe shell element formulations that include the transverse shear strain components are based on the first order shear deformation theory, which yield constant through thickness transverse shear strains. This violates the condition of zero traction on the top and bottom surfaces of the shell. Normally, this is corrected by the use of a shear correction factor. The shear correction factor is 5/6 for isotropic materials; however, this value is incorrect for sandwich and laminated shells. Not accounting for the correct transverse shear strain and stress could yield a very stiff behavior in sandwich and laminated shells. This problem is addressed here by the use of the equilibrium equations without gradient in the y-direction as described by what follows. Consider the stresses in a layered shell:

(6-348)

Assume that the bending center is known. Then

(6-349)

Δε33

Δε33

Δεi j

Δεi j12--- Δεij ζd

1–

1

∫=

ε fYTΔεi j Y=

han 1+

han

1 εaf

+( )=

σ x

i( )C11

i( )εx

° zχx+( ) C12i( )

εy° zχy+( )+ C11

i( )εx

° C12i( )

εy° z C11

i( )χx C12

i( )χy+( )+ += =

σ y

i( )C12

i( )εx

° C22

i( )εy

° z++ C12

i( )χx C22

i( )χy+( )=

τ xy

i( )C44

i( )εxy

° zχxy+( )=

zx

σ x

i( )z zx–( ) C11

i( )χx C12

i( )χy+( ) C11

i( )εx zx( ) C12

i( )εy zx( )+ +=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideTransverse Shear Treatment for Layered Shell

212

The bending moment is given by the following equation:

(6-350)

or

(6-351)

where “ ” is the number of layers in the material.

Assume and , then

(6-352)

and

(6-353)

(6-354)

Therefore, the stress becomes

(6-355)

Now considering the first equilibrium equation, one can write the following:

(6-356)

(6-357)

where is the shear force and is the constant of integration. This constant is obtained from the

transverse shear stress continuity requirement at the interface of each layer.

Mxx χx C11i( )

z2 zd

zi 1–

zi

∫i 1=

NL

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

χy C12i( )

z2 zd

zi 1–

zi

∫i 1=

NL

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

+=

Mxx13--- χx C11

i( )zi

3 zi 1–3–( ) χy C12

i( )zi

3 zi 1–3–( )

i 1=

NL

∑+

i 1=

NL

∑=

NL

εy 0= σx Eχεχ=

εx

z zx–

ρ------------- z zx–( )χx= =

Mxx13--- χx Ex

i( )zi

3 zi 1–3

–( )∑( )=

χx

3Mxx

Ex

i( )zi

3 zi 1–3–( )∑

---------------------------------------------=

σx

i( ) 3MxxEx

i( )z zx–( )

Ex

i( )zi

3 zi 1–3–( )

i 1=

NL

∑

------------------------------------------------=

∂τxz

∂z----------

∂σx

∂x---------–

3QxzE x

j( )z zx–( )

E x

i( )zi

3 zi 1–3–( )

i 1=

NL

∑

------------------------------------------------–= =

τxzj( )

3QxzE x

j( ) z2

2---- zzx–⎝ ⎠⎛ ⎞

E x

i( )zi

3 zi 1–3–( )

i 1=

NL

∑

------------------------------------------------– Cj+=

Qxz Cj


Let

(6-358)

Then

(6-359)

and

(6-360)

For the first layer

(6-361)

for subsequent layers

(6-362)

Here is the stress in previous layer at the interface with the current layer. The shear stress can also

be expressed as follows:

(6-363)

where

(6-364)

and

EI( )x13---E

x

i( )zi

3 zi 1–3–( )

i 1=

NL

∑=

Cj

3QxzE x

i( ) zi 1–2

2----------- zi 1– zx–⎝ ⎠⎛ ⎞

EI( )x

-------------------------------------------------------------- τxzi 1–( )+=

τxzi( ) τxz

i 1–( ) Qxz E x

i( )

EI( )x------------------

zi 1–2

2----------- zi 1– zx–

z2

2----– zzx++=

τxz

3QxzC11

1( )

C11i( )

zi3 zi 1–

3+( )i 1=

NL

∑

-------------------------------------------------z2 zo

2–

2---------------- zx z zo–( )––=

τxz τxzi 1–( ) 3QxzC11

i( )

C11j( )

zj3 zj 1–

3–( )j 1=

NL

∑

------------------------------------------------z2 zo

2–

2---------------- zx z zo–( )– zi 1– z zi≤ ≤,–=

τxzi 1–( )

τxz

3QxzC11

i( )

C11j( )

zj3 zj 1–

3–( )j 1=

NL

∑

------------------------------------------------ f x

i( ) z2 zi 1–2–

2---------------------- zx z zi 1––( )–+=

f x

i( ) 1

C11

i( )--------- C11

j( )hj

zj zj 1++

2---------------------- zx–

j 1=

NL

∑=

hj zj zj 1––=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideTransverse Shear Treatment for Layered Shell

214

To find , the shear force, assume that the strain energy expressed through average shear modules, ,

is equal to the strain energy expressed through the derived expressions as follows:

(6-365)

(6-366)

then

to calculate use for last layer at surface

(6-367)

where

(6-368)

Algorithm:

Qxz C66

U12---

Qxz

C66h------------ 1

2---

τxz2

C66-------- zd∫= =

1

C66

-------- 9h

C11

i( )zi

3 zi 1–3–( )

j 1=

NL

∑2

---------------------------------------------------------C11

2

C66--------- f x

i( ) z2 zi 1–2–( )

2--------------------------- zx z zi 1––( )–+

2

zd∫=

9h

C11i( )

zi3 zi 1–

3–( )j 1=

NL

∑2

---------------------------------------------------------C11

i( )( )2

C66i( )

----------------- f x

i( ) z2 zi 1–2–

2---------------------- zx z zi 1––( )–+

2

zd

zi 1–

zi

∫i 1=

NL

∑=

160------ 9h

C11i( ) zi

3 zi 1–3–( )

j 1=

NL

∑2

---------------------------------------------------------C11

i( )( )2h

C66i( )-------------------- f x

i( ) 60 f xi( ) 20hi zi 2zi 1– 3zx–+( )+[ ] +{

i

NL

∑=

zxhi 20zxhi 35zi 1–2 10zi 1– zi zi 1–+( )– 15zi

2–+[ ]

+zi zi zi 1–+( ) 3zi2 7zi 1–

2–( ) 8zi 1–4 }+

1

C66

--------=

Qxz τxzh C66γxzh= =

zx τxz z 0=

C11i( ) zi

2 zi 1–2–

2-----------------------⎝ ⎠⎛ ⎞ zx zi zi 1––( )–

i 1=

NL

∑ 0=

zx

C11i( )

hi zi zi 1++( )i 1=

NL

∑

2 C11i( )

hi

i 1=

NL

∑

----------------------------------------------------=


The following algorithm is used in the implementation of the transverse shear treatment.

1. Calculate according to (6-368)

2. Calculate according to (6-364)

3. Calculate

4. Calculate

5. Calculate according to equation (6-366)

6. Calculate

7. Calculate

Steps 1-5 are performed at the initialization stage. Step 6 is performed in the shell formulation subroutine, and Step 7 is performed in the stress calculation inside the constitutive subroutine.

zx

f xi( )

13--- C11

i( )zi

3 zi 1–3–( )

i 1=

NL

∑

h13--- C11

i( ) zi3 zi 1–

3–( )i 1=

NL

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ 2

C66

Qxz C66

γxzh=

τxz

τxz

i( ) 13--- C11

i( )zi

3 zi 1–3–( )Qxz C11

i( )f x

i( ) z2 zi 1–2–

2---------------------- zx z zi 1––( )–+ z, i 1– z zi≤ ≤

i 1=

NL

∑–=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCROD, Truss Element

216

CROD, Truss Element

One of the simplest elements is the pin-jointed truss element shown in Figure 6-22. This element has three degrees of freedom at each node and carries an axial force. The displacements and velocities measured in the local system are interpolated along the axis according to

(6-369)

(6-370)

where at , , and at , .

Figure 6-22 Truss Element

Incremental strains are found from

(6-371)

and are computed using

(6-372)

The normal force is then incrementally updated using a tangent modulus according to

(6-373)

Two constitutive models are implemented for the truss element: elastic and elastic-plastic with kinematic hardening.

Note: Unlike the CBEAM, this does not support torsion.

u u1xL--- u2 u1–( )+=

u· u· 1xL--- u· 2 u· 1–( )+=

x 0= u u1= x L= u u2=

u1

L

N1

N2

A

u2

Δεu· 2 u· 1–( )

L----------------------Δ t=

Δεn 1 2⁄+2 u· 2

n 1 2⁄+ u· 1n 1 2⁄+–( )

Ln Ln 1++-----------------------------------------------------Δ tn 1 2⁄+=

N Et

Nn 1+ NnAEt Δεn 1 2⁄++=

217Chapter 6: ElementsCQUAD4 - DYSHELLFORM = 9, Membrane Element

CQUAD4 - DYSHELLFORM = 9, Membrane ElementThe Belytschko-Lin-Tsay shell element {Belytschko and Tsay [1981], Belytschko et al., [1984a]} is the basis for this very efficient membrane element. In this section we briefly outline the theory employed which, like the shell on which it is based, uses a combined co-rotational and velocity-strain formulation. The efficiency of the element is obtained from the mathematical simplifications that result from these two kinematical assumptions. The co-rotational portion of the formulation avoids the complexities of nonlinear mechanics by embedding a coordinate system in the element. The choice of velocity strain or rate of deformation in the formulation facilitates the constitutive evaluation, since the conjugate stress is the more familiar Cauchy stress.

In membrane elements the rotational degrees of freedom at the nodal points may be constrained, so that only the translational degrees of freedom contribute to the straining of the membrane. A triangular membrane element may be obtained by collapsing adjacent nodes of the quadrilateral.

Co-rotational CoordinatesThe mid-surface of the quadrilateral membrane element is defined by the location of the element’s four corner nodes. An embedded element coordinate system (Figure 6-9) that deforms with the element is defined in terms of these nodal coordinates. The co-rotational coordinate system follows the development in equations (6-166) to (6-171).

Velocity-Strain Displacement RelationsThe co-rotational components of the velocity strain (rate of deformation) are given by:

(6-374)

The above velocity-strain relations are evaluated only at the center of the shell. Standard bilinear nodal interpolation is used to define the mid-surface velocity, angular velocity, and the element’s coordinates (isoparametric representation). These interpolation relations are given by

(6-375)

(6-376)

where the subscript is summed over all the element’s nodes and the nodal velocities are obtained by differentiating the nodal coordinates with respect to time, i.e., . The bilinear shape functions are

defined in (6-185).

The velocity strains at the center of the element, i.e., at , and , are obtained as in Co-rotational Technique, giving:

(6-377)

dˆ

i j12---

∂υi

∂ xj

--------∂υj

∂ xi

--------+⎝ ⎠⎜ ⎟⎛ ⎞

=

vm NI ξ η,( )vI=

xm NI ξ η,( )xI=

I

υI x· I=

ξ 0= η 0=

dˆ

x B1 I υxI=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCQUAD4 - DYSHELLFORM = 9, Membrane Element

218

(6-378)

(6-379)

where

(6-380)

(6-381)

Stress Resultants and Nodal ForcesAfter suitable constitutive evaluations using the above velocity strains, the resulting stresses are multiplied by the thickness of the membrane, , to obtain local resultant forces. Therefore,

(6-382)

where the superscript indicates a resultant force and the Greek subscripts emphasize the limited range of the indices for plane stress plasticity.

The above element centered force resultants are related to the local nodal forces by

(6-383)

(6-384)

where is the area of the element.

The above local nodal forces are then transformed to the global coordinate system using the transformation relations given in equation (6-173).

Membrane Hourglass ControlHourglass deformations need to be resisted for the membrane element. The hourglass control for this element is discussed in Hourglass Control (Belytschko-Lin-Tsay).

dˆ

y B2I υyI=

2dˆ

xy B2 I vxI B1I vyI+=

B1I

∂NI

∂ x---------=

B2I

∂NI

∂ y---------=

h

fˆαβR

h σαβ=

R

fˆxI A B1 I f

ˆxxR

B2 I fˆxyR

+( )=

fˆyI A B2 I f

ˆyyR

B1 I fˆxyR

+( )=

A

219Chapter 6: ElementsCELAS1D, Discrete Elements and CONM2, Masses

CELAS1D, Discrete Elements and CONM2, MassesThe discrete elements and masses in SOL 700 provide a capability for modeling simple spring-mass systems as well as the response of more complicated mechanisms. Occasionally, the response of complicated mechanisms or materials needs to be included in the models, e.g., energy absorbers used in passenger vehicle bumpers. These mechanisms are often experimentally characterized in terms of force-displacement curves. SOL 700 provides a selection of discrete elements that can be used individually or in combination to model complex force-displacement relations.

The discrete elements are assumed to be massless. However, to solve the equations of motion at unconstrained discrete element nodes or nodes joining multiple discrete elements, nodal masses must be specified at these nodes. SOL 700 provides a direct method for specifying these nodal masses in the model input.

All of the discrete elements are two-node elements, i.e., three-dimensional springs or trusses. A discrete element may be attached to any of the other SOL 700 continuum, structural, or rigid body element. The force update for the discrete elements may be written as

(6-385)

where the superscript indicates the time increment and the superposed caret indicates the force

in the local element coordinates; i.e., along the axis of the element. In the default case(i.e., no orientation vector is used), the global components of the discrete element force are obtained by using the element’s direction cosines:

(6-386)

where:

(6-387)

is the length

(6-388)

and are the global coordinates of the nodes of the spring element. The forces in Equation (6-386)

are added to the first node and subtracted from the second node.

fˆ i 1+ f

ˆ i Δ fˆ

+=

i 1+ ˆ( )

Fx

Fy

Fz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

l-

Δ lx

Δ ly

Δ lz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

nx

ny

nz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆn˜

= = =

Δ˜

l

Δ ly

Δ ly

Δ lz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

x2 x1–

y2 y1–

z2 z1–⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

= =

l

l Δ lx2 Δ ly

2 Δ lz2

+ +=

xi yi zi, ,( )

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCELAS1D, Discrete Elements and CONM2, Masses

220

For a node tied to ground we use the same approach but for the coordinates in equation (6-366),

the initial coordinates of node 1, i.e., are used instead; therefore,

(6-389)

The increment in the element force is determined from the user specified force-displacement relation. Currently, nine types of force-displacement/velocity relationships may be specified:

• Linear elastic

• Linear viscous

• Nonlinear elastic

• Nonlinear viscous

• Elasto-plastic with isotropic hardening

• General nonlinear

• Linear viscoelastic

• Inelastic tension and compression only

• Muscle model

Orientation VectorsAn orientation vector,

(6-390)

can be defined to control the direction the spring acts. If orientation vectors are used, it is strongly recommended that the nodes of the discrete element be coincident and remain approximately so throughout the calculation. If the spring or damper is of finite length, rotational constraints will appear in the model that can substantially affect the results. If finite length springs are needed with directional vectors, then the discrete beam elements, CROD, should be used with the coordinate system flagged for the finite length case.

We will first consider the portion of the displacement that lies in the direction of the vector. The displacement of the spring is updated based on the change of length given by

(6-391)

x2 y2 z2, ,( )

x0 y0 z0, ,( )

Fx

Fy

Fz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

l-

x0 x1–

y0 y1–

z0 z1–⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

nx

ny

nz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

= =

m˜

m1

m2

m3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

Δ I I I0–=


where is the initial length in the direction of the vector and is the current length given for a node to

node spring by

(6-392)

and for a node to ground spring by

(6-393)

The latter case is not intuitively obvious and can affect the sign of the force in unexpected ways if the user is not familiar with the relevant equations. The nodal forces are then given by

(6-394)

The orientation vector can be either permanently fixed in space as defined in the input or acting in a direction determined by two moving nodes which must not be coincident but may be independent of the nodes of the spring. In the latter case, we recompute the direction every cycle according to:

(6-395)

In equation (6-395) the superscript, , refers to the orientation nodes.

For the case where we consider motion in the plane perpendicular to the orientation vector we consider

only the displacements in the plane, , given by,

(6-396)

We update the displacement of the spring based on the change of length in the plane given by

(6-397)

where is the initial length in the direction of the vector and is the current length given for a node to

node spring by

(6-398)

I0 I

I m1 x2 x1–( ) m2 y2 y1–( ) m3 z2 z1–( )+ +=

I m1 x0 x1–( ) m2 y0 y1–( ) m3 z0 z1–( )+ +=

Fx

Fy

Fz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

m1

m2

m3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

m1

m2

m3⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1ln----

x2n x1

n–

y2n y1

n–

z2n z1

n–⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

n

Δ˜

lp

Δ˜

lp

Δ˜

l m˜

m˜

Δ˜

l⋅( )–=

Δ lp

lp

l0

p–=

l0

pl

lp

m1p

x2 x1–( ) m2p

y2 y1–( ) m3p

z2 z1–( )+ +=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCELAS1D, Discrete Elements and CONM2, Masses

222

and for a node to ground spring by

(6-399)

where

(6-400)

After computing the displacements, the nodal forces are then given by

(6-401)

Dynamic Magnification “Strain Rate” EffectsTo account for “strain rate” effects, we have a simple method of scaling the forces based to the relative velocities that applies to all springs. The forces computed from the spring elements are assumed to be the static values and are scaled by an amplification factor to obtain the dynamic value:

(6-402)

where

For example, if it is known that a component shows a dynamic crush force at 15m/s equal to 2.5 times the static crush force, use and .

= a user defined input value

= absolute relative velocity

= dynamic test velocity

lp

m1p

x0 x1–( ) m2p

y0 y1–( ) m3p

z0 z1–( )+ +=

m1p

m2p

m3p

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1

Δ lx

p2

Δ ly

p2

Δ lz

p2

+ +

----------------------------------------------------

Δ lx

p

Δ ly

p

Δ lz

p

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

Fx

Fy

Fz⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

fˆ

m1p

m2p

m3p

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

Fdynamic 1. kdVV0------+⎝ ⎠

⎛ ⎞ Fstatic=

kd

V

V0

kd 1.5= V0 15=


Deflection Limits in Tension and CompressionThe deflection limit in compression and tension is restricted in its application to no more than one spring per node subject to this limit, and to deformable bodies only. For example in the former case, if three spring are in series either the center spring or the two end springs may be subject to a limit but not all three. When the limiting deflection is reached momentum conservation calculations are performed and a common acceleration is computed:

(6-403)

An error termination will occur if a rigid body node is used in a spring definition where compression is limited.

acommonfˆ1 f

ˆ2+

m1 m2+--------------------=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideCDAMP2D, Linear Elastic or Linear Viscous

224

CDAMP2D, Linear Elastic or Linear ViscousThese discrete elements have the simplest force-displacement relations. The linear elastic discrete element has a force-displacement relation of the form

(6-404)

where is the element’s stiffness and is the change in length of the element. The linear viscous element has a similar force-velocity (rate of displacement) relation:

(6-405)

where is a viscous damping parameter and is the time step increment.

fˆ

KΔ l=

K Δ l

fˆ

CΔ lΔ t-----=

C Δ t

225Chapter 6: ElementsEulerian Elements

Eulerian Elements

Element DefinitionIn the Eulerian solver, the mesh is defined by grid points and solid elements. The elements are specified as being (partially) filled with certain materials or with nothing (VOID), and initial conditions are defined.

As the calculation proceeds, the material moves relative to the Eulerian mesh. The mass, momentum, and energy of the material is transported from element to element depending on the direction and velocity of the material flow. The solver then calculates the impulse and work done on each of the faces of every Eulerian element.

Eulerian elements can only be solid but have a general connectivity and therefore are defined in exactly the same way as Lagrangian elements.

Solid ElementsThere are three types of Euler elements, a six-sided CHEXA with eight grid points defining the corners, a CPENTA with six grid points, and a CTETRA with four grid points. The connectivity of the element is defined in exactly the same manner as a Lagrangian element, that is, with a CHEXA, CPENTA, or CTETRA entry. However, in order to differentiate between Lagrangian and Eulerian solid elements, the property entry for Euler is PEULERn rather than PSOLID.

Unlike Lagrangian solid elements, the CPENTA and CTETRA elements perform just as well as the CHEXA element. They can be used, therefore, wherever meshing demands such use.

The PEULERn entry references a MATDEUL material entry that is used to define the material filling the elements at the start of the calculation. When no material entry is referenced (the field contains a zero), the element is initially void.

CPENTA CTETRA

CHEXA

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideGraded Meshes

226

Graded MeshesNonuniform Euler meshes can easily be created by MD Patran. This is particularly useful when the elements do not need to be orthogonal. But in orthogonal meshes, the nonuniformity propagates to the boundaries of the mesh. Even at the boundary, there will be elements that have a small size in at least one direction. To allow for large element sizes in all three coordinate directions at the boundary, block structured meshing has to be used. This type of meshing is very effective when modeling the flow over bodies and is often used in CFD. Usually, the Euler elements are fine near the body and become coarser as the distance is increased away from the body. By using the graded mesh capability, a block of fine elements is used in the area of interest and a coarse block is used in other areas. To use this method, the blocks need to be glued together. This is done by adding PARAM,GRADMESH.

Graded meshes are supported by all Eulerian solvers except by the single material Euler solver with Strength. Intersection of a coupling surface segment with the interface between a coarse and fine block is not allowed. Graded meshes are not supported by multiple Euler domains. Figure 1 shows an example of graded meshes. This mesh is be used in Example 4.14.

Consider a blast wave simulation. Close to the ignition point elements need to be fine, but at some distance the wave blast becomes larger in radius and it becomes less steep allowing coarser mesh elements. To reduce the number of elements and the limit the problem size, part of the fine mesh can be replaced by a coarse mesh. For modeling, only part of the fine mesh is constructed and the coarse mesh is created such that it covers the whole problem domain. Next, the fine mesh is glued to the coarse mesh. This gluing is activated by PARAM,GRADED-MESH. The algorithm identifies the elements of the coarse mesh that are covered by the fine mesh. They are deactivated and removed from Euler archive output requests.

Requirements for Gluing MeshesConnecting meshes with varying mesh sizes was already possible by using multiple Euler domains with porous coupling surfaces. Using this method, two connected meshes are surrounded by a coupling surface together with a fully porous subsurface that connects the two domains. The domains are exclusively setup by using the MESH entry. In this approach there are no requirements on the size and location of the meshes. In output requests, only one domain can be examined. Making plots of the whole domain is not possible in one Patran session. This makes the approach cumbersome.

With the graded mesh functionality there is no longer any need to create coupling surfaces. An additional advantage is that there is no restriction on how the elements are created for the simulation. Any pre-processor may be used, defining the CHEXA elements in an input file, but also blocks of meshes can be created by means of the MESH entry. The interface between the fine and coarse mesh is identified in the solver when PARAM,GRADMESH is activated. However, one restriction applies to graded meshes. An Euler element of the coarse mesh has to be fully active or fully inactive. This means that the coarse element should not intersect elements of the fine mesh or it should be fully covered by the fine elements. Fine elements are not allowed to cover any part of the coarse elements. In practice, this means that the fine mesh has to fit nicely in the coarse mesh. As shown in the meshes in Figure 6-23, the four marked

227Chapter 6: ElementsGraded Meshes

locations a grid point of the fine mesh coincides with a grid point of the coarse mesh. This matching does not need to be exact, since the solver uses a tolerance to find the coinciding grid points. Visualization of the results of the complete Euler domain can be done in one session in a post-processor like MD Patran.

Figure 6-23 Graded Mesh with Structure

Gluing MeshesGluing of fine and coarse meshes is activated by PARAM,GRADMESH. The algorithm removes coarse elements that are completely covered by fine elements. Here the criterion for removal is based on the element volume. The element with the largest volume will be removed. It is also possible to remove the elements with the lowest element numbers. These two approaches are activated by, respectively, the MINVOL and ELNUM option of the GRADMESH PARAM.

The gluing algorithm performs the following steps:

• The Euler elements are sorted such that the connected elements are grouped together.

• For each group of elements, the algorithm determines which elements have to be removed. As mentioned the criterion for removal is based on the fact if these elements are completely covered by elements of another group that are smaller or have a larger element number.

• Next the groups of elements are connected at boundary faces and at locations where elements have been removed. Special faces will be created that connect an element of one element group to an element of another group.

If the fine mesh is fully surrounded by the coarse mesh, the interface between the two meshes consists of the boundary faces of the fine mesh. The geometry formed by these faces will be used to construct special faces that connect an element of the coarse mesh to an element of the fine mesh.


228

Using Graded MeshesBlocks of Euler elements can be defined by either MD Patran or the MESH,BOX entry. It is important that at the interface a face of a coarse element can be matched to several parts of faces of fine elements. This matching does not need to be exact since a tolerance is used.

One way to construct graded meshes is:

• Make the coarse mesh by using MESH,BOX.

• Run the simulation and read in the Euler mesh into MD Patran. For a part of this mesh, the coarse elements have to be replaced by finer elements. Select a part of this mesh by selecting two Eulerian grid points.

• Create the fine mesh by MESH,BOX, by using for the reference grid point as the first grid point. The width of the box is given by subtracting the coordinates of the second grid point from the coordinates of the first. To finish the definition of the fine mesh, the number of elements in each direction has to be defined. This ensures that the fine mesh fits nicely in the coarse mesh.

• Add PARAM,GRADMESH which activates the algorithm that is described above.

If needed a part of fine mesh can be replaced by an even finer mesh, by iterating through the steps above.

To construct graded meshes by MD Patran, a utility called “the break up of element” can be used.

Visualization with MD PatranElements that are fully covered by the fine mesh will not be included in the Euler archives. This allows for visualization of all euler elements in one MD Patran session. The fringe plots can lack smoothness at the interfaces. The reason is that Patran determines colors on the basis of grid point values. They are computed by averaging over the elements that are connected to the grid points. In graded meshes, there can be grid points that belong only to fine elements and not to coarse elements. These grid points are called hanging nodes. An example of a hanging node is shown in Figure 6-24. At the hanging node, the value only reflects the fine mesh. This results in loss of smoothness. An example is shown in Figure 6-25. If the results are postprocessed by using the element values this problem does not appear (see Figure 6-26).

229Chapter 6: ElementsGraded Meshes

Figure 6-24 Hanging Node

Figure 6-25 Lack of Smoothness

Figure 6-26 Using Element Values


230

Chapter 7: MaterialsMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

7 Materials

Lagrangian Material Models for SOL 700 232

Eulerian Material Models for SOL 700 325

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideLagrangian Material Models for SOL 700

232

Lagrangian Material Models for SOL 700In addition to most materials available in MD Nastran Solutions 101 - 200, the following material models which are suitable for explicit applications are also made available for SOL 700.

About ninety new material models are currently implemented in SOL 700. In this chapter, we will highlight the theoretical background of these material models. These are the material models that are most commonly used for typical structural applications. Additional materials will be added in future releases.

1 Elastic

2 Orthotropic Elastic

3 Kinematic/Isotropic Elastic-Plastic

5 Soil and Crushable/Non-crushable Foam

6 Viscoelastic

7 Blatz-Ko Rubber

9 Null Hydrodynamics

10 Isotropic-Elastic-Plastic-Hydrodynamic

12 Isotropic-Elastic-Plastic

13 Elastic-Plastic with Failure Model

14 Soil and Crushable Foam with Failure Model

15 Johnson/Cook Strain and Temperature Sensitive Plasticity

18 Power Law Isotropic Plasticity

19 Strain Rate Dependent Isotropic Plasticity

20 Rigid

22 Composite Damage Model

24 Piecewise Linear Isotropic Plasticity

26 Honeycomb

27 Compressible Mooney-Rivlin Rubber

28 Resultant Plasticity

29 Forced Limited Resultant Fomulation

30 Shape-Memory Superelastic Material

31 Slightly Compressible Rubber Model

32 Laminated Glass Model

34 Fabric

40 Nonlinear Orthotropic

54-55 Composite Damage Model

233Chapter 7: MaterialsLagrangian Material Models for SOL 700

57 Low Density Urethane Foam

58 Laminated Composite Fabric

59 Composite Failure Model - Plasticity Based

62 Viscous foam

63 Isotropic Crushable Foam

64 Strain Rate Sensitive Power-Law Plasticity

66 Linear Elastic Discrete Beam

67 Nonlinear Elastic Discrete Beam

68 Nonlinear Plastic Discrete Beam

69 SID Damper Discrete Beam

70 Hydraulic Gas Damper Discrete Beam

71 Cable Discrete Beam

72 Concrete Damage

72R Concrete Damage Release III

73 Low Density Viscoelastic Foam

74 Elastic Spring Discrete Beam

76 General Viscoelastic

77 Hyperviscoelastic Rubber

80 Ramberg-Osgood Plasticity

81 Plastic with Damage

83 Fu-Chang’s Foam with Rate Effects

87 Cellular Rubber

89 Plastic Polymer

93 Elastic Six Degrees of Freedom Spring Discrete Beam

94 Inelastic Spring Discrete Beam

95 Inelastic Spring Six Degrees of Freedom Discrete Beam

97 General Joint Discrete Beam

98 Simplified Johnson-Cook

99 Simplified Johnson-Cook Orthotropic Damage

100 Spot weld

112 Finite Elastic Strain Plasticity

114 Layered Linear Plasticity

119 General Nonlinear Six Degrees of Freedom Discrete Beam

121 General Nonlinear One Degree of Freedom Discrete Beam


234

In the table below, a list of the available material models and the applicable element types are given. Some materials include strain rate sensitivity, failure, equations of state, and thermal effects and this is also noted. General applicability of the materials to certain kinds of behavior is suggested in the last column.

123 Modified Piecewise Linear Plasticity

126 Modified Honeycomb

127 Arruda-Boyce rubber

158 Composite Fabric

181 Simplified Rubber

196 General Spring Descrete Beam

B01 Seatbelt

S01 Spring Elastic (Linear)

S02 Damper Viscous (Linear)

S03 Spring Elastoplastic (Isotropic)

S04 Spring Nonlinear Elastic

S05 Damper Nonlinear Viscous

S06 Spring General Nonlinear

S07 Spring Maxwell (Three parameter Viscoelastic)

S08 Spring Inelastic (Tension or Compression)

S13 Spring Tri-linear Degrading

S14 Spring Squat Shearwall

S15 Spring Muscle

SW1 Spot Weld (Simple Damage-Failure)

SW2 Spot Weld (Resultant-based Failure Criteria)

SW3 Spot Weld (Stress-based Failure)

SW4 Spot Weld (Rate Dependent Stress-based Failure)

SW5 Spot Weld (Additional Failure)


Material Title

Gn GeneralCm Composites

Cr CeramicsFl FluidsFm FoamGl Glass

Hy Hydro-dynMt MetalPl Plastic

Rb RubberSl Soil/Cone

1 Elastic Y Y Y Y Gn, Fl

2 Orthotropic Elastic (Anisotropic - solids) Y Y Y Cm, Mt

3 Plastic Kinematic/Isotropic Y Y Y Y Y Y Cm, Mt, Pl

5 Soil and Foam Y Fm, Sl

6 Linear Viscoelastic Y Y Y Y Y Y Y Rb

7 Blatz-Ko Rubber Y Y Rb, Polyurethane

9 Null Material Y Y Y Y Fl, Hy

10 Elastic Plastic Hydro (dynamics) Y Y Y Hy, Mt

12 Isotropic Elastic Plastic Y Y Y Y Mt

13 Isotropic Elastic-Plastic with Failure Y Y Mt

14 Soil and Foam with Failure Y Y Fm, Sl

15 Johnson/Cook Plasticity Model Y Y Y Y Y Y Hy, Mt

18 Power Law Plasticity (Isotropic) Y Y Y Y Y Mt, Pl

19 Strain Rate Dependent Rate Plasticity Y Y Y Y Y Mt, Pl

20 Rigid Y Y Y Y

22 Composite Damage Y Y Y Y Cm

24 Piecewise Linear Isotropic Plasticity Y Y Y Y Y Y Mt, Pl

26 Honeycomb Y Y Y Cm, Fm, Sl

27 Mooney-Rivlin Rubber Y Y Rb

28 Resultant Plasticity Y Y Mt

29 Forced Limited Resultant Formaultion Y

30 Shaped Memory Alloy Y Mt

31 Slightly Compressible Rubber Y Rb

32 Laminated Glass (Composite) Y Y Y Cm, Gl

34 Fabric Y

40 Nonlinear Orthotropic Y Y Y Cm

Mat

eria

l Nu

mb

er

Bri

cks

Bea

ms

Th

in S

hel

lsT

hic

k S

hel

ls

Str

ain

-Rat

e E

ffec

ts

Fai

lure

Eq

uat

ion

-of-

Sta

te

Th

erm

al E

ffec

ts


236

54 Composite Damage with Chang Failure Y Y Cm

55 Composite Damage with Tsai-Wu Failure Y Y Cm

57 Low Density Urethane Foam Y Y Y Fm

58 Laminated composite Fabric Y

59 Composite Failure - Plasticity Based Y Y Y Cm, Cr

62 Viscous foam (Crash Dummy) Y Y Fm

63 Isotropic Crushable Foam Y Y Fm

64 Rate Sensitive Power-Law Plasticity Y Y Y Y Mt

66 Linear Elastic Discrete Beam Y Y

67 Nonlinear Elastic Discrete Beam Y Y

68 Nonlinear Plastic Discrete Beam Y Y Y

69 SID Damper Discrete Beam Y Y

70 Hydraulic Gas Damper Discrete Beam Y Y

71 Cable Discrete Beam Y

72 Concrete Damage Y Y Y Y Sl

72R Concrete Damage Release III Y Y Y Y Sl

73 Low Density Viscous Foam Y Y Y Fm

74 Elastic Spring Discrete Beam Y

76 General Viscoelastic (Maxwell model) Y Y Rb

77 Hyperelastic and Ogden Rubber Y Rb

79 Hysteretic Soil (Elasto-Perfectly Plastic) Y Y Sl

80 Ramberg-Osgood Y Y Y Y Y Y Y Y

81 Plastic with Damage (Elasto-Plastic) Y Y Y Y Y Y Mt, Pl

83 Fu-Chang’s Foam Y Y Y Fm

87 Cellular Rubber Y Y Rb

89 Plastic Polymer Y

Material Title




Rb RubberSl Soil/ConeM

ate r

ial N

um

be r

Bri

cks

Bea

ms

Th

in S

hel

lsT

hic

k S

hel

ls

Str

ain

-Rat

e E

ffec

ts

Fai

lure

Eq

uat

ion

-of-

Sta

te

Th

erm

al E

ffec

ts


93Elastic Six Degrees of Freedom Spring Discrete Beam

Y

94 Inelastic Spring Discrete Beam Y

95Inelastic Six Degrees of Freedom Spring Discrete Beam

Y

97 General Joint Discrete Beam Y

98 Simplified Johnson-Cook Y Y Y Y

99Simplified Johnson-Cook Orthotropic Damage

Y Y Y Y

100 Spot weld Y

112 Finite Elastic Strain Plasticity Y

114 Layered Linear Plasticity Y Y

119General Nonlinear Six Degrees of Freedom Discrete Beam

Y

121General Nonlinear One Degree of Freedom Discrete Beam

Y

123 Modified Piecewise Linear Plasticity Y Y

126 Modified Honeycomb Y

127 Arruda-Boyce rubber Y

158 Composite Fabric Y Y Y Y Cm

181 Simplified Rubber Y RB

196 General Spring Discrete Beam Y

B01 Seatbelt

S01 Spring Elastic (Linear) Y

S02 Damper Viscous (Linear) Y Y

S03 Spring Elastoplastic (Isotropic) Y

S04 Spring Nonlinear Elastic Y Y

S05 Damper Nonlinear Viscous Y Y

Material Title





ater

ial N

um

ber

Bri

cks

Bea

ms

Th

in S

hel

lsT

hic

k S

hel

ls

Str

ain

-Rat

e E

ffec

ts

Fai

lure

Eq

uat

ion

-of-

Sta

te

Th

erm

al E

ffec

ts


238

Material Model 1: ElasticIn this elastic material we compute the co-rotational rate of the deviatoric Cauchy stress tensor as

(7-1)

and pressure

(7-2)

where and are the elastic shear and bulk moduli, respectively, and is the relative volume; i.e., the ratio of the current volume to the initial volume.

For standard MD Nastran solution sequence, this would be the same as using MAT1 to define a linear elastic material.

S06 Spring General Nonlinear Y

S07Spring Maxwell (Three Parameter Viscoelastic)

Y Y

S08 Spring Inelastic (Tension or Compression Y

S13 Spring Tri-linear Degrading

S14 Spring Squat Shearwall

S15 Spring Muscle

SW1 Spot Weld (Simple Damage-Failure) Y

SW2Spot Weld (Resultant-based Failure Criteria)

Y

SW3 Spot Weld (Stress-based Failure) Y

SW4Spot Weld (Rate Dependent Stress-based Failure)

Y

SW5 Spot Weld (Additional Failure) Y

Material Title





ate r

ial N

um

be r

Bri

cks

Bea

ms

Th

in S

hel

lsT

hic

k S

hel

ls

Str

ain

-Rat

e E

ffec

ts

Fai

lure

Eq

uat

ion

-of-

Sta

te

Th

erm

al E

ffec

ts

si j∇n 1 2⁄+

2Gε'· i jn 1 2⁄+

=

pn 1+ K Vn 1+ln=

G K V


Material Model 2: Orthotropic ElasticThe material law that relates second Piola-Kirchhoff stress to the Green-St. Venant strain is

(7-3)

where is the transformation matrix [Cook 1974].

(7-4)

, , are the direction cosines

(7-5)

and denotes the material axes. The constitutive matrix is defined in terms of the material axes as

(7-6)

where the subscripts denote the material axes; i.e.,

and (7-7)

Since is symmetric

, ect. (7-8)

The vector of Green-St. Venant strain components is

(7-9)

S E

S C E⋅ TtClT E⋅= =

T

T

l12 m1

2 n12 l1m1 m1n1 n1 l1

l22 m2

2 n22 l2m2 m2n2 n2 l2

l32 m3

2 n32 l3m3 m3n3 n3 l3

2 l1 l2 2m1m2 2n1n2 l1m2 l1m1+( ) m1n2 m2n1+( ) n1 l2 n2 l1+( )



=

li mi ni

x'i lix1 mix2 nix3+ += for i = 1,2,3

x'i Cl

Cl1–

1E11--------

υ21

E22--------–

υ31

E33--------– 0 0 0

υ12

E11--------–

1E22--------

υ32

E33--------– 0 0 0

υ13

E11--------–

υ23

E22--------–

1E33-------- 0 0 0

0 0 01

G12--------- 0 0

0 0 0 01

G23--------- 0

0 0 0 0 01

G31---------

=

υi j υx'ix'j= Eii Ex'i

=

Cl

υ12

E11--------

υ21

E22--------=

Et E11 E22 E33 E12 E23 E31,,,,,=


240

(7-10)

After computing , we use Equation (7-10) to obtain the Cauchy stress. This model will predict realistic

behavior for finite displacement and rotations as long as the strains are small.

For standard MD Nastran solution sequences, this would be the same as using MAT9 to define a linear anisotropic material. For shell elements, you would have used the MAT2 or MAT8 option.

Material Model 3: Elastic Plastic with Kinematic HardeningIsotropic, kinematic, or a combination of isotropic and kinematic hardening may be obtained by varying a parameter, called between 0 and 1. For equal to 0 and 1, respectively, kinematic and isotropic hardening are obtained as shown in Figure 7-1 where and are the undeformed and deformed length

of uniaxial tension specimen, respectively. Krieg and Key [1976] formulated this model and the implementation is based on their paper.

In isotropic hardening, the center of the yield surface is fixed but the radius is a function of the plastic strain. In kinematic hardening, the radius of the yield surface is fixed but the center translates in the direction of the plastic strain. Thus the yield condition is

(7-11)

where

(7-12)

(7-13)

The co-rotational rate of is

(7-14)

Hence,

(7-15)

σi jρρo-----

∂xi

∂Xk---------

∂xj

∂Xl

---------Skl=

Sij

β βl0 l

φ 12---ξijξi j

σy2

3------– 0= =

ξij Si jαij=

σy σ0 βEpεef fp+=

αi j

αi j∇ 1 β–( )2

3---Epε· i j

p=

αi jn 1+ α

ijn α

ij∇n 1 2⁄+ α

ikn Ω

ikn 1 2⁄+ α

jkn 1 2⁄+ Ω

kin 1 2⁄+++( )Δ tn 1 2⁄++=


Figure 7-1 Elastic-plastic Behavior with Isotropic and Kinematic Hardening

Strain rate is accounted for using the Cowper and Symonds [Jones 1983] model which scales the yield stress by a strain rate dependent factor

(7-16)

where and are user-defined input constants and is the strain rate defined as:

(7-17)

The current radius of the yield surface, , is the sum of the initial yield strength, , plus the growth

, where is the plastic hardening modulus

(7-18)

and is the effective plastic strain

Kinematic HardeningIsotropic Hardening

β 0=

β 1=

ll0----⎝ ⎠⎛ ⎞ln

YieldStress

E

Et

ρy 1ε·

C----⎝ ⎠⎛ ⎞

1 p⁄+ σ0 βEpεeff

p+( )=

p C ε·

ε· ε· i jε·

ij+

σy σ0

βEpεeffp Ep

Ep

EtE

E Et–---------------=

εef fp


242

(7-19)

The plastic strain rate is the difference between the total and elastic (right superscript) strain rates:

(7-20)

In the implementation of this material model, the deviatoric stresses are updated elastically, as described for model 1, but repeated here for the sake of clarity:

(7-21)

where

is the trial stress tensor,

is the stress tensor from the previous time step,

is the elastic tangent modulus matrix,

is the incremental strain tensor.

and, if the yield function is satisfied, nothing else is done. If, however, the yield function is violated, an increment in plastic strain is computed, the stresses are scaled back to the yield surface, and the yield surface center is updated.

Let represent the trial elastic deviatoric stress state at

(7-22)

and

(7-23)

Define the yield function,

(7-24)

For plastic hardening then:

(7-25)

Scale back the stress deviators:

(7-26)

εef fp 2

3---ε· ij

p ε· i jp

⎝ ⎠⎛ ⎞ 1 2/

td

0

t

∫=

ε· i jp ε· i j ε· i j

e–=

σi j* σi j

n CijklΔεkl+=

σi j*

σi jn

Cijkl

Δεkl

si j*

n 1+

si j* σi j

* 13---σkk

* σ ij–=

ξij* sij

* αi j–=

φ 32---ξij

* ξi j* σy

2– Λ2 σy2–= =

< 0 for elastic or neutral loading> 0 for plastic harding

εef fpn 1+

εeffpn Λ σy–

3G Ep+---------------------+ εef f

pnΔεef f

p+= =

σi jn 1+ σi j

*3GΔεef f

p

Λ---------------------ξi j

*–=


and update the center:

(7-27)

Plane Stress Plasticity

The plane stress plasticity options apply to beams, shells, and thick shells. Since the stresses and strain increments are transformed to the lamina coordinate system for the constitutive evaluation, the stress and strain tensors are in the local coordinate system.

The application of the Jaumann rate to update the stress tensor allows for the possibility that the normal stress, , will not be zero. The first step in updating the stress tensor is to compute a trial plane stress

update assuming that the incremental strains are elastic. In the above, the normal strain increment

is replaced by the elastic strain increment

(7-28)

where and are Lamé’s constants.

When the trial stress is within the yield surface, the strain increment is elastic and the stress update is completed. Otherwise, for the plastic plane stress case, secant iteration is used to solve Equation (7-26) for the normal strain increment required to produce a zero normal stress:

(7-29)

Here, the superscript indicates the iteration number.

The secant iteration formula for (the superscript is dropped for clarity) is

(7-30)

where the two starting values are obtained from the initial elastic estimate and by assuming a purely plastic increment; i.e.,

(7-31)

These starting values should bound the actual values of the normal strain increment.

The iteration procedure uses the updated normal stain increment to update first the deviatoric stress and then the other quantities needed to compute the next estimate of the normal stress in Equation (7-29). The

iterations proceed until the normal stress is sufficiently small. The convergence criterion requires

convergence of the normal strains:

αi jn 1+ αij

n 1 β–( )EpΔεef fp

Λ--------------------------------------+=

σ33

Δε33

Δε33

σ33 λ Δε11 Δε22+( )+

λ 2μ+------------------------------------------------------–=

λ γ

Δε33( )

σ33i σ33

*3GΔεef f

pi ξ33

Λ-----------------------------–=

i

Δε33 p

Δε33i 1+ Δε33

i 1– Δε33i Δε33

i 1––

σ33i σ

33i 1––

----------------------------------σ33i 1––=

Δε331 Δε11 Δε22–( )–=

σ33i


244

(7-32)

After convergence, the stress update is completed using the relationships given in Equations (7-25) and (7-27).

For SOL SEQ 106, 129, or 600, this material model is the same as using MATS1, with the hardening rule (HR) set to either Isotropic or Kinematic. These models do not allow between 0 and 1.

Material Model 5: Soil and Crushable FoamThis model, due to Krieg [1972], provides a simple model for foam and soils whose material properties are not well characterized. We believe the other foam models (such as material models 57, 62, and 63) in SOL 700 are superior in their performance and are recommended over this model which simulates the crushing through the volumetric deformations. If the yield stress is too low, this foam model gives nearly fluid like behavior.

A pressure-dependent flow rule governs the deviatoric behavior:

(7-33)

where , , and are user-defined constants. Volumetric yielding is determined by a tabulated curve

of pressure versus volumetric strain. Elastic unloading from this curve is assumed to a tensile cutoff as illustrated in Figure 7-2.

Implementation of this model is straightforward. One history variable, the maximum volumetric strain in compression, is stored. If the new compressive volumetric strain exceeds the stored value, loading is indicated. When the yield condition is violated, the updated trial stresses, , are scaled back using a

simple radial return algorithm:

(7-34)

If the hydrostatic tension exceeds the cutoff value, the pressure is set to the cutoff value and the deviatoric stress tensor is zeroed.

Δε33i Δε33

i 1––

Δε33i 1+

------------------------------------- 10 4–<

β

φs12--- si jsij a0 a1p a2p2+ +( )–=

a0 a1 a2

si j*

si jn 1+ a0 a1p a2p2+ +

12--- sijsi j

----------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ 1 2⁄

sij*=


Figure 7-2 Volumetric Strain Versus Pressure Curve for Soil and Crushable Foam Model

Material Model 6: ViscoelasticIn this model, linear viscoelasticity is assumed for the deviatoric stress tensor [Herrmann and Peterson 1968]:

(7-35)

where

(7-36)

is the shear relaxation modulus. A recursion formula is used to compute the new value of the hereditary integral at time from its value at time . Elastic bulk behavior is assumed:

(7-37)

where pressure is integrated incrementally.

Tension Cutoff Value

Tension

VV0------⎝ ⎠⎛ ⎞ln

Pressure

Volumetric Strain(Compression)

The bulk unloading modulus is used if thevolumetric crushing option is on (VCR = 0).

Loading and unloading follows the input curve ifthe volumetric crushing option is off (VCR = 1.0).

si j 2 φ t τ–( )∂ε'i j

∂τ---------- τd

0

t

∫=

φ t( ) G∞ G0 G∞–( )e β t–+=

tn 1+ tn

p K Vln=


246

Material Model 7: Continuum RubberThe hyperelastic continuum rubber model was studied by Blatz and Ko [1962]. In this model, the second Piola-Kirchhoff stress is given by

(7-38)

where is the shear modulus, is the relative volume, is Poisson’s ratio, and is the right

Cauchy-Green strain:

(7-39)

after determining , it is transformed into the Cauchy stress tensor, :

(7-40)

where and are the initial and current density, respectively. The default value of is 0.463.

Material 27 and 31 better represent incompressible materials.

Material Model 9: Null MaterialFor solid elements equations of state can be called through this model to avoid deviatoric stress calculations. A pressure cutoff may be specified to set a lower bound on the pressure. This model has been very useful when combined with the reactive high explosive model where material strength is often neglected. The null material should not be used to delete solid elements.

A optional viscous stress of the form

(7-41)

is computed for nonzero where is the deviatoric strain rate.

Sometimes it is advantageous to model contact surfaces via shell elements which are not part of the structure, but are necessary to define areas of contact within nodal rigid bodies or between nodal rigid bodies. Beams and shells that use this material type are completely bypassed in the element processing. The Young’s modulus and Poisson’s ratio are used only for setting the contact interface stiffnesses, and it is recommended that reasonable values be input.

Material Model 10: Elastic-Plastic-HydrodynamicFor completeness we give the entire derivation of this constitutive model based on radial return plasticity.

The pressure, ; deviatoric strain rate, ; deviatoric stress rate, ; and volumetric strain rate, , are

defined in Equation (7-42):

Sij G V 1– Cij V1

1 2υ–----------------–

δij–⎝ ⎠⎛ ⎞=

G V υ Cij

Cij

∂xk

∂Xi--------

∂xk

∂Xj--------=

Sij σi j

σi jρρ0-----

∂xi

∂Xk---------

∂xj

∂Xl--------Skl=

ρ0 ρ υ

σi j με′ij=

μ ε′ij

ρ ε'· i j s· i j ε· v


(7-42)

The Jaumann rate of the deviatoric stress, , is given by:

(7-43)

First we update to elastically

(7-44)

where the left superscript, *, denotes a trial stress value. The effective trial stress is defined by

(7-45)

and if exceeds yield stress , the von Mises flow rule:

(7-46)

is violated and we scale the trial stresses back to the yield surface; i.e., a radial return

(7-47)

The plastic strain increment can be found by subtracting the deviatoric part of the strain increment that

is elastic, , from the total deviatoric increment, , i.e.,

(7-48)

Recalling that,

(7-49)

and substituting Equation (7-49) into Equation (7-48) we obtain,

(7-50)

p13---σ i jδij–=

si j σ i j pδi j+=

si j∇ 2με'· i j 2Gε'· i j= =

ε'· ij ε· ij13--- ε· v–=

ε· v ε· i jδij=

si j∇

si j∇ s· i j sipΩpj– sjpΩpi–=

si jn si j

n 1+

si jn 1+* sij

n sipΩpj sjpΩpi 2Gε'· i jdt+ + + sijn Rij 2Gε'· ij d t+ += =

sijRn

2GΔε'i j

s* 32--- si j

n 1+ si jn 1+**

⎝ ⎠⎛ ⎞ 1 2⁄

=

s* σy

φ 12--- sij si j

σy2

3------ 0≤–=

si jn 1+ σy

s*----- si j

n 1+* m*sijn 1+= =

12G------- sij

n 1+ si jRn

–( ) Δε'i j

Δεi jp Δε'i j

12G------- sij

n 1+ si j

Rn–( )–=

Δε'i j

sijn 1+ sij

Rn–*

2G-----------------------------=

Δεi jp sij

n 1+ sijn 1+–*( )

2G---------------------------------------=


248

Substituting Equation (7-47)

(7-51)

into Equation (7-50) gives,

(7-52)

By definition an increment in effective plastic strain is

(7-53)

Squaring both sides of Equation (7-52) leads to:

(7-54)

or from Equations (7-45) and (7-53):

(7-55)

Hence,

(7-56)

where we have substituted for from Equation (7-47)

If isotropic hardening is assumed then:

(7-57)

and from Equation (7-56)

(7-58)

Thus,

(7-59)

and solving for the incremental plastic strain gives

(7-60)

si jn 1+ m*sij

n 1+=

Δεi jp 1 m–( )

2G------------------ sij

n 1+*1 m–2Gm------------- si j

n 1+ dλsijn 1+= = =

Δεp 23--- Δεij

p Δεijp

⎝ ⎠⎛ ⎞ 1 2⁄

=

Δεi jp Δεi j

p 1 m–2G

-------------⎝ ⎠⎛ ⎞ 2

si jn 1+* si j

n 1+*=

32---Δεp2 1 m–

2G-------------⎝ ⎠⎛ ⎞ 22

3---s*2=

Δεp∴ 1 m–3G

------------- s∗s∗ σy–

3G-----------------= =

m

mσy

s∗-----=

σyn 1+ σy

n EpΔεp+=

Δεps∗ σy

n 1+–( )3G

------------------------------s∗ σy

n– EpΔεp–( )3G

----------------------------------------------= =

3G Ep+( )Δεp s∗ σyn–( )=

Δεps∗ σy

n–( )3G Ep+( )

--------------------------=


The algorithm for plastic loading can now be outlined in five simple stress. If the effective trial stress exceeds the yield stress then

1. Solve for the plastic strain increment:

2. Update the plastic strain:

3. Update the yield stress:

4. Compute the scale factor using the yield strength at time :

5. Radial return the deviatoric stresses to the yield surface:

Material Model 12: Isotropic Elastic-PlasticThe von Mises yield condition is given by:

(7-61)

where the second stress invariant, , is defined in terms of the deviatoric stress components as

(7-62)

and the yield stress, , is a function of the effective plastic strain, , and the plastic hardening modulus,

:

(7-63)

The effective plastic strain is defined as:

(7-64)

where:

(7-65)

Δεps∗ σy

n–( )3G Ep+( )

--------------------------=

εpn 1+ εpn Δεp+=

σyn 1+ σy

n EpΔεp+=

n 1+

mσy

n 1+

s∗-------------=

si jn 1+ m sij

n 1+*=

φ J2

σy2

3------–=

J2

J212--- sij si j=

σy εef fp

Ep

σy σ0 Epεeffp+=

εef fp εeff

pd

0

t

∫=

dεeffp 2

3---dεij

p dεi jp=


250

and the plastic tangent modulus is defined in terms of the input tangent modulus, , as:

(7-66)

Pressure is given by the expression

(7-67)

where is the bulk modulus. This is perhaps the most cost effective plasticity model. Only one history

variable, , is stored with this model.

This model is not recommended for shell elements. In the plane stress implementation, a one-step radial return approach is used to scale the Cauchy stress tensor to if the state of stress exceeds the yield surface. This approach to plasticity leads to inaccurate shell thickness updates and stresses after yielding. This is the only model in SOL 700 for plane stress that does not default to an iterative approach.

For MD Nastran SOL SEQ 106, 129, and 600, this material model is similar to using the MATS1 option to define an elastic-plastic material.

Material Model 13: Isotropic Elastic-Plastic with FailureThis highly simplistic failure model is occasionally useful. Material model 12 is called to update the stress tensor. Failure is initially assumed to occur if either

(7-68)

or

(7-69)

where and are user-defined parameters. Once failure has occurred, pressure may never be

negative and the deviatoric components are set to zero:

(7-70)

for all time. The failed element can only carry loads in compression.

Material Model 14: Soil and Crushable Foam With FailureThis material model provides the same stress update as model 5. However, if pressure ever reaches its cutoff value, failure occurs and pressure can never again go negative. In material model 5, the pressure is limited to its cutoff value in tension.

Et

Ep

EEt

E Et–---------------=

pn 1+ K1

Vn 1+------------- 1–⎝ ⎠⎛ ⎞=

K

εef fp

pn 1+ pmin<

εef fp εmax

p>

pmin εmaxp

si j 0=


Material Model 15: Johnson and Cook Plasticity ModelJohnson and Cook express the flow stress as

(7-71)

where , , , , and are user-defined input constants, and:

effective plastic strain rate for

Constants for a variety of materials are provided in Johnson and Cook [1983].

Due to the nonlinearity in the dependence of flow stress on plastic strain, an accurate value of the flow stress requires iteration for the increment in plastic strain. However, by using a Taylor series expansion with linearization about the current time, we can solve for with sufficient accuracy to avoid iteration.

The strain at fracture is given by

(7-72)

where , , are input constants and is the ratio of pressure divided by effective stress:

(7-73)

Fracture occurs when the damage parameter,

(7-74)

reaches the value 1.

A choice of three spall models is offered to represent material splitting, cracking, and failure under tensile loads. The pressure limit model limits the minimum hydrostatic pressure to the specified value, .

If pressures more tensile than this limit are calculated, the pressure is reset to . This option is not

strictly a spall model since the deviatoric stresses are unaffected by the pressure reaching the tensile cutoff and the pressure cutoff value remains unchanged throughout the analysis. The maximum

principal stress spall model detects spall if the maximum principal stress, , exceeds the limiting

value . Once spall is detected with this model, the deviatoric stresses are reset to zero and no

hydrostatic tension is permitted. If tensile pressures are calculated, they are reset to 0 in the spalled material. Thus, the spalled material behaves as rubble. The hydrostatic tension spall model detects spall

σy A Bεpn+( ) 1 C ε·∗ln+( ) 1 T∗m–( )=

A B C n m

εp effective plastic strain=

ε·∗ ε· p

ε· 0

-----= ε· 0 1s 1–=

T∗T Troom–

Tmelt Troom–---------------------------------=

σy

εf D1 D2 D3s∗exp+[ ] 1 D4 ε∗ln+[ ] 1 D5T∗+[ ]=

Di i 1 … 5,,= σ∗

σ∗ pσeff---------=

D Δεp

εf---------∑=

p pmin≥

pmin

pmin

σmax

σp


252

if the pressure becomes more tensile than the specified limit, . Once spall is detected, the deviatoric

stresses are set to zero and the pressure is required to be compressive. If hydrostatic tension is calculated then the pressure is reset to 0 for that element.

In addition to the above failure criterion, this material model also supports a shell element deletion criterion based on the maximum stable time step size for the element, . Generally, goes down

as the element becomes more distorted. To assure stability of time integration, the global time step is the minimum of the values calculated for all elements in the model. Using this option allows the

selective deletion of elements whose time step has fallen below the specified minimum time step,

. Elements which are severely distorted often indicate that material has failed and supports little

load, but these same elements may have very small time steps and therefore control the cost of the analysis. This option allows these highly distorted elements to be deleted from the calculation, and, therefore, the analysis can proceed at a larger time step, and, thus, at a reduced cost. Deleted elements do not carry any load, and are deleted from all applicable slide surface definitions. Clearly, this option must be judiciously used to obtain accurate results at a minimum cost.

Material type 15 is applicable to the high rate deformation of many materials including most metals. Unlike the Steinberg-Guinan model, the Johnson-Cook model remains valid down to lower strain rates and even into the quasistatic regime. Typical applications include explosive metal forming, ballistic penetration, and impact.

This material is similar to the use of the ISOTROPIC option with the Johnson-Cook hardening rule.

Material Type 18: Power Law Isotropic PlasticityElastoplastic behavior with isotropic hardening is provided by this model. The yield stress, , is a

function of plastic strain and obeys the equation:

(7-75)

where is the elastic strain to yield and is the effective plastic strain (logarithmic).

A parameter, SIGY, in the input governs how the strain to yield is identified. If SIGY is set to zero, the strain to yield if found by solving for the intersection of the linearly elastic loading equation with the strain hardening equation:

(7-76)

which gives the elastic strain at yield as:

(7-77)

pmin

Δ tmax Δ tmax

Δ tmax

Δ tmax

Δ tcri t

σy

σy kεn k εyp εp+( )n= =

εyp εp

σ Eε=

σ lεn=

εypEk---⎝ ⎠⎛ ⎞

1n 1–------------

=


If SIGY yield is nonzero and greater than 0.02 then:

(7-78)

Strain rate is accounted for using the Cowper and Symonds model which scales the yield stress with the factor

(7-79)

where is the strain rate. A fully viscoplastic formulation is optional with this model which incorporates the Cowper and Symonds formulation within the yield surface. An additional cost is incurred but the improvement is results can be dramatic.

This material model is a subset of what may be specified through the MATEP option for SOL600.

Material Type 19: Elastic Plastic Material Model with Strain Rate Dependent YieldIn this model, a TABLED1 is used to describe the yield strength as a function of effective strain rate

where

(7-80)

and the prime denotes the deviatoric component. The yield stress is defined as

(7-81)

where is the effective plastic strain and is given in terms of Young’s modulus and the tangent

modulus by

(7-82)

Both Young's modulus and the tangent modulus may optionally be made functions of strain rate by specifying a TABLED1 ID giving their values as a function of strain rate. If these TABLED1 ID's are input as 0, then the constant values specified in the input are used.

εyp

σy

k-----⎝ ⎠⎛ ⎞

1n---

1ε·

C----⎝ ⎠⎛ ⎞

1 p/

+

ε·

σ0

ε·

ε· 2

3---ε'· ijε'· ij⎝ ⎠⎛ ⎞ 1 2⁄

=

σy σ0 ε·

( ) Epεp+=

εp Ep

Ep

EEt

E Et–---------------=

Note: All TABLED1s used to define quantities as a function of strain rate must have the same number of points at the same strain rate values. This requirement is used to allow vectorized interpolation to enhance the execution speed of this constitutive model.


254

This model also contains a simple mechanism for modeling material failure. This option is activated by specifying a TABLED1 ID defining the effective stress at failure as a function of strain rate. For solid elements, once the effective stress exceeds the failure stress the element is deemed to have failed and is removed from the solution. For shell elements the entire shell element is deemed to have failed if all integration points through the thickness have an effective stress that exceeds the failure stress. After failure the shell element is removed from the solution.

In addition to the above failure criterion, this material model also supports a shell element deletion criterion based on the maximum stable time step size for the element, . Generally, goes down

as the element becomes more distorted. To assure stability of time integration, the global time step is the minimum of the values calculated for all elements in the model. Using this option allows the

selective deletion of elements whose time step has fallen below the specified minimum time step,

. Elements which are severely distorted often indicate that material has failed and supports little

load, but these same elements may have very small time steps and therefore control the cost of the analysis. This option allows these highly distorted elements to be deleted from the calculation, and, therefore, the analysis can proceed at a larger time step, and, thus, at a reduced cost. Deleted elements do not carry any load, and are deleted from all applicable slide surface definitions. Clearly, this option must be judiciously used to obtain accurate results at a minimum cost.

This material model is a subset of what may be specified through the MATEP option for SOL600.

Material Type 20: RigidThe rigid material type 20 provides a convenient way of turning one or more parts comprised of beams, shells, or solid elements into a rigid body. Approximating a deformable body as rigid is a preferred modeling technique in many real world applications. For example, in sheet metal forming problems the tooling can properly and accurately be treated as rigid. In the design of restraint systems the occupant can, for the purposes of early design studies, also be treated as rigid. Elements which are rigid are bypassed in the element processing and no storage is allocated for storing history variables; consequently, the rigid material type is very cost efficient.

Two unique rigid part IDs may not share common nodes unless they are merged together using the rigid body merge option. A rigid body may be made up of disjoint finite element meshes, since this is a common practice in setting up tooling meshes in forming problems.

All elements which reference a given part ID corresponding to the rigid material should be contiguous, but this is not a requirement. If two disjoint groups of elements on opposite sides of a model are modeled as rigid, separate part ID's should be created for each of the contiguous element groups if each group is to move independently. This requirement arises from the fact that SOL 700 internally computes the six rigid body degrees-of-freedom for each rigid body (rigid material or set of merged materials), and if disjoint groups of rigid elements use the same part ID, the disjoint groups will move together as one rigid body.

Δ tmax Δ tmax

Δ tmax

Δ tmax

Δ tcri t


Inertial properties for rigid materials may be defined in either of two ways. By default, the inertial properties are calculated from the geometry of the constituent elements of the rigid material and the density specified for the part ID. Alternatively, the inertial properties and initial velocities for a rigid body may be directly defined, and this overrides data calculated from the material property definition and nodal initial velocity definitions.

Young's modulus, , and Poisson's ratio, , are used for determining sliding interface parameters if the rigid body interacts in a contact definition. Realistic values for these constants should be defined since unrealistic values may contribute to numerical problem in contact.

Material Model 22: Chang-Chang Composite Failure ModelFive material parameters are used in the three failure criteria based upon Chang and Chang 1987a, 1987b:

• , longitudinal tensile strength

• , transverse tensile strength

• , shear strength

• , transverse compressive strength

• , nonlinear shear stress parameter.

and are obtained from material strength measurement. is defined by material shear stress-strain

measurements. In plane stress, the strain is given in terms of the stress as

(7-83)

The third equation defines the nonlinear shear stress parameter .

A fiber matrix shearing term augments each damage mode:

(7-84)

which is the ratio of the shear stress to the shear strength.

The matrix cracking failure criteria is determined from

(7-85)

E υ

S1

S2

S12

C2

α

C2 α

ε11

E1------ σ1 υ1σ2–( )=

ε21

E2------ σ2 υ2σ1–( )=

2ε121

G12---------τ12 ατ12

3+=

α

τ

τ122

2G12------------ 3

4---ατ12

4+

S122

2G12------------ 3

4---αS12

4+

------------------------------------=

Fmatrix

σ2

S2------⎝ ⎠⎛ ⎞

2τ+=


256

where failure is assumed whenever . If , then the material constants , , , and

are set to zero.

The compression failure criteria is given as

(7-86)

where failure is assumed whenever . If , then the material constants , , and are

set to zero.

The final failure mode is due to fiber breakage.

(7-87)

Failure is assumed whenever . If , then the constants , , , , and are set

to zero.

Material Model 24: Piecewise Linear Isotropic PlasticityThis plasticity treatment in this model is quite similar to Model 3 but only isotropic hardening occurs. Deviatoric stresses are determined that satisfy the yield function

(7-88)

where

(7-89)

where the hardening function can be specified in tabular form as an option. Otherwise, linear

hardening of the form

(7-90)

is assumed where and are given in Equations (7-16) and (7-17), respectively. The parameter

accounts for strain rate effects. For complete generality a table defining the yield stress versus plastic strain may be defined for various levels of effective strain rate.

In the implementation of this material model, the deviatoric stresses are updated elastically (see material model 1), the yield function is checked, and if it is satisfied the deviatoric stresses are accepted. If it is not, an increment in plastic strain is computed:

(7-91)

Fmatrix 1> Fmatrix 1> E2 G12 υ

υ2

Fcomp

σ2

2S12-----------⎝ ⎠⎛ ⎞

2 C2

2S12-----------⎝ ⎠⎛ ⎞

21–

σ2

C2------ τ+ +=

Fcomp 1> Fcomp 1> E2 υ1 υ2

Ffiber

σ1

S1------⎝ ⎠⎛ ⎞

2τ+=

Ffiber 1> Ffiber 1> E1 E2 G12 υ1 υ2

φ 12--- sijsi j

σy2

3------ 0≤–=

σy β σ0 fh εeffp( )+[ ]=

fh εeffp( )

fh εeffp( ) Ep εef f

p( )=

Ep εef fp β

Δεef fp

32---si j

* sij*

⎝ ⎠⎛ ⎞ 1 2⁄

σy–

3G Ep+------------------------------------------=


is the shear modulus and is the current plastic hardening modulus. The trial deviatoric stress state

is scaled back:

(7-92)

For shell elements, the above equations apply, but with the addition of an iterative loop to solve for the normal strain increment, such that the stress component normal to the mid surface of the shell element approaches zero.

Three options to account for strain rate effects are possible:

• Strain rate may be accounted for using the Cowper and Symonds model which scales the yield stress with the factor

(7-93)

where is the strain rate.

• For complete generality a TABLED1, defining , which scales the yield stress may be input instead. In this curve the scale factor versus strain rate is defined.

• If different stress versus strain curves can be provided for various strain rates, the option using the reference to a table definition can be used.

A fully viscoplastic formulation is optional which incorporates the different options above within the yield surface. An additional cost is incurred over the simple scaling but the improvement is results can be dramatic.

If a TABLE ID is specified a curve ID is given for each strain rate. Intermediate values are found by interpolating between curves.

Material Model 26: Crushable FoamThis orthotropic material model does the stress update in the local material system denoted by the subscripts, , , and . The material model requires the following input parameters:

• , Young’s modulus for the fully compacted material;

• , Poisson’s ratio for the compacted material;

• , yield stress for fully compacted honeycomb;

• LCA, TABLED1 number for sigma-aa versus either relative volume or volumetric strain (see Figure 7-3);

Ep sij*

si jn 1+

σy

32--- si j

* sij*

⎝ ⎠⎛ ⎞ 1 2⁄------------------------------ si j

*=

β 1ε·

C----⎝ ⎠⎛ ⎞

1 p⁄+=

ε·

β

a b c

E

ν

σy

Note: In Figure 7-3, the “yield stress” at a volumetric strain of zero is nonzero. In the TABLED1 definition, the “time” value is the volumetric strain and the “function” value is the yield stress.


258

Figure 7-3 Stress Quantity Versus Volumetric Strain

• LCB, TABLED1 number for sigma-bb versus either relative volume or volumetric strain (default: LCB = LCA);

• LCC, the TABLED1 number for sigma-cc versus either relative volume or volumetric strain (default: LCC = LCA);

• LCS, the TABLED1 number for shear stress versus either relative volume or volumetric strain (default LCS = LCA);

• , relative volume at which the honeycomb is fully compacted;

• , elastic modulus in the uncompressed configuration;



• , elastic shear modulus in the uncompressed configuration;



• LCAB, TABLED1 number for sigma-ab versus either relative volume or volumetric strain (default: LCAB = LCS);

• LCBC, TABLED1 number for sigma-bc versus either relative volume or volumetric strain default: LCBC = LCS);

σi j

εi j–Strain0

Curve extends into negative strain quadrantsince SOL 700 extrapolates using the twoend points. It is important that the extrapolationdoes not extend into the negative stress region.

Unloading is based on the interpolatedYoung’s moduli which must provide anunloading tangent that exceeds theloading tangent.

Unloading andreloading path

Vf

Eaau

Ebbu

Eccu

Gabu

Gbcu

Gcau


• LCCA, TABLED1 number for sigma-ca versus either relative volume or volumetric strain (default: LCCA = LCS);

• LCSR, optional TABLED1 number for strain rate effects.

The behavior before compaction is orthotropic where the components of the stress tensor are uncoupled; i.e., an a component of strain will generate resistance in the local a direction with no coupling to the local b and c directions. The elastic moduli vary linearly with the relative volume from their initial values to the fully compacted values:

(7-94)

where

(7-95)

and is the elastic shear modulus for the fully compacted honeycomb material

(7-96)

The relative volume is defined as the ratio of the current volume over the initial volume; typically, at the beginning of a calculation. The relative volume, , is the minimum value reached during

the calculation.

The TABLED1s define the magnitude of the average stress as the material changes density (relative volume). Each curve related to this model must have the same number of points and the same abscissa values. There are two ways to define these curves: as a function of relative volume , or as a function of volumetric strain defined as:

(7-97)

In the former, the first value in the curve should correspond to a value of relative volume slightly less than the fully compacted value. In the latter, the first value in the curve should be less than or equal to zero corresponding to tension and should increase to full compaction.

Eaa Eaau β E Eaau–( )+=

Ebb Ebbu β E Ebbu–( )+=

Ecc Eccu β E Eccu–( )+=

Gab Gabu β G Gabu–( )+=

Gbc Gbcu β G Gbcu–( )+=

Gca Gcau β G Gcau–( )+=

β max min1 Vmin–

1 Vf–--------------------- 1,⎝ ⎠⎛ ⎞ 0,=

G

GE

2 1 ν+( )---------------------=

V

V 1= Vmin

V

εV 1 V–=

Note: When defining the curves, care should be taken that the extrapolated values do not lead to negative yield stresses.


260

At the beginning of the stress update we transform each element’s stresses and strain rates into the local element coordinate system. For the uncompacted material, the trial stress components are updated using the elastic interpolated moduli according to:

(7-98)

Then we independently check each component of the updated stresses to ensure that they do not exceed the permissible values determined from the TABLED1s; e.g., if

(7-99)

then

(7-100)

The parameter is either unity or a value taken from the TABLED1 number, LCSR, that defines as a function of strain rate. Strain rate is defined here as the Euclidean norm of the deviatoric strain rate tensor.

For fully compacted material we assume that the material behavior is elastic-perfectly plastic and updated the stress components according to:

(7-101)

Where the deviatoric strain increment is defined as:

(7-102)

We next check to see if the yield stress for the fully compacted material is exceeded by comparing:

(7-103)

σaan 1+

trial

σaan

EaaΔεaa+=

σbbn 1+

trial

σbbn

EbbΔεbb+=

σccn 1+

trial

σccn

EccΔεcc+=

σabn 1+

trial

σabn

EabΔεab+=

σbcn 1+

trial

σbcn

EbcΔεbc+=

σcan 1+

trial

σcan

EcaΔεca+=

σi jn 1t rial+ λσi j Vmin( )>

σi jn 1+ σi j Vmin( )

λσijn 1t rial+

σijn 1trial+

---------------------------=

λ λ

si jtrial sij

n 2GΔεijdevn 1 2⁄+

+=

Δεi jdev Δεij

13---Δεkkδi j–=

sefftrial 3

2--- sij

trialsijtrial

⎝ ⎠⎛ ⎞ 1 2⁄

=


the effective trial stress, to the yield stress . If the effective trial stress exceeds the yield stress, we

simply scale back the stress components to the yield surface:

(7-104)

We can now update the pressure using the elastic bulk modulus, :

(7-105)

and obtain the final value for the Cauchy stress:

(7-106)

After completing the stress update, we transform the stresses back to the global configuration.

Material Model 27: Incompressible Mooney-Rivlin RubberThis material model, available for solid elements only, provides an alternative to the Blatz-Ko rubber model. The strain energy density function is defined as in terms of the input constants , ,, and as:

(7-107)

where

(7-108)

(7-109)

= Poisson’s ratio

= shear modulus of linear elasticity

= strain invariants in terms of the principal stretches:

(7-110)

Recommended values for Poisson’s ratio are between .490 and .495 or higher. Lower values may lead to instabilities. In the derivation of the constants and incompressibility is assumed.

σy

si jn 1+ σy

sefftrial

------------ si jtrial=

K

pn 1+ pn KΔεkkn 1 2⁄+–=

KE

3 1 2ν–( )------------------------=

σi jn 1+ sij

n 1+ pn 1+ δij–=

A B υ

W I1 I2 I2,,( ) A I1 3–( ) B I2 3–( ) C1

I32

---- 1–⎝ ⎠⎛ ⎞ D I3 1–( )2+ + +=

C .5∗A B+=

DA 5υ 2–( ) B 11υ 5–( )+

2 1 2υ–( )-------------------------------------------------------------=

υ

G 2 A B+( )=

I1 I2 I3,,

I1 λ12 λ2

2 λ32

+ +=

I2 λ12λ2

2 λ22λ3

2 λ32λ1

2+ +=

I3 λ12λ2

2λ32

=

C D


262

Material Model 28: Resultant PlasticityThis plasticity model, based on resultants as illustrated in Figure 7-4, is very cost effective but not as accurate as through-thickness integration. This model is available only with the triangular, Belytschko-Tsay shell, and the Belytschko beam element since these elements, unlike the Hughes-Liu elements, lend themselves very cleanly to a resultant formulation. The elements are set by the SHELLFORM parameter.

Figure 7-4 Full Section Yield using Resultant Plasticity

In applying this model to shell elements the resultants are updated incrementally using the midplane

strains and curvatures :

(7-111)

(7-112)

where the plane stress constitutive matrix is given in terms of Young’s Modulus and Poisson’s ratio as:

(7-113)

Defining

(7-114)

(7-115)

(7-116)

the Ilyushin yield function becomes

(7-117)

C0

Membrane σy

σyBending

my σyζ ζdh2---–

h2---

∫h2

4-----σy= =

ny σy ζdh2---–

h2---

∫ hσy= =

εm κ

Δn Δ tCεm=

Δm Δ th3

12------Cκ=

E

ν

m mxx2 mxxmyy– myy

2 3mxy2+ +=

n nxx2 nxxnyy– nyy

2 3nxy2+ +=

m mxx2 mxxmyy– myy

2 3mxy2+ +=

mn mxxnxx12--- mxxnyy–

12---nxx myy– myny 3mxynxy+ +=

f m n,( ) n4 mn

h 3-------------- 16m

h2---------- ny

2≤+ + h2σy2= =


In our implementation we update the resultants elastically and check to see if the yield condition is violated:

(7-118)

If so, the resultants are scaled by the factor :

(7-119)

We update the yield stress incrementally:

(7-120)

where is the plastic hardening modulus which in incremental plastic strain is approximated by

(7-121)

Kennedy, et. al., report that this model predicts results that may be too stiff; users of this model should proceed cautiously.

In applying this material model to the Belytschko beam, the flow rule changes to

(7-122)

have been updated elastically according to Equations (6-77) through (6-79). The yield condition is checked Equation (7-118), and if it is violated, the resultants are scaled as described above.

This model is frequently applied to beams with nonrectangular cross sections. The accuracy of the results obtained should be viewed with some healthy suspicion. No workhardening is available with this model.

Material Model 29: FORCE LIMITED Resultant FormulationThis material model is available for the Belytschko beam element only. Plastic hinges form at the ends of the beam when the moment reaches the plastic moment. The momentversus- rotation relationship is specified by the user in the form of a TABLED1 and scale factor.

The point pairs of the TABLED1 are (plastic rotation in radians, plastic moment). Both quantities should be positive for all points, with the first point pair being (zero, initial plastic moment). Within this constraint any form of characteristic may be used including flat or falling curves. Different TABLED1s and scale factors may be specified at each node and about each of the local s and t axes.

f m n,( ) ny2>

α

αny

2

f m n,( )-----------------=

σyn 1+ σy

n EpΔεplast icef f+=

Ep

Δεplast icef f

f m n,( ) ny–

h 3G Ep+( )----------------------------------=

f m n,( ) fx2 4my

2

3 Iyy----------

4mz2

3 Izz---------- ny

2≤+ + A2σy2= =


264

Figure 7-5 Full Section Yield using Resultant Plasticity

Axial collapse occurs when the compressive axial load reaches the collapse load. The collapse load-versus-collapse deflection is specified in the form of a TABLED1. The points of the TABLED1 are (true strain, collapse force). Both quantities should be entered as positive for all points, and will be interpreted as compressive i.e., collapse does not occur in tension. The first point should be the pair (zero, initial collapse load).

The collapse load may vary with end moment and with deflection. In this case, several load-deflection curves are defined, each corresponding to a different end moment. Each TABLED1 should have the same number of point pairs and the same deflection values. The end moment is defined as the average of the absolute moments at each end of the beam, and is always positive.

It is not possible to make the plastic moment vary with axial load.

Figure 7-6 The Force Magnitude Limited Applied End Moment

Membrane σy

σyBending

my σyζ ζdh2---–

h2---

∫h2

4-----σy= =

ny σy ζdh2---–

h2---

∫ hσy= =

For

ce

Displacement

M1

M2

M3

M4

M5

M6

M7

M8


For an intermediate value of the end moment, MD Nastran SOL 700 interpolates between the curves to determine the allowable force.

A co-rotational technique and moment-curvature relations are used to compute the internal forces. The co-rotational technique will not be treated here as we will focus solely on the internal force update and computing the tangent stiffness. For this, we use the notation:

We emphasize that the local y and z base vectors in the reference configuration always coincide with the corresponding nodal vectors. The nodal vectors in the current configuration are updated using the Hughes-Winget formula while the base vectors are computed from the current geometry of the element and the current nodal vectors.

Internal Forces

Elastic Update

In the local system for a beam connected by nodes I and J, the axial force is updated as

(7-123)

where

(7-124)

.. (7-125)

The torsional moment is updated as

= Young’s modulus

= Shear modulus

= Cross sectional area

= Effective area in shear

= Reference length of beam

= Current length of beam

= Second moment of inertia about y

= Second moment of inertia about z

= Polar moment of inertia

= ith local base vector in the current configuration

= nodal vector in y direction at node I in the current configuration

= nodal vector in z direction at node I in the current configuration

E

G

A

As

ln

ln 1+

Iyy

Izz

J

ei

yI

zI

fael fa

n Kaelδ+=

Kael EA

ln-------=

δ ln 1+ ln–=


266

(7-126)

where

(7-127)

. (7-128)

The bending moments are updated as

(7-129)

(7-130)

where

(7-131)

(7-132)

(7-133)

(7-134)

In the following, we refer to as the (elastic) moment-rotation matrix.

Plastic Correction

After the elastic update the state of force is checked for yielding as follows. As a preliminary note we emphasize that whenever yielding does not occur the elastic stiffnesses and forces are taken as the new stiffnesses and forces.

The yield moments in direction i at node I as functions of plastic rotations are denoted . This

function is given by the user but also depends on whether a plastic hinge has been created. Whenever the elastic moment exceeds the plastic moment, the plastic rotations are updated as

(7-135)

and the moment is reduced to the yield moment

mtel mt

n Ktelθt+=

Ktel GJ

ln-------=

θt12---el

T yI yJ zI zJ×+×( )=

myel my

n Ayel θy+=

mzel mz

n Azel θz+=

A*el 1

1 φ*+---------------

EI**

ln-----------

4 φ*+ 2 φ*–

2 φ*– 4 φ*+⎝ ⎠⎜ ⎟⎛ ⎞

=

φ*

12E I**

GA* ln tn--------------------=

θyT e3

TyI zI× yJ zJ×⎝ ⎠

⎛ ⎞=

θzT e2

TyI zI× yJ zJ×⎝ ⎠

⎛ ⎞=

A*el

miIY θi I

P( )

θiIP n 1+( ) θi I

P n( ) miIel

miIY

–

max 0.001 Ai II( )el ∂miI

Y

∂θiIP

-----------+,⎝ ⎠⎜ ⎟⎛ ⎞

------------------------------------------------------------------+=


(7-136)

The corresponding diagonal component in the moment-rotation matrix is reduced as

(7-137)

where is a parameter chosen such that the moment-rotation matrix remains positive definite.

The yield moment in torsion is given by and is provided by the user. If the elastic torsional

moment exceeds this value, the plastic torsional rotation is updated as

(7-138)

and the moment is reduced to the yield moment

The torsional stiffness is modified as

(7-139)

where again is chosen so that the stiffness is positive.

Axial collapse is modeled by limiting the axial force by ; i.e., a function of the axial strains and

the magnitude of bending moments. If the axial elastic force exceeds this value it is reduced to yield

(7-140)

and the axial stiffness is given by

(7-141)

We neglect the influence of change in bending moments when computing this parameter.

miIn 1+ miI

Y θiIP n 1+( )( ) miI

el( )sgn=

Ai II( )n 1+

Ai II( )el

1 αAi II( )

el

max 0.001 Ai II( )el ∂miI

Y

∂θiIP

-----------+,⎝ ⎠⎜ ⎟⎛ ⎞

------------------------------------------------------------------–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

α 1≤

mtY θt

P( )

θtP n 1+( ) θt

P n( ) mtel

mtY

–

max 0.0001 Ktel ∂mt

Y

∂θtP

----------+,⎝ ⎠⎜ ⎟⎛ ⎞

---------------------------------------------------------------+=

mtn 1+

mtY θt

P n 1+( )( ) mtel( )sgn=

Ktn 1+

Ktel

1 αKt

el

Ktel ∂mt

Y

∂θtP

----------+

--------------------------–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

α 1≤

faY ε m,( )

fan 1+ fa

Y εn 1+ mn 1+,( ) fael( )sgn=

Kan 1+ max 0.05Ka

el ∂ faY

∂ε--------,⎝ ⎠

⎛ ⎞=


268

Damping

Damping is introduced by adding a viscous term to the internal force on the form

(7-142)

(7-143)

where is a damping parameter.

Transformation

The internal force vector in the global system is obtained through the transformation

(7-144)

where

(7-145)

(7-146)

Tangent Stiffness

Derivation

The tangent stiffness is derived from taking the variation of the internal force

(7-147)

which can be written

fv Dddt-----

δθt

θy

θz

=

D γ

Kael

Ktel

Ayel

Azel

=

γ

fgn 1+ Sft

n 1+=

S

e1– 0 e3 ln 1+⁄– e3 ln 1+⁄– e2 ln 1+⁄ e2 ln 1+⁄

0 e1– e2 0 e3 0

e1 0 e3 ln 1+⁄ e3 ln 1+⁄ e2 ln 1+⁄– e2 ln 1+⁄–

0 e1 0 e2 0 e3

=

fln 1+

fan 1+

mtn 1+

myn 1+

mzn 1+

=

δfgn 1+ δSfl

n 1+ Sδfln 1++=

δfgn 1+ Kgeoδu Kmatδu+=


where

. (7-148)

There are two contributions to the tangent stiffness: geometrical and material. The geometrical contribution is given (approximately) by

(7-149)

where

(7-150)

(7-151)

(7-152)

(7-153)

and I is the 3 by 3 identity matrix. We use as the outer matrix product and define

. (7-154)

The material contribution can be written as

(7-155)

where

(7-156)

δu δxIT δωI

T δxJT δωj

T⎝ ⎠⎛ ⎞ T

=

Kgeo R fln 1+ I⊗( )W

1ln 1+ ln 1+------------------------Tfl

n 1+ L–=

R

R1 0 R3 ln 1+⁄ R3 ln 1+⁄ R2 ln 1+⁄– R2 n 1+( )⁄–

0 R1 R2– 0 R3– 0

R1– 0 R3 ln 1+⁄– R3 ln 1+⁄– R2 ln 1+⁄ R2 ln 1+⁄

0 R1– 0 R2– 0 R3–

=

W R1 ln 1+⁄– e1e1T 2⁄ R1 ln 1+⁄ e1e1

T 2⁄⎝ ⎠⎛ ⎞=

T

0 0 e3– e3– e2 e2

0 0 0 0 0 00 0 e3 e3 e2– e2–

0 0 0 0 0 0

=

L e1T 0 e1

T 0⎝ ⎠⎛ ⎞=

⊗

Riv ei v×=

Kmat SKST=

K

Kan 1+

Ktn 1+

Ayn 1+

Azn 1+

=


270

Material Model 30: Shape Memory AlloyThis section presents the mathematical details of the shape memory alloy material in MD Nastran SOL 700.

The description closely follows the one of Auricchio and Taylor [1997] with appropriate modifications for this particular implementation.

Mathematical Description of the Material Model

The Kirchhoff stress in the shape memory alloy can be written

(7-157)

where i is the second order identity tensor and

. (7-158)

Here K and G are bulk and shear modulii, and e are volumetric and shear logarithmic strains and and are constant material parameters. There is an option to define the bulk and shear modulii as

functions of the martensite fraction according to

(7-159)

in case the stiffness of the martensite differs from that of the austenite. Furthermore, the unit vector n is defined as

(7-160)

and a loading function is introduced as

(7-161)

where

. (7-162)

τ

τ p i t+=

p K θ 3αξSεL–( )=

t 2G e ξSεLn–( )=

θ α

εL

K KA ξS KS KA–( )+=

G GA ξS GS GKA–( )+=

n e e 10 12–+( )⁄=

F 2G e 3αKθ βξS–+=

β 2G 9α2K+( )εL=


For the evolution of the martensite fraction in the material, the following rule is adopted

. (7-163)

Here , , , and are constant material parameters. The Cauchy stress is finally obtained as

(7-164)

where J is the Jacobian of the deformation.

Algorithmic Stress Update

For the stress update, we assume that the martensite fraction and the value of the loading function

is known from time and the deformation gradient at time , , is known. We form the left Cauchy-

Green tensor as which is diagonalized to obtain the principal values and directions

and . The volumetric and principal shear logarithmic strains are given by

where

is the total Jacobian of the deformation. Using Equation (7-161) with , a value of the loading

function can be computed. The discrete counterpart of Equation (7-163) becomes

(7-165)

ξS

F RsAS

0>–

F·

0>ξS 1<

⎭⎪⎪⎬⎪⎪⎫

ξS⇒ 1 ξS–( ) F·

F RfAS

–-------------------–=

F RsSA

0<–

F·

0>ξS 0>

⎭⎪⎪⎬⎪⎪⎫

ξS⇒ ξSF·

F RfSA

–-------------------=

RsAS

RfAS

RsSA

RfSA

σ τ J⁄=

ξsn

Fn

tn tn 1+ F

B FFT= Λdiag λi( )

Q

θ J( )log=

ei λi J1 3⁄⁄( )log=

J λ1λ2λ3=

ξS ξSn

= Ftrial

Ftrial RsAS

0>–

Ftrial Fn 0>–

ξSn

1<⎭⎪⎪⎬⎪⎪⎫

ΔξS⇒ 1 ξSn

– ΔξS–( )Ftrial βΔξS– min max Fn Rs

AS,( ) Rf

AS,( )–

Ftrial βΔξS– RfAS

–------------------------------------------------------------------------------------------------------–=

Ftrial RsSA 0>–

Ftrial Fn 0>–

ξSn 1<

⎭⎪⎪⎬⎪⎪⎫

ΔξS⇒ ξSn ΔξS+( )

Ftrial βΔξS– min max Fn RsSA

,( ) RfSA

,( )–

Ftrial βΔξS– RfSA–

------------------------------------------------------------------------------------------------------=


272

If none of the two conditions to the left are satisfied, set , , and compute the stress

using Equations (7-157), (7-158), (7-159), (7-164) and . When phase transformation occurs

according to a condition to the left, the corresponding equation to the right is solved for . If the bulk

and shear modulii are constan,t this is an easy task. Otherwise, as well as depends on this parameter and makes things a bit more tricky. We have that

.

where and are Young’s modulii for martensite and austenite, respectively. The subscript n is

introduced for constant quantities evaluated at time . To simplify the upcoming expressions, these

relations are written

(7-166)

respectively, where we have for simplicity set

Inserting these expressions into Equation (7-165) results in

(7-167)

and

respectively, where we have for simplicity set

.

The solutions to these equations are approximated with two Newton iterations starting in the point

.. Now set and compute and according to Equations

(7-157), (7-158), (7-159), (7-160), (7-162) and .

ξSn 1+ ξS

n= Fn 1+ Ftrial=

σn 1+ ξS ξSn

=

ΔξS

Ftrial β

Ftrial Fntrial 1

ES EA–

En-------------------ΔξS+⎝ ⎠

⎛ ⎞=

β βn 1ES EA–

En-------------------ΔξS+⎝ ⎠

⎛ ⎞=

ES EA

tn

Ftrial Fntrial ΔFtrialΔξS+=

β βn ΔβΔξS+=

f ΔξS( ) Δβ 1 ξSn

–( )ΔξS2

RfAS

F˜

ASn

– βn ΔFtrial–( ) 1 ξSn

–( )+( )ΔξS+ +=

1 ξSn–( ) F

˜ASn

Fntrial–( ) 0=

f ΔξS( ) ΔβξSnΔξS

2F˜

SAn

RfSA

– βn ΔFtrial–( )ξSn

+( )ΔξS+ +=

ξSn F

˜SAn

Fntr ial–( ) 0=

F˜

ASn

min max Fn RsAS,( ) Rf

AS,( )=

F˜

SAn

min max Fn RfSA,( ) Rs

SA,( )=

ΔξS 0= ξSn 1+

min 1 max 0 ξSn ΔξS+,( ),( )= σn 1+ Fn 1+

ξS ξSn 1+

=


Tangent Stiffness Matrix

An algorithmic tangent stiffness matrix relating a change in true strain to a corresponding change in Kirchhoff stress is derived in the following. Taking the variation of Equation (7-158) results in

. (7-168)

The variation of the unit vector in Equation (7-159) can be written

where I is the fourth order identity tensor. For the variation of martensite fraction, we introduce the indicator parameters and that should give information of the probability of phase transformation occurring in the next stress update. Set initially and change them according to

using the quantities computed in the previous stress update. For the variation of the martensite fraction we take variations of Equations(7-167) and (7-168) with

which results in

where

.

As can be seen, we use the value of obtained in the previous stress update since this is easier to implement and will probably give a good indication of the current value of this parameter.

δp K δθ 3αδξSεL–( ) δK θ 3αξSεL–( )+=

δ t 2G δe δξSεLn– ξSεLδn–( ) 2δG e ξSεLn–( )+=

δn1

e 10 12–+---------------------------- I n n⊗–( )δe=

HAS HSA

HAS HSA 0= =

Ftrial RsAS

0>–

Ftrial Fn 0>–

ξSn ΔξS 1≤+

⎭⎪⎪⎬⎪⎪⎫

HAS⇒ 1=

Ftrial RsSA 0>–

Ftrial Fn 0>–

ξSn ΔξS 0≥+

⎭⎪⎪⎬⎪⎪⎫

HSA⇒ 1=

δFntrial 2Gn :δe 3αKδθ+=

δξS γ 2Gn :δe 3αKδθ+( )=

γ1 ξS

n–( )HAS

RfAS F

˜ASn

β( n ΔFntrial ) 1 ξS

n–( )–+–------------------------------------------------------------------------------------------

ξSnHSA

F˜

SAn

RfSA– βn ΔFn

trial–( )ξSn+

--------------------------------------------------------------------------+=

γ


274

The variation of the material parameters K and G results in

and, finally, using the identities

results in

where is the fourth order deviatoric identity tensor. In general this tangent is not symmetric because of the terms on the second line in the expression above. We simply use a symmetrization of the tangent stiffness above in the implementation. Furthermore, we transform the tangent to a tangent closer related to the one that should be used in the MD Nastran SOL 700 implementation,

Material Model 31: Slightly Compressible Rubber ModelThis model implements a modified form of the hyperelastic constitutive law first described in [Kenchington 1988].

The strain energy functional, , is defined in terms of the input constants as:

(7-169)

δK KS KA–( )δξS=

δG GS GA–( )δξS=

n :δe n :δε=

δθ i :δε=

δτ iδp δ t+=

δτ 2G 1ξSεL

e 10 12–+----------------------------–⎝ ⎠

⎛ ⎞ Idev K 1 9α2KγεL– 3αγ KS KA–( ) θ 3αξSεL–( )+( ) i i⊗+⎩⎨⎧

+=

2γG KS KA–( ) θ 3αξSεL–( ) i n 6γαK GS GA–( ) e ξSεL–( )n i +⊗+⊗

2GξSεL

e 1 10 12––+------------------------------------- 2GγεL– 2γ GS GA–( ) e ξDεL–( )+⎝ ⎠⎛ ⎞

n n 6KGαγεL i n n i⊗+⊗( ) }δε–⊗

Idev

C J 1– 2G 1ξSεL

e 10 12–+----------------------------–⎝ ⎠

⎛ ⎞ Idev K 1 9α2KγεL– 3αγ KS KA–( ) θ 3αξSεL–( )+( ) i i +⊗+⎩⎨⎧

=

γG KS KA–( ) θ 3αξSεL–( ) 3γαK GS GA–( ) e ξSεL–( ) 6KGαγεL–+( ) i n n i⊗+⊗( ) +

2GξSεL

e 10 12–+---------------------------- 2GγεL– 2γ GS GA–( ) e ξSεL–( )+⎝ ⎠⎛ ⎞ n n⊗

⎭⎬⎫

δε

U

U C100I1 C200I12 C300I1

3 C400I14 C110I1I2 C210I1

2I2 C010I2 C020I22 j J( )+ + + + + + + +=


where the strain invariants can be expressed in terms of the deformation gradient matrix, , and the

Green-St. Venant strain tensor, :

. (7-170)

The derivative of with respect to a component of strain gives the corresponding component of stress

(7-171)

where, , is the second Piola-Kirchhoff stress tensor which is transformed into the Cauchy stress tensor:

(7-172)

where and are the initial and current density, respectively.

Material Model 32: Laminated Glass ModelThis model is available for modeling safety glass. Safety glass is a layered material of glass bonded to a polymer material which can undergo large strains.

The glass layers are modeled by isotropic hardening plasticity with failure based on exceeding a specified level of plastic strain. Glass is quite brittle and cannot withstand large strains before failing. Plastic strain was chosen for failure since is increases monotonically and, therefore, is insensitive to spurious numberical noise in the solution.

The material to which the glass is bonded is assumed to stretch plastically without failure.

The user defined integration rule option must be used with this material. The user defined rule specifies the thickness of the layers making up the safety glass. Each integration point is flagged with a zero if the layer is glass and with a one if the layer is polymer.

An iterative plane stress plasticity algorithm is used to enforce the plane stress condition.

Material Model 34: FabricThe fabric model is a variation on the Layered Orthotropic Composite material model (Material 22) and is valid for only 3 and 4 node membrane elements. This material model is strongly recommended for modeling airbags and seatbelts. In addition to being a constitutive model, this model also invokes a special membrane element formulation that is better suited to the large deformations experienced by fabrics. For thin fabrics, buckling (wrinkling) can occur with the associated inability of the structure to

Fij

Eij

J Fij=

I1 Eii=

I212---δpq

i jEpiEqj=

U

Sij∂U∂Eij----------=

Sij

σi jρρo-----

∂xi

∂Xk---------

∂xj

∂Xl--------Skl=

ρo ρ


276

support compressive stresses; a material parameter flag is included for this option. A linear elastic liner is also included which can be used to reduce the tendency for these material/elements to be crushed when the no-compression option is invoked.

If the airbag material is to be approximated as an isotropic elastic material, then only one Young’s modulus and Poisson’s ratio should be defined. The elastic approximation is very efficient because the local transformations to the material coordinate system may be skipped. If orthotropic constants are defined, it is very important to consider the orientation of the local material system and use great care in setting up the finite element mesh.

If the reference configuration of the airbag is taken as the folded configuration, the geometrical accuracy of the deployed bag will be affected by both the stretching and the compression of elements during the folding process. Such element distortions are very difficult to avoid in a folded bag. By reading in a reference configuration, such as the final unstretched configuration of a deployed bag, any distortions in the initial geometry of the folded bag will have no effect on the final geometry of the inflated bag. This is because the stresses depend only on the deformation gradient matrix:

(7-173)

where the choice of may coincide with the folded or unfold configurations. It is this unfolded

configuration which may be specified here. When the reference geometry is used, then the no-compression option should be active. With the reference geometry, it is possible to shrink the airbag and then perform the inflation. Although the elements in the shrunken bag are very small, the time step can be based on the reference geometry so a very reasonable time step size is obtained. The reference geometry based time step size is optional in the input.

The parameters fabric leakage coefficient, FLC, fabric area coefficient, FAC, and effective leakage area, ELA, for the fabric in contact with the structure are optional for the Wang-Nefske and hybrid inflation models. It is possible for the airbag to be constructed of multiple fabrics having different values of porosity and permeability. The gas leakage through the airbag fabric then requires an accurate determination of the areas by part ID available for leakage. The leakage area may change over time due to stretching of the airbag fabric or blockage when the outer surface of the bag is in contact with the structure. MD Nastran SOL 700 can check the interaction of the bag with the structure and split the areas into regions that are blocked and unblocked depending on whether the regions are in contact or not, respectively. Typically, the parameters, FLC and FAC, must be determined experimentally and their variation with time and pressure are optional inputs that allow for maximum modeling flexibility.

Material Model 36: Barlat’s 3-Parameter Plasticity ModelThis model was developed by Barlat and Lian [1989] for modeling sheets under plane stress conditions. The anisotopic yield criterion for plane stress is defined as:

(7-174)

Fij

∂xi

∂Xj--------=

Xj

Φ

Φ a K1 K2+ ∗ a K1 K2– ∗ c 2Km

+ + 2σYm

= =


where is the yield stress and are given by:

(7-175)

The anisotropic material constants a, c, h, and p are obtained through , and :

(7-176)

The anisotropy parameter is calculated implicitly. According to Barlat and Lian the value, width to thickness strain ratio, for any angle can be calculated from:

(7-177)

where is the uniaxial tension in the direction. This expression can be used to iteratively calculate

the value of . Let and define a function as

(7-178)

An iterative search is used to find the value of .

For face-centered-cubic (FCC) materials m=8 is recommended and for body-centered-cubic (BCC) materials m=6 may be used. The yield strength of the material can be expressed in terms of and :

(7-179)

where is the elastic strain to yield and is the effective plastic strain (logarithmic). If is set to

zero, the strain to yield if found by solving for the intersection of the linearly elastic loading equation with the strain hardening equation:

(7-180)

which gives the elastic strain at yield as:

σY Ki 1 2,=

K1

σx hσy–

2---------------------=

K2

σx hσy–

2---------------------⎝ ⎠⎛ ⎞

2

p2τxy2+=

R00 R45, R90

a 2 2R00

1 R00+------------------

1 R90+

R90------------------–=

c 2 a–=

hR00

1 R00+------------------

1 R90+

R90------------------=

p R

Φ

Rφ2mσY

m

∂Φ∂σx--------- ∂Φ

∂σy---------+⎝ ⎠

⎛ ⎞ σφ

-------------------------------------- 1–=

σφ Φ

p Φ 45= g

g p( )2mσY

m

∂Φ∂σx--------- ∂Φ

∂σy---------+⎝ ⎠

⎛ ⎞ σφ

-------------------------------------- 1– R45–=

p

k n

σy kεn k εyp εp+( )n= =

εyp εp σY

σ Eε=

σ kεn=


278

(7-181)

If yield is nonzero and greater than 0.02 then:

(7-182)

Material Type 37: Transversely Anisotropic Elastic-PlasticThis fully iterative plasticity model is available only for shell elements. The input parameters for this model are: Young’s modulus ; Poisson’s ratio ; the yield stress; the tangent modulus ; and the

anisotropic hardening parameter .

Consider Cartesian reference axes which are parallel to the three symmetry planes of anisotropic behavior. Then the yield function suggested by Hill [1948] can be written

(7-183)

where , and are the tensile yield stresses and , and are the shear yield stresses.

The constants , and are related to the yield stress by

. (7-184)

The isotropic case of von Mises plasticity can be recovered by setting

and .

εypEk---⎝ ⎠⎛ ⎞

1n 1–------------

=

σY

εyp

σy

k-----⎝ ⎠⎛ ⎞

1n---

=

E υ Et

R

F σ22 σ33–( )2G σ33 σ11–( )2

H σ11 σ22–( )22Lσ23

2 2Mσ312 2Nσ12

2 1–+ + + + + 0=

σy1 σy2, σy3 σ12 σ23, σy31

F G H L M, , , , N

2L1

σ232

--------=

2M1

σy312

-----------=

2N1

σy122

-----------=

2F1

σy22

-------- 1

σy32

-------- 1

σy12

--------–+=

2G1

σy32

-------- 1

σy12

-------- 1

σy22

--------–+=

2H1

σy12

-------- 1

σy22

-------- 1

σy32

--------–+=

F G H1

2σy2

----------= = =

L M N1

2σy2

----------= = =


For the particular case of transverse anisotropy, where properties do not vary in the plane, the

following relations hold:

(7-185)

where it has been assumed that .

Letting , the yield criterion can be written

, (7-186)

where

The rate of plastic strain is assumed to be normal to the yield surface so is found from

. (7-187)

Now consider the case of plane stress, where . Also, define the anisotropy input parameter as

the ratio of the in-plane plastic strain rate to the out-of-plane plastic strain rate:

. (7-188)

It then follows that

. (7-189)

Using the plane stress assumption and the definition of , the yield function may now be written

(7-190)

x1 x2–

2F 2G1

σy32

--------= =

2H2

σy2

------ 1

σy32

--------–=

N2

σy2

------ 1

2---–

1

σy32

--------–=

σy1 σy2 σy= =

Kσy

σy3--------=

F σ( ) σe σy= =

F σ( ) σ11 σ222

K2σ33

2K

2σ33 σ11 σ22+( )– 2 K2

–( )σ11σ22– 2Lσy2 σ23

2 σ312

+( ) 2 212---K

2–⎝ ⎠

⎛ ⎞ σ122

+ + ++1 2⁄

≡

ε· yp

ε· yp

λ ∂F∂σi j----------=

σ33 0= R

Rε· 22

p

ε· 33p

-------=

R2

K2------ 1–=

R

F σ( ) σ112 σ22

2 2RR 1+-------------σ11σ22– 2

2R 1+R 1+

----------------- σ122+ +

1 2⁄=


280

Material Type 38: Blatz-Ko Compressible Foam

(7-191)

where is the shear modulus and , and are the strain invariants. Blatz and Ko [1962] suggested

this form for a 47 percent volume polyurethane foam rubber with a Poisson’s ratio of 0.25. The second Piola-Kirchhoff stresses are given as

(7-192)

where

after determining , it is transformed into the Cauchy stress tensor:

where and are the initial and current density, respectively.

Material Model 39: Transversely Anisotropic Elastic-Plastic With FLDSee Material Model 37 for the similar model theoretical basis. The first history variable is the maximum strain ratio defined by:

(7-193)

corresponding to . This history variable replaces the effective plastic strain in the output. Plastic strains can still be obtained but one additional history variable must be written into the D3PLOT database.

The strains on which these calculations are based are integrated in time from the strain rates:

(7-194)

and are stored as history variables. The resulting strain measure is logarithmic.

W I1 I2 I3, ,( ) μ2---

I2

I3---- 2 I3 5–+⎝ ⎠⎛ ⎞=

μ I1 I2, I3

Sij μ Iδi j Gij–( ) 1I3---- I3

I2

I3----–⎝ ⎠

⎛ ⎞ Gij+=

Gij

∂xk

∂Xi

--------∂xk

∂Xj

--------=

Gij∂Xi

∂xk--------

∂Xj

∂xk--------=

Sij

σi jρρo-----

∂xi

∂Xk---------

∂xj

∂Xl--------Skl=

ρo ρ

εmajorworkpiece

εmajorfld

--------------------------------

min workpieceorε

εi jn 1+ εi j

n ε∇n

12---+Δ t

n12---+

+=


Figure 7-7 Flow Limit Diagram

Material Model 53: Low Density Closed Cell Polyurethane FoamA rigid, low density, closed cell, polyurethane foam model developed at Sandia Laboratories [Neilsen et. al., 1987] has been recently implemented for modeling impact limiters in automotive applications. A number of such foams were tested at Sandia and reasonable fits to the experimental data were obtained.

In some respects this model is similar to the crushable honeycomb model type 26 in that the components of the stress tensor are uncoupled until full volumetric compaction is achieved. However, unlike the honeycomb model this material possesses no directionality but includes the effects of confined air pressure in its overall response characteristics.

(7-195)

where is the skeletal stress and is the air pressure computed from the equation:

(7-196)

-50 -40 -30 -20 -10 0 +10 +20 +30 +40 +50

% MINOR STRAIN

10

20

30

40

50

εmjr

εmnr 0=

Plane Strain

εmjr

εmjr εmnrεmnr

StretchDraw

% M

AJO

R S

TR

AIN

60

70

80

σi j σi jsk δijσ

air–=

σi jsk σair

σairp0γ

1 γ φ–+----------------------–=


282

where is the initial foam pressure usually taken as the atmospheric pressure and defines the

volumetric strain

(7-197)

where is the relative volume and is the initial volumetric strain which is typically zero. The yield

condition is applied to the principal skeletal stresses which are updated independently of the air pressure. We first obtain the skeletal stresses:

(7-198)

and compute the trial stress,

(7-199)

where is Young’s modulus. Since Poisson’s ratio is zero, the update of each stress component is uncoupled and where is the shear modulus. The yield condition is applied to the principal

skeletal stresses such that if the magnitude of a principal trial stress component, , exceeds the yield

stress, , then

. (7-200)

The yield stress is defined by

(7-201)

where , and are user defined input constants. After scaling the principal stresses they are transformed back into the global system and the final stress state is computed

.

Material Models 54 and 55: Enhanced Composite Damage ModelThese models are very close in their formulations. Material 54 uses the Chang matrix failure criterion (as Material 22), and Material 55 uses the Tsai-Wu criterion for matrix failure.

Arbitrary orthothropic materials, e.g., unidirectional layers in composite shell structures can be defined. Optionally, various types of failure can be specified following either the suggestions of [Chang and Chang, 1984] or [Tsai and Wu, 1981]. In addition special measures are taken for failure under compression. See [Matzenmiller and Schweizerhof, 1990]. This model is only valid for thin shell elements.

The Chang/Chang criteria is given as follows:

p0 γ

γ V 1– γ0+=

V γ0

σi jsk σi j σi jσair+=

σi jskt

σi jskt σ ij

skEε· ijΔ t+=

E

2G E= G

σiskt

σy

σisk min σy σi

sk,( )σi

skt

σiskt

-------------=

σy a b 1 cγ+( )+=

a b, c

σi j σi jsk δijσair–=


for the tensile fiber mode,

then , (7-202)

for the compressive fiber mode,

then , (7-203)

for the tensile matrix mode,

then , (7-204)

and for the compressive matrix mode,

then (7-205)

In the Tsai/Wu criteria the tensile and compressive fiber modes are treated as in the Chang/Chang criteria. The failure criterion for the tensile and compressive matrix mode is given as:

(7-206)

For , we get the original criterion of Hashin [1980] in the tensile fiber mode.

For , we get the maximum stress criterion which is found to compare better to experiments.

Failure can occur in any of four different ways:

1. If DFAILT is zero, failure occurs if the Chang/Chang failure criterion is satisfied in the tensile fiber mode.

2. If DFAILT is greater than zero, failure occurs if the tensile fiber strain is greater than DFAILT or less than DFAILC.

3. If EFS is greater than zero, failure occurs if the effective strain is greater than EFS.

4. If TFAIL is greater than zero, failure occurs according to the element time step as described in the definition of TFAIL above.

σaa 0> ef2 σaa

Xt--------⎝ ⎠⎛ ⎞

2β

σab

Sc--------⎝ ⎠⎛ ⎞ 1–+=

> 0 failed< 0 elastic

Ea Eb Gab νba νab 0= = = = =

σaa 0> ec2 σaa

Xc--------⎝ ⎠⎛ ⎞

21–=


Ea νba νab 0= = =

σbb 0> em2 σbb

Yt

--------⎝ ⎠⎛ ⎞

2 σab

Sc

--------⎝ ⎠⎛ ⎞

21–+=


Eb νba 0. Gab→ 0= = =

σbb 0< ed2 σbb

2Sc--------⎝ ⎠⎛ ⎞

2 Yc

2St--------⎝ ⎠⎛ ⎞

21–

σbb

Yc--------

σab

Sc--------⎝ ⎠⎛ ⎞

21–+ +=


Eb νba νab 0. Gab→ 0= = = =

Xc 2Yc= for 50% fiber volume

emd2 σbb

2

YcYt-----------

σab

Sc--------⎝ ⎠⎛ ⎞

2 Yc Yt–( )σbb

YcYt------------------------------- 1–+ +=


β 1=

β 0=


284

When failure has occurred in all the composite layers (through-thickness integration points), the element is deleted. Elements which share nodes with the deleted element become “crashfront” elements and can have their strengths reduced by using the SOFT parameter with TFAIL greater than zero.

Information about the status in each layer (integration point) and element can be plotted using additional integration point variables. The number of additional integration point variables for shells written to the database is input by the PARAM, DYNEIPS definition as variable NEIPS. For Models 54 and 55 these additional variables are tabulated below (i = shell integration point):

These variables can be plotted in MD Patran as element components 81, 82, ..., 80+ NEIPS. The following components, defined by the sum of failure indicators over all through-thickness integration points, are stored as element component 7 instead of the effective plastic strain:

Material Model 57: Low Density Urethane FoamThe urethane foam model is available to model highly compressible foams such as those used in seat cushions and as padding on the Side Impact Dummy (SID). The compressive behavior is illustrated in Figure 7-8 where hysteresis on unloading is shown. This behavior under uniaxial loading is assumed not to significantly couple in the transverse directions. In tension the material behaves in a linear fashion until tearing occurs. Although our implementation may be somewhat unusual, it was first motivated by

History Variable Description Value

d3plot Component

1. ef(i) tensile fiber mode 81

2. ec(i) compressive fiber mode 1 - elastic 82

3. em(i) tensile matrix mode 0 - failed 83

4. ed(i) compressive matrix mode 84

5. efail max[ef(ip)] 85

6. dam damage parameter-1 - element intact

10-8 - element in crashfront+1 - element failed

86

Description Integration Point

1

2

3

1nip-------- ef i( )

i 1=

nip

∑

1nip-------- ec i( )

i 1=

nip

∑

1nip-------- cm i( )

i 1=

nip

∑


Shkolnikov [1991] and a paper by Storakers [1986]. The recent additions necessary to model hysteretic unloading and rate effects are due to Chang, et. al., [1994]. These latter additions have greatly expanded the usefulness of this model.

Figure 7-8 Behavior of the Low-density Urethane Foam Model

The model uses tabulated input data for the loading curve where the nominal stresses are defined as a function of the elongations, , which are defined in terms of the principal stretches, , as:

(7-207)

The stretch ratios are found by solving for the eigenvalues of the left stretch tensor, , which is obtained

via a polar decomposition of the deformation gradient matrix, :

(7-208)

The update of follows the numerically stable approach of Taylor and Flanagan [1989]. After solving

for the principal stretches, the elongations are computed and, if the elongations are compressive, the corresponding values of the nominal stresses, , are interpolated. If the elongations are tensile, the

nominal stresses are given by

(7-209)

The Cauchy stresses in the principal system become

(7-210)

The stresses are then transformed back into the global system for the nodal force calculations.

Typical unloading curves determinedby the hysteric unloading factor. Withthe shape factor equal to unity.

Unloading

Strain

σ

Strain

Typical unloading for a large shapefactor; e.g., 5.-8 and a small hystericfactor; e.g., .010.

σ

Curves

εi λi

εi λi 1–=

Vij

Fij

Fij RikUkj VikRkj= =

Vij

τi

τi Eεi=

σi

τi

λiλk----------=


286

When hysteretic unloading is used, the reloading will follow the unloading curve if the decay constant, , is set to zero. If is nonzero the decay to the original loading curve is governed by the expression:

(7-211)

The bulk viscosity, which generates a rate dependent pressure, may cause an unexpected volumetric response and, consequently, it is optional with this model.

Rate effects are accounted for through linear viscoelasticity by a convolution integral of the form

(7-212)

where is the relaxation function. The stress tensor, , augments the stresses determined from

the foam, ; consequently, the final stress, , is taken as the summation of the two contributions:

(7-213)

Since we wish to include only simple rate effects, the relaxation function is represented by one term from the Prony series:

(7-214)

given by,

(7-215)

This model is effectively a Maxwell fluid which consists of a damper and spring in series. We characterize this in the input by a Young's modulus, , and decay constant, . The formulation is

performed in the local system of principal stretches where only the principal values of stress are computed and triaxial coupling is avoided. Consequently, the one-dimensional nature of this foam material is unaffected by this addition of rate effects. The addition of rate effects necessitates twelve additional history variables per integration point. The cost and memory overhead of this model comes primarily from the need to “remember” the local system of principal stretches.

Material Type 58: Laminated Composite FabricParameters to control failure of an element layer are: ERODS, the maximum effective strain; i.e., maximum 1 = 100 % straining. The layer in the element is completely removed after the maximum effective strain (compression/tension including shear) is reached. The stress limits are factors used to limit the stress in the softening part to a given value,

, (7-216)

β β

1 e β t––

σi j gijkl t τ–( )∂εkl

∂τ----------

0

t∫=

gijkl t τ–( ) σi jr

σ i jf σi j

σi j σi jf σi j

r+=

g t( ) α0 αme β t–

m 1=

N

∑+=

g t( ) Edeβ1t–

=

Ed β1

σmin SLIMxx s trength⋅=


thus, the damage value is slightly modified such that elastoplastic like behavior is achieved with the threshold stress. As a factor for a number between 0.0 and 1.0 is possible. With a factor of 1.0, the stress remains at a maximum value identical to the strength, which is similar to ideal elastoplastic behavior. For tensile failure a small value for is often reasonable; however, for compression

is preferred. This is also valid for the corresponding shear value. If is smaller than 1.0, then localization can be observed depending on the total behavior of the lay-up. If the user is intentionally using , it is generally recommended to avoid a drop to zero and set the value to something in between 0.05 and 0.10.

Then elastoplastic behavior is achieved in the limit which often leads to less numerical problems. Defaults for .

The crashfront-algorithm is started if and only if a value for TSIZE (time step size, with element elimination after the actual time step becomes smaller than TSIZE) is input

The damage parameters can be written to the postprocessing database for each integration point as the first three additional element variables and can be visualized.

Material models with FS=1 or FS=-1 are favorable for complete laminates and fabrics, as all directions are treated in a similar fashion.

For material model FS=1, an interaction between normal stresses and shear stresses is assumed for the evolution of damage in the a- and b-directions. For the shear damage is always the maximum value of the damage from the criterion in a- or b- direction is taken.

For material model FS=-1, it is assumed that the damage evolution is independent of any of the other stresses. A coupling is present only via the elastic material parameters and the complete structure.

In tensile and compression directions and in a- as well as in b- direction, different failure surfaces can be assumed. The damage values, however, increase only when the loading direction changes.

Special Control of Shear Behavior of Fabrics

For fabric materials a nonlinear stress strain curve for the shear part of failure surface FS=-1 can be assumed as given below. This is not possible for other values of FS. The curve, shown in Figure 19.58.1, is defined by three points:

a. a) the origin (0,0) is assumed,

b. the limit of the first slightly nonlinear part (must be input), stress (TAU1) and strain (GAMMA1), see below.

c. the shear strength at failure and shear strain at failure.

In addition, a stress limiter can be used to keep the stress constant via the SLIMS parameter. This value must be less than or equal to 1.0 and positive, which leads to an elastoplastic behavior for the shear part. The default is 1.0E-08, assuming almost brittle failure once the strength limit SC is reached.

SLIMxx

SLIMTx

SLIMCx 1.0= SLIMxx

SLIMxx 1.0<

SLIMxx 1.0E-8=


288

Figure 7-9 Stress-strain Diagram for Shear

Material Type 62: Viscous FoamThis model was written to represent the energy absorbing foam found on certain crash dummies, i.e., the ‘Confor Foam’ covering the ribs of the Eurosid dummy.

The model consists of a nonlinear elastic stiffness in parallel with a viscous damper. A schematic is shown in Figure 7-10. The elastic stiffness is intended to limit total crush while the viscous damper absorbs energy. The stiffness prevents timestep problems.

Figure 7-10 Schematic of Material Model 62

Both and are nonlinear with crush as follows:

(7-217)

where is the relative volume defined by the ratio of the current to initial volume. Typical values are (units of )

SLIMS*SC

GMSGAMMA1

TAU1

SC

τ

γ

E2

E1

V2 E2

E1 V2

E1t E1 V

n1–( )=

V2t V2 abs 1 V–( )( )

n2=

V

N mm s,,


Material Type 63: Crushable FoamThe intent of this model is model crushable foams in side impact and other applications where cyclic behavior is unimportant.

This isotropic foam model crushes one-dimensionally with a Poisson’s ratio that is essentially zero. The stress versus strain behavior is depicted in Figure 7-11 where an example of unloading from point a to the tension cutoff stress at b then unloading to point c and finally reloading to point d is shown. At point the reloading will continue along the loading curve. It is important to use nonzero values for the tension cutoff to prevent the disintegration of the material under small tensile loads. For high values of tension cutoff the behavior of the material will be similar in tension and compression.

In the implementation we assume that Young’s modulus is constant and update the stress assuming elastic behavior.

(7-218)

The magnitudes of the principal values, , are then checked to see if the yield stress, , is

exceeded and if so they are scaled back to the yield surface:

if then (7-219)

After the principal values are scaled, the stress tensor is transformed back into the global system. As seen in Figure 7-11, the yield stress is a function of the natural logarithm of the relative volume, ; i.e., the volumetric strain.

E1 0.0036=

n1 4.0=

V2 0.0015=

E2 100.0=

n2 0.2=

ν 0.05=

σi jt rial σi j

n Eε· ijn 1 2⁄+ Δ tn 1 2⁄++=

σit rial i, 1 3,= σy

σy σit rial< σi

n 1+ σy

σit rial

σit rial

-----------------=

V


290

Figure 7-11 Yield Stress Versus Volumetric Strain Curve for the Crushable Foam

Material Model 64: Strain Rate Sensitive Power-Law PlasticityThis material model follows a constitutive relationship of the form:

(7-220)

where is the yield stress, is the effective plastic strain, is the effective plastic strain rate, and the constants, , , and can be expressed as functions of effective plastic strain or can be constant with respect to the plastic strain. The case of no strain hardening can be obtained by setting the exponent of the plastic strain equal to a very small positive value; i.e., 0.0001.

This model can be combined with the superplastic forming input to control the magnitude of the pressure in the pressure boundary conditions in order to limit the effective plastic strain rate so that it does not exceed a maximum value at any integration point within the model.

A fully viscoplastic formulation is optional. An additional cost is incurred but the improvement is results can be dramatic.

Material Model 65: Modified Zerilli/ArmstrongThe Armstrong-Zerilli Material Model expresses the flow stress as follows.

For fcc metals,

(7-221)

effective plastic strain

σi j

d

a

E

bc

Volumetric Strain Vln–

σ kεmε· n=

σ ε ε·

k m n

σ C1 C2 εp( )1 2⁄ eC3– C4 ε· *( )ln+( )τ

C3+⎩ ⎭⎨ ⎬⎧ ⎫ μ T( )

μ 293( )------------------⎝ ⎠⎛ ⎞+=

εp =


effective plastic strain rate where , 1e-3, le-6 for time units

of seconds, milliseconds, and microseconds, respectively.

For bcc metals,

(7-222)

where

(7-223)

The relationship between heat capacity (specific heat) and temperature may be characterized by a cubic polynomial equation as follows:

(7-224)

A fully viscoplastic formulation is optional. An additional cost is incurred but the improvement in results can be dramatic.

Material Model 67: Nonlinear Stiffness/Viscous 3-D Discrete BeamThe formulation of the discrete beam (Type 6) assumes that the beam is of zero ngth and requires no orientation node. A small distance between the nodes joined by the beam is permitted. The local coordinate system which determines (r, s, t) is given by the coordinate ID in the cross-sectional input where the global system is the default. The local coordinate system axes rotate with the average of the rotations of the two nodes that define the beam.

For null TABLED1 IDs, no forces are computed. The force resultants are found from TABLED1s (see Figure 7-12) that are defined in terms of the force resultant versus the relative displacement in the local coordinate system for the discrete beam. The resultant forces and moments are determined by a table lookup, if the origin of the TABLED1 is at [0,0], as shown in Figure 7-12b, and the tension and compression responses are symmetric.

ε· * ε·

ε· 0

-----= ε· 0 1=

σ C1 C2eC3– c4lm ε· *( )+( )τ

C5 εp( )n C6+[ ] μ T( )μ 293( )------------------⎝ ⎠⎛ ⎞+ +=

μ T( )μ 293( )------------------ B1 B2T B3T2+ +=

Cp G1 G2T G3T2 G4T3+ + +=


292

Figure 7-12 Resultant Forces and Moments Determined by Table Lookup

Material Model 68: Nonlinear Plastic/Linear Viscous 3-D Discrete Beam The folmulation of the discrete beam (Type 6) assumes that the beam is of zero length and requires no orientation node. A small distance between the nodes joined by the beam is permitted. The local coordinate system, which determines (r, s, t) is given by the coordinate ID in the cross-sectional input where the global system is the default. The local coordinate system axes rotate with the average of the rotations of the two nodes that define the beam. Each force resultant in the local system can have a limiting value defined as a function of plastic displacement by using a TABLED1 (see Figure 7-13). For the degrees of freedom where elastic behavior is desired, the TABLED1 ID is simply set to zero.

Figure 7-13 Resultant Forces and Moments Limited by the Yield Definition

DISPLACEMENT

RESULTANT

RESULTANT

DISPLACEMENT(a) (b)

RESULTANT

PLASTIC DISPLACEMENT


Catastrophic failure, based on force resultants, occurs if the following inequality is satisfied:

(7-225)

Likewise, catastrohic failure based on displacement resultants occurs if:

(7-226)

After failure, the discrete element is deleted. If failure is included, either one or both of the criteria may be used.

Material Model 69: Side Impact Dummy Damper (SID Damper)The side impact dummy uses a damper that is not adequately treated by nonlinear force versus relative velocity curves, since the force characteristics are also dependent on the displacement of the piston. As the damper moves, the fluid flows through the open orifices to provide the necessary damping resistance. While moving as shown in Figure 7-14, the piston gradually blocks off and effectively closes the orifices. The number of orifices and the size of their openings control the damper resistance and performance. The damping force is computed from the equation.

(7-227)

where is a user defined constant or a tabulated function of the absolute value of the relative velocity,

is the piston's relative velocity, is the discharge coefficient, is the piston area, is the total

open areas of orifices at time , is the fluid density, is the coefficient for the linear term, and

is the coefficient for the quadratic term.

In the implementation, the orifices are assumed to be circular with partial covering by the orifice controller. As the piston closes, the closure of the orifice is gradual. This gradual closure is taken into account to insure a smooth response. If the piston stroke is exceeded, the stiffness value, , limits further movement; i.e., if the damper bottoms out in tension or compression, the damper forces are calculated by replacing the damper by a bottoming out spring and damper, and , respectively. The piston stroke must exceed the initial length of the beam element. The time step calculation is based in part on the stiffness value of the bottoming out spring. A typical force versus displacement curve at constant relative velocity with only the linear velocity term active is shown in Figure 7-15. The factor, , which scales the force defaults to 1.0 and is analogous to the adjusting ring on the damper.

Fr

Frfail

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Fs

Fsfai l

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Ft

Ftfail

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Mr

Mrfail

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Ms

Msfai l

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Mt

Mtfai l

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1 0≥–+ + + + +

ur

urfai l

----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 us

usfai l

----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 ut

utfai l

----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θr

θrfail

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θs

θsfai l

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θt

θtfail

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1 0≥–+ + + + +

F SF KApVp

Cl

A0t

------ C2 Vp ρfluid

Ap

CA0t

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1–+⎩ ⎭⎨ ⎬⎧ ⎫

f s s0+( )– Vpg s s0+( ) }+⎩⎨⎧

=

K

Vp C Ap A0t

t ρfluid C1 C2

k

k c

SF


294

Figure 7-14 Mathematical Model for Side Impact Dummy Damper

Figure 7-15 Force Versus Displacement

FORCE

Displacement

Force increases as orficeis gradually covered

Last orficecloses

Linear loading afterofifices close


Material Model 70: Hydraulic/Gas DamperThis special purpose element represents a combined hydraulic and gas-filled damper which has a variable orifice coefficient. A schematic of the damper is shown in Figure 7-16. Dampers of this type are sometimes used on buffers at the end of railroad tracks and as aircraft undercarriage shock absorbers. This material can be used only as a discrete beam element.

Figure 7-16 Schematic of Hydraulic/Gas Damper

As the damper is compressed two actions contribute to the force that develops. First, the gas is adiabatically compressed into a smaller volume. Secondly, oil is forced through an orifice. A profiled pin may occupy some of the cross-sectional area of the orifice; thus, the orifice area available for the oil varies with the stroke. The force is assumed proportional to the square of the velocity and inversely proportional to the available area. The equation for this element is:

(7-228)

where is the element deflection and is the relative velocity across the element.

Material Model 71: CableThis material can be used only as a discrete beam element. The force, , generated by the cable is nonzero only if the cable is in tension. The force is given by:

(7-229)

where is the change in length

(7-230)

and the stiffness is defined as:

(7-231)

F SCLF KhVa0-----⎝ ⎠⎛ ⎞ 2

P0

C0

C0 S–----------------⎝ ⎠⎛ ⎞

nPa– Ap⋅+

⎩ ⎭⎨ ⎬⎧ ⎫

⋅=

S V

F

F K max ΔL 0.,( )⋅=

ΔL

ΔL current length ini tia l length of fset–( )–=

KE area⋅

in it ial length offset–( )-----------------------------------------------------------------=


296

The area and offset are defined on either the cross section or element cards in the SOL 700 input. For a slack cable the offset should be input as a negative length. For an initial tensile force the offset should be positive. If a TABLED1 is specified, the Young’s modulus will be ignored and the TABLED1 will be used instead. The points on the TABLED1 are defined as engineering stress versus engineering strain; i.e., the change in length over the initial length. The unloading behavior follows the loading.

Material Model 73: Low Density Viscoelastic FoamThis viscoelastic foam model is available to model highly compressible viscous foams. The hyperelastic formulation of this model follows that of material 57.

Rate effects are accounted for through linear viscoelasticity by a convolution integral of the form

(7-232)

where is the relaxation function. The stress tensor, , augments the stresses determined from

the foam, ; consequently, the final stress, , is taken as the summation of the two contributions:

(7-233)

Since we wish to include only simple rate effects, the relaxation function is represented by up to six terms of the Prony series:

(7-234)

This model is effectively a Maxwell fluid which consists of a dampers and springs in series. The formulation is performed in the local system of principal stretches where only the principal values of stress are computed and triaxial coupling is avoided. Consequently, the one-dimensional nature of this foam material is unaffected by this addition of rate effects. The addition of rate effects necessitates 42 additional history variables per integration point. The cost and memory overhead of this model comes primarily from the need to “remember” the local system of principal stretches and the evaluation of the viscous stress components

Material Model 74: Elastic Spring for the Discrete BeamThis model permits elastic springs with damping to be combined and represented with a discrete beam element type 6. Linear stiffness and damping coefficients can be defined and, for nonlinear behavior, a force versus deflection and force versus rate curves can be used. Displacement based failure and an initial force are optional

If the linear spring stiffness is used, the force, , is given by:

(7-235)

σi jr gijkl t τ–( )

∂εkl

∂τ---------- τd

0

t∫=

gijkl t τ–( ) σ i jr

σ i jf σi j

σi j σi jf σi j

r+=


m 1=

N

∑+=

F

F F0 FΔL DΔL+ +=


where is the stiffness constant, and is the viscous damping coefficient.

If the TABLED1 ID for . is specified, nonlinear behavior is activated. For this case the force is given by:

(7-236)

where and are damping coefficients for nonlinear behavior, is a factor to scale time units, and . is an optional TABLED1 defining a scale factor versus deflection for TABLED1 ID, .

In these equations, is the change in length

.

Failure can occur in either compression or tension based on displacement values of CDF and TDF, respectively. After failure no forces are carried. Compressive failure does not apply if the spring is initially zero length.

The cross sectional area is defined on the beam property card for the discrete beam elements. The square root of this area is used as the contact thickness offset if these elements are included in the contact treatment.

Material Model 76: General ViscoelasticRate effects are taken into account through linear viscoelasticity by a convolution integral of the form:

(7-237)

where is the relaxation function.

If we wish to include only simple rate effects for the deviatoric stresses, the relaxation function is represented by six terms from the Prony series:

(7-238)

We characterize this in the input by shear modulii, , and decay constants, . An arbitrary number of

terms, up to 6, may be used when applying the viscoelastic model.

K D

f ΔL( )

F F0 Kf ΔL( ) 1 C1 ΔL· C2 ΔL·( ) max 1.ΔL·

DLE------------,

⎩ ⎭⎨ ⎬⎧ ⎫

⎝ ⎠⎜ ⎟⎛ ⎞

lnsgn⋅+⋅++=

+DΔL· g ΔL( )h ΔL·( )+

C1 C2 DLE

g ΔL( ) h ΔL dt⁄( )

ΔL

ΔL currentlength initiallength–=


∂τ---------- τd

0

t∫=

gijkl t τ–( )

g t( ) Gmeβmt–

m 1=

N

∑=

Gi βi


298

Figure 7-17 Relaxation Curve

For volumetric relaxation, the relaxation function is also represented by the Prony series in terms of bulk modulii:

(7-239)

Material Model 77: Hyperviscoelastic RubberRubber is generally considered to be fully incompressible since the bulk modulus greatly exceeds the shear modulus in magnitude. To model the rubber as an unconstrained material a hydrostatic work term,

, is included in the strain energy functional which is function of the relative volume, ,

[Ogden, 1984]:

Note: This curve defines stress versus time where time is defined on a logarithmic scale. For best results, the points defined in the TABLED1 should be equally spaced on the logarithmic scale. Furthermore, the TABLED1 should be smooth and defined in the positive quadrant. If nonphysical values are determined by least squares fit, SOL 700 will terminate with an error message after the initialization phase is completed. If the ramp time for loading is included, then the relaxation which occurs during the loading phase is taken into account. This effect may or may not be important.

k t( ) Kmeβkm

t–

m 1=

N

∑=

WH J( ) J( )


(7-240)

In order to prevent volumetric work from contributing to the hydrostatic work the first and second invariants are modified as shown. This procedure is described in more detail by Sussman and Bathe [1987]. For the Ogden model the energy equation is given as:

(7-241)

where the asterisk indicates that the volumetric effects have be eliminated from the principal

stretches, . See Ogden [1984] for more details.

Rate effects are taken into account through linear viscoelasticity by a convolution integral of the form:

(7-242)

or in terms of the second Piola-Kirchhoff stress, , and Green's strain tensor, ,

(7-243)

where and are the relaxation functions for the different stress measures. This stress

is added to the stress tensor determined from the strain energy functional.

If we wish to include only simple rate effects, the relaxation function is represented by six terms from the Prony series:

(7-244)

given by,

(7-245)

This model is effectively a Maxwell fluid which consists of a dampers and springs in series. We characterize this in the input by shear moduli, , and decay constants, . The viscoelastic behavior is

optional and an arbitrary number of terms may be used.

The Mooney-Rivlin rubber model is obtained by specifying . In spite of the differences in formulations with Model 27, we find that the results obtained with this model are nearly identical with those of Model 27 as long as large values of Poisson’s ratio are used.

W J1 J2 J,,( ) Cpq J1 3–( )p J2 3–( )q WH J( )+

p q, 0=

n

∑=

J1 I1I31 2⁄–=

J2 I2I31 2⁄–=

W∗μj

αj----- λi

*α1 1–( ) 12---K J 1–( )2+

j 1=

n

∑i 1=

3

∑=

*( )

λj*


∂τ---------- τd

0

t

∫=

Sij Eij

Si j Gijkl t τ–( )∂Ekl

∂τ----------- τd

0

t∫

gijkl t τ–( ) Gijkl t τ–( )


m 1=

N

∑+=

g t( ) Gieβi t–

j 1=

n

∑=

Gi βi

n 2+


300

When viscoelastic terms are not included, this model is similar to the use of the Ogden model in solution 600, defined on the MATHE option.

Material Model 78: Soil/ConcreteConcretePressure is positive in compression. Volumetric strain is defined as the natural log of the relative volume and is positive in compression where the relative volume, , is the ratio of the current volume to the initial volume. The tabulated data should be given in order of increasing compression. If the pressure drops below the cutoff value specified, it is reset to that value and the deviatoric stress state is eliminated.

If the TABLED1 ID is provided as a positive number, the deviatoric perfectly plastic pressure dependent yield function , is described in terms of the second invariant, , the pressure, , and the table, , as

where is defined in terms of the deviatoric stress tensor as:

assuming that. If the ID is given as negative, the yield function becomes:

being the deviatoric stress tensor.

If cracking is invoked, the yield stress is multiplied by a factor which reduces with plastic stain according to a trilinear law as shown in Figure 7-18.

Figure 7-18 Strength Reduction Factor

V

φ J2 p F p( )

φ 3J2 F p( )– σy F p( )–=

J2

J212---SijSi j=

φ J2 Fp–=

f

f

1.0

b

ε1 ε2 εp


and are tabulated functions of pressure that are defined by TABLED1s (see Figure 7-19). The values

on the curves are pressure versus strain and should be entered in order of increasing pressure. The strain values should always increase monotonically with pressure.

By properly defining the TABLED1s, it is possible to obtain the desired strength and ductility over a range of pressure. (see Figure 7-20).

Figure 7-19 Cracking Strain Versus Pressure

Figure 7-20 Strength and Ductility Over a Range of Pressures

= residual strength factor

= plastic stain at which cracking begins.

= plastic stain at which residual strength is reached.

b

ε1

ε2

ε1 ε2

ε

ε2

ε2

P

Yield Stress

Plastic Strain

p3

p2

p1


302

Material Model 79: Hysteretic SoilThis model is a nested surface model with five superposed “layers” of elasto-perfectly plastic material, each with its own elastic modulii and yield values. Nested surface models give hysteretic behavior, as the different “layers” yield at different stresses.

The constants ( , , ) govern the pressure sensitivity of the yield stress. Only the ratios between

these values are important - the absolute stress values are taken from the stressstrain curve.

The stress strain pairs ( , ) define a shear stress versus shear strain curve. The first point on the curve is assumed by default to be (0,0) and does not need to be entered. The slope of the curve must decrease with increasing . Not all five points need be to be defined. This curve applies at the reference pressure; at other pressures, the curve variesaccording to , , and as in the soil and

crushable foam model, Material 5.

The elastic moduli G and K are pressure sensitive.

where and are the input values, is the current pressure, the cut-off or referencepressure (must

be zero or negative). If attempts to fall below (i.e., more tensile) the shear stresses are set to zero

and the pressure is set to . Thus, the material has no stiffness orstrength in tension. The pressure in

compression is calculated as follows:

where is the relative volume; i.e., the ratio between the original and current volume.

Material Model 80: Ramberg-Osgood PlasticityThe Ramberg-Osgood equation is an empirical constitutive relation to represent the one-dimensional elastic-plastic behavior of many materials, including soils. This model allows a simple rate independent representation of the hysteretic energy dissipation observed in soils subjected to cyclic shear deformation. For monotonic loading, the stress-strain relationship is given by:

(7-246)

a0 a1 a2

γ1 τ1,( ) … γ5 τ5,( )

γ

a0 a1 a2

G G0 p p0–( )b=

K K0 p p0–( )b=

G0 K0 p po

p po

po

p K0 V( )ln–[ ]1

1 b–------------

=

V

γγy---- τ

τy---- α τ

τy----

r+= i f γ 0≥

γγy

---- ττy

---- α ττy----

r–= i f γ 0<


where is the shear strain and is the shear stress. The model approaches perfect plasticity as the stress exponent . These equations must be augmented to correctly model unloading and reloading

material behavior. The first load reversal is detected by . After the first reversal, the stress-strain relationship is modified to

(7-247)

where and represent the values of strain and stress at the point of load reversal. Subsequent load

reversals are detected by .

The Ramberg-Osgood equations are inherently one-dimensional and are assumed to apply to shear components. To generalize this theory to the multidimensional case, it is assumed that each component of the deviatoric stress and deviatoric tensorial strain is independently related by the one-dimensional stress-strain equations. A projection is used to map the result back into deviatoric stress space if required. The volumetric behavior is elastic, and, therefore, the pressure is found by

(7-248)

where is the volumetric strain.

Material Model 81 and 82: Plasticity with Damage and Orthotropic OptionWith this model an elasto-viscoplastic material with an arbitrary stress versus strain curve and arbitrary strain rate dependency can be defined. Damage is considered before rupture occurs. Also, failure based on a plastic strain or a minimum time step size can be defined.

An option in the keyword input, ORTHO, is available, which invokes an orthotropic damage model. This option, which is implemented only for shell elements with multiple integration points through thickness, is an extension to include orthotropic damage as a means of treating failure in aluminum panels. Directional damage begins after a defined failure strain is reached in tension and continues to evolve until a tensile rupture strain is reached in either one of the two orthogonal directions.

The stress versus strain behavior may be treated by a bilinear stress strain curve by defining the tangent modulus, ETAN. Alternately, a curve similar to that shown in Figure 7-21 is expected to be defined by (EPS1,ES1) - (EPS8,ES8); however, an effective stress versus effective plastic strain curve (LCSS) may be input instead if eight points are insufficient. The cost is roughly the same for either approach. The most general approach is to use the table definition (LCSS) discussed here.

γ τr ∞→

γγ· 0<

γ γ0–( )2γy

-------------------τ τ0–( )2τy

------------------- α τ τ0–( )2τy

-------------------r

+= i f γ 0≥

γ γ0–( )2γy

-------------------τ τ0–( )2τy

------------------- α τ τ0–( )2τy

-------------------r

–= i f γ 0<

γ0 τ0

( )0 0γ γ γ− <&

p

p Kεv=

εv


304

Figure 7-21 Stress Strain Behavior When Damage is Included.

Two options to account for strain rate effects are possible. Strain rate may be accounted for using the Cowper and Symonds model which scales the yield stress with the factor,

(7-249)

where is the strain rate, . If the viscoplastic option is active, VP=1.0, and if SIGY is > 0 then

the dynamic yield stress is computed from the sum of the static stress, , which is typically given

by a TABLED1 ID, and the initial yield stress, SIGY, multiplied by the Cowper-Symonds rate term as follows:

(7-250)

where the plastic strain rate is used. If SIGY=0, the following equation is used instead where the static

stress, , must be defined by a TABLED1:

(7-251)

This latter equation is always used if the viscoplastic option is off.

0Failure Begins

Rupture

Damage increases linearly withplastic strain after failure

Nominal stressafter failure

Yield stress versuseffective plastic strainfor undamaged material

σyield

ω 0=ω 1=

εeffp

1ε·

C----⎝ ⎠⎛ ⎞

1 p⁄+

ε· ε· εi jε·

ij=

σys εeff

p( )

σy εeffp ε· eff

p,( ) σy

s εef fp( ) SIGY

ε· ef fp

C--------⎝ ⎠⎛ ⎞

1 p⁄⋅+=

σys εeff

p( )

σy εeffp ε· ef f

p,( ) σy

s 1ε· ef f

p

C--------⎝ ⎠⎛ ⎞

1 p⁄+=


For complete generality a TABLED1 (LCSR) to scale the yield stress may be input instead. In this curve the scale factor versus strain rate is defined.

The constitutive properties for the damaged material are obtained from the undamaged material properties. The amount of damage evolved is represented by the constant, , which varies from zero if no damage has occurred to unity for complete rupture. For uniaxial loading, the nominal stress in the damaged material is given by:

(7-252)

where is the applied load and A is the surface area. The true stress is given by:

(7-253)

where is the void area. The damage variable can then be defined:

(7-254)

In this model damage is defined in terms of plastic strain after the failure strain is exceeded:

if (7-255)

After exceeding the failure strain softening begins and continues until the rupture strain is reached.

By default, deletion of the element occurs when all integration points in the shell have failed.

Note in Figure 7-22 that the origin of the curve is at (0,0). It is permissible to input the failure strain, , as zero for this option. The nonlinear damage curve is useful for controlling the softening behavior after the failure strain is reached.

Figure 7-22 A Nonlinear Damage Curve (Optional)

ω

σnominalPA---=

P

σt rueP

A Aloss–----------------------=

Aloss

ωAloss

A------------= 0 ω 1≤ ≤

ωεef f

p εfai lurep–

εrapturep εfai lure

p–-------------------------------------------= εfai lure

p εeffp εrupture

p≤ ≤

f s

1

Dam

age

Failureεeff

pfs–


306

Material Model 83: Fu-Chang’s Foam With Rate EffectsThis model allows rate effects to be modeled in low and medium density foams, see Figure 7-23. Hysteretic unloading behavior in this model is a function of the rate sensitivity with the most rate sensitive foams providing the largest hysteresis and visa versa. The unified constitutive equations for foam materials by Fu-Chang [1995] provide the basis for this model. This implementation incorporates the coding in the reference in modified form to ensure reasonable computational efficiency. The mathematical description given below is excerpted from the reference.

Figure 7-23 Rate Effects in Fu Chang's Foam Model

The strain is divided into two parts: a linear part and a non-linear part of the strain

(7-256)

and the strain rate become

(7-257)

is an expression for the past history of . A postulated constitutive equation may be written as:

(7-258)

where is the state variable and is a functional of all values of in and

(7-259)

E t( ) EL t( ) EN t( )+=

E· t( ) E· L t( ) E· N t( )+=

E· N EN

σ t( ) EtN τ( ) S t( ),[ ] τd

τ = 0

∞

∫=

S t( )τ = 0

∞

∫ τ Tτ : 0 τ ∞≤ ≤

EtN τ( ) EN t τ–( )=


where is the history parameter:

(7-260)

It is assumed that the material remembers only its immediate past, i.e., a neighborhood about .

Therefore, an expansion of in a Taylor series about yields:

(7-261)

Hence, the postulated constitutive equation becomes:

(7-262)

where we have replaced by , and is a function of its arguments.

For a special case,

(7-263)

we may write

(7-264)

which states that the nonlinear strain rate is the function of stress and a state variable which represents the history of loading. Therefore, the proposed kinetic equation for foam materials is:

(7-265)

where and are material constants, and is the overall state variable. If either or

then the nonlinear strain rate vanishes.

(7-266)

(7-267)

(7-268)

(7-269)

where and are material constants and:

(7-270)

τEt

N τ = ∞( ) the virgin material⇔

τ 0=

EtN τ( ) τ 0=

EtN τ( ) EN 0( )

∂EtN

∂ t---------- 0( )dt+=

σ t( ) σ∗ EN t( ) E· N t( ) S t( ),,( )=

∂EtN

∂ t---------- E· N σ∗

σ t( ) σ∗ E· N t( ) S t( ),( )=

E· tN f S t( ) s t( ),( )=

E· N σσ

---------D0 c0t r σS( )

σ 2( )-----------------⎝ ⎠⎛ ⎞ 2n0–exp=

D0 c0, n0 S D0 0=

c0 ∞→

S· i j c1 aijR c2Sij–( )P c3Wn1 W· N( )

n2Ii j+[ ]R=

R 1 c4E· N

c5------------ 1–⎝ ⎠⎛ ⎞

n3+=

P tr σE· N( )∫=

W tr σ Ed( )∫=

c1 c2 c3 c4 c5 n1 n2 n3, , , , , , , , aij

σ σijσi j( )1 2/=

E· E· ijE·

i j( )1 2/=

EN· E· i jN

E· i jN( )1 2/=


308

In the implementation by Fu Chang the model was simplified such that the input constants and the

state variables are scalars.

Material Model 87: Cellular RubberThis material model provides a cellular rubber model combined with linear viscoelasticity as outlined by Christensen [1980].

Rubber is generally considered to be fully incompressible since the bulk modulus greatly exceeds the shear modulus in magnitude. To model the rubber as an unconstrained material a hydrostatic work term,

, is included in the strain energy functional which is function of the relative volume, ,

[Ogden, 1984]

(7-271)

In order to prevent volumetric work from contributing to the hydrostatic work the first and second invariants are modified as shown. This procedure is described in more detail by Sussman and Bathe [1987].

The effects of confined air pressure in its overall response characteristics are included by augmenting the stress state within the element by the air pressure.

(7-272)

where is the bulk skeletal stress and is the air pressure computed from the equation:

(7-273)

where is the initial foam pressure usually taken as the atmospheric pressure and defines the

volumetric strain

(7-274)

where is the relative volume of the voids and is the initial volumetric strain which is typically zero.

The rubber skeletal material is assumed to be incompressible.


(7-275)


aij

Si j

WH J( ) J

W J1 J2 J,,( ) Cpq J1 3–( )p J2 3–( )q WH J( )+

p q, 0=

n

∑=

J1 I1I31 3⁄–

=

J2 I2I32 3⁄–

=

σi j σi jsk δijσair–=

σ i jsk σair

σairp0γ

1 γ φ–+----------------------–=

p0 γ

γ V 1– γ0+=

V γ0


∂τ---------- td

0

t

∫=

Sij Eij


(7-276)



Since we wish to include only simple rate effects, the relaxation function is represented by one term from the Prony series:

(7-277)

given by

. (7-278)

This model is effectively a Maxwell fluid which consists of a damper and spring in series. We characterize this in the input by a shear modulus, , and decay constant, .

The Mooney-Rivlin rubber model is obtained by specifying . In spite of the differences in formulations with Model 27, we find that the results obtained with this model are nearly identical with those of 27 as long as large values of Poisson’s ratio are used. By setting the initial air pressure to zero, an open cell, cellular rubber can be simulated as shown in Figure 7-24.

Figure 7-24 Cellular Rubber with Entrapped Air

Sij Gijkl t τ–( )∂Ekl

∂τ----------- td

0

t

∫=


g t( ) a0 αm e β t–+

m 1=

N

∑+=

g t( ) Edeβ1 t–

=

G β1

n 1=


310

Material Model 89: Plasticity PolymerUnlike other MD Nastran SOL 700 material models, both the input stress-strain curve and the strain to failure are defined as total true strain, not plastic strain. The input can be defined from uniaxial tensile tests; nominal stress and nominal strain from the tests must be converted to true stress and true strain. The elastic component of strain must not be subtracted out.

The stress-strain curve is permitted to have sections steeper (i.e. stiffer) than the elastic modulus. When these are encountered the elastic modulus is increased to prevent spurious energy generation.

Material Model 94: Inelastic Spring Discrete BeamThe yield force is taken from the TABLED1:

(7-279)

where is the plastic deflection. A trial force is computed as:

(7-280)

and is checked against the yield force to determine :

(7-281)

The final force, which includes rate effects and damping, is given by:

(7-282)

where , are damping coefficients, is a factor to scale time units.

Unless the origin of the curve starts at (0,0), the negative part of the curve is used when the spring force is negative where the negative of the plastic displacement is used to interpolate, . The positive part of

the curve is used whenever the force is positive. In these equations, is the change in length

Material Model 97: General Joint Discrete BeamFor explicit calculations, the additional stiffness due to this joint may require additional mass and inertia for stability. Mass and rotary inertia for this beam element is based on the defined mass density, the volume, and the mass moment of inertia defined in the beam property input.

The penalty stiffness applies to explicit calculations. For implicit calculations, constraint equations are generated and imposed on the system equations; therefore, these constants, RPST and RPSR, are not used.

FY Fy ΔLplastic( )=

Lplastic

FT Fn K ΔL· Δ t⋅⋅+=

F

F FY

FT⎩⎨⎧

=if FT FY>

if FT FY≤

Fn 1+ F 1 C1 ΔL· C2 ΔL·( ) max 1.ΔL·

DLE------------,

⎩ ⎭⎨ ⎬⎧ ⎫

⎝ ⎠⎜ ⎟⎛ ⎞

lnsgn⋅+⋅+ DΔL· g ΔL( )h ΔL·( )+ +⋅=

C1 C2 DLE

Fy

ΔL

ΔL current length initial length–=


Material Model 98: Simplified Johnson CookJohnson and Cook express the flow stress as

where

The maximum stress is limited by sigmax and sigsat by:

Failure occurs when the effective plastic strain exceeds psfail.

If the viscoplastic option is active, VP=1.0, the SIGMAX and SIGSAT parameters are ignored since these parameters make convergence of the viscoplastic strain iteration loopdifficult to achieve. The viscoplastic option replaces the plastic strain in the forgoing equationsby the viscoplastic strain and the strain rate by the viscoplastic strain rate. Numerical noise issubstantially reduced by the viscoplastic formulation.

Material Model 100: Spot WeldThis material model applies to beam element type 9 for spot welds. These beam elements may be placed between any two deformable shell surfaces, see Figure 7-25, and tied with type 7 constraint contact which eliminates the need to have adjacent nodes at spot weld locations. Beam spot welds may be placed between rigid bodies and rigid/deformable bodies by making the node on one end of the spot weld a rigid body node which can be an extra node for the rigid body. In the same way, rigid bodies may also be tied together with this spot weld option.

It is advisable to include all spot welds which can be arbitrarily placed within the structure, which provide the slave nodes, and spot welded materials, which define the master segments, within a single type 7 tied interface. As a constraint method, multiple type 7 interfaces are treated independently which can lead to significant problems if such interfaces share common nodal points. The offset option, “o 7”, should not be used with spot welds.

The weld material is modeled with isotropic hardening plasticity coupled to two failure models. The first model specifies a failure strain which fails each integration point in the spot weld independently. The second model fails the entire weld if the resultants are outside of the failure surface defined by:

(7-283)

A, B, C and n are input constants

ellective plastic strain

effective strain rate for

σy A Bεpn+( ) 1 c ε· *ln+( )=

εp

ε· *ε·

ε· 0

-----=ε· 0 1s 1–=

σy min min A Bεpnsigmax,+[ ] 1 c ε· *ln+( ) sigsat,{ }=

Nrr

NrrF

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Nrs

NrsF

-----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Nrt

NrtF

----------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Mrr

MrrF

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 Mss

MssF

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1Trr

TrrF

----------⎝ ⎠⎜ ⎟⎛ ⎞ 2

–+ + + + 0=


312

Figure 7-25 Deformable Spotwelds

where the numerators in the equation are the resultants calculated in the local coordinates of the cross section, and the denominators are the values specified in the input. If the user defined parameter, NF, which the number of force vectors stored for filtering, is nonzero the resultants are filtered before failure is checked. The default value is set to zero which is generally recommended unless oscillatory resultant forces are observed in the time history databases. Even though these welds should not oscillate significantly, this option was added for consistency with the other spot weld options. NF affects the storage since it is necessary to store the resultant forces as history variables.

If the failure strain is set to zero, the failure strain model is not used. In a similar manner, when the value of a resultant at failure is set to zero, the corresponding term in the failure surface is ignored. For example, if only is nonzero, the failure surface is reduced to . None, either, or both of the failure

models may be active depending on the specified input values.

The inertias of the spot welds are scaled during the first time step so that their stable time step size is Δt. A strong compressive load on the spot weld at a later time may reduce the length of the spot weld so that stable time step size drops below . If the value of is zero, mass scaling is not performed, and the spot welds will probably limit the time step size. Under most circumstances, the inertias of the spot welds are small enough that scaling them will have a negligible effect on the structural response and the use of this option is encouraged.

Spotweld force history data is written into the SWFORC ASCII file. In this database the resultant moments are not available, but they are in the binary time history database.

NrrFNrr NrrF

=

Δ t Δ t


The constitutive properties for the damaged material are obtained from the undamaged material properties. The amount of damage evolved is represented by the constant, , which varies from zero if no damage has occurred to unity for complete rupture. For uniaxial loading, the nominal stress in the damaged material is given by

(7-284)

where is the applied load and is the surface area. The true stress is given by:

(7-285)

where is the void area. The damage variable can then be defined:

(7-286)

In this model damage is defined in terms of plastic strain after the failure strain is exceeded:

(7-287)

After exceeding the failure strain softening begins and continues until the rupture strain is reached.

Material Model 116: Composite LayupThis material is for modeling the elastic responses of composite lay-ups that have an arbitrary number of layers through the shell thickness. A pre-integration is used to compute the extensional, bending, and coupling stiffness for use with the Belytschko-Tsay resultant shell formulation. The angles of the local material axes are specified from layer to layer in the PCOMP or PSHELLD input. This material model must be used with the user-defined integration rule for shells, which allows the elastic constants to change from integration point to integration point. Since the stresses are not computed in the resultant formulation, the stress output to the binary databases for the resultant elements are zero.

This material law is based on standard composite lay-up theory. The implementation, [See Jones 1975], allows the calculation of the force, , and moment, , stress resultants from:

(7-288)

ω

σnominalPA---=

P A

σt rueP

A Aloss–----------------------=

Aloss

ωAloss

A------------= 0 ω 1≤ ≤

ω εeff

p εfailurep

–

εrupturep εfai lure

p–

------------------------------------------ i f εfai lurep εeff

p εrupturep≤ ≤=

N M

Nx

Ny

Nxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

A11 A12 A16

A21 A22 A26

A16 A26 A66

εx0

εy0

εz0

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

B11 B12 B16

B21 B22 B26

B16 B26 B66

κx

κy

κxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

+=

Mx

My

Mxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

B11 B12 B16

B21 B22 B26

B16 B26 B66

εx0

εy0

εz0

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

D11 D12 D16

D21 D22 D26

D16 D26 D66

κx

κy

κxy⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

+=


314

where is the extensional stiffness, is the bending stiffness, and is the coupling stiffness, which

is a null matrix for symmetric lay-ups. The mid-surface strains and curvatures are denoted by and ,

respectively. Since these stiffness matrices are symmetric, 18 terms are needed per shell element in addition to the shell resultants, which are integrated in time. This is considerably less storage than would typically be required with through thickness integration which requires a minimum of eight history variables per integration point; e.g., if 100 layers are used, 800 history variables would be stored. Not only is memory much less for this model, but the CPU time required is also considerably reduced.

Material Model 119: General Nonlinear Six Degrees of Freedom Discrete BeamCatastrophic failure, which is based on displacement resultants, occurs if either of the following inequalities are satisfied:

(7-289)

After failure the discrete element is deleted. If failure is included either the tension failure or the compression failure or both may be used.

Figure 7-26 Load and Unloading Behavior

Aij Dij Bij

εi j0 κij

ur

urt fail

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 us

ustfai l

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 ut

utt fail

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θt

θrt fail

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θs

θst fail

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θt

θtt fa il

------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1. 0≥–+ + + + +

ur

urcfai l

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 us

uscfai l

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 ut

utcfail

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θt

θrcfai l

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θs

θscfai l

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2 θt

θtcfail

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

1. 0≥–+ + + + +

Unload = 0 Unload = 1

Unload = 2 Unload = 3

RESULTANT

RESULTANT

RESULTANT

RESULTANT

DISPLACEMENT

DISPLACEMENT DISPLACEMENT

DISPLACEMENT

Unloading Curve

Unloading Curve

Loading-unloading Curve

Unloading Curve

OFFSET x UminUmin


Material Model 124: Tension-Compression PlasticityThis is an isotropic elastic-plastic material where a unique yield stress versus plastic strain curve can be defined for compression and tension. Failure can occur based on plastic strain or a minimum time step size. Rate effects are modeled by using the Cowper-Symonds strain rate model.

The stress-strain behavior follows one curve in compression and another in tension. The sign of the mean stress determines the state where a positive mean stress (i.e., a negative pressure) is indicative of tension. Two TABLED1s, and , are defined, which give the yield stress, , versus effective plastic strain

for both the tension and compression regimes. The two pressure values, and , when exceeded,

determine if the tension curve or the compressive curve is followed, respectively. f the pressure, , falls between these two values, a weighted average of the two curves are used:

if

Strain rate is accounted for using the Cowper and Symonds model, which scales the yield stress with the factor

(7-290)

where is the strain rate .

Material Model 126: Modified HoneycombFor efficiency it is strongly recommended that the TABLED1 ID’s: LCA, LCB, LCC, LCS, LCAB, LCBC, and LCCA, contain exactly the same number of points with corresponding strain values on the abscissa. If this recommendation is followed the cost of the table lookup is insignificant. Conversely, the cost increases significantly if the abscissa strain values are not consistent between TABLED1s.

The behavior before compaction is orthotropic where the components of the stress tensor are uncoupled; i.e., a component of strain will generate resistance in the local α-direction with no coupling to the local b and c directions. The elastic modulii vary from their initial values to the fully coaaumpacted values linearly with the relative volume:

(7-291)

where

(7-292)

ft p( ) fc p( ) σy

pt pc

p

pt p pc

scalepc p–

pc pt+----------------=

σy scale ft p( ) 1 scale–( ) fc p( )⋅+⋅=⎩⎪⎨⎪⎧

≤ ≤–

1ε·

C----⎝ ⎠⎛ ⎞

1 p/

+

ε· ε· ε· i jε·

ij=

Eaa Eaau βaa E Eaau–( )+=

Ebb Ebbu βbb E Ebbu–( )+=

Ecc Eccu βcc E Eccu–( )+=

Gab Gabu β G Gabu–( )+=

Gbc Gbcu β G Gbcu–( )+=

Gca Gcau β G Gcau–( )+=

B max min1 v–1 vf–------------- 1,⎝ ⎠⎛ ⎞ 0,⎝ ⎠⎛ ⎞=


316

and is the elastic shear modulus for the fully compacted honeycomb material

(7-293)

The relative volume, , is defined as the ratio of the current volume over the initial volume, and typically, at the beginning of a calculation.

The TABLED1s define the magnitude of the stress as the material undergoes deformation. The first value in the curve should be less than or equal to zero corresponding to tension and increase to full compaction. Care should be taken when defining the curves so the extrapolated values do not lead to negative yield stresses.

At the beginning of the stress update we transform each element’s stresses and strain rates into the local element coordinate system. For the uncompacted material, the trial stress components are updated using the elastic interpolated modulii according to:

(7-294)

We then independently check each component of the updated stresses to ensure that they do not exceed the permissible values determined from the TABLED1s, e.g., if

(7-295)

then

(7-296)

The components of are defined by TABLED1s. The parameter is either unity or a value taken

from the TABLED1 number, LCSR, that defines as a function of strain-rate. Strain-rate is defined here as the Euclidean norm of the deviatoric strain-rate tensor.

For fully compacted material we assume that the material behavior is elastic-perfectly plastic and updated the stress components according to:

(7-297)

where the deviatoric strain increment is defined as

(7-298)

G

GE

2 1 υ+( )----------------------=

ν

V 1=

σaan 1t rial+ σaa

n EaaΔεaa+=

σbbn 1t rial+ σbb

n EbbΔεbb+=

σccn 1t rial+ σcc

n EccΔεcc+=

σabn 1t rial+ σab

n 2GabΔεab+=

σbcn 1t rial+ σbc

n 2GbcΔεbc+=

σcan 1t rial+ σca

n 2GcaΔεca+=

σi jn 1t rial+ λσi j εi j( )>

σi jn 1+ σi j εij( )

λσ ijn 1trial+

σin 1t rial+

---------------------------=

σi j εij( ) λ

λ

si jtrial sij

n 2GΔεijdevn 1+ 2⁄

+=

Δεi jdev Δεij

13---Δεkkδi j–=


We now check to see if the yield stress for the fully compacted material is exceeded by comparing

(7-299)

the effective trial stress to the yield stress, . If the effective trial stress exceeds the yield stress, we

simply scale back the stress components to the yield surface

(7-300)

We can now update the pressure using the elastic bulk modulus,

(7-301)

and obtain the final value for the Cauchy stress

(7-302)

After completing the stress update we transform the stresses back to the global configuration.

In Figure 7-27, note that the "yield stress" at a strain of zero is nonzero. In the TABLED1 definition, the "time" value is the directional strain and the "function" value is the yield stress.

Figure 7-27 Stress Quantity Versus Strain

sefftrial 3

2--- sij

trialsijtrial

⎝ ⎠⎛ ⎞ 1 2⁄

=

σy

si jn 1+ σy

sefftrial

------------ si jtrial=

K

pn 1+ pn KΔεkkn 1 2⁄+–=

KE

3 1 2υ–( )-------------------------=

σi jn 1+ sij

n 1+ pn 1+ δij–=

Curve extends into negative strain quadrant since SOL 700 will extrapolate using the two end points. It is important that the extrapolation does not extend into the negative stress region.

Unloading is based on the interpolated Young’s moduli which must provide an unloading tangent that exceeds the loading tangent.


318

Material Model 127: Arruda-Boyce Hyperviscoelastic RubberThis material model, described in the paper by Arruda and Boyce [1993], provides a rubber model that is optionally combined with linear viscoelasticity. Rubber is generally considered to be fully incompressible since the bulk modulus greatly exceeds the shear modulus in magnitude; therefore, to model the rubber as an unconstrained material, a hydrostatic work term, , is included in the strain

energy functional which is function of the relative volume, :

(7-303)

The hydrostatic work term is expressed in terms of the bulk modulus, , and , as:

(7-304)


(7-305)


(7-306)



If we wish to include only simple rate effects, the relaxation function is represented by six terms from the Prony series:

(7-307)

given by,

(7-308)

This model is effectively a Maxwell fluid which consists of a dampers and springs in series. We characterize this in the input by shear moduli, , and decay constants, . The viscoelastic behavior is

optional and an arbitrary number of terms may be used.

WH J( )

J

W J1 J2 J, ,( ) nkθ 12--- J1 3–( ) 1

20N---------- J1

2 9–( ) 111050N2------------------- J1

3 27–( )+ +

nkθ 197000N----------------⎝ ⎠⎛ ⎞ 3

J14 81–( ) 519

673750N4------------------------- J1

5 243–( )+ WH J( )+ +

=

J1 I1J 1 3⁄–=

J2 I2J=

K J

WH J( ) K2---- H 1–( )2=


∂τ---------- τd

0

t

∫=

Sij ijE

Sij Gijkl t τ–( )∂Ekl

∂τ----------- τd

0

t

∫=



m 1=

N

∑+=

g t( ) Gieβit–

i 1=

N

∑=

Gi βi


When viscoelastic terms are not included, this model is similar to the use of the Arruda Boyce model in solution 600, defined in the MATHE option. When viscoelasticity is included, the formulation of these two models are different.

Material Model 158: Rate Sensitive Composite FabricSee material type 58, Laminated Composite Fabric, for the treatment of the composite material. Rate effects are taken into account through a Maxwell model using linear viscoelasticity by a convolution integral of the form:

(7-309)

where is the relaxation function for different stress measures. This stress is added to the stress

tensor determined from the strain energy functional. Since we wish to include only simple rate effects, the relaxation function is represented by six terms from the Prony series:

We characterize this in the input by the shear moduli, , and the decay constants, . An arbitrary

number of terms, not exceeding 6, may be used when applying the viscoelastic model. The composite failure is not directly affected by the presence of the viscous stress tensor.

Material Model 181: Simplified Rubber FoamMaterial type 181 in SOL 700is a simplified “quasi”-hyperelastic rubber model defined by a single uniaxial TABLED1 or by a family of curves at discrete strain rates. The term “quasi” is used because there is really no strain energy function for determining the stresses used in this model. However, for deriving the tangent stiffness matrix we use the formulas as if a strain energy function were present. In addition, a frequency independent damping stress is added to model the energy dissipation commonly observed in rubbers.

Hyperelasticity Using the Principal Stretch Ratios

A hyperelastic constitutive law is determined by a strain energy function that here is expressed in terms of the principal stretches; i.e., ) , . To obtain the Cauchy stress , as well as the

constitutive tensor of interest, , they are first calculated in the principal basis after which they are

transformed back to the “base frame”, or standard basis. The complete set of formulas is given by Crisfield [1997] and is for the sake of completeness recapitulated here.

The principal Kirchhoff stress components are given by

(7-310)


∂τ---------- τd

0

t∫=

gijkl t τ–

g t( ) Gmeβmt–

m 1=

N

∑=

Gi βi

W W λ1 λ2 λ3,,( )= σi j

CijklTC

τijE λi

∂W∂λi--------= no sum( )


320

that are transformed to the standard basis using the standard formula

(7-311)

The are the components of the orthogonal tensor containing the eigenvectors of the principal basis.

The Cauchy stress is then given by

(7-312)

where is the relative volume change.

The constitutive tensor that relates the rate of deformation to the Truesdell (convected) rate of Kirchhoff stress can in the principal basis be expressed as

(7-313)

These components are transformed to the standard basis according to

(7-314)

and finally the constitutive tensor relating the rate of deformation to the Truesdell rate of Cauchy stress is obtained through

(7-315)

Stress and Tangent Stiffness

The principal Kirchhoff stress is in material model 181 given by

(7-316)

where is a TABLED1 determined from uniaxial data (possible at different strain rates). Furthermore, is the bulk modulus and is the relative volume change of the material. This stress cannot be deduced

from a strain energy function unless for some constitutive parameter . A consequence of this is that when using the formulas in the previous section the resulting tangent stiffness matrix is not necessarily symmetric. We remedy this by symmetrizing the formulas according to

τi j qikqjlτklE=

qij

σi j J 1– τ i j=

J λ1λ2λ3=

CiijjTKE λj

∂i iE

∂λj-------- 2τi i

E δij–=

CijijTKE λj

2τi iE λi

2τj jE–

λ12 λj

2–--------------------------------=

CijijTKE λi

2----

∂τiiE

∂λi---------

∂τi iE

∂λj---------–⎝ ⎠

⎛ ⎞=

i j λi λj≠( ) (no sum),≠

i j λi,≠ λj=

CijklTKE qipqjgqkrqlsCpqrs

TKE=

CijklTC J 1– Cijkl

TK=

τiE f λi( ) K J 1–( ) 1

3--- f λk( )

k 1=

3

∑–+=

f K

J

f λ( ) E λln= E


(7-317)

Two Remarks

The function introduced in the previous section depends not only on the stretches but for some choices of input also on the strain rate. Strain rate effects complicate things for an implicit analyst and here one also has to take into account whether the material is in tension/compression or in a loading/unloading stage. We believe that it is of little importance to take into account the strain rate effects when deriving the tangent stiffness matrix and therefore this influence has been disregarded.

For the fully integrated brick element, we have used the approach in material model 77 to account for the constant pressure when deriving the tangent stiffness matrix. Experiments have shown that this is crucial to obtain a decent implicit performance for nearly incompressible materials.

Modeling of the Frequency Independent Damping

An elastic-plastic stress is added to model the frequency independent damping properties of rubber.

This stress is deviatoric and determined by the shear modulus and the yield stress . This part of the

stress is updated incrementally as

(7-318)

where . is the strain increment. The trial stress is then radially scaled (if necessary) to the yield surface according to

where is the effective von Mises stress for the trial stress

The elastic tangent stiffness contribution is given by

(7-319)

and if yield has occurred in the last time step the elastic-plastic tangent is used

(7-320)

Here is the deviatoric fourth order identity tensor.

λj

∂τiiE

∂λj---------⎝ ⎠

⎛ ⎞symn

12--- λj

∂τiiE

∂λj--------- λi

∂τjjE

∂λi---------+⎝ ⎠

⎛ ⎞ KJ

23--- λi f' λi( )

16--- λi f

· λi( ) λj f' λj( )+( )–⎩⎪⎪⎨⎪⎪⎧

+= =if i j=

otherwise

f

σd

G σY

σdn 1+ σd

n 2GIdevΔε+=

Δε

σdn 1+ σd

n 1+min 1

σY

σeff---------,⎝ ⎠

⎛ ⎞=

σef f σdn 1+

Cd 2GIdev=

Cd 2GIdev 3G

σY2

-------σd σd⊗–=

Idev


322

Material Model 196: General Spring Discrete BeamIf TYPE=0, elastic behavior is obtained. In this case, if the linear spring stiffness is used, the force, , is given by:

(7-321)

but if the TABLED1 ID is specified, the force is then given by:

In these equations, .L is the change in length

If TYPE=1, inelastic behavior is obtained. In this case, the yield force is taken from the TABLED1:

(7-322)

where is the plastic deflection. A trial force is computed as:

(7-323)

and is checked against the yield force to determine :

(7-324)

The final force, which includes rate effects and damping, is given by:

Unless the origin of the curve starts at (0,0), the negative part of the curve is used when the spring force is negative where the negative of the plastic displacement is used to interpolate, . The positive part of

the curve is used whenever the force is positive.

The cross-sectional area is defined on the section card for the discrete beam elements. See *SECTION_BEAM. The square root of this area is used as the contract thickness offset if these elements are included in the contact treatment.

F

F F0 KΔL DΔL·+ +=

F F0 Kf ΔL( ) 1 C1 ΔL· C2 ΔL·( )sgn max 1.ΔL·

DLE------------,

⎩ ⎭⎨ ⎬⎧ ⎫

⎝ ⎠⎜ ⎟⎛ ⎞

ln⋅+⋅+ DΔL· g ΔL( )h ΔL·( )+ + +=

ΔL current length initial length–=

FY Fy ΔLplastic( )=

Lplastic

FT Fn KΔL· Δ t+=

F

F FY

FT⎩⎨⎧

=if FT FY>

if FT FY≤

Fn 1+ F 1 C1 ΔL· C2 ΔL·( ) max 1.ΔL·

DLE------------,

⎩ ⎭⎨ ⎬⎧ ⎫

⎝ ⎠⎜ ⎟⎛ ⎞

lnsgn⋅+⋅+ DΔL· g ΔL( )h ΔL·( )+ +⋅=

Fy


Grunseisen Equation of State (EOSGRUN)The Gruneisen equation of state with cubic shock velocity-particle velocity defines pressure for compressed material as

(7-325)

where is the internal energy per initial volume, is the intercept of the curve, , and

are the coefficients of the slope of the curve, is the Gruneisen gamma, and a is the first order

volume correction to . Constants, , and , are user-defined input parameters. The

compression is defined in terms of the relative volume, , as:

.

For expanded materials as the pressure is defined by:

(7-326)

Tabulated Compaction Equation of State (EOSTABC)Pressure is positive in compression, and volumetric strain is positive in tension. The tabulated

compaction model is linear in internal energy per unit volume. Pressure is defined by

(7-327)

during loading (compression). Unloading occurs at a slope corresponding to the bulk modulus at the peak (most compressive) volumetric strain, as shown in Figure 7-28. Reloading follows the unloading path to the point where unloading began, and then continues on the loading path described by Equation (7-327)). In the compacted states, the bulk unloading modulus depends on the peak volumetric strain.

pρ0C2μ 1 1

γ0

2----–⎝ ⎠

⎛ ⎞ μ α---μ2–+

1 S1 1–( )μ– s2μ2

μ 1+-------------– s3

μ3

μ 1+( )2--------------------–

-------------------------------------------------------------------------------------------------- γ0 αμ+( )E+=

E C us up– S1 S2, S3

us up– γ0

γ0 C S1 S2 S3 γ0, , , , α

V

μ 1V--- 1–=

p ρ0C2μ γ0αμ( )E+=

εV

p C εV( ) γT εV( )E+=


324

Figure 7-28 Pressure vs Volumetric Strain Curve for EOSTABC

Tabulated Equation of State (EOSTAB)The tabulated equation of state model is linear in internal energy. Pressure is defined by

(7-328)

The volumetric strain is given by the natural algorithm of the relative volume. Up to 10 points and as

few as 2 may be used when defining the tabulated functions. The pressure is extrapolated if necessary. Loading and unloading are along the same curve unlike tabulated compaction equation of state (EOSTABC).

Volumetric Strain εV–( )

Pre

ssur

e p A

B

C

KεvCKεv

BKεvA

εvA εv

B εvC

p C εV( ) γT εV( )E+=

εV

325Chapter 7: MaterialsEulerian Material Models for SOL 700

Eulerian Material Models for SOL 700

MATDEUL – Eulerian Material

The MATDEUL material entry is a general material definition and provides a high degree of flexibility in defining material behavior. The basis of the MATDEUL entry is the reference of a combination of material descriptions: equation of state, yield model, shear model, failure model, and spall model. The only material parameter defined on the MATDEUL entry is the reference density.

The MATDEUL entry can be used to define all types of material behavior from materials with very simple linear equations of state to materials with complex yielding and shearing behavior and different failure criteria.

The required input is the reference density, the number of an EOSxxx entry defining the equation of state, and the number of an SHRxxx entry defining the shear properties of the material. The equation of state defines the bulk behavior of the material. It may be a polynomial equation, a gamma law gas equation, or an explosive equation. A single-term polynomial equation produces a linear elastic behavior.

Further material property definitions are optional. A referenced YLDxxx entry selects one of the following: a hydrodynamic response (zero yield stress), a von Mises criterion that gives a bilinear elastoplastic behavior, or a Johnson-Cook yield model where the yield stress is a function of plastic strain, strain rate, and temperature. If no YLDxxx model is referenced, the material is assumed to be fully elastic.

A FAILxxx entry can be referenced to define a failure model for the material. This failure model can be based on a maximum plastic strain limit or Johnson Cook failure model. If no FAILxxx entry is referenced, the material has no failure criterion.

A PMINxxx entry can be referenced to define the spall characteristics of the material. Currently, only the PMINC entry is available. The entry provides a constant spall limit for the material. When no PMINxxx entry is referenced, the material has a zero spall limit.

Shear ModelsThe shear model is referenced from a MATDEUL entry. It defines the shear behavior of the material. At present, an elastic shear model is available with a constant or polynomial shear modulus.

SHREL – Constant Modulus Shear Model

The SHREL entry defines a shear model with a constant shear modulus G. The model is referenced from a MATDEUL entry that defines the general material properties.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideEulerian Material Models for SOL 700

326

Figure 7-29 Elastic Shear as Function of Strain.

SHRPOL – Polynomial Shear Model

The SHRPOL model defines a polynomial shear model where the shear modulus is related to the effective plastic shear strain by a cubic equation.

where = effective plastic shear strain and and are constants.

Yield ModelsYield models is referenced by MATDEUL entries. The yield models can be used to model elastic perfectly plastic behavior, bilinear elastoplastic behavior, piecewise linear behavior, or hydrodynamic behavior (zero yield stress).

YLDHY – Hydrodynamic Yield Model

The YLDHY entry defines a yield model with constant zero yield stress. This model should be used for fluids that have no shear strength and are, therefore, hydrodynamic.

YLDVM – von Mises Yield Model

The YLDVM entry defines a von Mises yield model. The yield stress and hardening modulus are defined by giving either a bilinear or piecewise linear stress-strain curve. Only an elastic perfectly plastic yield model can be used. The hardening modulus is not used.

G

Strain

Shear Stress

G G0 G1γ G2γ2G3γ3

+ + +=

γ G0 G1 G2, , G3


Bilinear Representation

where the yield stress is given by

Piecewise Linear Representation

During every iteration, the stress s is determined from the current equivalent strain by interpolating from the stress-strain table

where and are the points in the table. The stress-strain characteristic used internally in MD Nastran

Nonlinear Explicit (SOL 700) is in terms of true stress and equivalent plastic strain. However, for convenience, the stress-strain characteristic can be input in any of the following ways:

• True stress/true strain

• Engineering stress/engineering strain

• True stress/plastic strain

• True stress/plastic modulus

where = yield stress

= Youngs modulus

= hardening modulus

= equivalent plastic strain

σ

ε

E

Ehσ0

σy

σy σ0

EEh

E Eh–----------------εp+=

σ0

E

Eh

εp

σ

ε

ε

σ σi σ i 1––( ) εi εi 1––( ) εiεi 1–( )⁄[ ] σi 1–+=

σj εj


328

True stress is defined as

where = current force, = current area.

Plastic strain is

where = true strain, = elastic strain

True strain is defined as

where = incremental change in length, = current length.

By comparison, engineering stress and strain are given by

where = original area

where = original length

True stress/true strain and engineering strain are related by the following formulas:

At small strains, there is little difference between true stress-strain and engineering stress-strain. However, at moderate and large strains there can be very large differences, and it is important that the correct stress-strain characteristic is input.

When defining the material properties using Young’s modulus, yield stress, and hardening modulus, the hardening modulus must be estimated from a plot of true stress versus true strain. This estimate may well require a measured material characteristic to be replotted.

Some simple examples follow:

σt rueFA---=

F A

εpl

εpl εtrue εel–=

εt rue εel

εt rued ll

-----∫=

dl l

σeng εeng

σengFA0------= A0

εeng

I I0–( )I0

------------------= I0

σt rue σeng 1 εeng+( )=

εt rue 1 εeng+( )ln=


True Stress Versus True Strain

Figure 7-30 True Stress vs True Strain Curve

The slope of the first segment of the curve gives the Young’s modulus for the material (when it is not defined explicitly) and the first nonzero stress point gives the yield stress . The point corresponding to

the origin can be omitted.

Engineering Stress Versus Engineering Strain

Figure 7-31 Engineering Stress vs Engineering Strain Curve

True Stress Versus Plastic Strain

Figure 7-32 True Stress vs Plastic Strain Curve

Since the curve is defined in terms of the equivalent plastic strain, there is no elastic part in the curve. The first point must be the yield stress of the material at zero plastic strain. Young’s modulus is defined separately.

σy

E

True Strain

True Stress

σy

σy

E

Engineering Strain

Engineering Stress

True Stress

Plastic Strain

σy


330

True Stress Versus Plastic Modulus

Figure 7-33 True Stress vs Plastic Strain Curve

This option is slightly different since the curve is specified as a series of pairs of stress and hardening moduli, rather than as a series of pairs of stress and strain. Young’s modulus and yield stress are defined explicitly so that the table consists of pairs of values with the hardening modulus (x-axis) and the true stress (y-axis) at the end of the segment.

Yielding occurs when the von Mises stress

exceeds the yield stress . The principal stresses are , and .

Isotropic hardening is assumed, which means that the yield surface increases in diameter as yielding occurs, but its center does not move.

This yield model can be used with beam, shell, and solid elements. When used with shell or solid elements, strain-rate sensitivity and failure can be included. Strain-rate sensitivity can be defined in two ways:

1. You can specify a table giving the variation of a scale factor with strain-rate . The scale factor is multiplied by the stress found from the stress-strain characteristic to give the actual stress. The failure criterion is based on plastic strain. When the plastic strain exceeds the specified value, the element fails. All the stresses are set to zero, and the element can carry no load. (This failure criterion is referred to from the DMATEP or the DYMAT24 entry.)

where is the dynamic yield stress, is the static yield stress and is the equivalent strain rate.

YLDJC – Johnson-Cook Yield Model

The YLDJC entry defines a Johnson-Cook yield model in which the yield stress is a function of the plastic strain, strain rate, and temperature

σ1

Eh2

σ2

Eh1

True Stress

Plastic Strain

σvm σ1 σ2–( )2 σ2 σ3–( )2 σ3 σ1–( )2+ +[ ] 2⁄=

σy σ σ2, σ3

S dε dt⁄

σd

σy------ 1

ε·

D----

⎩ ⎭⎨ ⎬⎧ ⎫

1 P⁄+=

σd σy ε·

σy A Bεpn+( ) 1 C ε· ε· o⁄( )ln+( ) 1 T*m–( )=


YLDTM – Tanimura-Mimura Yield Model

The YLDTM entry defines a Tanimura-Mimura yield model in which the yield stress is a function of the plastic strain, strain rate, and temperature

This yield model is suitable for a wide range of strain rates, strains and temperatures.

where =

= effective plastic strain

= effective strain rate

= reference strain rate

= temperature

= room temperature

= melt temperature

, and are constants

where =

= Temperature

= room temperature

= melt temperature

= effective plastic strain


= Quasi-static strain rate


= Critical yield stress

, and are constants

T*T Tr–( )

Tm Tr–( )-----------------------

εp

ε·

ε· 0

T

Tt

Tm

A B n C, , , m

σy A Bεp C Dεp+( ) 1A Bεp+

σcr--------------------–⎝ ⎠

⎛ ⎞ ε·

ε· s

----⎝ ⎠⎛ ⎞ln+ + 1 T*m–( ) E

ε·

ε· 0

-----⎝ ⎠⎛ ⎞

k+=

T*T Tr–( )

Tm Tr–( )-----------------------

T

Tr

Tm

εp

ε·

ε· s

ε· 0

σcr

A B C D m E, , , , , k


332

YLDZA – Zerilli-Armstrong Yield Model

The YLDZA entry defines a Zerilli-Armstrong yield model in which the yield stress is a function of the plastic strain, strain rate, and temperature

for Fcc metals

for Bcc metals

This yield model can be used for both Fcc type of metals, like iron and steels, as well as Bcc type of metals, like aluminum and alloys.

YLDRPL – Rate Power Law Yield Model

The YLDRPL entry defines a rate power law yield model in which the yield stress is a function of the plastic strain and strain rate.

YLDPOL – Polynomial Yield Model

The YLDPOL entry defines a polynomial yield model in which the yield stress is a function of the plastic strain

where = effective plastic strain



= Temperature

, and are constants

where =effective plastic strain

=effective strain rate

, and are constants


= Maximum yield stress

, and are constants

σy A Bεpn+( )e

mT– CTε·

ε· 0-----⎝ ⎠⎜ ⎟⎛ ⎞

ln+

=

σy A Bεpn+( )D e+

mT– CTε·

ε· 0-----⎝ ⎠⎜ ⎟⎛ ⎞

ln+

=

εp

ε·

ε· 0

T

A B n C m, , , , D

σy MAX C A Bεpnε· m+,( )=

εp

ε·

A B n m, , , C

σy MIN σmax A Bεp Cεp2 Dεp

3 Eεp4 Fεp

5+ + + + +,( )=

εp

σmax

A B C D E, , , , F


YLDSG – Steinberg-Guinan Yield Model

The YLDSG entry defines a Steinberg-Guinan yield model in which the yield stress is a function of the plastic strain, strain rate, and temperature.

YLDMSS – Multi-surface Yield Model for Snow

Defines the yield model for snow material. This card must be used in combination with MATDEUL, EOSPOL, and SHREL.

Snow is a very specific material that lies between water and ice. In micro term, the structure of snow looks like general porous materials where the degree of compaction can vary quite large. Therefore, from the constitutive equation point of view, snow belongs to the family of soil. One of plasticity models applied for snow is a multi-surface one.

The multi-surface plasticity model for snow, see references 1 and 2 (there are misprinting in those papers) is characterized by two independent hardening (softening) mechanisms and a set of yield functions as shown in the following form:

and are the first and second invariant of the stress tensor and . The material parameter

is related to the cohesion of snow. is the hardening (softening) parameter associated with the yield

surface . determines the shape of . Hence, it is a model parameter that may be set independently


= temperature

= room temperature

= melt temperature

= Pressure

= density

, and are constants

AT A1 1 A3εp+( )A4

=

σy min A2 AT,( ) 1 H T Tr–( )– Bpρ

ρref---------⎝ ⎠⎛ ⎞

13---

+ T Tm<,=

σy 0 T Tm≥,=

εp

T

Tr

Tm

p

ρ

A … A4 H, , , B

fc I1 J2 qc αc( ), ,( ) J2

cc

qc----- I1 qc+( )4

+ κc I1– ccqc3

+ 0= =

ft I1 qt αt( ),( ) I13---

qt– 0= =

I1 J2 I1 T I1–= T

qc qt( )

fc ft( ) cc fc


334

from the specific type of snow. is a material parameter related to the angle of friction. Figure 7-34

contains plots of the yield functions and in the meridian plane at different stages of the hardening

process of and the softening process of , respectively.

Figure 7-34 Snow Model: Plots of Loading Functions in the Meridian Plane

defines a so-called “tension-cut-off”-plane perpendicular to the hydrostatic axis. For simplicity, a

linear softening law is adopted:

where is the hydrostatic tensile strength of snow, is the softening modulus and represents the

accumulated plastic volumetric tensile strain.

For the implementation in MD Nastran Nonlinear Explicit (SOL 700), the accumulated plastic strain is updated if the tensile-pressure is bigger than the current . The incremental strain is calculated using the

difference of the pressure divided by the bulk modulus. Then the new is updated to be used in the next

cycle. Furthermore, the deviatoric stresses are brought to zero.

constitutes a smooth yield function closed along the compressive and the tensile branch of the

hydrostatic axis. Its shape in the stress space changes continuously in the course of hardening, see Figure 7-34. A specific hardening law, similar to the one used in the Cap Model (reference 3) was adopted for snow on the basis of results from hydrostatic compression tests:

κc

fc ft

fc ft

Hardening

Softeningg

-I1

2J Drucker-Prager failure surface

Tension cut-off planes

Yield surfaces

ft

qt ft t Dsαt–=

ftu Ds αt

qt

qt

fc

qc1

2ac-------- 1

αc

bc------–⎝ ⎠

⎛ ⎞ln=

qc

αc fcbc–

2 acbc 1 fc–( )( )--------------------------------------

1 fc–( )ln

2ac------------------------–=

if αc fcbc≤

αc fcbc≤if


are parameters determined from hydrostatic compression tests. is a parameter that avoids

singularity in above equations at . It will be set to 0.99. As grows, the model obtains a shape

similar to the Drucker-Prager failure criterion, see Figure 7-34. The following relation obtains the correlation between the proposed model for snow and the Drucker-Prager model.

The plasticity evolution is done using an additive plasticity model and associative flow rule (with isotropic hardening law) as follows:

The incremental plastic strains can be derived as follows:

They consist of deviatoric and volumetric plastic strains as follows:

is calculated according to the following procedure. First the trial stresses are updated using elastic assumption.

From the above formulation we can derive the following relation:

ac bc, fc

αc bc= αc

κc αDp=

ε εe εp+=

σ C : ε εp–( )=

ε· λ·∂ fv

∂σ-------=

Δεp Δ tλ·∂ fc

∂σ-------

n 1+

Δλ∂ fc

∂J2D------------

∂J2D

∂σ------------

∂ fc

∂I1--------

∂I1

∂σ--------+

n 1+Δλ S

2 J2

cc

qc----- I1 qc+( )+

----------------------------------------------- κc

4cc

qc----- I1 qc+( )3

2 J2

cc

qc----- I1 qc+( )4

+

--------------------------------------------------–

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

+= = =

Δep Δλ S

2 J2

cc

qc----- I1 qc+( )

4+

--------------------------------------------------=

Δενp 3Δλ κc

4cs

qc----- I1 qc+( )3

2 J2

cc

qc----- I1 qc+( )4

+

--------------------------------------------------–=

Δλ

σn 1+ Ktraceεn 1+e 2Gen 1+

e+ σE KtraceΔεn 1+p– 2Gen 1+

p–= =

σn 1+ σE– Δλ 3K κc

4cc

qc----- I1 qc+( )3

2 J2

cc

qc----- I1 qc+( )

4+

--------------------------------------------------–

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1– 2GS

2 J2

cc

qc----- I1 qc+( )

4+

--------------------------------------------------–=


336

Using the Newton iteration scheme as follows:

The following relation is obtained.

Therefore, can be calculated as follows:

In this way the volumetric equivalent plastic strain can be updated with the consequence that the yield surface is growing. Therefore a few iterations are needed to bring the trial stresses, , back to the

updated yield surface with a chosen accuracy.

Using this model an excellent agreement between simulation and the experiment results has been achieved as mentioned in reference 1 and 4.

References

1. “Friction Mechanisms of Tread Blocks on Snow Surfaces”, R. Mundl, G. Meschke and W. Liederer, Tire Science and Technology, TSTCA, vol. 25, no. 4, 1997, pp. 245-264.

2. “A New Visco-plastic Model for Snow at Finite Strains”, G. Meschke, 4th International Conference on Computational Plasticity, Barcelona, April 3-6, 1995, Pineridge Press, pp. 2295-2306

3. “Material Models for Granular Soils”, F.L. DiMaggio and I.S. Sandler, Journal of Engineering Mechanics A.S.C.E., 1971, pp. 935-950

4. “Implementation and Verification of Snow Material in Euler Scheme; Using the type-10 element (hydrodynamic material with strength)”, M. Mahardika, MSC.Software, 2001, to be published.

Equations of StateEquations of state are referenced from the MATDEUL entry. The equation of state for a material is of the basic form

Pressure = f (density, specific internal energy)

fc σn 1+( ) fc σE( )∂ fc

∂σ-------

J1E

: σn 1+ σE–( )+≈

fc σE( ) S

2 J2

cc

qc

----- I1 qc+( )4+

--------------------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

:Δλ σn 1+ σE–( )+ 0=

Δλ

Δλfc σE( )

GJ2D

J2

cc

qc

----- I1 qc+( )4+

----------------------------------------------

----------------------------------------------------fc σE( )

9K κc

4cc

qc----- I1 qc+( )3

2 J2

cc

qc----- I1 qc+( )4

+

--------------------------------------------------–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ 2

GJ2D

J2

cc

qc----- I1 qc+( )4

+

------------------------------------------+

--------------------------------------------------------------------------------------------------------------------------------------= =

σn 1+


The simplest equation of state is the gamma law equation of state, defined by the EOSGAM entry. The only input required is the ratio of specific heats for an ideal gas.

The EOSPOL entry defines a polynomial equation of state.

The EOSTAIT entry defines an equation of state based on the Tait model in combination with a cavitation model.

The EOSJWL entry defines an equation of state based on the JWL explosive model. It is used to calculate the pressure of the detonation products of high explosives. The JWL model is empirically based and requires the input of five constants.

The EOSIG entry defines the properties of Ignition and Growth equation of state and the reaction rate equation used to model high explosives.

EOSGAM – Gamma Law Equation of State

The EOSGAM model defines a gamma law equation of state for gases where the pressure is a function of

the density, the specific internal energy, and the ideal gas ratio of specific heats of an ideal gas

The EOSGAM equation of state can also be used to model viscous gases.

EOSPOL – Polynomial Equation of State

The EOSPOL model defines a polynomial equation of state where the pressure is related to the relative volume and specific internal energy by a cubic equation.

In compression

In tension

where = specific internal energy unit mass

= overall material density

= ratio of specific heats

where =

=


γ

p γ 1–( )ρe=

e

ρ

γ Cp Cv⁄( )

μ 0>( )

p a1μ a2μ2 a3μ3 b0 b1μ b2μ2 b3μ3+ + +( )ρ0e+ + +=

μ 0≤( )

p a1μ b0 b1μ÷( )ρ0e+=

μ η 1–

η ρ ρ0⁄

ρ


338

The EOSPOL equation of state can also be used to model viscous fluids.

EOSTAIT – Tait Equation of State

The EOSTAIT model defines a equation of state based on the Tait model in combination with a cavitation model where the pressure is defined as follows:

No cavitation ,

Cavitation ,

The pressure can not fall below the cavitation pressure , although the density

can continue to decrease below its critical value .

The EOSTAIT equation of state can als be used to model viscous fluids.

EOSJWL – JWL Equation of State

This equation of state can be used only with Eulerian elements.

These parameters are defined in Reference 3.

= reference density

= specific internal energy per unit mass

where =


= reference density

= critical density which produces the cavitation pressure

where = specific internal energy per unit mass

= reference density


=

, and are constants.

ρ0

e

p

ρ ρc>( )

p a0 a1 ηγ 1–( )+=

ρ ρc≤( )

p pc=

η ρ ρ0⁄

ρ

ρ0

ρc pc

pc a0 a1 ρc( ) ρ0( )⁄( )γ1–( )+=

ρc

E

ρ0

ρ

PA 1

ωηR1--------–⎝ ⎠

⎛ ⎞ eR1 η⁄–

B 1ωηR2--------–⎝ ⎠

⎛ ⎞ eR2 η⁄–

ωηρ0e+ +

η ρ ρ0⁄=

A B ω R1, , , R2


A DETSPH entry must be used to specify the detonation time, the location of the detonation point, and the velocity of a spherical detonation wave. When no DETSPH entry is present, all the material detonates immediately and completely.

EOSMG - Mie-Gruneisen Equation of State

The Mie-Gruneisen equation is useful in high-strain rate processes. The pressure is split in a part that only depends on density and a part that only depends on temperature.

The cold pressure is computed from the Rankine-Hugenot equations and is given by

Here is the reference density, is the speed of sound, is the volumetric compressive

strain. The defining equation for the parameter s is the linearization of the relationship between linear shock speed and particle velocity .

The thermal part of the pressure follows from thermodynamic considerations and reads

where is the specific internal energy and the parameter is given by

where is the isothermal bulk modulus, the specific heat at constant volume and the volumetric

thermal expansion coefficient. is the Gruneisen parameter at reference density. The Gruneisen

parameter at other densities is given by .

Material FailureOne of the nonlinear features of a material's behavior is failure. When a certain criterion -the failure criterion- is met, the material fails and can no longer sustain its loading and breaks. In a finite-element method this means that the element where the material reaches the failure limit, cannot carry any stresses anymore. The stress tensor is effectively zero. The element is flagged for failure and essentially is no longer part of the structure.

Failure criteria can be defined for a range of materials and element types. The failure models are referenced from the material definition entries.

p pc pT+=

pc

ρ0c02η

1 sη–( )2----------------------- 1

Γ0η2

----------–⎝ ⎠⎛ ⎞=

ρ0 c0 η 1ρ0

ρ-----–=

Us Up

Us c0 sUp+=

pT Γ0ρ0e=

e Γ0

Γ0ρ0

βKT

CV----------=

KT CV β

Γ0

Γ Γ0

ρ0

ρ-----=


340

There are several different failure models available:

FAILMPS – Maximum Plastic Strain Failure Model

The most commonly used failure model is the one that is based on a maximum equivalent plastic strain. The material fails completely when the plastic strain reaches beyond the defined limit.

FAILJC – Johnson-Cook Failure Model

Failure is determined from a damage model. Damage is an element variable and increments are given by the plastic strain increment divided by a fracture strain. In addition the damage variable is transported along with material as it move from one Euler element to the other.

It is only available for the Multi-material Euler solver with strength. The use of coupling surfaces is not supported.

Spallation ModelsA spallation model defines the minimum pressure prior to spallation. At present there is only one spallation model, PMINC, that defines a constant spallation pressure.

PMINC – Constant Minimum Pressure

A constant minimum pressure must be defined that must be less than or equal to zero. Note that the pressure is positive in compression. If the pressure in an element falls below the minimum pressure, the element spall and the pressure and yield stress are set to zero. The material then behaves like a fluid. When the pressure subsequently becomes positive, the material will no longer be in a spalled state. The pressure can then decrease again to the specified minimum (the spall limit) before spallation occurs again.

Figure 7-35 Minimum Pressure Cutoff

FAILMPS Constant, maximum plastic strain.

FAILJC The Johnson-cook failure model

PMINC

Pressure

Volume


Material ViscosityViscous fluid material models are available for the Roe fluid solver only. The viscous behavior is referenced from the entry to define the equation of state (EOSPOL or EOSTAIT). For these viscous materials, the stress tensor is defined as:

and

where denotes the bulk modulus, the density, the deviatoric stress tensor, the pressure, the

deviatoric strain tensor, and the coefficient of viscosity. The Roe solver computes the stresses directly from the velocity gradients.

tij

tij p δi j Sij+⋅=

dpdt------ K

ρ---- dρ

dt------=

Sij 2μdeij

d

dt---------⋅=

K ρ Sij p eijd

μ


342

Chapter 8: Contact Impact AlgorithmMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide Overview

8Contact Impact Algorithm

Overview 344

Penalty Methods 346

Preliminaries 346

Slave Search 347

Contact Force Calculation 351

Improvements to the Contact Searching 352

Bucket Sorting 353

Accounting For the Shell Thickness 356

Initial Contact Penetrations 357

Contact Energy Calculation 358

Friction 359

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewOverview

344

OverviewThe input of contact data, Contact for Explicit Nonlinear (SOL 700), is defined similarly as for Implicit Nonlinear (SOL 600). It is instructive to first read Chapter 12 of the SOL 600 User’s Guide where an overview is given of the nonlinear contact methods.

This section will summarize those parts of the contact methods applicable to SOL 700, and will describe the theory of the algorithms used.

To activate contact in SOL 700, the Case Control entry BCONTACT must be given. There are three methods available to define contact:

1. All elements in the model in one contact definition, using default settings

Case Control: BCONTACT=ALL

This will result in automatic contact detection between all elements in the model. For SOL 700, it is advisable to use this method.

2. User defined Contact Bodies, using default or non-default settings

Case Control: BCONTACT = n

Bulk Data: BCBODY, BSURF, or BCBOX or BCPROP or BCMATL, and BCTABLE

It is possible to define contact bodies (BCBODY), and specify which contact bodies need to be checked for contact (BCTABLE). This method using BCBODY definitions provides extreme flexibility and is compatible with the Implicit Nonlinear solution (SOL 600).

A contact body is defined by the Bulk Data entry BCBODY, which references a set of elements (BSURF), a set of elements inside a box (BCBOX), or certain property IDs (BCPROP) or with certain material IDs (BCMATL).

Often used definitions related to contact methods are:

Single Surface Contact: This refers to any contact definition where no master is defined.

Master Slave Contact: This refers to any contact definition where a master is defined.

SOL 600 Contact Capabilities not yet supported by SOL 700

• 2-D contact, since SOL 700 only applies to 3-D (DIM on BCBODY)

• A rigid BCBODY defined by patches or geometric entities

- Rigid BCBODY as a symmetry plane- Motion/Load controlled rigid BCBODY

(See note below on rigid body modeling in SOL 700)

(BEHAV=RIGID

(ISTYP(CONTROL

on BCBODY)

on BCBODY)on BCBODY)

• Body smoothing (IDSPL on BCBODY)

• User-defined distance below which a node is considered touching

(ERROR on BCTABLE)

• Separation Force (FNTOL on BCTABLE)

345Chapter 8: Contact Impact AlgorithmOverview

Notes on Rigid Body Modeling in SOL 700:Currently, only BEHAV = DEFORM is supported in SOL 700 on the BCBODY option. Rigid body modeling is possible, however, by defining a rigid material (MATD020).

• Elements belonging to a MATD020 are properly treated in the contact calculations, and it is allowed to include them in a BCBODY with BEHAV = DEFORM. The contact calculations will operate as if the material is rigid. The penalty based contact forces applied on the nodes are accumulated for the whole rigid body and applied as an external force and moment to the center-of-gravity of the rigid body.

• Rigid body motion is allowed and properly simulated by SOL 700.

• Initial velocities and boundary conditions acting on the nodes will be applied to the rigid body.

• Rigid body must be assigned in “Masters” field in BCTABLE.

With these capabilities, a faceted rigid body, similar to a BCBODY with BEHAV=RIGID can be easily modeled. The BEHAV=RIGID logic will be implemented in the next release of SOL 700.

SOL 600 Contact Limitations that do not apply to SOL 700To allow switching between SOL 700 and SOL 600, it is advisable to work within these limits.

In SOL 600, each node and element should be in at most one contact body. When using the penalty method of SOL 700 (the default), this limitation does not apply. Forces as calculated by each contact are simply accumulated and applied as an external force vector.

In SOL 600, only 1000 contact bodies are allowed. This limit does not apply to SOL 700.

In SOL 600, it is important to properly define the order in which contact bodies are defined for deformable-to-deformable contact. The order of contact body definition has no influence on the results in SOL 700.

• Interference closure (CINTERF on BCTABLE)

•Glue options

(IGLUE & JGLUE

on BCTABLE)

• Contact heat transfer (HEATC on BCTABLE)

• Searching order (ISEARCH on BCTABLE)

• Initial node movement for stress-free contact (SOL 700 contact offers similar capability by means of the IGNORE option on the BCTABLE)

(ICOORD on BCTABLE)

•Option to identify relevant nodes

(BCHANGE entry)

•Initial approach, release, motion until contact

(BCMOVE entry)

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewPenalty Methods

346

Penalty MethodsThe treatment of sliding and impact along interfaces is an important capability in SOL 700.

Internally, the interfaces are defined in three dimensions by listing in arbitrary order all triangular and quadrilateral segments that comprise each side of the interface. One side of the interface is designated as the slave side, and the other is designated as the master side. Nodes lying in those surfaces are referred to as slave and master nodes, respectively. When slave nodes penetrate the master, normal interface springs are placed between the penetrated nodes and the master surface. See Slave Search in this chapter for details on the contact force calculations.

PreliminariesConsider the time-dependent motion of two bodies occupying regions and in their undeformed configuration at time zero. Assume that the intersection

(8-1)

is satisfied. Let and denote the boundaries of and , respectively. At some later time, these

bodies occupy regions and bounded by and as shown in Figure 8-1. Because the deformed configurations cannot penetrate,

(8-2)

Figure 8-1 Reference and Deformed Configuration

As long as , the equations of motion remain uncoupled. In the foregoing and following

equations, the right superscript denotes the body to which the quantity refers.

B1 B2

B1 B2∩ φ=

∂B1 ∂B2 B1

B2

b1 b2 ∂b1 ∂b2

b1 ∂b1–( ) b2∩ φ=

∂b2

∂B1

∂b1

∂B2

B01

B02

b1 b

2

∂b1 ∂b2∩ φ=

α 1 2,=( )

347Chapter 8: Contact Impact AlgorithmSlave Search

Before a detailed description of the theory is given, some additional statements should be made concerning the terminology. The surfaces and of the discretized bodies and become the master and slave surfaces respectively. Choice of the master and slave surfaces is arbitrary when the symmetric penalty treatment is employed. Otherwise, the more coarsely meshed surface should be chosen as the master surface, nodal points that define are called master nodes and nodes that define

are called slave nodes. When , the constraints are imposed to prevent penetration.

Right superscripts are implied whenever a variable refers to either the master surface , or slave surface, ; consequently, these superscripts are dropped in the development which follows.

Slave SearchThe slave search is common to all interface algorithms implemented in SOL 700. This search finds for each slave node its nearest point on the master surface. Lines drawn from a slave node to its nearest point will be perpendicular to the master surface, unless the point lies along the intersection of two master segments, where a segment is defined to be a 3- or 4-node element of a surface.

Consider a slave node, , sliding on a piecewise smooth master surface and assume that a search of the

master surface has located the master node, , lying nearest to . Figure 8-2 depicts a portion of a

master surface with nodes and labeled. If and do not coincide, can usually be shown to

lie in a segment via the following tests:

(8-3)

where vector and are along edges of and point outward from . Vector s is the projection of

the vector beginning at , ending at , and denoted by , onto the plane being examined (see

Figure 8-3).

Figure 8-2 Four Master Segments can Harbor Slave Node given that is the Nearest Master Node

∂b1 ∂b2 b1 b2

∂b1

∂b2 ∂b1 ∂b2∩( ) φ≠

∂b1

∂b2

ns

ms ns

ms ns ms ns ns

ss

ci s×( ) ci ci 1+×( ) 0>⋅

ci s×( ) s ci 1+×( ) 0>⋅

ci ci 1+ s1 ms

ms ns g

ns ms

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewSlave Search

348

Figure 8-3 Projection of onto Master Segment

(8-4)

where for segment

(8-5)

Since the sliding constraints keep close but not necessarily on the master surface and since may lie

near or even on the intersection of two master segments, the inequalities of Equation (8-3) may be inconclusive; i.e., they may fail to be satisfied or more than one may give positive results. When this occurs is assumed to lie along the intersection which yields the maximum value for the quantity

(8-6)

When the contact surface is made up of badly shaped elements, the segment apparently identified as containing the slave node actually may not, as shown in Figure 8-4.

Assume that a master segment has been located for slave node and that is not identified as lying on

the intersection of two master segments. Then the identification of the contact point, defined as the point on the master segment which is nearest to , becomes nontrivial. Each master surface segment , is

given the parametric representation:

(8-7)

where

(8-8)

g s1

s g g m⋅( )m–=

s1

mci ci 1+×ci ci 1+×

--------------------------=

ns ns

ns

g ci⋅ci

------------ i 1 2 3 4 …, , , ,=

ns ns

ns s1

r f1 ξ η,( ) i1 f2 ξ η,( ) i2 f3 ξ η,( ) i3+ +=

fi ξ η,( ) φjxij

j 1=

4

∑–

349Chapter 8: Contact Impact AlgorithmSlave Search

When the nearest node fails to contain the segment that harbors the slave node, segments numbered 1-8 are searched in the order shown in Figure 8-4.

Figure 8-4 Search Order for Segments

Note that is at least once continuously differentiable and that

(8-9)

Thus, represents a master segment that has a unique normal whose direction depends continuously on

the points of .

Let be a position vector drawn to slave node and assume that the master surface segment has

been identified with . The contact point coordinates on must satisfy

(8-10)

(8-11)

The physical problem is illustrated in Figure 8-5, which shows lying above the master surface.

Equations (8-10) and (8-11) are readily solved for and . One way to accomplish this is to solve

Equation (8-10) for in terms of , and substitute the results into (8-11). This yields a cubic equation

in .

12

3

4

5

6 7

8

r1

∂r∂ξ------ ∂r

∂η-------× 0≠

r

s1

t ns s1

sn ξc ηc,( ) s1

∂r∂ξ------ ξc ηc,( ) t r ξc ηc,( )–[ ]⋅ 0=

∂r∂η------- ξc ηc,( ) t r ξc ηc,( )–[ ]⋅ 0=

ns

ξc ηc

ξc ηc

ηc

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewSlave Search

350

Figure 8-5 Location of Contact Point when lies above Master Segment

The equations are solved numerically. When two nodes of a bilinear quadrilateral are collapsed into a single node for a triangle, the Jacobian of the minimization problem is singular at the collapsed node. Fortunately, there is an analytical solution for triangular segments since three points define a plane. Newton-Raphson iteration is a natural choice for solving these simple nonlinear equations. The method diverges with distorted elements unless the initial guess is accurate. An expanded search procedure as discussed in Improvements to the Contact Searching section is used.

Three iterations with a least-squares projection are used to generate an initial guess:

(8-12)

followed by the Newton-Raphson iterations which are limited to ten iterations, but which usually converges in four or less.

(8-13)

In concave regions, a slave node may have isoparametric coordinates that lie outside of the range for all of the master segments, yet still have penetrated the surface. A simple strategy is used for handling this case, but it can fail. The contact segment for each node is saved every time step.

x3

x2

x1

t

r1

23

4

ns

η

ξ

∂r∂η-------

∂r∂ξ------

ns

ξ0 0 η0, 0,= =

r,ξ

r,η

r,ξr,η[ ] ΔξΔη⎩ ⎭

⎨ ⎬⎧ ⎫ r,ξ

r,η

r ξi ,ηi( ) t–{ } ,=

ξi 1+ ξi 1+ Δξ , ηi 1++ ηi Δη+= =

H[ ]ΔξΔη⎩ ⎭

⎨ ⎬⎧ ⎫ r ξ,

r η,⎩ ⎭⎨ ⎬⎧ ⎫

r ξi ηi,( ) t–{ }–=

H[ ]r χ,

r η,⎩ ⎭⎨ ⎬⎧ ⎫

r ξ, r η,[ ]0 r r ξη,⋅

r r ξη,⋅ 0+

ξi 1+ ξi Δξ+=

ηi 1+ ηi Δη+=

1 +1,–[ ]

351Chapter 8: Contact Impact AlgorithmContact Force Calculation

If the slave node contact point defined in terms of the isoparametric coordinates of the segment, is just outside of the segment, and the node penetrated the isoparametric surface, and no other segment associated with the nearest neighbor satisfies the inequality test, then the contact point is assumed to occur on the edge of the segment.

In effect, the definition of the master segments is extended so that they overlap by a small amount. In the hydrocode literature, this approach is similar to the slide line extensions used in two dimensions. This simple procedure works well for most cases, but it can fail in situations involving sharp concave corners.

Contact Force CalculationEach slave node is checked for penetration through the master surface. If the slave node does not penetrate, nothing is done. If it does penetrate, an interface force is applied between the slave node and its contact point. The magnitude of this force is proportional to the amount of penetration. This may be thought of as the addition of an interface spring.

Penetration of the slave node through the master segment which contains its contact point is

indicated if

(8-14)

where

(8-15)

is normal to the master segment at the contact point. The amount of penetration is equal to the value of l.

If slave node has penetrated through master segment , we add an interface force vector :

if l < (8-16)

to the degrees of freedom corresponding to and

if (8-17)

to the four nodes that comprise master segment . By default, SOFT=1 on the BCTABLE

entry, and the stiffness factor is given in terms of the nodal masses and the global timestep, as

(8-18)

With SOFT=1, it is mostly not needed to scale the contact stiffness, even if materials with very different stiffness properties come into contact.

When SOFT=0 on the BCTABLE entry, the stiffness factor for master segment is given in terms of

the bulk modulus , the volume , and the face area of the element that contains as

ns

l ni t r ξc ηc,( )–[ ] 0<( )=

ni ni ξc ηc,( )=

ns si fs

fs lki– ni= l 0<

ns

fmi φ2 ξc ηc,( ) fs= l 0<

i 1 2 3 4, , ,=( ) si

ki d t

ki fsi

mslavemmaster

mslave mmaster+----------------------------------------- 1

dt2

-------⋅ ⋅=

ki si

Ki Vi Ai si

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewImprovements to the Contact Searching

352

(8-19)

for brick elements and

(8-20)

for shell elements. When too much penetration is observed, the contact stiffness can be increased by the

interface stiffness scale factor, . This scale factor is 0.1 by default, and can be defined by FACT on the

BCTABLE entry. Larger values may cause instabilities unless the time step size is scaled back in the time step calculation.

Improvements to the Contact SearchingA number of recent changes have been made in the surface-to-surface contact including contact searching, accounting for thickness, and contact damping.

Sometimes problems with the closest master node contact searching were found. The nearest node algorithm described above can break down since the nearest node is not always anywhere near the segment that harbors the slave node as is assumed in Figure 8-4 (see Figure 8-6). Such distorted elements are commonly used in rigid bodies in order to define the geometry accurately.

Figure 8-6 Failure to find the Contact Segment can be caused by Poor Aspect Ratios in the Finite Element Mesh

To circumvent the problem caused by bad aspect ratios, an expanded searching procedure is used in which we attempt to locate the nearest segment rather than the nearest nodal point.

Nodes 2 and 4 share segments with node 3. Therefore, the two nearest nodes are 1 and 5. The nearest contact segment is not considered since its nodes are not members of the nearest node set.

ki

fsiKiAi2

Vi------------------=

ki

fsiKiAi

max shell diagonal( )--------------------------------------------------=

fsi

slave node

closest nodal point

353Chapter 8: Contact Impact AlgorithmBucket Sorting

The nearest contact segment to a given node, , is defined to be the first segment encountered when moving in a direction normal to the surface away from . A major deficiency with the nearest node search is depicted in Figure 8-7 where the nearest nodes are not even members of the nearest contact segment. Obviously, this would not be a problem for a more uniform mesh. To overcome this problem we have adopted segment based searching in both surface to surface and single surface contact.

Figure 8-7 Expanded Search

Bucket SortingBucket sorting is now used extensively in the SOL 700 contact algorithms.

The reasons for eliminating slave node tracking by incremental searching is illustrated in Figure 8-8 where surfaces are shown which cause the incremental searches to fail. With bucket sorting incremental searches may still be used but for reliability they are used after contact is achieved. As contact is lost, the bucket sorting for the affected nodal points must resume.

Figure 8-8 Examples of Models where Incremental Searching may Fail

k

k

1 2 3 4 5

Normal vectorat node 3

tied interfacenot yet supported

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewBucket Sorting

354

In a direct search of a set of nodes to determine the nearest node, the number of distance comparisons required is . Since this comparison needs to be made for each node, the total number of comparisons

is , with each of these comparisons requiring a distance calculation

(8-21)

that uses eight mathematical operations. The cumulative effect of these mathematical operations for compares can dominate the solution cost at less than 100 elements.

The idea behind a bucket sort is to perform some grouping of the nodes so that the sort operation need only calculate the distance of the nodes in the nearest groups. With this partitioning the nearest node will either reside in the same bucket or in one of the two adjoining buckets. The number of distance calculations is now given by

(8-22)

where is the number of buckets. The total number of distance comparisons for the entire one-dimensional surface is

(8-23)

Thus, if the number of buckets is greater than 3, then the bucket sort will require fewer distance comparisons than a direct sort. It is easy to show that the corresponding number of distance comparisons for two-dimensional and three-dimensional bucket sorts are given by

for 2-D (8-24)

for 3-D (8-25)

where and are the number of partitions along the additional dimension.

Incremental searching may fail on surfaces that are not simply connected. The contact algorithm in SOL 700 avoids incremental searching for nodal points that are not in contact and all these cases are considered (see Figure 8-6).

The cost of the grouping operations, needed to form the buckets, is nearly linear with the number of nodes . For typical SOL 700 applications, the bucket sort is 100 to 1000 times faster than the corresponding

direct sort. However, the sort is still an expensive part of the contact algorithm, so that, to further minimize this cost, the sort is performed every ten or fifteen cycles and the nearest three nodes are stored. This can be specified by BSORT on the BCTABLE entry. Typically, three to five percent of the calculational costs will be absorbed in the bucket sorting when most surface segments are included in the contact definition.

N

N 1–

N N 1–( )

12 xi xj–( )2 yi yj–( )2 zi zj–( )2+ +=

N N 1–( )

3Nα

------- 1–

α

N3Nα

------- 1–⎝ ⎠⎛ ⎞

N9Nαb------- 1–⎝ ⎠⎛ ⎞

N27Nαbc---------- 1–⎝ ⎠⎛ ⎞

b c

N

355Chapter 8: Contact Impact AlgorithmBucket Sorting

Bucket Sorting in Single Surface ContactWe set the number of buckets in the x, y, and z coordinate directions to NX, NY, and NZ, respectively.

The product of the number of buckets in each direction always approaches NSN or 5000 whichever is smaller,

(8-26)

where the coordinate pairs , , and span the entire contact surface. In this

procedure, we loop over the segments rather than the nodal points. For each segment we use a nested DO LOOP to loop through a subset of buckets from to , to , and to where:

(8-27)

and , , are the bucket pointers for the kth node. Figure 8-9 shows a segment passing through a volume that has been partitioned into buckets. The orthogonal distance of each slave node contained in the box from the segment is determined. The box is subdivided into sixty buckets.

Figure 8-9 The Orthogonal Distance of each Slave Node

NX NY NZ MIN NSN 5000,( )≤⋅⋅

xmin xmax,( ) ymin ymax,( ) zmin zmax,( )

IMIN IMAX JMIN JMAX KMIN KMAX

IMIN MIN PXI PX2 PX3 PX4,,,( )=

IMAX MAX PX1 PX2 PX3 PX4,,,( )=

JMIN MAX PY1 PY2 PY3 PY4,,,( )=

JMAX MAX PY1 PY2 PY3 PY4,,,( )=

KMIN MAX PZ1 PZ2 PZ3 PZ4,,,( )=

kMAX MAX PZ1 PZ2 PZ3 PZ4,,,( )=

PXk PYk PZk

z

x

y

Nodes in buckets shown are checkedfor contact with the segment

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewAccounting For the Shell Thickness

356

We check the orthogonal distance of all nodes in the bucket subset from the segment. As each segment is processed, the minimum distance to a segment is determined for every node in the surface and the two nearest segments are stored. Therefore the required storage allocation is still deterministic. This would not be the case if we stored for each segment a list of nodes that could possibly contact the segment.

We have now determined for each node, , in the contact surface the two nearest segments for contact. Having located these segments we permanently store the node on these segments which is nearest to node

. When checking for interpenetrating nodes we check the segments surrounding the node including the nearest segment since during the steps between bucket searches it is likely that the nearest segment may change. It is possible to bypass nodes that are already in contact and save some computer time; however, if multiple contacts per node are admissible then bypassing the search may lead to unacceptable errors.

Accounting For the Shell ThicknessShell thickness effects are important when shell elements are used to model sheet metal. Unless thickness is considered in the contact, the effect of thinning on frictional interface stresses due to membrane stretching will be difficult to treat. In the treatment of thickness we project both the slave and master surfaces based on the mid-surface normal projection vectors as shown in Figure 8-10. The surfaces, therefore, must be offset by an amount equal to 1/2 their total thickness (Figure 8-11). This allows the program to check the node numbering of the segments automatically to ensure that the shells are properly oriented.

Figure 8-10 Contact Surface Based Upon Midsurface Normal Projection Vectors

k

k

Length of projection vector is 1/2 the shell thickness

Projected Contact Surface

357Chapter 8: Contact Impact AlgorithmInitial Contact Penetrations

Figure 8-11 The Slave and Master Surfacess

Thickness changes in the contact are accounted for “if and only if” the shell thickness change option is flagged on the PARAM* DYCONTHKCHG. Each cycle, as the shell elements are processed, the nodal thicknesses are stored for use in the contact algorithms. The interface stiffness may change with thickness depending on the input options used.

To account for the nodal thickness, the maximum shell thickness of any shell connected to the node is taken as the nodal thickness and is updated every cycle. The projection of the node is done normal to the contact surface:

Initial Contact PenetrationsThe need to offset contact surfaces to account for the thickness of the shell elements contributes to initial contact penetrations. These penetrations can lead to severe numerical problems when execution begins so they should be corrected if SOL 700 is to run successfully. Often an early growth of negative contact energy is one sign that initial penetrations exist. Currently, warning messages are printed to the D3HSP file to report penetrations of nodes through contact segments and the modifications to the geometry made by SOL 700 to eliminate the penetrations. Sometimes such corrections simply move the problem elsewhere since it is very possible that the physical location of the shell mid-surface and possibly the shell thickness are incorrect. In the single surface contact algorithms any nodes still penetrating on the second time step are removed from the contact with a warning message.

In some geometry's, penetrations cannot be detected since the contact node penetrates completely through the surface at the beginning of the calculation. Such penetrations are frequently due to the use of coarse meshes. This is illustrated in Figure 8-12. Another case contributing to initial penetrations occurs when the edge of a shell element is on the surface of a solid material as seen in Figure 8-13. Currently, shell edges are rounded with a radius equal to one-half the shell thickness.

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewContact Energy Calculation

358

Figure 8-12 Undetected Penetration

Figure 8-13 Undetected Penetration due to rounding the Edge of the Shell Element

To avoid problems with initial penetrations, the following recommendations should be considered:

• Adequately offset adjacent surfaces to account for part thickness during the mesh generation phase.

• Use consistently refined meshes on adjacent parts which have significant curvatures.

• Be very careful when defining thickness on shell and beam section definitions - especially for rigid bodies.

• Scale back part thickness if necessary. Scaling a 1.5mm thickness to .75mm should not cause problems but scaling to .075mm might. Alternatively, define a smaller contact thickness by part ID. Warning: if the part is too thin contact failure will probably occur

• Use spot welds instead of merged nodes to allow the shell mid surfaces to be offset.

Contact Energy CalculationContact energy, , is incrementally updated from time to time for each contact interface as:

Detected Penetration Undetected Penetration

shell

brick

Inner penetration ifedge is too close

Econtact n n 1+

Econtactn 1+ Econtact

n ΔFislave Δdisti

slave ΔFimaster Δdisti

master×i 1=

nmn

∑+×i 1=

nsn

∑n

12---+

+=

359Chapter 8: Contact Impact AlgorithmFriction

Where is the number of slave nodes, is the number of master nodes, is the interface force

between the ith slave node and the contact segment is the interface force between the ith master

node and the contact segment, is the incremental distance the ith slave node has moved during

the current time step, and is the incremental distance the ith master node has moved during

the current time step. In the absence of friction the slave and master side energies should be close in magnitude but opposite in sign. The sum, , should equal the stored energy. Large negative contact

energy is usually caused by undetected penetrations. Contact energies are reported in the GLSTAT file. In the presence of friction and damping discussed below the interface energy can take on a substantial positive value especially if there is, in the case of friction, substantial sliding.

FrictionFriction in SOL 700 is based on a Coulomb formulation. Let be the trial force, the normal force,

k the interface stiffness, the coefficient of friction, and the frictional force at time n. The frictional algorithm, outlined below, uses the equivalent of an elastic plastic spring. The steps are as follows:

1. Compute the yield force, :

(8-28)

2. Compute the incremental movement of the slave node

(8-29)

3. Update the interface force to a trial value:

(8-30)

4. Check the yield condition:

if (8-31)

5. Scale the trial force if it is too large:

if (8-32)

An exponential interpolation function smooths the transition between the static, , and dynamic, ,

coefficients of friction where is the relative velocity between the slave node and the master segment:

(8-33)

where

(8-34)

is the time step size, and is a decay constant.

nsn nmn ΔFislave

ΔFimaster

Δdistislave

Δdistimaster

Econtact

f * fn

μ f n

Fy

Fy μ fn=

Δe rn 1+ ξcn 1+ ηc

n 1+,( ) rn 1+ ξcn ηc

n,( )–=

f∗ fn kΔe–=

fn 1+ f∗= f∗ Fy≤

fn 1+Fyf∗

f∗-----------= f∗ Fy>

μs μd

ν

μ μd μs μd–( )e c ν–+=

ν ΔeΔT---------------=

Δ t C

MD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide OverviewFriction

360

Typical values of friction, see Table 8-1, can be found in Marks Engineering Handbook.

Table 8-1 Typical Values of Coulomb Friction [Marks]

MATERIALS STATIC SLIDING

Hard steel on hard steel 0.78 (dry) .08 (greasy), .42 (dry)

Mild steel on mild steel 0.74 (dry) .10 (greasy), .57 (dry)

Aluminum on mild steel 0.61 (dry) .47 (dry)

Aluminum on aluminum 1.05 (dry) 1.4 (dry)

Tires on pavement (40psi) 0.90 (dry) .69(wet), .85(dry)

Chapter 9: Fluid Structure InteractionMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

9Fluid Structure Interaction

General Coupling 362

Multiple Coupling Surfaces with Multiple Euler Domains 367

Fluid- and Gas Solver for the Euler Equations 371

Modeling Fluid Filled Containers 372

Hotfilling 373

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362

General Coupling

Fluid-structure InteractionThe objective of fluid-structure interaction using the coupling algorithm is to enable the material modeled in Eulerian and Lagrangian meshes to interact. Initially, the two solvers are entirely separate. Lagrangian elements that lie within an Eulerian mesh do not affect the flow of the Eulerian material and no forces are transferred from the Eulerian material back to the Lagrangian structure. The coupling algorithm computes the interaction between the two sets of elements. It thus enables complex fluid-structure interaction problems to be analyzed.

The first task in coupling the Eulerian and Lagrangian sections of a model is to create a surface on the Lagrangian structure. This surface is used to transfer the forces between the two solver domains. The surface acts as a boundary to the flow of material in the Eulerian mesh. At the same time, the stresses in the Eulerian elements cause forces to act on the coupling surface, distorting the Lagrangian elements.

By means of a BSURF, BCPROP, BCMATL, or BCSEG entry, you can define a multifaceted surface on the Lagrangian structure. A set of BCSEGs, element numbers, property numbers, material numbers, or any combination of these identify the element faces in this surface. The method of defining of the surface is therefore extremely flexible and can be adapted to individual modeling needs.

The coupling algorithm is activated using the COUPLE entry. It specifies that the surface is used for Euler-Lagrange coupling. You can define whether the inside or the outside domain is covered by the coupling surface by setting the COVER field on the entry. This means that the Euler domain cannot contain material where it is covered by the outside or the inside of the Lagrangian structure. For problems where the Eulerian material is inside a Lagrangian structure (for example, an inflating air bag), COVER should be set to OUTSIDE since the Eulerian elements outside the coupling surface must be covered. For problems where the Eulerian material is outside the Lagrangian structure (for example a projectile penetrating soft material), the inside of the coupling surface must covered, and COVER should be set to INSIDE.

The coupling surface must have a positive volume. This means that the normals of all the segments of the surface must point outwards. By default, the solver will check the direction of the normal vectors and automatically reverse them when necessary. However, if you wish to switch of the check to save some computational time in the generation of the problem, you can define this using the REVERSE field on the COUPLE entry.

The coupling algorithm activated using the COUPLE entry is the most general interaction algorithm. It can handle any Euler mesh. There is an option, however, to switch to a faster algorithm by setting the parameter DYPARAM, FASTCOUP. This algorithm makes use of knowledge of the geometry of the Euler mesh. As a result, the requirement is that the Euler mesh must be aligned with the basic coordinate system axes.

363Chapter 9: Fluid Structure InteractionGeneral Coupling

Closed VolumeThe coupling surface must form a closed volume. This requirement is fundamental to the way the coupling works. It means that there can be no holes in the surface and the surface must be closed.

In order to create a closed volume, it may be necessary to artificially extend the coupling surface in some problems. In the example shown below, a plate modeled with shell elements is interacting with an Eulerian mesh. In order to form a closed coupling surface, dummy shell elements are added behind the plate. Dummy shell elements can be created using material MATD009. The shape of these dummy shell elements does not matter and it is best to use as few as possible to make the solution more efficient.

The closed volume formed by the coupling surface must intersect at least one Euler element otherwise the coupling surface is not recognized by the Eulerian mesh.

Care must be taken when doing so, however. The additional grid points created for the dummy elements do not move, since they are not connected to any structural elements. When the shell elements move so far that they pass beyond these stationary grid points, the coupling surface turns inside out and has a negative volume causing the solver to terminate.

Dummy Elementsto Form a ClosedCoupling Surface(Property 200)

LagrangianShell Elements(Property 200)

Eulerian Domain

COUPLE, 1, 10, INSIDEBCPROP, 10, 100, 200PSHELL, 100, 100, 0.05PSHELL, 200, 999, 1.0E-6MATD009, 999, 1.0E-20


364

PorosityThere is a general purpose capability to model porosity of a coupling surface. Porosity allows material to flow from the Eulerian region through the coupling surface or vice-versa. This method is addressed using the PORFLOW, PORFCPL entry. PORFLOW is used to model the interaction between an Eulerian region to the environment while PORFCPL is used for flow from an Eulerian region to another one. For air bags, a newer and better methodology for modeling porosity has been implemented.

The coupling surface or parts of it can be made porous by referring to a LEAKAGE entry from the COUPLE entry. This will be further explained by means of two examples. The first example models an air bag with porous material and two holes using PORFLOW:

The required input is:

PSHELL, 1, 100 , 1.E-3PSHELL, 100, 999, 1.E-6BCPROP, 11, 1, 100BCPROP, 100, 1COUPLE, 1, 11, OUTSIDE, , , 55LEAKAGE, 1, 55, , PORFLOW, 42, CONSTANT, 0.009LEAKAGE, 2, 55, 22, PORFLOW, 42, CONSTANT, 1.0PORFLOW, 42, , MATERIAL, 33, PRESSURE, 1.E-5, METHOD, PRESSURE

The second example models two chambers divided by a membrane with a hole. Two sets of Euler elements must be defined in which each set belongs to each coupling surface (COUPLE). The interaction between the two sets of Euler elements is defined by using a hole. The hole is modeled by using LEAKAGE that refers to PORFCPL.

Membrane Elements with PID = 1Dummy Elements with PID = 100

Dummy Elements with PID = 100

Hole

High Pressure

Low Pressure

365Chapter 9: Fluid Structure InteractionGeneral Coupling

The required input is:

chamber 1 (low pressure)BSURF,1,.....BSURF,10,1,....COUPLE,60,1,,,,70,,,++,,,,,,,,,++,,21LEAKAGE,80,70,10,PORFCPL,50, ,<coeffv>PORFCPL,50,,,,1020MESH,21,BOX,,,,,,,++,0.,0.,0.,1.,1.,1.,,,++,10,10,10,,,,EULER,500PEULER1,500,,HYDRO,19TICEUL1,19,1 (initialization to low pressure)

chamber 2 (high pressure)BSURF,2,.....COUPLE,1020,2,,,,,,,++,,,,,,,,,++,,22MESH,22,BOX,,,,,,,++,0.8,0.8,0.8,2.,2.,2.,,,++,10,10,10,,,,EULER,600PEULER1,600,,HYDRO,20TICEUL1,20,20 (initialization to high pressure)

Note that the porosity characteristics need to be defined for chamber 1 only. The gas automatically flows from chamber 2 into 1 and vice versa.

Two different algorithms are available to calculate the mass transport through the coupling surface. The desired method can be activated from the PORFLOW entry or by choosing PORFCPL with SIZE=LARGE for velocity method and PORFCPL with SIZE=SMALL for pressure method.

1. Velocity method

This algorithm is activated by:

PORFLOW, 42, , MATERIAL, 33, PRESSURE, 1.E-5, METHOD, VELOCITY

The transport of mass through the porous area is based on the velocity of the gas in the Eulerian elements, relative to the moving coupling surface.

.

Face of the couplingsurface that intersectsthe Eulerian element

v

Eulerian Element

Coupling Surface

SXint


366

The volume of the Eulerian material transported through the faces of the coupling surface that intersect an Eulerian element is equal to

The transported mass through the porous area is equal to the density of the gas times the transported volume.

2. Pressure Method

This algorithm is activated by:

PORFLOW, 42, , MATERIAL, 33, PRESSURE, 1.E5, METHOD, PRESSURE.

The transport of mass through the porous area is based on the pressure difference between the gas in the Eulerian element and the outside pressure. The outside pressure is the pressure as specified on the PORFLOW entry.

The volume of the Eulerian material transported through the faces of the coupling surface that intersect an Eulerian element is equal to:

where = transported volume during one time step ( for outflow;

for inflow)

= time step

= porosity coefficient

= velocity vector of the gas in the Eulerian mesh

= area of the face of the coupling surface that intersects the Eulerian element is equal to the area of the face that lies inside the Eulerian element.

Vtrans d t α v A⋅( )⋅ ⋅=

Vtrans Vtrans 0>

Vtrans 0<

dt

α

v

A

A


v

Eulerian Element

Coupling Surface

SXint

Vtrans d t α A2

γ 1–------------ p

ρ---

pexh

p----------⎝ ⎠⎛ ⎞

2γ---

pexh

p----------⎝ ⎠⎛ ⎞

γ 1+γ

------------

–⋅ ⋅ ⋅ ⋅=

367Chapter 9: Fluid Structure InteractionMultiple Coupling Surfaces with Multiple Euler Domains

The pressure at the face is approximated by the one-dimensional isentropic expansion of the gas to the critical pressure or the environmental pressure according to

where is the critical pressure:

In case the outside pressure is greater than the pressure of the gas, inflow through the coupling surface will occur. This porosity model can only be used for ideal gases; i.e., materials modeled with the gamma law equation of state (EOSGAM).

Multiple Coupling Surfaces with Multiple Euler DomainsMultiple coupling surfaces are available for HYDRO, MMHYDRO, and MMSTREN Euler Solvers in combination with the fast coupling algorithm. It is not available for the Strength Euler solver. The fast coupling algorithm is activated by the DYPARAM,FASTCOUP entry. Each couple surface is associated with an Eulerian domain. An Eulerian domain is a mesh that is aligned with the basic coordinate axes. You can define such a domain using the MESHID on the COUPLE entry.

Through a surface that is shared by two coupling surfaces, mass can flow from one coupling surface to the other. Such a surface is called a hole. A hole can be either a porous subsurface of a coupling surface or be part of a coupling surface with interactive failure.

where = transported volume during one time step ( for outflow;

for inflow)

= time step

= porosity coefficient

= area of the face of the coupling surface that intersects the Eulerian element is equal to the area of the face that lies inside the Eulerian element

= pressure of the gas in the Eulerian element

= density of the gas in the Eulerian element

= adiabatic exponent

= pressure at the face

Vtrans Vtrans 0>

Vtrans 0<

dt

α

A

A

p

ρ

γ Cp Cv⁄=

pexh

ppenv penv pc>( )

pc penv pc<( )⎩⎪⎨⎪⎧

=

pc

pc p2

γ 1+------------⎝ ⎠⎛ ⎞

γγ 1–------------

⋅=

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideMultiple Coupling Surfaces with Multiple Euler Domains

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If all Euler domains are defined by MESH, BOX, or MESH, ADAPT, then there are no restrictions on the use multiple coupling surfaces. In that case, a simulation may both contain porous holes that connect one Euler domain to another as well as coupling surfaces with interactive failure.

With the multi-material solver, Multiple Euler domains are only supported if all Euler domains are defined by either MESH,BOX or MESH,ADAPT.

Coupling Surface with FailureA coupling surface is always associated with a Lagrangian structure. When the material model used in the Lagrangian structure supports failure (for example, by defining a failure model for the material), the faces in the coupling surface can fail when the underlying material in the structure fails. You can define the failure mode for the coupling surface by specifying “DYPARAM,FASTCOUP, ,FAIL”.

When a Lagrangian element fails and the element is shared by two coupling surfaces, mass from one Eulerian domain flows to the other Eulerian domain through the hole. The interaction between these Eulerian domains is defined through a COUPINT entry. When you do not define the interaction between the Eulerian domains but the coupling surface fails, default ambient values for the state variables are used to compute the in- or outflow through the hole in the surface. The ambient values of the variables can be defined on the COUP1FL entry.

Coupling Surfaces with Porous HolesThe porous hole is a surface that is shared by two coupling surfaces and connects the two coupling surfaces to each other. By selecting either the porosity model PORFLCPL or PORFCP, flow is enabled from the Euler domain in one coupling surface to the Euler domain in the other coupling surface. The model PORFCPL with SIZE=LARGE uses the velocity method and is for general use whereas PORFCPL with SIZE=SMALL uses the pressure method and is only for small holes. The porous hole can be either partially or fully porous and can be a subsurface of the whole coupling surface. A major application is the flow inside multi-compartment air bags.

To activate flow between two coupling surfaces through a porous hole the following steps have to taken:

• Associate an Euler domain to each of the two coupling surfaces

• Make a subsurface of the elements of the coupling surfaces that models the hole. The elements in this subsurface should be shared by both coupling surfaces.

• Define a LEAKAGE entry for one of the coupling surfaces. This LEAKAGE references a PORFCPL entry.

• Create a PORCPL entry. The other coupling surface has to be referenced by this PORFCPL entry.

369Chapter 9: Fluid Structure InteractionMultiple Coupling Surfaces with Multiple Euler Domains

Flow Between DomainsThe facets in the coupling surfaces that represent an open area are subdivided into smaller facets, that each connect exactly to one Euler element in the first Euler domain and to exactly one Euler element in the second Euler domain. Material flow takes place across these smaller, subdivided facets which are called POLPACKs. This is the most accurate method.

This method has the following limitations:

• Flow faces and wallets are not supported. Note: flowdef is supported

• Viscosity is not supported

• All Euler domains have to be created by the mesh entry. Euler domains consisting of a set of Euler elements are not supported.

Euler meshes should have at least one element overlapping at the hole. When the meshes are created dynamically using MESH,ADAPT, this is taken care of automatically. When mesh sizes are comparable for two Euler meshes that are connected by a hole some reduction in costs is achieved by choosing the same mesh size and the same reference point for the two Euler meshes.

In general holes should not be precisely on Euler element faces.

DeactivationDeactivation is only supported by the Roe solver. In case you are using the multiple coupling surfaces functionality, it is also possible to deactivate a coupling surface and the associated Eulerian domain at a certain time using the TDEAC field on the COUPLE entry. The deactivation stops the calculation of the coupling algorithm and its associated Eulerian domain. The analysis of the Lagrangian structure continues. Activation of the coupling surface is not possible.

Initialization

To initialize Euler elements several PEULER1 and TICEUL1 entries are used as follows:

PEULER1,6,,HYDRO,19TICEUL1,19,19TICREG,1,19,SPHERE,1,3,5,1.0SPHERE,1,,0.0,0.0,0.0,500.0TICVAL,5,,SIE,400000.,DENSITY,0.2MESH,22,BOX,,,,,,,++,-0.01001,-0.01001,-0.01001,0.14002,0.12002,0.12002,25,OUTSIDE,++,14,12,12,,,,EULER,6$PEULER1,7,,HYDRO,20TICEUL1,20,20TICREG,2,20,SPHERE,2,3,6,2.0SPHERE,2,,0.0,0.0,0.0,500.0TICVAL,6,,SIE,400000.,DENSITY,1.9MESH,23,BOX,,,,,,,++,0.11499,-0.00501,-0.00501,0.1302,0.11002,0.11002,50,OUTSIDE,++,26,22,22,,,,EULER,7

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideMultiple Coupling Surfaces with Multiple Euler Domains

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The MESH entry references a unique property number that is also used by the PEULER1 entry. So the PEULER1 entry provides the link between the MESH entry and a TICEUL1 entry. In this way, each Euler domain references a unique TICEUL1 entry. The level indicators that occur on a TICEUL1 entry only apply to the Euler mesh that is linked to this TICEUL1 entry.

To initialize all meshes to one initial state, only one property set is used in combination with only one PEULER1 and TICEUL1 entry. Also, the PEULER entry may be used in this case.

OutputEuler archive output is restricted when this option facet has not been set. The restrictions are:

• The entry ELEMENTS on an Euler archive output request is required to be ALLEULHYDRO, ALLMULTIEULHYDRO or ALLMULTIEULSTREN. Specifying element numbers is not supported.

• For each mesh, a separate Euler archive file is created. Thus, one Euler archive output request gives multiple archive files. Each of these archive files may contain more than one cycle. To distinguish the Euler archive files from each other, the Euler archive files have a tag that specifies to what mesh they belong. This tag has the form: FV_(Mesh-ID). Here FV is an acronym for “Finite Volume”.

• When one of the Euler meshes is of TYPE ADAPT, all Euler archives will contain only one cycle.

Example: A mesh with ID=22 and a mesh with ID=23 have been defined. Then this output request would generate the archives ALLEULER_FV22_0 and ALLEULER_FV23_0.

Using the Preference of PatranFor an example, refer to workshop 11. The preference supports simulations with multiple coupling surfaces; each using a different Euler mesh entry. For Interactive failure, the preference also supports simulations with multiple coupling surfaces; each using a set of Euler elements. Interactive failure is activated by the entry COUPINT.

Creating multiple coupling surfaces with multiple coupling Euler domains with Patran is done as follows:

1. Create geometry for the coupling surfaces and mesh these geometries. Use Loads-BCs/Coupling or Loads-BCs/Airbag to define each coupling surface. The air bag option allows for porosity definitions.

2. Make an arbitrary solid for each Euler mesh using geometry/Create/Solid. This solid only serves to associate an Euler Mesh to a 3-D property set. The geometry of the solid will not be used and any solid will suffice.

3. Create a material.

4. Create a 3-D Eulerian property for each of the dummy solids of step 2 using Properties/Create/3D/EulerianSolid. In defining the property set select the material created in step 3.

5. Create the Euler meshes with Loads-BCs/Create/MeshGenerator. Use the coupling or air bag bc’s from step 1 and the 3-D properties from step 4.

371Chapter 9: Fluid Structure InteractionFluid- and Gas Solver for the Euler Equations

To define a porous hole between two closed surfaces, the surfaces should not be defined as coupling surfaces but as air bags. The first air bag is defined in one go using all the elements in the first surface. To define the second air bag, two subsurfaces will be created from the second surface. The first subsurface is the hole. The second subsurface consists of all the elements of the second surface that are not part of the hole. The second air bag is then the combination of these two subsurfaces.

1. Create the first air bag using all elements in the first surface with Loads-BCs/Create/Airbag/Surface.

2. Define a porous hole with Airbag/Subsurface and choose as porosity model PORFCPL.

3. Create a subsurface consisting of all elements of the second surface that are not part of the subsurface created in step 2.

4. Create the second air bag by selecting the two subsurfaces in step 2 and 3.

Fluid- and Gas Solver for the Euler EquationsFor gases and fluids flow, a state-of-the-art Eulerian solver is available that is based on the ideas of Professor Philip Roe. The fluid and gas Euler solver is based on the solution of so-called Riemann problems at the faces of the finite-volume elements. The mathematical procedure amounts to a decomposition of the problem into a discrete wave propagation problem. By including the physics of the local Riemann solution at the faces, a qualitatively better and physically sounder solution is obtained. The fluid- and gas solver is also known as an approximate Riemann solver.

The solver can be either first or second order accurate in space in the internal flow field. Second order spatial accuracy is obtained by applying a so-called MUSCL scheme in combination with a nonlinear limiter function. The MUSCL approach guarantees that no spurious oscillations near strong discontinuities in the flow field will occur. The scheme is total variation diminishing (TVD), meaning it does not produce new minima or maxima in the solution field. The original Roe solver can be activated by using the DYPARAM,LIMITER,ROE entry in the input file.

Improvements have been made to arrive at a full second order scheme in the fluid- and gas solver to further gain accuracy in the solution. All boundary conditions –that is, the flow boundary conditions, and the wall boundary conditions, are fully second order accurate in space. You can use the new and improved solver (either first- or second-order) by entering the keyword 2ndOrder or 1stOrder on the PEULER or PEULER1 entry.

Furthermore, a so-called “entropy fix” has been added to avoid the sharp discontinuities in those areas where the eigenvalues of the local Riemann problem vanish. In effect, the entropy fix ensures that an expansion shock (although mathematically sound) is broken down into a correct expansion fan. The expansion shock would yield a physical impossibility of decreasing entropy in the system. That is the reason for the name “entropy fix”. The entropy fix brings a very, almost unnoticeable, form of locally necessary dissipation into the solution to improve the differentiability of the equations where the eigenvalues vanish.

The time integration in the fluid- and gas solver is performed by a multi-stage time integrator, also know as a Runge-Kutta type scheme. Higher-order temporal accuracy can be achieved by applying multiple stages in the time integration. The required number of stages is automatically selected when you select

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideModeling Fluid Filled Containers

372

either a first order or a second order solution. A first order spatial accurate solution uses a one-stage time integration scheme; a second order spatial accurate solution applies a three-stage time integration scheme.

Multiple coupling surfaces with failure can be requested if the fast coupling algorithm is used by setting the DYPARAM,FASTCOUP, ,FAIL entry. You can also define interaction between multiple coupling surfaces, like in cases where you wish to model “chambers” that after structural failure will show a “connection” between them through which fluid or gas could flow. A typical example is an explosion in the cargo space of an aircraft after which the floor may be ruptured and the high-pressure gas can vent into the passenger cabin.

The fluid solver allows you to introduce viscosity into the solution. You use Tait’s equation of state to model the fluid with additional viscosity terms.

There are some limitations in the current implementation. Eulerian elements must be completely filled with materials, so void or partial void elements are not allowed. For fluid flows where voids sometimes may occur, we recommend that you use Tait’s equation of state. This equation of state is fully supported by the improved second order Euler solver, and allows you to define a so-called critical density. When the fluid’s density falls below the critical value, the fluid cavitates (i.e., the pressure retains the value associated with the critical density). The density can further drop, but the pressure remains constant. In this fashion, you avoid the creation of voids and still allow the fluid to cavitate. When you only have data available for the fluid that satisfies the simple bulk equation of state, you can still use Tait’s model by setting the parameter to zero, to one, and add the critical density value at which cavitation occurs.

Especially for blast wave types of problems, the full second order solution is recommended because of the accuracy it inherently brings. The JWL equation of state is not supported. A major advantage of using the blast wave approach is the speed at which the analysis can be performed. Especially spherical wave propagation through a Cartesian mesh is much more accurate in a full second order solution than in first order, or second order with (cheaper) first order accurate boundary conditions.

Modeling Fluid Filled ContainersContainers, for example plastic bottles, are often subjected to axial loading. Axial loading occurs when the bottle is crushed, or for example, stacked. It may concern both empty and (partially) filled bottles. Fluid filled containers or bottles can be modeled using a full multi-material Euler description. However, using full multi-material Euler fluid dynamics solver is a quite expensive method to solve the quasi-static behavior of the fluid. An alternative way of modeling is available through the FFCONTR (Fluid Filled CONTaineR) option. Using this option removes the need for a full fluid dynamics solution.

The FFCONTR option uses the uniform pressure algorithms to calculate the pressure increase due to the compression of the container. The pressure is uniformly distributed but may change in time due to volume changes that occur when the container deforms.

You need to define a surface to indicate the boundary of the container. The volume enclosed by the surface then equals the volume of the container.

A0 γ

373Chapter 9: Fluid Structure InteractionHotfilling

The normals of the faces of the surface must point outwards in order to compute the correct (positive) volume. When the normals point inwards, they are automatically reversed such that the resulting volume is positive. The surface must be closed to ensure a correct volume calculation.

You must define the amount of fluid in the container or bottle. The volume of the gas above the fluid then follows immediately from the difference between the volume of the container and the fluid volume in the container. Obviously, the fluid volume cannot exceed the volume enclosed by the surface. The fluid is assumed to be incompressible. Thus, any volume change directly translates to a volume change of the gas above the fluid. The gas above the fluid is assumed to behave as an ideal gas under iso-thermal conditions:

where is the pressure, is the volume, and represents a constant.

To define the constant , you have to specify the initial pressure of the gas above the fluid on the FFCONTR entry. The initial pressure is only used for the calculation of the pressure changes and does not have an effect on other boundary conditions that you may have applied. If an over-pressure (for example, a carbonated soft drink) or an under-pressure (for example, a hot filled container) is present, you must model this separately using a PLOAD definition. The values for the pressure on the PLOAD entry and the pressure generated by the FFCONTR are superimposed for the calculation.

HotfillingFilling bottles with hot liquid can cause large deformations during cooling. To simulate these deformations the Fluid filled functionality container option can be used. Since MD Nastran Nonlinear Explicit (SOL 700) has only a limited cooling functionality the temperature of the fluid has to be specified by the user. In addition, the volume of the fluid will depend on temperature. A temperature versus time table and water density versus temperature table are input options for the fluid filled container. If these are set the gas is no longer iso-thermal but satisfies

Here is the volume of the gas. This volume is computed as the difference between the total volume and the volume of fluid. The volume of the fluid is given by

,

Here the fluid density depends on the temperature as specified by table entry.

p V⋅ C=

p V C

C

PV T⁄ C=

V

VMwater

ρ-----------------=

ρ

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374

Chapter 10: Eulerian SolversMD Nastran R3 Explicit Nonlinear (SOL 700) User’s Guide

10Eulerian Solvers

The Standard Euler Solver 376

Approximate Riemann Euler Solver 386

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376

The Standard Euler SolverIn the Eulerian approach material is not attached to elements but can move from one Euler element to the other. Mass, momentum and energy are element averages and are defined in the centers of the elements. This property is called cell-centered.

The equations solved are the conservation laws of mass, momentum and energy as given in equation (10-1). Here, is the material density, are the velocity components, is the pressure, is the bulk

viscosity, is gravity and is the specific total energy. is a volume and is its boundary.

(10-1)

For Eulerian materials with strength the pressure is replaced by the stress tensor. The volume integrals represent the total mass, momentum and energy in the Volume V. The surface integrals on the left signify transport out of the volume through parts of the area A. The surface integrals on the right represent the momentum and energy increase caused by forces acting on the boundary of the volume. The numerical scheme is a finite volume method. It is obtained by applying equation (10-1) to the material inside an Euler element and by specifying how transport terms are computed. The first equation signifies that the decrease of mass in an element equals the loss of mass trough the element boundary. In transporting mass between elements mass should be conserved globally. This is achieved by looping across the element interfaces and adding the transported mass, momentum and energy to the acceptor element and subtracting it from the donor element. In this way the finite volume scheme conserves mass, momentum and energy.

In applying equation (10-1), it is assumed that density, velocity, and specific energy are constant across an Euler element and only depend on time. In addition, they are constant within one time step. This is consistent with a first-order approach. The evolved time at cycle n will be denoted by . Element density,

velocity, and specific total energy inside an element at cycle n will be denoted by , , and , respectively.

ρ ui p q

g e V A

ddt----- ρ V ρ u n⋅( )

A∫+d Ad

V∫ 0=

ddt----- ρui V ρui u n⋅( ) Ad

A∫+d

V∫ p q+( )ni A ρge3V–d

A∫–=

ddt----- ρe V ρe u n⋅( )

A∫+d Ad

V∫ uipni Ad

A∫–=

p

tn

ρn un en

377Chapter 10: Eulerian SolversThe Standard Euler Solver

(10-2)

Applying first-order time integration gives equation (10-2). The integration is from to . Here, denotes the mass inside the element, momentum, and energy.

For the surface integrals that represent transport, the forward Euler method is used. Consequently, surface integrals are evaluated at the beginning of the time step. The surface integral with the pressure terms is evaluated using the new density and specific total energy.

The transport velocity depends on both the donor as well as the acceptor element and is given by the average of the donor and acceptor velocity. Multiplying the transport velocity with surface area and time step yields the transport volume. This volume is filled up with mass of the donor element. Multiplying this volume with the density of the donor element gives the transported mass. Likewise the transport volume times the donor velocity gives the transported momentum.

Since the fluid-structure interaction forms an integral part of the numerical scheme we first discuss fluid-structure interaction.

Fluid-structure InteractionMaterial in an Euler mesh can interact with Lagrangian structures. Eulerian material can exert forces on a structure causing displacement and deformation. On the other hand structures provide a barrier to Eulerian material. That is Eulerian material cannot penetrate the structure and the structural surface determines which Euler element have the capacity to hold mass. Consider for example a tank shell surface. Euler Elements that are outside the tank surface cannot hold material and only elements that are partially or completely inside the surface have the capacity to contain mass. This surface defines the effective boundary of the Euler domain and will be called the coupling surface. In most cases the coupling surface will consist of Lagrangian Elements. But also the interaction of Eulerian material with a Lagrangian solid is possible. Then the coupling surface consists of surface elements that have no Lagrangian model attached but only serve to enable interaction. In the following we shall assume that the coupling surface is a Lagrangian shell surface. The coupling surface will also be referred to as the structural surface.

Mn 1+ Mn– ρ u n⋅( ) AΔ td

A∫

tn

–=

Pn 1+ Pn– ρui u n⋅( ) Ad

A∫

tn

Δ t– p ρ s,( )ni A ρge3V+d

A∫

tn 1+

Δ t–=

En 1+ En– ρe u n⋅( ) A uip ρ e,( )ni Ad

A∫+d

A∫–=

ρn 1+ Mn 1+

Vn 1+--------------- en 1+,

En 1+

Mn 1+--------------- sn 1+, en 1+ 1

2--- uk

2

k 1=

3

∑–= = =

tn tn 1+ M

P E

u n⋅


378

The coupling surface consists of shell elements that deform under pressure loads from material inside the Eulerian domain. An explicit finite element solver solves the shell dynamics, and an explicit Euler solver solves the fluid dynamics for the inside region of the coupling surface. The interaction between these two solvers takes place in two ways.

• The mass in the Euler elements exerts a pressure load on the lagrangian elements associated with the structural surface. These loads constitute an additional set of boundary conditions for the finite element solver, resulting in new grid point accelerations and velocities for the structure. From the updated plastic strain or updated stresses of the shell elements it is determined which elements are failing. Finally the structural grid points are moved using the new velocities

• The structural grid points move and so the Euler mesh has a new effective boundary. Consequently, the volume of mass in each element may change. Since density is mass divided by the volume of the mass, densities will also change, and so will the pressures. In the following we shall assume that the material is insides the coupling surface.

The Numerical SchemeThe Euler elements are integrated in time by applying a finite volume method directly to the physical domain, avoiding the use of coordinate transformations. Therefore the finite volume method is applied to the 3D object that consists of that part of the Euler element that is inside the coupling surface. This is in general not a cube but a multi-faceted object. For the 2D case this is sketched in Figure 10-1.

Figure 10-1 The Boundary of an Euler Element

In this figure the square represents an Euler element that is intersected by the coupling surface. Only that part of the square that is inside the coupling surface can contain mass. Therefore this part is the effective volume of material in the element. The boundary of this effective volume consists of two types of surfaces:

• Euler element boundaries that connect two neighboring elements called ‘Euler faces’.


• Parts of the coupling surface that are within the Euler element. They will be called ‘polpacks’, which is short for “polyhedron packets”.

The effective boundary of an Euler element consists of Euler faces and polpacks.

A polpack is the intersection of a coupling surface shell element with an Euler element and is completely inside an Euler element and completely inside a coupling surface shell element. An algorithm is available that computes these polpacks for any given, closed 3-D faceted surface, and any 3-D Euler domain.

Faces refer to two Euler elements, whereas polpacks refer to only one Euler element. For both faces and polpacks, areas and normals are computed.

The finite volume method results from applying equation (10-2) to these 3-D objects. The volume in equation (10-2) is the effective volume of the Euler element. Furthermore, the surface integrals are computed by summing over the faces and polpacks. A contribution of a polpack or a face to integrals signifying transport is called a flux.

When there is more than one material present in the simulation the mass conservation law applies to each material separately. This means that for each material inside an Euler element the density has to be monitored. In applying the momentum law, it does not matter whether there are several materials since all materials inside an Element are assumed to have the same velocity. The energy equation is also applied to each material separately.

First consider simulations with only one material present. In the mass conservation law, the mass flux across a face gives:

Here is the mass in the Euler element, is the velocity vector, denotes the area vector of the face, is the time step, denotes the element supplying mass, and denotes the element

receiving mass. In most cases, the coupling surface is not permeable and there is no transport across the polpacks. However, in case coupling surface shell elements have a porosity model assigned, the flux equations take that into account.

The momentum in an element can increase by either transport of momentum or by a pressure load working on the polpacks and faces. The pressure load contribution to this momentum increase is the

surface integral . The force contribution of a face to the momentum increase of the element left

to the face and right to the face reads:

Vn 1+

ΔMDONOR ρ– DONOR V A⋅ Δ t=

ΔMACCEPTOR +ρDONOR V A⋅ Δ t=

M V A

Δ t DONOR ACCEPTOR

pni AΔ td

A∫–

ΔPLeft pFaceAΔ t–=

ΔPRight pFaceAΔ t=


380

Here is the momentum of an element, is the area vector pointing from the left element to the right

element and is a weighted average of the pressure in the two elements that are on the left and right of the face. These momentum updates clearly conserve the combined momentum of the left and right element.

For polpacks, the contribution is likewise, but now the pressure at the polpack is given by the pressure in the Euler element that contains the polpack. To conserve momentum, the negative of this momentum contribution is put as a force on the coupling surface shell element that hosts the polpack. This is the way boundary conditions are imposed on the Lagrangian element constituting the coupling surface.

In a similar way the energy equation is applied.

The procedure for advancing the Euler domain with one time step is as follows:

Do all finite element objects and contact. Move the finite element objects in accordance to their grid point velocities.

Using the new position of the finite element structures compute new polpacks. Using polpacks and faces compute for all Euler elements the volume of the portion that is inside the coupling surface.

Transport mass, momentum, and energy across all faces and permeable polpacks using the conservation laws. The flux velocity is the average of the left and right Euler element velocity. In case no right Euler element is available the flux velocity is determined from an inflow condition and in some cases the velocity of the Euler element. Examples are holes and parts of surfaces that enable flow into the inside region of the coupling surface as a means of filling the inside region of the coupling surface. At the end of this step, element masses are fully updated.

For each Euler element, compute density from the new mass and volume and compute pressure from the equation of state using the new density.

Compute the effect of Euler element pressures to both structure as well as other Euler elements by going over respectively polpacks and Euler faces. This effect will contribute to the Euler element momentum. The transport contribution to the momentum increase has already been computed in step 3. At the end of this step the element momentum and energy are fully updated.

Advance the Lagrangian shell elements associated with the coupling surface with one time step using the internal shell element forces, contact forces, and external forces from the Euler domain and compute new velocities on the grid points.

Compute a new stable time step based on the mesh size, speed of sound and velocity. The stability criterion used is the CFL condition and applies to both the tank surface as well as to the Euler elements.

The Time Step CriterionTo maintain stability of the explicit scheme the time step should not exceed:

(10-3)

P A

pface

Δ tmaxΔx

u c+------------=


Euler with StrengthDeviatoric stress is a property of mass and is transported along with mass. Deviatoric stress in an element changes because masses with different stresses can enter the element and because strain increments raise stresses. When moving along with a piece of material the change in deviatoric stress denoted by is given by:

(10-4)

Here the derivative is along the path of the moving mass and denotes the velocity in the Euler

element. Since the velocity of the moving mass equals the velocity in the Euler element the total derivative is given by

(10-5)

Therefore the change of deviatoric stress in an Euler element is given by

(10-6)

This equation is not in conservation form and using the equation as it is would require the additional computation of shear stress gradients. By putting the equation in conservation, gradient computations are not needed and the equation be solved using the divergence theorem.

To enable further use for other quantities like plastic strain consider

(10-7)

By using the continuity equation it can be written in conservative form.

(10-8)

This gives

(10-9)

For stresses this becomes:

(10-10)

s

dsij

dt--------- 2μ

deijdev

d t------------- μ

∂ui

∂xi--------

∂uj

∂xj-------- 1

3---

∂uk

∂xk--------δij–+⎝ ⎠

⎛ ⎞= =

uk( )

dsij

d t---------

∂si j

∂ t---------

∂si j

∂xk

---------∂xk

∂ t--------+

∂sij

∂ t--------- uk

∂si j

∂xk

---------+= =

∂sij

∂ t--------- uk

∂sij

∂xk---------+ 2μ

deijdev

dt-------------=

∂φ∂ t------ uk

∂φ∂xk--------+ D=

∂ ρφ( )∂ t

--------------- ∂ρ∂ t------ φ ρ∂φ

∂ t------+

∂ ρuk( )∂xk

-----------------φ– ρ D uk∂φ∂xk--------–⎝ ⎠

⎛ ⎞+∂ ρukφ( )

∂xk---------------------– ρD+= = =

∂ ρφ( )∂ t

---------------∂ ρφuk( )

∂xk---------------------+ ρD=

1ρ--- ∂ ρφ( )

∂ t---------------

∂ ρφuk( )∂xk

---------------------+⎝ ⎠⎛ ⎞ D=

1ρ---

∂ ρsi j( )∂ t

-----------------∂ ρuksij( )

∂xk

-----------------------+⎝ ⎠⎛ ⎞ 2μeij

dev=


382

In this way, transport of stresses can be computed in close analogy to mass and momentum by transporting mass times shear stress. In the same way transport of plastic strain is carried out.

Strain rates are computed from velocity gradients. They are obtained by use of the divergence theorem as follows:

(10-11)

Pressure can be either computed from density or updated from volume strain rates. The first corresponds to splitting the stress tensor computation into a hydrostatic part and a deviatoric part. The second computes the stress tensor without any splitting and uses the isotropic Hooke’s law in terms of strain rates. We show that the two approaches are equivalent. Consider computing pressure from the volume strain rates. Differentiation of the isotropic Hooke’s law gives

(10-12)

with the bulk modulus. Using the continuity equation in the form

(10-13)

yields

(10-14)

The pressure in an element can be traced back by using equation (10-14).

(10-15)

So to first-order in density, Hooke’s law and the equation of state give the same pressure. Basing the pressure on the logarithm of the density ratio is expensive and the linearization is sufficiently accurate. Pressures are computed using the linearization.

To account for rotation of material the Jaumann correction is applied.

The Multi-material SolverIn simulations with multiple materials, it is important to keep track of the interfaces between materials. For example, in fuel tank sloshing simulations, there is an interface visible between regions filled with fuel and regions filled with air. To handle these interfaces, several extensions of the transport logic and pressure computation are necessary. The extended transport logic is known as preferential transport and

V∂ui

∂xj--------

∂ui

∂xj-------- Vd∫ div ejui( ) Vd∫ uiej Sd∫ ui

FaceSj

Face

faces∑= = = =

dpdt------ K

∂ui

∂xi

--------–=

K

dρdt------ ∂ρ

∂ t------ ui

∂ρ∂xi-------+ ρ

∂ui

∂xi--------–= =

dpdt------ K

ρ---- dρ

dt------=

dpdt------ d

dt----- K ρln( )=

p d K ρln( )ρref

ρ

∫ Kρ

ρref---------⎝ ⎠⎛ ⎞ln K 1

ρ ρref–

ρref-------------------+⎝ ⎠

⎛ ⎞ Kρ ρref–

ρref-------------------⎝ ⎠⎛ ⎞≈ln= = =


tries to maintain interfaces between materials. Consider, for example, the case of a blast wave of air in water. It is important that the interface between water and air during the expansion of the blast wave is maintained and does not deteriorate by the unphysical mixing of water and air.

To enable a multi-material simulation, a certain amount of bookkeeping is needed. For every Euler element, the following information is available:

• The number of materials inside the Euler elements

• For each material, the volume fraction, the material ID, the density, the mass, the specific energy, the total energy, and volume strain rates are stored. The volume fraction of a material is defined as the fraction of Element volume that is filled with that material.

The transport logic for multi-material amounts to:

• Compute the volume to be transported. This is and this volume flux gives rise to a mass flux. If there is only one material, the mass flux is the density times the volume flux, but now the donor element has several materials and each material has a distinct density and, therefore, the mass flux is split into several mass fluxes. Each material in the donor element has a distinct mass flux and this material specific mass flux is easily converted into a volume flux by using the material density. Using this conversion, the mass fluxes should give rise to volume fluxes that add up to a total volume flux that equals . Materials are transported out of the element until the prescribed total volume flux is reached. The only remaining issue is which materials are transported first.

• Determine for both donor element as well as acceptor element which materials are present in the element.

• Look which materials are common to both elements.

• First transport any material that is common to both elements. Transport these common materials in proportion to their acceptor material fraction. A material is transported with the material density of the donor element and this material density translates a volume flux into a mass flux and vice versa. Subtract any mass that is transported from the flux volume. If there is sufficient mass of the common materials in the donor element the whole flux volume will be used to transport the common materials.

• If after transport of the common materials the flux volume is not fully used yet, transport materials in ratio to their donor material fraction.

To illustrate how this procedure aims at preserving material interfaces, consider two adjacent Euler elements and assume that flow is from the left element to the right element. The left element is filled with fuel and air and the right one is filed with only air. Since air is the only material common to both elements, it is transported first. If there is sufficient air, only air will be transported. If during transport there is no air left, transport of this common material is not able to use the full flux volume and fuel has to also be transported. In both cases, the interface between fuel and air is maintained.

The pressure computation for Euler elements with only one material is straightforward: the pressure readily follows from the equation of state and the density. For elements with more than one material, each material has a distinct equation of state and a distinct density and this results in a distinct pressure for each material. The pressure computation for these elements is based on the thermodynamic principle of

V A⋅ Δ t

V A⋅ Δ t


384

pressure equilibrium. Since masses of materials in Euler elements are only changed by the transport computation, these masses are fixed during the pressure computation. The volume taken up by each material in an Euler element is not known but determines the pressure inside the material. By adjusting the volumes of the materials simultaneously, pressure equilibrium is achieved. Therefore, the pressure computation amounts to an iterative process that iterates on the volumes of the materials inside the Euler element.

To understand the influence of the material volumes consider an element with fuel and air. Suppose that at the start of a cycle, there is pressure equilibrium and that during transport, air enters the element. Because of the surplus of air there is no longer pressure equilibrium. Physically, it is expected that the air very slightly compresses the fuel until pressure equilibrium is achieved. The compression of the air is just the adjustment of the material volumes of fuel and air. The material volume of air increases while the material volume of fuel decreases.

Viscosity

Viscous stresses only contribute to the momentum balance:

(10-16)

Here, the deviator shear stress tensor is given by

(10-17)

The contribution of viscous dissipation to the energy balance law is small and is not taken into account. Velocity gradients are computed by Gauss’s law as given by equation (10-18). For boundary contributions, the imposed velocity boundary condition is used. Material in one element exerts a viscous force on material in the adjacent elements and leads to changes in momentum. The momentum transferred across an Euler element face is given by

(10-18)

The second term is most significant and requires a special treatment. This term is proportional to the normal velocity derivative at the face. It can be obtained by averaging the normal derivative over the left and right Euler element:

(10-19)

This leads to decoupling. To show this, consider an Euler element and two of its opposing faces. For both opposing faces, a viscous flux that is proportional to equation (10-18) is added to the Euler element. The net contribution is proportional to the difference of equation (10-18) for the two faces. In this subtraction, the normal velocity derivative of the element itself drops out. As a result, the contribution of viscous

ddt----- ρui V ρui u n⋅( ) Ad

A∫+d

V∫ pni A sijnj Ad

A∫+d

A∫–=

si j

si j μ eij13--- ekk–⎝ ⎠

⎛ ⎞=

eij12---

∂ui

∂xj--------

∂ui

∂xj--------+⎝ ⎠

⎛ ⎞=

si jnj Ad td∫23---μuk k, ni– μ

∂ui

∂n-------- μ

∂un

∂xi

---------+ +⎝ ⎠⎛ ⎞ FACEAREA*Δ t=

∂ui

∂n--------

Face

12---

∂ui

∂n--------

L

∂ui

∂n--------

R+⎝ ⎠

⎛ ⎞=


fluxes to the momentum increase of the Euler elements is only weakly coupled to the velocity gradient in the Euler element. In practice, decoupling follows. To avoid this decoupling, the gradient is computed directly using the velocity difference between across the face, giving:

(10-20)

Furthermore, walls exert viscous stress on material and, at the wall, a no-slip condition is applied. Computing these in a specific local system enables a straightforward use of the no-slip condition. In this local system, the x-axis is along the normal of the boundary. The no-slip condition ensures that all tangential velocity derivatives are zero. In addition, normal derivatives are computed directly from the velocity difference between element and wall. This leads to shear stresses at the wall:

(10-21)

These shear stresses are added to the momentum balance.

Fluid-structure Interaction with Interactive Failure

Consider a box filled with gas. If a blast wave is initiated inside the box, some parts of the box may fail and gas can escape through ruptures. To simulate this flow, the gas inside the box is modeled by an Euler domain and the box surface by shell elements. These shell elements form the coupling surface for this Euler domain. Once shell elements of this box have failed, gas flows from the inner domain to an outer Euler domain that models the ambient. The shell surface also forms the coupling surface for this outer Euler domain and is, therefore, able to connect Euler elements inside the shell surface to elements outside the shell surface. Flow from one Euler domain to another is also possible through fully or partly porous segments.

Figure 10-2 The Overlapping Mesh

∂ui

∂n--------

Face

uiR ui

L–

Δx-------------------=

sxxloc 4

3---μ∂u

∂x------ 4

3---μuwall uelement–

xwall xelement–--------------------------------------= =

sxyloc μ∂v

∂x------ μvwall velement–

xwall xelement–--------------------------------------= =

szxloc μ∂w

∂x------- μwwall welement–

xwall xelement–----------------------------------------= =

MD Nastran R3 Explicit Nonlinear (SOL 700) User’s GuideApproximate Riemann Euler Solver

386

Flow Between Domains

Flow from one Euler domain to another takes place through either shell elements that are porous or that have failed. Flow through a segment can only take place if it is inside both Euler domains. In the following, let us assume that both Euler domains are sufficiently large. In general, a segment can intersect several Euler elements of the first Euler domain and the same applies to the second domain. Therefore, the segment connects several Euler elements in the first Euler domain to several Euler elements in the second domain. For an accurate and straightforward computation of flow through the segment, it is partitioned into sub segments such that each sub segment is in exactly one Euler element of each Euler domain. To carry out this partitioning, an overlapping Euler mesh is created that is the union of both Euler meshes. Then, for each element in this overlapping domain, the intersection with the segment is determined. Each intersection gives one sub segment that refers to both the original segment and to the element in the overlap domain. Since an element in the overlap domain is in exactly one element of both domains, the sub segment connects exactly one element in the first Euler domain to exactly one element in the second domain. Transport across this sub segment is straightforward because it closely resembles transport that takes place between the Euler elements that are within an Euler domain. In computing transport across the sub segment, the velocity of the segment has to be taken into account. If the segment is moving with the same velocity as the material on either side, no material will flow through the segment.

Approximate Riemann Euler Solver

Euler Equations of MotionThe analysis of the physical behavior of fluids and gases is best solved using a Eulerian approach. The nature of the behavior of these types of materials is represented in a natural way using a finite volume description based on the Euler equations of motion. An accurate solver is available that allows you to analyze the behavior of fluids and gases, coupled to structures if necessary. The solution approach is based on a so-called Riemann solution at the element faces that defines the fluxes of mass, momentum and energy, the conserved problem quantities.

This section gives a more detailed explanation of the theory behind the Riemann-based Euler solver, its boundary condition treatment, and accuracy in time and space.

The inviscid flow of a fluid or a gas is fully governed by the Euler equations of motion. We will use the equations in their conservative form:

(10-22)

where is the state vector and , and represent the fluxes of the conserved state variables. They are defined as follows:

∂q∂ t------ ∂ f q( )

∂x------------- ∂g q( )

∂y--------------- ∂h q( )

∂z---------------+ + + 0=

q f q( ) g q( ), h q( )

387Chapter 10: Eulerian SolversApproximate Riemann Euler Solver

(10-23)

Equation (10-22) describes the conservation of mass, momentum and energy. In equation (10-23), is the material density, the velocity components, the pressure, and the total energy. For a gas, we can close the system (note that we have five equations, with six unknowns) by adding the equation of state for a calorically perfect gas (the “gamma law equation of state in MD Nastran Nonlinear Explicit (SOL 700)).

(10-24)

In equation (10-24), denotes the specific internal energy of the gas and is the ratio of specific heats. There exist more equations of state for gases, but most gases can be described as calorically perfect gases, in which case equation (10-24) applies.

For a fluid in its simplest form, we may use a so-called “simple bulk” equation of state:

(10-25)

In equation (10-24), is the material bulk modulus and is the reference density at which the material

has no pressure. Also, for fluids, there are more equations of state, like a full polynomial or Tait’s equation of state. Both are implemented in the approximate Riemann solver that MD Nastran Nonlinear Explicit (SOL 700) uses, but the method of implementation is similar to the simple bulk equation of state and is not described in detail here.

Numerical ApproachThe conservation laws as described by equation (10-23) are numerically solved by an upwind, cell-centered finite volume method on unstructured 3-D meshes. We will briefly describe the solution method here.

When the conservation laws are written in integral form, by integrating over an arbitrary volume, the finite volume (discretized) method becomes apparent when we consider each element in an Eulerian mesh as a finite volume on which we have to solve the conservation laws as described by equation (10-23). The integral form of equation (10-23)) when using Gauss’ integral theorem:

(10-26)

From equation (10-26), it becomes apparent that the fluxes of mass, momentum, and energy have to be integrated normal to the boundary of the volume or its surface. When we use the rotational invariance of the Euler equations of motion, the integral form can be rewritten using the transformation matrix that describes the transformation of the state variables in a direction normal to the surface:

q

ρρu

ρv

ρw

E⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

= f q( )

ρu

ρu2 p+

ρuv

ρuw

E p+( )u⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

= g q( )

ρv

ρuv

ρv2 p+

ρvw

E p+( )v⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

= h q( )

ρw

ρuw

ρvw

ρw2 p+

E p+( )w⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

ρ

u v w, , p E

p γ 1–( ) ρ e⋅ ⋅=

e γ

p Kρρ0----- 1–⎝ ⎠⎛ ⎞=

K ρ0

∂∂ t----- q V f q( ) nx g q( ) ny h q( ) nz⋅+⋅+⋅( ) dS⋅

∂v∫+d

V∫ 0=


388

(10-27)

where denotes the state vector transformed to a coordinate system with the local x-axis in the direction of the normal to the surface. When we then make the step to a discretized form, by defining the volume as the volume of a finite element (an element of the Euler mesh), and the surface defined by the faces spanning the element, equation (10-27) becomes a local one-dimensional system of equations for each face of the element with the local x-axis in the direction of the normal to the element’s face. Note that the fluxes in the local y- and z-direction do not contribute to the change of the state variable. The system of equations to solve for each element face thus becomes:

(10-28)

where defines the x-direction normal to the element’s face. Considering the fact that each face has a left and a right element connected to it, we can view the state variables in the left- and right connected element as initial conditions for the solution of the flux normal to the face:

(10-29)

Equations (10-28) and (10-29) describe a so-called Riemann problem. Thus, the solution for the fluxes at the element faces amounts to solving a local 1-D Riemann problem for each of the faces of the element, considering the left and right state of the fluid or the gas. The contribution of the face fluxes result in the state change in the element as a function of time as denoted by the first term in equation (10-28). The fluxes on the faces are determined using a flux function, by which, using equations (10-27),

(10-28), and (10-29), the discretization becomes:

(10-30)

In equation (10-30), denotes the element number, the element volume, the face numbers of the

element, and the associated face area.

Using a flux difference scheme, the flux function can be written as:

(10-31)

The flux difference terms in equation (10-31) are defined as:

(10-32)

∂∂ t----- q V f q( ) dS⋅

∂V∫+d

V∫ 0=

q

∂ q∂ t------ ∂ f q( )

∂ x-------------+ 0=

x

q x 0,( ) qL x 0<

qR x 0>⎩⎨⎧

=

fR qL qR,( )

dqi

dt-------- 1

Vi

----- fR qL qR,( ) An⋅n 1=

6

∑–=

i Vi n

An

fR qL qR,( ) 12--- f qL( ) f qR( )+{ } 1

2--- Δ f+ Δ f-–{ }–=

Δ f+ fR qL qR,( ) f qL( )–=

Δ f- f qR( ) fR qL qR,( )–=


When we use the eigenvectors, the eigenvalues, and the wave strengths that can be found from a diagonalization of the Jacobian matrix of the Euler equations, we arrive at a simple definition of the flux function. Note that the shape of the eigenvectors, eigenvalues, and the wave strengths depend on the type of equation of state the flux function is constructed for. In general terms, the numerical flux function used in the scheme is defined as:

(10-33)

After some rewriting of equation (10-33), we find:

(10-34)

Using the ideal gas equation of state (gamma-law equation of state), we find for the wave strengths:

(10-35)

And for the associated eigenvectors:

(10-36)

In the above equations, the quantities denoted by a “tilde” are weighted quantities according to:

(10-37)

All quantities are averaged using the above definition, except for the density:

(10-38)

The above described flux evaluation scheme is called an approximate Riemann scheme due to the fact a linearization using the weighted quantities at the element faces is applied. As a result, the scheme exhibits an artifact, namely that it does not satisfy the entropy inequality. The entropy inequality states that the entropy of a system can only remain constant or increase. Due to the artifact, the scheme is able to also capture mathematically sound, but physically impossible discontinuities like expansion shocks. This is easily “repaired” by adding a so-called entropy fix to the scheme as described in the next section.

fR qL qR,( ) 12--- f qL( ) f qR( )+{ } 1

2--- αi λi Ri⋅ ⋅

i 1=

5

∑⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

–=

fR qL qR,( ) 12--- f qL( ) f qR( ) u Δq u a– u–( ) α1 R1 u a+ u–( ) α2 R2⋅ ⋅+⋅ ⋅+⋅[ ]–+{ }=

α1Δp ρ aΔu⋅–

2 a2---------------------------------=

α2Δp ρ aΔu⋅+

2 a2---------------------------------=

R1 1 u a– v w H u a–⎝ ⎠⎛ ⎞ T

=

R2 1 u a+ v w H u a+⎝ ⎠⎛ ⎞ T

=

φ θ φL 1 θ–( ) φR⋅+⋅=

θ ρL

ρL ρR+----------------------------=

ρ ρL ρR⋅=


390

Entropy Fix for the Flux Difference Riemann SchemeAs described in the earlier section, a so-called entropy fix must be added to the scheme in order to have the scheme correctly decompose a expansion discontinuity into a physically correct expansion fan. The entropy fix in fact amounts to adding some numerical viscosity or dissipation to sonic points, shocks and contact discontinuities. The dissipation is added only at those points where any one of the system’s eigenvalues vanishes.

The entropy fix can be written in terms of a simple function.

(10-39)

The function works automatically on the eigenvalues of the system, represented in equation (10-39) by , and is governed by a single parameter that depends on the flow field:

(10-40)

Second Order Accuracy of the SchemeWhen we consider the flux function as given in general by equation (10-34), it does not say anything about the order of accuracy at which the face fluxes are computed. The accuracy is governed by the way in which the left and the right state variables are determined.

A first order scheme results when the left and the right state variables are taken as the values the state variables have at the left and the right element center; a so-called first order extrapolation to the face.

When we increase the stencil by which we determine the left and right state variable values at the face by including the left-left and the right-right element, we arrive at a second order accurate scheme in space. A so-called nonlinear limiter that avoids the creation of new minimum or maximum values limits the second order left- and right face values of the state variables. Such a scheme is called total variation diminishing, (TVD). Near sharp discontinuities the scheme reverts to locally first order as to introduce the necessary numerical viscosity to avoid oscillations in the solution near the discontinuity.

The second order approximate Riemann solver in MD Nastran Non-linear explicit (SOL 700) applies the Superbee limiter. The second order scheme can be formally written as:

(10-41)

for the left side of the face, with:

(10-42)

ψ z( )z

z2 δ12+( )

2δ1----------------------

⎩⎪⎨⎪⎧

=

a δ1≥

a δ1<

z δ1

δ1 δ ua--- v

a--- w

a----+ +⎝ ⎠

⎛ ⎞⋅=

qi 1 2⁄+L qi

12---ΨL qi qi 1––( )⋅+=

ΨL 12--- 1 κ–( ) Φ rL( ) 1 κ+( ) rL Φ 1

rL-----⎝ ⎠⎛ ⎞⋅ ⋅+⋅=

rLqi 1+ qi–

qi qi 1+–-----------------------=


For the right side of the face, the second order approximation is defined by:

(10-43)

with:

(10-44)

The upwind scheme is defined for and the limiter function is the Superbee limiter:

(10-45)

Time IntegrationThe set of equations is integrated in time using a multi-stage scheme. For the second order accurate solution, a three-stage time integration scheme is used:

(10-46)

In equation (10-46), denotes the state variable value in the kth integration stage, the stage

coefficients, and the flux contributions are defined as

(10-47)

using the state variables at each stage of the integration. The final step gives the solution of the state variables at the new time level.

qi 1 2⁄+R qi 1+

12---ΨR qi 2+ qi 1+–( )⋅–=

ΨR 12--- 1 κ–( ) Φ rR( ) 1 κ+( ) rR Φ 1

rR-----⎝ ⎠⎛ ⎞⋅ ⋅+⋅=

rRqi 1+ qi–

qi 2+ qi 1+–------------------------------=

κ 1–= Φ

Φ r( ) max 0 min 2r 1,( ) min r 2,( ),,[ ]=

q 0( ) qin=

q 1( ) q 0( )–α1 Δ t⋅

Vi----------------- F 0( )⋅⎝ ⎠⎛ ⎞–=

q 2( ) q 0( )–α2 Δ t⋅

Vi----------------- F 1( )⋅⎝ ⎠⎛ ⎞–=

q 3( ) q 0( )–α3 Δ t⋅

Vi----------------- F 2( )⋅⎝ ⎠⎛ ⎞–=

qin 1+ q 3( )=

q k( ) αk

F k( )

F k( ) fR qLk( ) q

Rk( ),( ) An⋅

n 1=

6

∑=


392

Chapter 11: Airbags and Occupant Safety MD Nastran R3 Explicit Nonlinear (SOL 700) User’s Guide

11 Airbags and Occupant Safety

Introduction 394

Airbag Definition 395

Seatbelts 405

Occupant Dummy Models 413

Pre- and Postprocessing 414

MD Nastran R3 Explicit Nonlinear (SOL 700) User’s GuideIntroduction

394

IntroductionThe MD Nastran r2 release, SOL 700 includes a Fluid Structure Interaction (FSI) capability that is based on the advanced Finite Volume (Eulerian) and General Coupling Technology available in MSC.Dytran.

The objective of fluid-structure interaction using the coupling algorithm is to enable the material modeled in Eulerian and Lagrangian meshes to interact. Initially, the two solvers are entirely separate. Lagrangian elements that lie within an Eulerian mesh do not affect the flow of the Eulerian material and no forces are transferred from the Eulerian material back to the Lagrangian structure. The coupling algorithm computes the interaction between the two sets of elements. It thus enables complex fluid-structure interaction problems to be analyzed.

The first task in coupling the Eulerian and Lagrangian sections of a model is to create a surface on the Lagrangian structure. This surface is used to transfer the forces between the two solver domains. The surface acts as a boundary to the flow of material in the Eulerian mesh. At the same time, the stresses in the Eulerian elements cause forces to act on the coupling surface, distorting the Lagrangian elements.

The method used for airbag simulation is full gas dynamics and is based on General Coupling with adaptive Euler. Unlike other techniques such as the ALE (Arbitrary Lagrange Euler) where the Eulerian mesh is fixed in space or GBAG method where the gas flow is modeled by applying a pre-determined pressure profile to inflate the bag, in the General Coupling technique, the Eulerian mesh will “adapt” itself to the Lagrangian fabric model as the airbag is inflated. In other words, when the airbag is initially at the folded stage, there is a small Eulerian domain encapsulating the Lagrangian mesh. When the airbag is inflated, the Eulerian mesh expands as the gas jet flows through the airbag compartments and adapts itself to follow the airbag fabric. This technique is unique in MD Nastran r2 and is considered the most accurate method to predict the complex airbag behavior such as Out-of-Position (OOP) simulation, as required by FMVSS 208, where the occupant is already leaning forward when the airbag is inflated. In addition, CFD deployment of multi-compartmented airbags can easily be modeled with this technique by using multiple, fully automatic, adaptive Euler domains. The following capabilities are available:

• Analyze multiple compartments with the CFD approach.

• Simulate flow from one CFD domain into another CFD domain.

• The individual CFD domains are dynamic and adaptive. The user does not need to mesh the CFD domains, nor does he have to worry about the size, since the CFD domains will automatically follow the deploying airbag compartments.

• Flow through both small and large holes is accurately calculated.

• Inflator models

395Chapter 11: Airbags and Occupant SafetyAirbag Definition

Figure 13-1 Multi-compartment Side Curtain Airbag

Airbag DefinitionThe airbags are defined by using the AIRBAG entry in SOL 700. Airbags can be automatically inflated by defining a gas flow rate in Eulerian domain using full gas dynamics method. A second method, the conventional uniform pressure method, is also available but it is not as accurate as the full gas dynamics.

All related airbag input definitions such as inflator and porosity models, environmental parameters, are grouped in entries that can be directly input in the AIRBAG entry. Every entity has certain parameters associated with it that has to be input immediately following the name of the entity. The order of which entity is entered first in the AIRBAG card is immaterial as long as the associated parameters are defined right after the name of the entity. The following entity groups are available:

(a) Shape of Side Curtain airbag at Start of Simulation

(b) Euler Meshes for all Six Regions

(c) Shape of Bag after 20 milliseconds

“CFD” Entries for this entity describe the properties of the Eulerian domain. Only one “CFD” section can be defined. If this section is not defined, the uniform pressure method is used.

“ENVIRONM” Entries for this entity describe the properties of the environmental conditions for the airbag. Only one “ENVIRONM” section can be defined. If not defined, the values for “INITIAL” will be used.

“INITIAL” Entries for this entity describe the properties of the environmental conditions for the airbag. Only one “INITIAL” section can be defined and is required.

“INFLATOR” Entries for this entity describe the properties of an inflator that is attached to the airbag. More than one inflator may be defined.

MD Nastran R3 Explicit Nonlinear (SOL 700) User’s GuideAirbag Definition

396

There are other entries in SOL 700 to define airbag properties. These are:

Please see MD Nastran r2 QRG for more details.

“CGINFLTR” Entries for this entity describe the properties of a cold gas inflator that is attached to the airbag. More than one inflator may be defined.

“SMALHOLE”

Entries for this entity describe the properties of a small hole in the airbag. More than one SMALHOLE may be defined. A small hole should be used when the size of the hole is of the same order as the size of the elements of the Euler mesh.

“LARGHOLE” Entries for this entity describe the properties of a large hole in the airbag. More than one LARGHOLE may be defined. A large hole should be used when the size of the hole is of the larger than the size of the elements of the Euler mesh.

“PERMEAB” Entries for this entity describe the properties of the permeability of the airbag fabric. More than one PERMEAB may be defined.

“CONVECT” Entries for this entity describe the properties of the loss of energy of the gas in the airbag by means of convection through the airbag surface. More than one CONVECT may be defined.

“RADIATE” Entries for this entity describe the properties of the loss of energy of the gas in the airbag by means of radiation through the airbag surface. More than one RADIATE may be defined. When this option is used, the Stephan-Boltzmann constant must be defined by PARAM, SBOLTZ.

“GAS” Entries for this entity describe the properties of gases. More than 1 GAS may be defined and the gases defined in one AIRBAG entry can be referenced by other AIRBAG entries.

PARAM, UGASC defines a value for the universal gas constant.

PARAM, SBOLTZ defines a value for the Stephan-Boltzmann constant

PARAM, DYDEFAUL

controls the default setting of the simulation

INFLFRC defines the hybrid inflator gas fraction

EOSGAM Gamma Law Gas Equation of State

GRIA grid point in airbag reference geometry


Inflator Models in Airbags

There are several methods available to define an inflator in airbag analyses. The most general inflator

definitions are:

Figure 13-2 Airbag and Occupant Safety using SOL 700

For both the uniform pressure model and the full gas dynamics (CFD) method, the inflator location and area are defined by means of a subsurface created by a BSURF, BCPROP, BCMATL, or BCSEG parameters. These parameters are referenced by BFID field on the INFLTR entity. It can only reference shell elements that belong to the airbag surface, as defined by BFID. The characteristics of the inflator are specified on an INFLTR entity on AIRBAG card. This entry references tables for the mass flow rate and the temperature of the inflowing gas.

A model can be defined containing both (CFD) and uniform pressure model for the airbag. These two options can be defined with identical inflator characteristics. When the CFD entity and its associated parameters are omitted from the AIRBAG card, the uniform pressure method will be used. In case the CFD entity is present, the airbag will use the Euler method from the start of the simulation. In case the value of SWITCH is nonzero, the airbag will switch from an Eulerian representation to a Uniform Pressure formulation.

Hybrid Inflator Model

The hybrid inflator supports the inflow of multiple gases through an inflator subsurface, as well as providing a type of thermally ideal gas for which the specific heat at constant pressure can be

dependent on temperature.

In addition, the properties of the gas contained in an air bag will be changed based on the gas composition and temperature. Updating of gas constants is available for use together with INFLTR, INFLHB, INFLTNK, and INFLCG inflator definitions, and with PORHOLE, PERMEAB, PERMGBG, PORFCPL, and PORFGBG porosity definitions.

INFLTR Standard inflator defined by mass flow rate and static temperature of a single inflowing gas.

INFLHB Hybrid inflator defined by mass flow rate and static temperature of multiple inflowing gasses.

INFLCG Coldgas inflator behaves like a reservoir filled with high pressure gas that flows into the airbag.

cp


398

Ideal gas description

A thermally ideal gas is specified by the specific gas constant and the variation of specific heat at constant pressure with temperature.

The specific gas constant for a gas is defined as:

where is the universal gas constant and the gas molar weight.

Using the specific heat as function of temperature, the specific internal energy of a gas as a function of temperature is found as:

We can now define so that:

Mixture of gas

A hybrid inflator is specifically meant to give an inflow of several gases with different properties. To account for the properties of these gases, it is necessary to keep track of the composition of a gas at a certain time, not only for the inflator but also for the gas mixture inside the air bag. For use with hybrid inflators, it is assumed that instantaneous mixing takes place. This means the gas composition are the same throughout the volume.

For an inflator, gas fractions are given as user input. For gasbags and Eulerian, gas fractions are based on the total mass of each gas at a certain time. Gas fractions are defined as follows:

Properties of the gas mixture inside a surface are based upon the principle that a mixture of (thermally) ideal gases is itself an ideal gas. This yields for the properties of the mixture:

R Runi M⁄=

Runi M

e T( ) cp T( ) R–( ) dT eref+⋅Tref

T

∫=

cv

e T( ) cv T( ) T⋅=

mfraci t( )massi t( )

masstot t( )--------------------------=

R* mfraci t Ri⋅i 1=

m

∑=

cv* mfraci t( ) cvi

T( )⋅i 1=

m

∑=

cp* cv

* R*+=

γ*cp

*

cv*

-----=


Here is the temperature of the gas mixture. This may be the inflow temperature of the inflator, the temperature inside a constant pressure gasbag, or the average temperature of all Eulerian elements that are not covered by the coupling surface. The latter is found as:

Energy/work

A formulation for the change of energy in a closed volume can be found when and are a function

of temperature and gas composition. This takes into account inflow (by hybrid inflators), outflow (by porosity), and energy loss (through convection and radiation).

We know:

where:

For inflow of a certain mixture, we find:

Similarly, for outflow:

For the work done by the gas mixture, the following holds:

Given the expressions above for a thermally ideal gas and a general mixture of these gases, this finally yields for the rate of temperature change in an enclosed volume:

T

TEuler

ecel l∑cvEuler

-----------------=

cp cv

pdVdUdQ +=

dQ dqin dqloss– dqout–=

dU m de e dm⋅+⋅=

dqin

dt---------- M· in t( ) cpin

* Tin( )⋅[ ] Tin⋅=

dqout

d t------------- M· out t( ) cpout

* Tout( )⋅[ ] Tout⋅=

pdVdt------- M cp

* cv*–( ) T

V·

V---⋅ ⋅ ⋅=

1Tcv

*-----

dcv*

dT---------+

⎝ ⎠⎜ ⎟⎛ ⎞ T·

T---⋅ 1

M e⋅------------ M· in cpin

* t Tin,( )⋅[ ] Tin M· out cpout* t Tout,( )⋅[ ] Tout Qloss–⋅–⋅( )+⋅=

cviT( ) m· i⋅( )

i∑

M cv*⋅

--------------------------------------– γ* 1–( ) V·

V---⋅–


400

Constant Volume Tank TestsConstant volume tank tests are used to characterize inflators. The inflator is ignited within the tank and, as the propellant burns, gas is generated. The inflator temperature is assumed to be constant. From experimental measurements of the time history of the tank pressure it is straightforward to derive the mass flow rate, .

From energy conservation, where and are defined to be the temperature of the inflator and tank,

respectively, we obtain:

For a perfect gas under constant volume, , hence,

and, finally, we obtain the desired mass flow rate:

Porosity in AirbagsPorosity is defined as the flow of gas through the airbag surface. There are two ways to model this:

1. Holes: The airbag surface contains a discrete hole.

2. Permeability: The airbag surface is made from material that is not completely sealed.

The same porosity models are available for both the uniform pressure airbag model as the Eulerian coupled (CFD) airbag model. The porous flow can be either to and from the environment or into and from another uniform pressure model.

Holes

Flow through holes as defined on the SMALHOLE entries is based on the theory of one-dimensional gas flow through a small orifice. LARGHOLE entries define flow through a hole with the velocity method.

The velocity method can only be active for Eulerian airbags. When the SMALHOLE is used on the AIRBAG card, the theory of one-dimensional gas flow through a small orifice is applied.

Velocity Method

The transport of mass through the porous area is based on the velocity of the gas in the Eulerian elements, relative to the moving of coupling surface (airbag fabric).

m·

Ti Tt

cpm· Ti cvm· Tt cvm· T· t+=

V· 0=

p· V m· RTt mRT· t+=

m·cvp· V

cpRTt--------------=


The volume of the Eulerian materian transported through the faces of the coupling surface that intersect an Eulerian element is equal to

where

The transport mass through the porous area is equal to the density of the gas times the transported volume.

Pressure Method

The transport of mass through the porous area is based on the pressure difference between the gas in the Eulerian element and the outside pressure. The outside pressure is the pressure as specified on the ENVIRONM section in the AIRBAG entry.

= transported volume during one time step ( for the outflow; for

the inflow).

= time step.

= porosity coefficient.

= velocity vector of the gas in the Eulerian mesh

= area of the face of the coupling surface that intersects the Eulerian element is equal to the area of the face that lies inside the Eulerian element.

SXint

v


Coupling Surface

Eulerian Element

Vtrans d– t α v A⋅( )⋅ ⋅=

Vtrans Vtrans 0> Vtrans 0<

dt

α

v

A A

SXint

v


Coupling Surface

Eulerian Element


402

The volume of the Eulerian material transported through the faces of the coupling surface that intersect an Eulerian element is equal to:

where

The pressure at the face is approximated by the one-dimensional isentropic expansion of the gas to the critical pressure or the environmental pressure ascending to

where is the critical pressure:

In case the outside pressure is greater than the pressure of the gas, inflow through the coupling surface occurs. This porosity model can only be used for ideal gases; i. e., materials modeled with the gamma law equation of state (EOSGAM).

Permeability

Permeability is defined as the velocity of gas through a surface area depending on the pressure difference over that area.

= transported volume during one time step ( for outflow; for inflow).

= time step.

= porosite coefficient.

= velocity vector of the gas in the Eulerian mesh.

= area of the face of the coupling surface that intersects the Eularian element is equal to the area of the face that lies inside the Eulerian element.

= = pressure of the gas in the Eulerian element.

= density of the gas in the Eulerian element.

= adiabatic exponent = .

= pressure of the face.

Vtrans dt α A A⋅( ) 2pργγ 1–-------------

pexh

p---------⎝ ⎠⎛ ⎞

2 γ/ pexh

p---------⎝ ⎠⎛ ⎞

γ 1+γ

------------–⋅ ⋅ ⋅=

Vtrans Vtrans 0> Vtrans 0<

dt

α

v

A A

p

ρ

γ Cp Cv⁄

pexh

ppenv pc>

penv pc<⎩⎪⎨⎪⎧

=

pc

pcp2

γ 1+------------⎝ ⎠⎛ ⎞

rr 1–-----------

⋅


On the PERMEAB entity in AIRBAG card, permeability can be specified by either a coefficient or a pressure dependent table:

1. Coefficient: Massflow = coeff * pressure_difference

2. Table

The velocity of the gas flow can never exceed the sonic speed:

where is the gas constant of in- or outflowing gas and is the critical temperature.

The critical temperature can be calculated as follows:

where is the temperature of outflowing gas.

Initial Metric Method for AirbagsIf the reference configuration of the airbag is taken as the folded configuration, the geometrical accuracy of the deployed bag will be affected by both the stretching and the compression of elements during the folding process. Such element distortions are very difficult to avoid in a folded bag. By reading in a reference configuration such as the final unstretched configuration of a deployed bag, any distortions in the initial geometry of the folded bag will have no effect on the final geometry of the inflated bag. This is because the stresses depend only on the deformation gradient matrix:

Coeff

press_diff

coeffδ massflow)( )δ pressdiff( )

---------------------------------=

Gas Velocity

press_diff

Pressure Dependent Table

Gas Velocity

Vmax Vsonic– γRTcrit–

γ Tcrit

Tcrit

Tgas--------- 2

γ 1+( )-----------------–

Tgas

Fij

∂xi

∂Xj--------=


404

where the choice of may coincide with the folded or unfold configurations. It is this unfolded

configuration which may be specified here.

Note that a reference geometry which is smaller than the initial airbag geometry will not induce initial tensile stresses.

If a liner is included and the LNRC parameter set to 1 in MATD034, compression is disabled in the liner until the reference geometry is reached; i.e., the fabric element becomes tensile.

Heat Transfer in AirbagsFor airbags with high temperature, energy is exchanged with the environment. There are two ways to define heat transfer in airbags, convection (CONVECT) and radiation (RADIATE).

The heat-transfer rates due to convection and radiation are defined by:

1. Convection:

where is the time-dependent heat transfer coefficient, is the (sub)surface area for heat transfer, is the temperature inside the airbag, and is the environment temperature.

2. Radiation:

where is the gas emissivity, is the (sub)surface area for heat transfer, is the temperature inside the airbag, and is the environment temperature.

Xj

qconv h t( )A T Tenv–( )–

h t( ) A

T Tenv

qrad eAs TA TenvA–[ ]=

e A T

Tenv

405Chapter 11: Airbags and Occupant SafetySeatbelts

SeatbeltsBelt elements are single degree of freedom elements connecting two nodes. When the strain in an element is positive (i.e. the current length is greater then the unstretched length), a tension force is calculated from the material characteristics and is applied along the current axis of the element to oppose further stretching. The unstretched length of the belt is taken as the initial distance between the two nodes defining the position of the element plus the initial slack length.

Seatbelt shell elements must be used with caution. The seatbelt shells distribute the loading on the surface of the dummy more realistically than the two node belt elements. For the seatbelt shells to work with sliprings and retractors it is necessary to use a logically regular mesh of quadrilateral elements.

Figure 13-3 Seatbelt Shell Elements Definition

The ordering of the nodes and elements are important for seatbelt shells.

Slipring

Retractor

Top View:

RE3

RE4

RE2

RN5 SN5

SN4RN4

RN3

RN2

RN1RE1

SN3

SN2

SN1

SRE14 SRE24

SRE13 SRE23

SRE12 SRE22

SRE11 SRE21

MD Nastran R3 Explicit Nonlinear (SOL 700) User’s GuideSeatbelts

406

Seatbelt PretensionerPretensioners allow modeling of five types of active devices which tighten the belt during the initial stages of a crash. Types 1 and 5 represent a pyrotechnic device which spins the spool of a retractor, causing the belt to be reeled in. The user defines a pull-in versus time curve which applies once the pretensioner activates. Types 2 and 3 represent preloaded springs or torsion bars which move the buckle when released. The pretensioner is associated with any type of spring element including rotational. Note that when the preloaded spring, locking spring, and any restraints on the motion of the associated nodes are defined in the normal way; the action of the pretensioner is merely to cancel the force in one spring until (or after) it fires. With the second type, the force in the spring element is canceled out until the pretensioner is activated. In this case, the spring in question is normally a stiff, linear spring which acts as a locking mechanism, preventing motion of the seat belt buckle relative to the vehicle. A preloaded spring is defined in parallel with the locking spring. This type avoids the problem of the buckle being free to ‘drift’ before the pretensioner is activated. Type 4, a force type, is described below.

To activate the pretensioner, the following sequence of events must occur:

1. Any one of up to four sensors must be triggered.

2. Then a user-defined time delay occurs.

3. Then the pretensioner acts.

Type 1 pretensioner is intended to simulate a pyrotechnic retractor. Each retractor has a loading (and optional unloading) curve that describes the force on the belt element as a function of the amount of belt that has been pulled out of the retractor since the retractor locked. The type 1 pretensioner acts as a shift of this retractor load curve. An example will make this clear. Suppose at a particular time that 5mm of belt material has left the retractor. The retractor responds with a force corresponding to 5mm pull-out on it's loading curve. But suppose this retractor has a type 1 pretensioner defined, and, at this instant of time, the pretensioner specifies a pull-in of 20mm. The retractor then responds with a force that corresponds to (5mm + 20mm) on it's loading curve. This results in a much larger force. The effect can be that belt material will be pulled in, but there is no guarantee. The benefit of this implementation is that the force vs. pull-in load curve for the retractor is followed and no unrealistic forces are generated. Still, it may be difficult to produce realistic models using this option, so two new types of pretensioners have been added. These are available in MD Nastran r2 and later versions.

Types 2 and 3 are simple triggers for activating or deactivating springs, which then pull on the buckle. No changes have been made to these, and they are not discussed here.

The type 4 pretensioner takes a force vs. time curve (see Figure 13-4). Each time step, the retractor computes the desired force without regard to the pretensioner. If the resulting force is less than that specified by the pretensioner load curve, then the pretensioner value is used instead. As time goes on, the pretensioner load curve should drop below the forces generated by the retractor, and the pretensioner is then essentially inactive. This provides for good control of the actual forces, so no unrealistic values are generated. The actual direction and amount of belt movement is unspecified, and depends on the other forces being exerted on the belt. This is suitable when the force the pretensioner exerts over time is known.


Figure 13-4 Force versus Time Pretensioner. At the intersection, the retractor locks.

The type 5 pretensioner is essentially the same as the old type 1 pretensioner, but with the addition of a force limiting value. The pull-in is given as a function of time, and the belt is drawn into the retractor exactly as desired. However, if at any point the forces generated in the belt exceed the pretensioner force limit, then the pretensioner is deactivated and the retractor takes over. In order to prevent a large discontinuity in the force at this point, the loading curve for the retractor is shifted (in the abscissa) by the amount required to put the current (pull-out, force) on the load curve. For example, suppose the current force is 1000, and the current pull-out is -10 (10mm of belt has been pulled in by the pretensioner). If the retractor would normally generate a force of 1000 after 25mm of belt had been pulled OUT, then the load curve is shifted to the left by 3, and remains that way for the duration of the calculation. So that at the current pull-in of 10, it generates the force normally associated with a pull out of 25. If the belt reaches a pull out of 5, the force is as if it were pulled out 40 (5 + the shift of 35), and so on. This option is included for those who liked the general behavior of the old type 1 pretensioner, but has the added feature of the force limit to prevent unrealistic behavior.

The type 6 pretensioner is a variation of the type 4 pretensioner, with features of the type 5 pretensioner. A force vs. time curve is input and the pretensioner force is computed each cycle. The retractor linked to this pretensioner should specify a positive value for PULL, which is the distance the belt pulls out before it locks. As the pretensioner pulls the belt into the retractor, the amount of pull-in is tracked. As the pretensioner force decreases and drops below the belt tension, belt will begin to move back out of the retractor. Once PULL amount of belt has moved out of the retractor (relative to the maximum pull in encountered), the retractor will lock. At this time, the pretensioner is disabled, and the retractor force curve is shifted to match the current belt tension. This shifting is done just like the type 5 pretensioner. It is important that a positive value of PULL be specified to prevent premature retractor locking which could occur due to small outward belt movements generated by noise in the simulation.

Force

Defined Forcevs.

Time Curve

Retractor Lock Time

Retractor Pull-out Force

Time


408

Seatbelt RetractorThe unloading curve should start at zero tension and increase monotonically (i.e., no segments of negative or zero slope).

Retractors allow belt material to be paid out into a belt element. Retractors operate in one of two regimes: unlocked when the belt material is paid out, or reeled in under constant tension and locked when a user defined force-pullout relationship applies.

The retractor is initially unlocked, and the following sequence of events must occur for it to become locked:

1. Any one of up to four sensors must be triggered. (The sensors are described below.)

2. Then a user-defined time delay occurs.

3. Then a user-defined length of belt must be paid out (optional).

4. Then the retractor locks and once locked, it remains locked.

In the unlocked regime, the retractor attempts to apply a constant tension to the belt. This feature allows an initial tightening of the belt and takes up any slack whenever it occurs. The tension value is taken from the first point on the force-pullout load curve. The maximum rate of pull out or pull in is given by 0.01 × fed length per time step. Because of this, the constant tension value is not always achieved.

In the locked regime, a user-defined curve describes the relationship between the force in the attached element and the amount of belt material paid out. If the tension in the belt subsequently relaxes, a different user-defined curve applies for unloading. The unloading curve is followed until the minimum tension is reached.

The curves are defined in terms of initial length of belt. For example, if a belt is marked at 10mm intervals and then wound onto a retractor, and the force required to make each mark emerge from the (locked) retractor is recorded, the curves used for input would be as follows:

Pyrotechnic pretensions may be defined which cause the retractor to pull in the belt at a predetermined rate. This overrides the retractor force-pullout relationship from the moment when the pretensioner activates.

If desired, belt elements may be defined which are initially inside the retractor. These will emerge as belt material is paid out, and may return into the retractor if sufficient material is reeled in during unloading. Elements e2, e3 and e4 are initially inside the retractor, which is paying out material into element e1. When the retractor has fed into e1, where

0 Minimum tension (should be > zero)

10mm Force to emergence of first mark

20mm Force to emergence of second mark

.

.

.

.

.

.

Lcrit

Lcrit fed length 1.1 minimum length×–=


(minimum length defined on belt material input)

(fed length defined on retractor input)

Element e2 emerges with an unstretched length of ; the unstretched length of element e1 is reduced by the same amount. The force and strain in e1 are unchanged; in e2, they are set equal to those in e1. The retractor now pays out material into e2.

If no elements are inside the retractor, e2 can continue to extend as more material is fed into it.

As the retractor pulls in the belt (for example, during initial tightening), if the unstretched length of the mouth element becomes less than the minimum length, the element is taken into the retractor.

To define a retractor, the user enters the retractor node, the ‘mouth’ element (into which belt material will be fed), e1 in Figure 13-5, up to 4 sensors which can trigger unlocking, a time delay, a payout delay (optional), load and unload curve numbers, and the fed length. The retractor node is typically part of the vehicle structure; belt elements should not be connected to this node directly, but any other feature can be attached including rigid bodies. The mouth element should have a node coincident with the retractor but should not be inside the retractor. The fed length would typically be set either to a typical element initial length, for the distance between painted marks on a real belt for comparisons with high speed film. The fed length should be at least three times the minimum length.

If there are elements initially inside the retractor (e2, e3 and e4 in the Figure 13-5) they should not be referred to on the retractor input, but the retractor should be identified on the element input for these elements. Their nodes should all be coincident with the retractor node and should not be restrained or constrained. Initial slack will automatically be set to 1.1 × minimum length for these elements; this overrides any user-defined value.

Weblockers can be included within the retractor representation simply by entering a ‘locking up’ characteristic in the force pullout curve (see Figure 13-6. The final section can be very steep (but must have a finite slope).

1.1∞minimum length


410

Figure 13-5 Elements in a Retractor

Figure 13-6 Retractor Force Pull Characteristics

BeforeElement 1

Element 2

Element 3

Element 4

All nodes within this area are coincident

Element 1

Element 2

Element 3

Element 4

After

without weblockers

with weblockers

PULLOUT

FO

RC

E


Seatbelt SensorSensors are used to trigger locking of retractors and activate pretensioners. Four types of sensors are available which trigger according to the following criteria:

By default, the sensors are inactive during dynamic relaxation. This allows initial tightening of the belt and positioning of the occupant on the seat without locking the retractor or firing any pretensioners. However, a flag can be set in the sensor input to make the sensors active during the dynamic relaxation phase.

Seatbelt SlipringSliprings allow continuous sliding of a belt through a sharp change of angle. Two elements (1 & 2 in Figure 13-7) meet at the slipring. Node B in the belt material remains attached to the slipring node, but belt material (in the form of unstretched length) is passed from element 1 to element 2 to achieve slip. The amount of slip at each time step is calculated from the ratio of forces in elements 1 and 2. The ratio of forces is determined by the relative angle between elements 1 and 2 and the coefficient of friction, . The tension in the belts are taken as and , where is on the high tension side and T1 is the force

on the low tension side. Thus, if is sufficiently close to , no slip occurs; otherwise, slip is just

sufficient to reduce the ratio to . No slip occurs if both elements are slack. The out-of-balance

force at node B is reacted on the slipring node; the motion of node B follows that of slipring node.

If, due to slip through the slipring, the unstretched length of an element becomes less than the minimum length (as entered on the belt material card), the belt is remeshed locally: the short element passes through the slipring and reappears on the other side (see Figure 13-7). The new unstretched length of e1 is

. Force and strain in e2 and e3 are unchanged; force and strain in e1 are now equal to those in e2. Subsequent slip will pass material from e3 to e1. This process can continue with several elements passing in turn through the slipring.

To define a slipring, the user identifies the two belt elements which meet at the slipring, the friction coefficient, and the slipring node. The two elements must have a common node coincident with the slipring node. No attempt should be made to restrain or constrain the common node for its motion will

Type 1 – When the magnitude of x-, y-, or z- acceleration of a given node has remained above a given level continuously for a given time, the sensor triggers. This does not work with nodes on rigid bodies.

Type 2 – When the rate of belt payout from a given retractor has remained above a given level continuously for a given time, the sensor triggers.

Type 3 – The sensor triggers at a given time.

Type 4 – The sensor triggers when the distance between two nodes exceeds a given maximum or becomes less than a given minimum. This type of sensor is intended for use with an explicit mass/spring representation of the sensor mechanism.

μ

T1 T2 T2

T2 T1

T2 T1⁄ eμΘ

1.1 minimum length×


412

automatically be constrained to follow the slipring node. Typically, the slipring node is part of the vehicle body structure and, therefore, belt elements should not be connected to this node directly, but any other feature can be attached, including rigid bodies.

Figure 13-7 Elements Passing Through Slipring

Before After

Slipring

Element 1

Element 2Element 1

Element 2

B

Element 3

Element 3

413Chapter 11: Airbags and Occupant SafetyOccupant Dummy Models

Occupant Dummy ModelsThe occupant dummy models, also known as ATDs (Anthropomorphic Test Devices) were introduced in the previous release of SOL 700 for those applications that airbag was not needed. Many applications such as sled test and aircraft seat design, or armored vehicle design where the occupant behavior is studied when a land mine is detonated do not require airbag simulation.

SOL 700 supports occupant dummies that are readily available in LS-DYNA *key file format. These include:

1. LS-DYNA public domain dummies:

• 5th percentile deformable female dummy

• 50th percentile deformable male dummy


• 5th percentile rigid female dummy

• 50th percentile rigid male dummy


2. ETA (Engineering Technology Associates) calibrated dummies. The calibrated dummies are similar to LS-DYNA dummies except that they are validated against a set of standard tests. The calibration tests are conducted on all Hybrid III, SID and EUROSID models to assure model fidelity.

• 5th percentile deformable female dummy



• 5th percentile rigid female dummy



• EUROSID - 1 (Euro Side Impact Dummy)

• US DOT SID

• EEVC Upper Legform

• FMVSS 201 Headform

Figure 13-8 Typical Occupant Dummies in VPG (courtesy of ETA)

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There are many other ATD models available through ETA VPG (Virtual Proving Ground) pre-post processor. In addition VPG supports the following barriers:

Barriers

• FMVSS & ECE Side Impact Barriers

• 0-Degree Front Rigid Barrier

• 30deg Rigid Front Barrier

• Front Offset Deformable Barrier

• Rear Impact Barrier

Impactors

• Rams

• Pendulums

• Head Forms

For more details, contact ETA in Troy, Michigan.

3. FTSS (First Technology Safety Systems) ATDs.

The FTSS ATD’s are high fidelity dummies and are available with additional licensing. The following FTSS dummies are supported through SimX Crash or ETA – VPG:

• 5% Female dummy

• SID II

• Hybrid III - 3% Child Dummy (W.I.P.)

• Hybrid III - 6% Child Dummy (W.I.P.)

• BIOSID (W.I.P.)

Pre- and PostprocessingEven though MD Patran supports SOL 700, it lacks a versatile dummy positioner or airbag folder. Dummy positioning can be done using pre-post processing tools such as MSC Software SimX Crash and ETA-VPG, both of which support SOL 700.

Chapter 12: System Information and Parallel ProcessingMD Natran R3 Explicit Nonlinear (SOL 700) User’s Guide

12System Information and Parallel Processing

Introduction 416

General Information 416

User Notes 419

Additional information of Different Platforms 421

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IntroductionThis chapter supports

• A general description about the hardware and software requirements and definitions.

• How to use parallel operation for different platforms.

General InformationRelease Platforms

MD Nastran SOL 700 was built and tested on the following hardware with the listed software installed as given in Table 12-1.

Table 12-1 Supported Hardware Configuration

Platform Operating System

Intel (32-bit) Windows 2000-SP4Windows XP-SP2

SGI ** R10K/R12K IRIX64 6.5.22m

HP-UX – PA RISC 2.0 ** HPUX B.11.00

HP-UX Itanium2 HPUX B.11.23

Compaq Alpha Tru64 UNIX v5.1/1885

Sun Sparc Solaris (**) Solaris 8, 64-bit( = SunOS 5.8 )

IBM RS/6000 AIX 5.1

Linux Itanium2 IA64 *) RedHat Advanced Server 3

Linux EM64T x86_64 RedHat Enterprise Linux WS 3

Intel Linux ** Red Hat Linux release 9

*For correct operation of the Intel Fortran compiler, MS Visual Studio .NET must be installed prior to installing the Intel 8.1 compiler. For SOL 700, these installs are not mandatory.

**The SOL 700 in MD Nastran 2006r2 does not support the following configurations:

SGI: R4k and R5k; HP: HP-UX 10.20; SUN: Solaris 7; Linux: Redhat 7.3, Windows NT

417Chapter 12: System Information and Parallel ProcessingGeneral Information

MPI for MD Nastran SOL 700SOL 700 requires MPI to be installed on every machine used. This is true even for single processor machines.

SOL 700 expects hardware-specific native MPI to have been installed at default locations. When MPI is not properly installed on your Unix/Linux machine or is not installed at the expected default location, a job submission will exit with an error message to this effect. To avoid problems of this nature, or problems caused by different versions of MPI, on several of the supported platforms the MPI version is now part of the release and will be installed at a defined location. See Table 12-2 for details.

Hardware and Software RequirementsFor the Linux platforms, the LAM 6.5.9 is required on all the machines that are clustered for the job submission. LAM 6.5.9 is made available and is installed with the Nastran SOL 700 installation.

Table 12-2 MPI Version and Expected Location

Platform Operating System MPI Version MPI Location

Intel (32-bit)Windows 2000-SP4Windows XP-SP2

MPIch V.1.2.5 1

SGI R10K/R12K IRIX64 6.5.22m SGI MPT 1.2 2

HP-UX – PA RISC 2.0 HPUX B.11.00 HP MPI 1.08 3

HP-UX Itanium2 HPUX B.11.23 HP MPI 1.08 3

Compaq Alpha Tru64 Unix v5.1/1885 Compaq dmpirun 1.9.6 /usr/opt/MPI196

Sun Sparc Solaris Solaris 8, 64-bit ( = SunOS 5.8 ) SUN HPC mprun 5.0 /opt/SUNWhpc/HPC5.0

IBM RS/6000 AIX 5.1 POE 3.2 /usr/lpp/ppe.poe

Linux Itanium2 IA64 RedHat Advanced Server 3 LAM 6.5.9 3

Linux EM64T x86_64 RedHat Enterprise Linux WS 3 LAM 6.5.9 3

Intel Linux Red Hat 9 LAM 6.5.9 3

1. Exact location not important as long as it is installed. Install location will be picked up from the registry.

2. Proper install will provide soft links to /usr/bin, /usr/lib.

3. MPI is part of the release and automatically installed in $installdir/bin/exe/mpi

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For HP (HPUX Risc2 and HPUX Itanium2), the HP MPI is required on all the machines that are clustered for the job submission. HP MPI 02.00.00.00 is made available and is installed with the MD Nastran SOL 700 installation.

For all other Unix platforms, it is assumed that a native MPI program is pre-installed. For SGI IRIX64, "Array Services 3.3 MPT 1.3" should be available. For Sun/Solaris, it is "HPC 5.0". For Dec/Alpha "Compaq MPI 1.9.6" and for IBM AIX "POE 3.2".

Although no specific hardware requirements exist for MD Nastran SOL 700 to run in distributed memory parallel mode, it is preferable to have fast network connections between the machines if more than one machine is used. It is recommended that the network should have a speed of at least 100 MBit per second. The appropriate licenses are required in all machines used.

For windows, "MP-MPICH" from University of Aachen, Lehrstuhl für Betriebssysteme is required on all the machines that are clustered for the job submission. MPICH is made available on the Nastran 2005 installation CD. Install MPIch version 1.2.5, by "default" and then the MPIch service will be installed. And, make sure you define MSC_LICENSE_FILE and/or LM_LICENSE_FILE environment variables correctly. If you make any changes make sure you always reboot your machine. The Windows Services manager that is running the MPIch service needs to get these new settings and that is why a reboot is necessary.

One workaround is to make a registry entry setting for MSC_LICENSE_FILE in the Flexlm key location. This change is immediate and doesn’t need a reboot.

The head node is used to check your licenses. So make sure the head node has correctly defined license settings.

If only two machines are to be used, you can use a hub or a cross-over cable to connect them. If more than two machines are to be used, a switch is preferable. TCP/IP is used for communications.

CompatibilityVersion 2006r2 only support connection of homogeneous networks with machines of the same type. Two machines are compatible if they can both use the same executables. Some examples of compatible machines are:

1. Several machines with exactly the same processor type and O/S.

2. One SGI R8000/Irix 6.5 and one SGI R10000/Irix 6.5 machine.

3. One HP J-Class/HPUX-11.0 and one HP C-Class/HPUX-11.0.

Definition1. Root machine: The machine on which the job is started.

2. Remote machine: Any machine other than the root machine that is part of a distributed parallel run on the network.

419Chapter 12: System Information and Parallel ProcessingUser Notes

User NotesThis section assumes that MD Nastran has been successfully installed on at least one of the machines that are to be used in a distributed analysis, and that the appropriate SOL 700 licenses are in order. Assume that host1 is the host name of the machine on which the job is to be started (the root machine). The host names of the other machines (remote machines) are host2, host3, etc.

MD Nastran SOL 700 will always use one of the CPU's of the root machine. Additional slave nodes can be used in a single calculation by using a hostfile. The hostfiles between windows and Linux/Unix are different.

Specification of the Host file for WindowsSee below for an example of a hostfile for windows:

segers 1 D:\MSC.Software\nastran\2006r2\dytran-lsdyna\dytran-lsdyna.exeweenix 2 C:\MSC.Software\nastran\2006r2\dytran-lsdyna\dytran-lsdyna.exe

Specification of the Host file for UNIX and LINUXTo use more than one platform, a host file is must be used. The host file has the following general format.

hostname1 [number 1 of CPUs] [working directory] [executable location 1]hostname2 [number 2 of CPUs] [working directory] [executable location 2]hostname3 [number 3 of CPUs] [working directory] [executable location 3]

...

For example,

Start run on four CPU’s: two on the root machine, bari and two on the remote machine, pisa. Both machines have exactly the same location of executable. Then hostfile has the following lines:

bari 2pisa 2

hostnamei =name of the machine in the network. Each new hostname must be on a new line.

number i CPUs = number of CPU’s to be used on each machine. Default is 1.

working directory =working directory on each machine. This is dummy input for this version.

executable location =

location of executable on each machine. Default for executable location 1 is the script will figure out MSC installation on hostname1. Default for other executable location is executable location 1.

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And, if two CPU’s on bari and one CPU on pisa are used and pisa has different executable location, then the following lines must be specified in hostfile:

bari 2pisa 1 /tmp /app2/msc/nastran/msc2006r2/dyna (note the blank after /tmp)

How to Run MD Nastran SOL 700 in ParallelThis section explains how to run SOL 700 using mutiparallel processors.

If test.dat file is submitted to analyze and host file name is hostlist.txt in UNIX, the command is:

nastran test1.dat hlist=hostfile

If more than one processor on one machine is used, sol700.pth file can be used using PATH=3. sol700.pth includes how many number of processors are used, the full path of Dytran-LSDYNA executable, amount of memory, etc. The details of this file are found in MD Nastran Quick Reference Guide.

For example, sol700.pth includes the commands below if four processors, 20 Mb memories are used in this machine:

/users/joe/sol700/run_dytranexe=/users/joe/sol700/dytran-lsdynanproc=4memory=20m

The name of script is run_dytran located to /users/joe/sol700/ and the executive dytran-lsdyna located to /users/joe/sol700/dytran-lsdyna.

In this case, SOL 700 entry in the Executive Control Statement must have PATH=3.

SOL700, 129, PATH=3

Or without sol700.pth, the command can control the number of processors:

/users/joe/sol700/run_dyna exe=/users/joe/sol700/dytran-lsdyna jid=abcd.dat nproc=4 memory=20m

In the case of Windows, the sequence for multiparallel processing is similar to that of UNIX and Linux.

For example, if PC’s named bari and pisa will use two and four processors respectively, sol700.pth must include following lines (additional options are shown in the MD Nastran Quick Reference Guide for the SOL 700 entry).

E:\users\joe\sol700\ryn_dytrannproc=6 steps=2memory=90mmachine=bari#2+pisa#4

For SOL 700 parallel runs, PATH=3 must be set on the SOL 700 executive control entry and a file named sol700.pth similar to the above must be used. The option to use a separate hostfile for PC’s is not presently available.

421Chapter 12: System Information and Parallel ProcessingAdditional information of Different Platforms

How to Run Fluid Structure Interaction (FSI) in ParallelSee the following steps to run an FSI appplication on multiple processors:

1. In the sol700.pth file, add fsidmp=yes.

2. Add PARAM,FSIDMP,YES in the input deck.

Parallel FSI is supported on Windows 32, Linux 32, and Linux x8664 platforms.

Additional information of Different PlatformsHPUX11 & HPUX IA64

The MD Nastran installation will create a full installation of HP MPI in the MSC installation directory. It is important to note that MD Nastran version 2006r2 is used on all the platforms in the cluster. Using different versions of MD Nastran for DMP calculations is not allowed.

Linux X8664 & Linux i386 & Linux Itanium21. First check that rsh is installed by doing:

rpm -qa | grep rsh

The answer should be:

rsh-server-0.17-14rsh-0.17-14

2. If installed then check the following two files:

/etc/xinetd.d/rsh/etc/xinetd.d/rlogin

There is a line that should say:

disable = no

If it says yes, change it to no.

3. Check the following file:

/etc/nsswitch.conf

It should say:

hosts: files dns nis

files should come before anything else

4. Restart after making changes:

/etc/init.d/xinetd restart

5. Add in .rhosts file to your home directory that contains the following entries:

{machine1} {user}{machine1 full name} {user}{machine2} {user}{machine2 full name} {user}+

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For example:

murano waltermurano.adams.com walterusher walterusher.adams.com waltermiller waltermiller.adams.com walter+

6. Make sure that rsh does not show any echo's from the .cshrc file. You must modify your .cshrc file such with if (! $?prompt) that no echo's or clear's are done. You can check that by:

rsh {remote system} date

7. Set the path such that hboot can be executed. This means that the path must be set correctly by the .cshrc on the remote machine. You can test this by:

rsh {remote system} 'echo $path'rsh {remote system} 'which hboot'

Do not omit the quotes from the command line. Without them, the actual path of the machine you are on will be echoed.

The same working directory as used on the root machine does not have to exist on the remote machines.

On Linux systems, if multiple jobs are to be run by a particular user simultaneously, a manual lamboot from the command line (or a script) must be done prior to running the first job. In addition, PATH=3 must be specifid on the SOL 700, ID entry and a line lamboot=no added. A user can determine the proper location of lam by running a small job, examining the f06 file and searching for the string “execut” The command shown in the example below should then be executed.

A typical f06 file shows the following:

dytran-lsdyna will be executed using the command/usr/msc/programs/nastran/dytran-lsdyna/run_dytran

For this case, the following commands should be placed in a script and the script executed to establish the lamboot:

#!/bin/cshsetenv LAMHOME /usr/msc/programs/nastran/dytran-lsdyna/mpi/binset path = ($LAMHONE $path)

$LAMHOME/lamboot

To check if lam is running, the user should execute the following command:

ps -ef | grep lam

As long as the lam processes is not “killed”, it will remain running unless the machine is rebooted. To terminate the process, either kill it from the command line or enter the following command:

/usr/msc/programs/nastran/dytran-lsdyna/mpi/bin/lamhalt


AIX

The same working directory on the remote machines (the full path) must exist as the one used on the root machine.

Alpha

The same working directory as used on the root machine does not have to exist on the remote machines.

Solaris

The same working directory on the remote machines (the full path) must exist as the one used on the root machine.

SGI Altix & Irix64

Before you can start the MPI on SGI, you first need to setup and start the array services. This array keeps track of which machines belong to the MPI Universe. Unfortunately you need to be root to define one:

<> su root

First check if the array services are enabled:

# vi /usr/lib/array/arrayd.auth

The variable AUTHENTICATION must be set from NOREMOTE to NONE.

Create the array:

# vi /usr/lib/array/arrayd.conf[add in the following lines:]array {array name}machine {machine1}machine {machine2}

Add as many as desired. For instance:

array irixaamachine zephyrmachine tigra

Restart (or start) the array:

# /etc/init.d/array restart

You also need to add a special login name:

#vi /etc/passwdguest:x:81:99:array services guest:/dev/null:/bin/tcsh#vi /etc/groupguest:x:99:Exit root login:# exit


424

To check which arrays are active you can do:<> /usr/sbin/ainfo arrayArrays known to array services daemonARRAY meIDENT 0x78ea

ARRAY {array name}IDENT 0x78f2

The first array is the one for the machine itself (so you can DMP on the machine itself if it has more than 1 CPU). The second one is the one we defined ourselves and can be used for DMP with other machines. To see which machines are defined in the array:

<> /usr/sbin/ainfo -a {array name} machines

For instance:

<> /usr/sbin/ainfo -a irixaa machinesMachines in array "irixaa" per default serverMACHINE zephyrHOSTNAME zephyrPORT 5434IP_ADDR 192.168.22.196IDENT unknown

MACHINE tigraHOSTNAME tigraPORT 5434IP_ADDR 192.168.22.194IDENT unknown

So this array has two machines available: zephyr and tigra. To check if the array was able to access the remote machines:

<> /usr/sbin/array -a {array name} who

The return should be "who" from all machines. For instance:

<> /usr/sbin/array -a irixaa who

The same working directory on the remote machines (the full path) must exist as the one used on the root machine. To start a SOL 700 calculation, you must specify the array to use:

Add array=irixaa to sol700.pth file and use PATH=3 on the SOL 700 executive control entry

User Machine From What

walter tigra.dtw.macsch.com 172.31.188.227 /bin/tcsh

walter zephyr 72.31.188.227 array -a irixaa who


For Altix, the user has two options. He may use SGI MPI or LAM MPI. The LAM is default. To use the SGI MPI the user must use the option: sgimpi=yes. When using SGI MPI the user should not forget to specify also the array. For example:

Add sgimpi=yes array=irixaa to sol700.pth file and use PATH=3 on the SOL 700 executive control entry

Windows XP and 2000MPIch (service) needs to be running on all the machines you are planning to use.

You should either start MPIch manually using “%MPI_ROOT%\mpd\bin\mpd -install” or make sure the service runs by default. Automatically starting the MPIch daemon is not supported.

Run MPIRegister to allow your account to be used to run MPIch."%MPI_ROOT%\mpd\bin\MPIRegister.exe"Define your specific account settings DOMAIN\user. Supply your domain password.Option "Do you want this action to be persistent (y/n)?" ->> y(not sure what happens once your password will be changed though)If ever needed, to clean your registry, use: "%MPI_ROOT%\mpd\bin\MPIRegister.exe -remove"

The most important thing is that you define two shares correctly.

Share 1: %NASTRAN_INSTALLDIR%\%NASTRAN_VERSION%\dynaPreferably this share should have full read access for everyone.

Share 2: %YOUR_RUN_DIR%This should be the directory where your input file is located. So it might be a good idea to define a single location to run all your jobs. Access should be rather open on this share. You will need to have write permissions and most likely the MPIch "service" too. All tests have been performed with Full Access for everyone.

Extra MPIch informationFor more information please visit:

http://www-unix.mcs.anl.gov/mpi/mpich/mpich-nt/mpich.nt.faq.html

Why do I get this error "LaunchProcess failed, CreateProcessAsUser failed, No more connections can be made to this remote computer at this time because there are already as many connections as the computer can accept."?

This error usually occurs when you try to launch an executable from a shared directory on WindowsNT Workstation, Windows 2000 Professional, or WindowsXP Professional. The professional versions of Windows as apposed to the server editions have limitations on the file sharing capabilities. Place the executable on a network share on a server machine or copy the executable to the local drive of each machine.


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Why do I get this error, "Logon failure: unknown user name or bad password"?

You must have the same account credentials on all the nodes participating in the mpich job. If your cluster is set up with a domain controller then you can use a domain account to launch an mpich job. If you do not have a domain controller then you must set up user accounts on all the nodes individually with the same credentials on each node. Each user can have whatever password they choose, but they must use the same password on all the nodes. In other words, UserA-PasswordA must be the same on all the nodes and UserB-PasswordB must be the same on all the nodes, etc.

Why do I get this error, "LaunchProcess failed, CreateProcessAsUser failed, The system cannot find the file specified."?

The executable used in an mpich job must be available to all the nodes participating in the job. The path to the executable must be valid on all the nodes. This can be accomplished by copying the executable to a common location on all the nodes or copying it to a shared location. For example, you could copy cpi.exe to c:\temp\cpi.exe on all the nodes and run "mpirun -np 3 c:\temp\cpi.exe". Or you could copy cpi.exe to a shared directory \\myhost\myshare\cpi.exe and then execute.

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13Examples

Crash 428

Airbags and Occupant Safety 444

Bird Strike and Fan Blade Out (FBO) 459

Drop Test 510

Defense 524

Time Domain NVH 555

Prestress 564

Smooth Particle Hydrodynamics (SPH) 576

Sheet Metal Forming 582

Miscellaneous 596

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Crash

Pick-up Truck Frontal Crash

Description

Auto companies perform crash tests simulation to increase safety of the passengers and comply with government regulations such as those of FMVSS (Federal Motor Vehicle Safety Standards). This is an example of a pick-up truck frontal crash at 15 m/s (34 m.p.h.) against a rigid wall. To model the simulation, contact was defined between the truck and the rigid wall to predict the stress and deformations of the structure.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDTOUTPARAM, DYSHELLFORMPARAM, DYSHINPBCTABLEBCBODYBCPROPBSURFMATD020MATD024CSPOTMATD20M

Model

429Chapter 13: ExamplesCrash

The model has a total of 62984 grid points and 55572 elements as follow:

• 346 Bars

• 1815 Trias

• 49954 Quads

• 3458 Hexas

The rigid wall and the ground are modeled by 500 and 25 shell elements, respectively. All shell elements are Belytschko-Tsay formulation. The crash speed of the truck is modeled by defining an initial velocity of 15 m/s, applied on all the grid points of the truck in a horizontal direction towards the wall. Gravity load is included to take into account the mass of the truck. The simulation time is 0.09 seconds. The unit system is Newtons, seconds, and millimeters.

Input

All nodes of the truck have an initial velocity specified by the TIC entry. All nodes of the rigid wall and the ground have been constrained in all the degrees of freedom. Contact is defined between:

1. The truck and rigid wall surface

2. The truck tires and the ground surface

3. Self contact for the truck to avoid penetration among various components.

Input file:

SOL 700 is a executive control that activates an explicit nonlinear transient analysis.


TIME 10000CEND ECHO = NONE DISPLACEMENT(SORT1,print,PLOT) = ALL Stress(SORT1,PLOT) = ALL Strain(SORT1,PLOT) = ALL accel(print,plot)= ALL velocity(print,plot)= ALL echo=bothSET 990009 = 105843 105655....SET 990619 = 74752 77110


430

SET is an executive control entry that defines a set that contains some grid points. The set will later be referenced by the CSPOT entry in the bulk entry section.

The bulk entry section starts…

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (10) and Time Increment (9e-3 seconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

PARAM, DYDTOUT is a SOL 700 bulk data entry which describes the time interval of d3plot outputs.

PARAM*, DYSHELLFORM is a SOL 700 bulk data entry to define the shell formulation. If DYSHELLFORM = 2, the default shell formulation is Belytschko-Tsay.

PARAM,DYSHINP is a SOL 700 bulk data entry that specifies the number of integration points for SOL 700 shell elements.

LOADSET = 1 SPC = 2 DLOAD = 2 IC=1 TSTEPNL = 20 BCONTACT = 1 weightcheck=yes page

BEGIN BULKINCLUDE 1.truck.bdfINCLUDE rconn.bdfTSTEPNL 20 10 9e-3 1 5 10 ++ ++ 0

PARAM,OGEOM,NOPARAM,AUTOSPC,YESPARAM,GRDPNT,0 PARAM,dydtout,0.001

PARAM*,DYSHELLFORM,2

PARAM,DYSHNIP,2


Bulk data entry that defines gravity load effects:

GRAV is a bulk data entry that defines acceleration vectors for gravity or other acceleration loading.

Bulk data entry that defines Contact relations and Contact bodies:

BCTABLE is a SOL 700 bulk data entry that is also used by SOL 600 and it is meant to define Master-Slave as well as self contact.

BCBODY is a bulk data entry that is used by MD Nastran Implicit Nonlinear (SOL 600) and SOL 700 only, which defines a flexible or rigid contact body in 2-D or 3-D.

$ ------- GRAVITATION ----- LSEQ 1 5 4TLOAD1 4 3 100GRAV 4 0 9806. 0. 0. -1.DLOAD 2 1. 1. 4TABLED1 100 0. 1. 1. 1. ENDT

BCTABLE 1 4 SLAVE 1 MASTERS 2 SLAVE 3 MASTERS 4 SLAVE 5

BCBODY 1 3 DEFORM 1 0BCBODY 2 3 DEFORM 2 0BCBODY 3 3 DEFORM 3 0BCBODY 4 3 DEFORM 4 0BCBODY 5 3 DEFORM 5 0

BCPROP,1,1,2,3,4,10,11,12,+....+,268,269,270,271,24,67,70,195$ rigid wallBCPROP,2,266$ tiresBCPROP,3,168,169,185,187$ groundBCPROP,4,272


432

BCPROP is a bulk data entry that is used by MD Nastran Implicit Nonlinear (SOL 600) and SOL 700 only, which defines a 3-D contact region by element properties. All elements with the specified properties define a contact body.

BSURF is a bulk data entry that is used by MD Nastran Implicit Nonlinear (SOL 600) and SOL 700 only, which defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

Bulk data entry that defines properties for shell and beam elements:

Bulk data entry that defines concentrated mass at grid points:

$ Single Surface ContactBSURF 5 5 6 7 8 9 10 11....

$ ========== PROPERTY SETS ========== PSHELL 1 63 3.137 63 63....PBEAM 224 208 452.4 16290. 16290. 32570.....

$ * conm2 *$CONM2 1990624 91344 1e-06....CONM2 1990693 983105 1e-06


Bulk data entry that continues defining properties for elements, spring, damper, rigid wall and ground:

Bulk data entry that defines material properties:

MATD020 is a SOL 700 bulk data entry that is used to model rigid materials.

MATD024 is a SOL 700 bulk data entry. It is used to model an elasto-plastic material with an yield stress versus strain curve and arbitrary strain rate dependency. Failure can also be defined based on the plastic strain or a minimum time step size.

$ * mpc158 *PBAR 226 158 .01 1e-05 1e-05....$ * mpc408 *PBAR 263 224 .01 1e-05 1e-05$$ * spring *PELAS 264 14.4$$ * damper *PVISC 265 2.935$$ * rigid *PSHELL 266 70 2. 70 70....* ground_prop *PSHELL 272 71 2.0 71 71

$ ========= MATERIAL DEFINITIONS ==========$$$ -------- Material MATRIG.21 id =21MATD020 217.89e-09 210000. .3....

MATD024 637.89e-09 210000. .3 0.9 1....


434

Bulk data entry that defines tables:

Bulk data entry that defines rigid body elements:

Bulk data entry that defines boundary conditions and initial velocity of the pick-up truck:

TIC is a bulk data entry that defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees of freedom.

Bulk data entry that defines spot-weld with failure:

CSPOT is a SOL 700 bulk data entry and was used to define spot-weld with failure. The number of a specific SET defined in the executive control section was referred in the entry.

$ ================ TABLES =================$$ ------- TABLE 1: table270 -------TABLES1 1+B000710+B000710 0. 270. .04879 320.3 .09531 366.3 .1398 402.5+B000711+B000711 .2231 438.8 .2624 448.5 2.398 449. ENDT....

$ ------- Nodal Rigid Body: NodalRigid_5 ----- RBE2 5 104247 123456 104272 104614 104615 105038 105039+B000181+B000181 105043 105044....

SPC1 1 123456 990803 THRU 991384SPCADD 2 1TIC 1 1 1 15000.....TIC 1 510439 1 15000.

CSPOT,990009,990009+,,,1e+08....CSPOT,990619,990619+,,,1e+08


Bulk data entry that merges and defines joint between rigid bodies:

MATD20M is a SOL 700 bulk data entry. It was used to merges MATD020 rigid bodies into one assembly for SOL 700 only.

RBJOINT is a supported bulk data entry in SOL 700, which defines a Joint between two rigid bodies.

End of input file.

MATD20M,181,180,221,182,183....

RBJOINT,1,REVOLUTE,903453,903456,101294,903455....RBJOINT,18,UNIVERS,983102,983103,983104,983105ENDDATA


436

Results

t = 0.0 seconds

t = 0.035 seconds

t = 0.09 seconds


Train-barrier Impact

Description

This is an example of a train frontal crash at 30.559 meter/second against two rigid barrier cylinders to represent an inability to stop at the end of the line. To model the simulation, Contact was defined between the train and the rigid barrier to predict the stress and deformations of the structure.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM,DYDTOUTPARAM*,DYTSTEPDT2MSBCTABLEBCBODYBCPROPMATD024MATD001MATD020CSPOT

Model

Because of symmetry, only half of the actual structures (train and barrier) were modeled in this example. Boundary conditions were applied along the center line of the structures to ensure symmetric behavior. The model has a total of 117820 grid points and 113770 elements as follow:

• 586 Bars

• 1533 Trias

• 107047 Quads

The train model contains 112530 elements in the form of shell and beam. The rigid barrier cylinder model contains 1240 shell elements. All shell elements are Belytschko-Wong-Chiang formulation. The crash speed of the train is modeled by defining an initial velocity of 30.559 meter/second, applied on all the grid points of the train in a horizontal direction towards the barrier. The simulation time is 0.35 seconds. The unit system is Kilonewton, millimeters, and milliseconds.


438

Input

All nodes of the train have an initial velocity specified by the TIC card. All nodes along the center line have a boundary condition that ensures symmetric behavior of the structure. Adaptive contact relation was defined as follows:

1. Contact between the train and the barrier

2. Self-Contact of the train components

Input file:


SET is an executive control entry that defines a set that contains some grid points. The set will later be referenced by the CSPOT entry in the Bulk Entry Section.


TIME 10000CEND ECHO = NONE DISPLACEMENT(SORT1,print,PLOT) = ALL Stress(SORT1,PLOT) = ALL Strain(SORT1,PLOT) = ALL accel(print,plot)= ALL velocity(print,plot)= ALL echo=bothSET 26148 = 92304 92319 92334....SET 84302 = 98670 103916

$ LOADSET = 1 SPC = 2$ DLOAD = 2 IC=1 TSTEPNL = 20 BCONTACT = 1 weightcheck=yes page


The bulk entry section starts...

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (10) and Time Increment (35 milliseconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

PARAM, DYDTOUT is a SOL 700 bulk data entry which describes the time interval for d3plot outputs.

PARAM*,DYTSTEPDT2MS is a SOL 700 bulk data entry card. When it is specified, mass scaling will be activated. The value given will be the minimum time step with which the model will run. If the stable time step of an element is smaller than this value, its density will be increased.


BEGIN BULKINCLUDE modelini.datINCLUDE cspot_new.datINCLUDE cspot_old.dat$TSTEPNL 20 10 35.0 1 5 10 ++ ++ 0

PARAM,OGEOM,NOPARAM,AUTOSPC,YESPARAM,GRDPNT,0 PARAM,dydtout,5.0

PARAM*,DYTSTEPDT2MS,-5e-3

BCTABLE 1 4 SLAVE 1 0. 0. 0.7 0. 0 0. 0 0 0 0.3 0.0 full+ 0 1 SLAVE 2 0. 0. 0.7 0. 0 0. 0 0 0 0.3 0.0 MASTERS 3


440




Bulk data entry that defines grid points and elements:

Bulk data entry that defines concentrated mass at grid points:

BCBODY 1 3 DEFORM 1 0BCBODY 2 3 DEFORM 2 0BCBODY 3 3 DEFORM 3 0

$ Single Surface ContactBCPROP,1,585,772,586,587,634,635,636,+....BCPROP,2,585,731,1079,4915,4916,4364,4368,+....BCPROP,3,4381,4382

GRID 1 -12750. 586.2983.919....GRID 117820 -6002. 1145.5 950.....$DBLOCK FS_AB_SIDEPOSTS$ CQUAD4 ELEMENTS IN PART - FS_AB_SIDEPOSTS (PID = 584)$CQUAD4 1914 584 6594 6597 6544 6545....

$DBLOCK Added Masses$ CONM2 ELEMENTS IN PART - Added Masses (PID = 5242)CONM2 101538 6729 1.972 Added Ma....CONM2 107566 104763 30000. Added Ma


Bulk data entry that defines boundary conditions along the center line:

Bulk data entry that defines material properties.

MATD024 is a SOL 700 bulk data entry that is used to model an elasto-plastic material with an yield stress versus strain curve and arbitrary strain rate dependency. Failure can also be defined based on the plastic strain or a minimum time step size.

MATD001 is a SOL 700 bulk data entry that is used to model an isotropic elastic material available for beam, shell, and solid elements.


Bulk data entry that defines properties for bar, elastic and shell elements:

$ SPC BCS IN BCSET - Nodal SPC (SID = 2)SPC 2 11 246 0.0.... SPC 2 104764 23456 0.0

MATD024,59,7.900-6 ,200.000, 0.30, 0.340,,,,++,,,10....

MATD001, 61,7.800-6 ,200.000, 0.30,....

MATD020,62, 7.800-6, 200.000, 0.30,,,,,++1.0,0,0....

PBAR 687 61 156. 24760. 24760.....PELAS 5235 5.0 Section_....PSHELL 5241 60 3. 60 60 FS_FR_FR


442

Bulk data entry that defines tables:

Bulk data entry that defines initial velocities for train model:






End of input file.

TABLES1,5,,,,,,,,++,0.0,3.494E-1,5.00E-02,4.58E-01,2.0E-01,5.749E-1,3.02E-01,6.119E-1,++,4.0E-01,6.33E-01,ENDT....

TIC,1,1,1,,30.559....TIC,1,117820,1,,30.559

CSPOT,26148,26148,,,1e+20 ,0.5....CSPOT,48348,48348,,,1e+20 ,0.5

CSPOT,1099,1099,,,40,100,1,1....CSPOT,84302,84302,,,40,100,1,1


Results

t = 0.0 seconds

t = 0.12 seconds

t = 0.24 seconds

t = 0.35 seconds

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideAirbags and Occupant Safety

444

Airbags and Occupant Safety

Simulation of Multi-compartment Airbag

Description

The purpose of this example is to demonstrate the simulation of an multi-compartment airbag in MD R2 Nastran SOL 700.

Multi-compartment airbag analysis is a new capability that is introduced in MD R2 SOL 700. New entries (AIRBAG, GRIA, EOSGAM) are added in Bulk Data entries to support the capability.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDTOUTAIRBAGEOSGAMGRIA

Analysis Scheme

Airbag Model (BDF)

MD Nastran SOL 700

Obtain Binary Results

• Deformation (AIRBAG)

• CFD results (GAS)

445Chapter 13: ExamplesAirbags and Occupant Safety

Model

There are five compartments in the airbag as shown in the figure below. These compartments were folded, and each compartment was connected to the gas supply bag through a large hole. An inflator was modeled next to the gas supply bag. The gas jet is initiated from the inflator and running into the gas supply bag. Fixed boundary conditions were applied to the brackets attached to the gas supply bag. The simulation time was 0.02 seconds.

Unit used for this analysis was kg for weight, meter for length, second for time, Kelvin for temperature.

Input

Detail for AIRBAG card was described below:

AIRBAG , 1 , 25 , , , , , , ,++ ,CFD , 3 , , 1.527 ,0.009 ,0.009 ,0.009 , ,+

Within the CFD keyword line, CFD related data was defined. Gamma low equation of state was defined referring to the EOSGAM card as shown below.

+ ,INITIAL , 101325., 293. , 1.4 , 294. , , , ,+

AIRBAG , 1 , 25 , , , , , , ,++ ,CFD , 3 , , 1.527 ,0.009 ,0.009 ,0.009 , ,++ ,INITIAL , 101325., 293. , 1.4 , 294. , , , ,++ ,INFLATOR , 1001 , 1 , 350. , , ,0.7 , ,++ , , 1.557 , 243. , , , , , ,++ ,LARGHOLE , 301 , 2 , , 1.0 , , , ,++ ,LARGHOLE , 302 , 3 , , 1.0 , , , ,++ ,LARGHOLE , 303 , 4 , , 1.0 , , , ,++ ,LARGHOLE , 304 , 5 , , 1.0 , , , ,++ ,LARGHOLE , 305 , 6 , , 1.0

Inflator

Compartment

Fix

Gas supply bag


446

Within the INITIAL keyword line, initial conditions of gas property inside an airbag were defined. Initial pressure was , initial temperature was 293 K, initial gamma gas constant was 1.4,

initial R gas constant was .

+ ,INFLATOR , 1001 , 1 , 350. , , ,0.7 , ,++ , , 1.557 , 243. , , , , , ,+

Within the INFLATOR keyword line, gas property from an inflator was defined. Mass flow rate was defined referring a table data (TABLED1). Temperature of inflowing gas was 350 K, a scale factor of available inflow area was 0.7, and the gamma gas constant of the inflator gas was 1.557, and the R gas constant of the inflator gas was

+ ,LARGHOLE , 301 , 2 , , 1.0 , , , ,+

Within the LARGEHOLE keyword line, the airbag into which gas flowed was defined. Here, the airbag ID 2 was the airbag to which this airbag gave gas. A scale factor of available inflow area was 1.0

Each compartment airbag must be defined with AIRBAG card. The reference density must be the same for all AIRBAGs that were defined in one simulation.

The new card which describes gas state was provided. EOSGAM was the card which defined the gamma low gas equation of state where the pressure p was defined as:

where was a constant, was specific internal energy per unit mass, was overall material density.

A gamma constant was 1.517, R gas constant was in this model.

GRIA card is a SOL 700 Bulk Data entry which defines the final unstretched configuration of a deployed bag. All IDs of GRIA cards must exist in GRID cards and the same as the IDs of GRID cards.

AIRBAG , 1 , 25 , , , , , , ,+AIRBAG , 2 , 35 , , , , , , ,++ ,CFD , 3 , , 1.527 ,0.011 ,0.011 ,0.011 , ,++ ,INITIAL , 101325., 293. , 1.4 , 294. , , , ,

EOSGAM,3,1.517,226.4

GRIA 1 .0009375-.626128 .230000GRIA 2 .0009375-.626128 .220000GRIA 61 .0005000-.414100 .450000GRIA 62 .0005000-.339100 .450000...

101,325N m2⁄

294m2 s2 K⁄⁄

243m2 s2 K⁄⁄

p γ 1–( ) ρe⋅=

γ e ρ

226.4m2 s2 K⁄⁄


Result

There are two types of results files, ARC and d3plot. The ARC file which is the original MSC.Dytran result file format is for the results of Euler elements (fluid) and d3plot file which is the native LS-DYNA result file format. The deformation results between SOL 700 and MSC.Dytran are compared to be identical. These results were from the binary result files (ARC file for both cases).

Deformation Result of Airbag (d3plot)

Time MD R2 Nastran SOL 700 MSC.Dytran 2005r3

0.000

0.002

0.004


448

0.006

0.008

0.010



Euler Regions for Multi-compartment (ARC)

0.0148

0.020


0.020



450

Airbag with Dummy

Description

Automotive companies perform crash simulations including airbags and dummies to predict the forces that would be exerted on the passenger. For people of average size the airbag can be computed using a uniform gas bag method. In other cases the airbag can hit the passenger before the airbag is fully deployed. Then the flow is not uniform yet and to get an accurate force prediction the flow inside the airbag has to be computed by a CFD approach. The following simulation demonstrates the CFD capability of the SOL 700 to predict the interaction of the airbag and the occupant model during a crash event.

SOL 700 Entries IncludedSOL 700

TSTEPNL

PARAM, DYDTOUT

AIRBAG

BCTABLE

BCBODY

BCGRID

BCPROP

BCSEG

BSURF

EOSGAM

MATD001

MATD009

MATD020

SPC1

SPCADD

SPCD2


Model

The model has a total of 10868 grid points and 9594 elements as follows:

• 3 Bars

• 23 Beams

• 2078 Quads

• 3626 Trias

• 3410 Hexas

• 454 Pentas

The dummy is modeled by several types of elements and the floor, chair and window are modeled by shell elements. The airbag surface is meshed with membrane elements. To simulate the gas jet inside the airbag, no elements need to be defined because that is done by the adaptive hex mesher of the Eulerian solver of SOL 700. In other words, the adaptive Euler solver automatically creates the elements inside the bag. As the airbag is inflated, these elements are expanded and “adapt” to follow the fabric surface of the bag. This mesher provides an Euler mesh that is large enough to cover the whole airbag membrane but at the same time keeps the elements that are outside the airbag surface limited. Too many empty elements outside the airbag surface increases runtime without benefit.

Input

Several contacts are defined between airbag, dummy and parts of the car.

Input file

SOL 700,NLTRAN stop=1


452

SOL 700 is an executive control entry similar to SOL 600. It activates an explicit nonlinear transient analysis using MD Nastran solver.

This parameter set defaults to original LS-Dyna values.

The bulk entry section begins with...

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (50) and Time Increment (0.0012 sec) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

Bulk data entry that defines the airbag:

AIRBAG is a SOL 700 bulk data entry that instructs SOL 700 to create an airbag using either the CFD solver or a uniform gas bag method. Here, the CFD solver will be used. Inflow of gas into the airbag is defined by the entries following the INFLATOR key word. Outflow is defined by adding SMALHOLE.

Bulk data entries defining materials:

MATD001 is a SOL 700 bulk data entry that models an isotropic elastic material. It represents the membrane material.

SUBCASE 1 TITLE=This is a default subcase. TSTEPNL = 1 SPC = 2 BCONTACT = 1 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALLPARAM,DYDEFAUL,DYNA

TSTEPNL 1 50 .0012 1 ADAPT 2 10

AIRBAG 10 25 ON ++ CFD 1 1.025-12 20.0 20.0 20.0 ++ NONE 11. 11. 11. ++ INITIAL .101325 297. 1.4 226.45E6 + + INFLATOR 82 100 102 ++ 1.4 286.E6 +

+ SMALHOLE 81 OUT 0.9

$ PERMEAB OUT 20 0.8

MATD001,111,6.E-10,6.E1,0.3


EOSGAM is a SOL 700 bulk data entry that defines the ideal gas inside the airbag.

MATD009 is a SOL 700 bulk data entry that models materials without resistance to deviatoric stresses.

MATD020 is a SOL 700 bulk data entry that models rigid objects.

All other MATD** entries model the various parts of the dummy.

Bulk data entries entry that defines geometric properties of the various surfaces

All properties that are not part of the dummy are defined by the PSHELL1 entry. The dummy is modeled by using many element types and joints: CPENTA, CHEXA , RBJOINT, RBJSTIFF, CBAR, CBEAM, HGSUPPR, CSPR, PSPRMAT, MAT1, and MATRIG.

EOSGAM,1,1.4,226.45E6

MATD009,222,1.E-12MATD020,444,7.8E-9, 2.1E05,0.3MATD020,185,7.8E-9, 2.1E05,0.3

Surface Elements Grid Points Properties

Airbag membrane 20001-32132 20001- 32690 601-613 651-663 701-713 751-763 (CTRIA3)

Inflator 23225-23514 605 (CTRIA3)

Holes in the airbag 20655,20656,21461,

21462,22267,22268,

23073,23074

610,660,710,760 (CTRIA3)

Car Window +frame 31863-32102 2100 (CQUAD4)

Floor 31683-32132 2101 (CQUAD4)

dummy 1-5678 1 – 9015 All other properties. Various element types

Gas inside the airbag Elements created by adaptive Euler solver


454

Bulk data entries that define Contact relations and Contact bodies:


BCBODY is a bulk data entry that is used by SOL 600 and SOL 700 only. It defines a flexible or rigid contact body in 2-D or 3-D. It could be specified with a BSURF, BCBOX, BCPROP, or BCMATL entry.

BPROP is a SOL 700 bulk data entry that is used by SOL 600 and SOL 700 only, which defines 3-D contact regions by element properties.

BSURF is a SOL 700 bulk data entry that is used by SOL 600 and SOL 700 only. It defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

BCTABLE*1 7 ** *$$ contact 1 :$ ID = 1$ TITLE = CHEST TO RIB$* SLAVE 1 ** 0.9000000 ** ** ** 0.9000000 SS2WAY ** ** 0.1000000 1.0000000E+20 ** 1.000000 1.000000 ** ** 2 1 ** ** 20.00000 ** 5.000000 5.000000 ** 1 1.000000 YES ** *

$ Deform Body Contact LBC set: airbagBCBODY 21 26 0BCPROP 3 26 601 602 2119 2120 2121 2122

$ Deform Body Contact LBC set: plateBCBODY 4 3D DEFORM 4 0BSURF 4 2801 2802 2803 2804 2805 2806 2807....


BCSEG is a bulk data entry that is used by SOL 600 and SOL 700 only. It specifies grid points to be used in a contact analyses.

Using the BCTABLE and several BCBODY, BCSEG, and BCSURF entries the following contacts are defined

Bulk data entries that defines output at specific positions

BCSEG* 1 1 2680 3399 ** 3400

Slave Master

1 Chest rib

2 neck Self contact

3 Rib shoulder

4 Rib Jacket

5 Neck cable

6 Shoulder belt chest

7 Lap pelvic

8 Airbag Self contact

9 Airbag Dummy front

10 Dummy Chair

11 Dummy shoes Ground

$ Number of Seatbelt elements = 6$$ to compare simulation results with actual measurementsACCMETR*1 9001 8316 8317 ** 0 0....$ five more


456

Bulk data entry that defines constraints:

End of input file.

$ constraint for inflatorSPCADD 2 1 2 5 6$ Displacement Constraints of Load Set : spc1.1SPC1 1 123456 20253 20254 20255 20256 20257 20258....

Frame chair floor

SPCD2, 2,RIGID,MR185,1,2,101, 0.,,++, 0., 10.SPCD2, 2,RIGID,MR185,2,2,101, 0.,,++, 0., 10.SPCD2, 2,RIGID,MR185,3,2,101, 0.,,++, 0., 10.SPCD2, 2,RIGID,MR185,5,2,101, 0.,,++, 0., 10.SPCD2, 2,RIGID,MR185,6,2,101, 0.,,++, 0., 10.SPCD2, 2,RIGID,MR185,7,2,101, 0.,,++, 0., 10.

RBE2A* 2 MR2 ** ** 3922 3923 8297 8319 ** 8463 8465


Results

t = 0 sec

t = 0.02999 sec


458

t = 0.0444 sec

t = 0.0444 sec

459Chapter 13: ExamplesBird Strike and Fan Blade Out (FBO)

Bird Strike and Fan Blade Out (FBO)

Bird Strike Simulation on Composite Glass Panel

Description

Aerospace companies perform bird strike test simulation to predict the impact-resistance properties of the aircraft structure. This is an example of a 3.8 lbs bird, impacting against the composite glass panel of an aircraft canopy. The bird’s velocity is 7874 in/s (447 m.p.h.).

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDTOUTBCTABLEBCBODYBCPROP BSURF PCOMPMATD054MATD010PSOLIDDEOSPOL

Model

To model the bird, a cylinder with 2.42 inch radius and 6 inch length was built. Material model MATD010 was used to simulate the elastic-plastic hydrodynamic properties of the bird with parameters as follows:

The relation between density and pressure ( ) of the bird material was specified through an equation of linear polynomial:

ρ 9E-5lb-mass/in3=

σy 2.9 psi=

G 145 psi=

Eh 0.145 psi=

P

P aeμ3 b0 b1μ b2μ2 b3μ3+ + +( )ρ0E+=

μ ρ ρ0 1–⁄=

ρ0 reference density=

a1 0.0=

a3 4.25 E6 psi=

ρ overall material density=

E specific internal energy per unit mass=

a2 0.0=

b1 b2 b3 0.0= = =

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideBird Strike and Fan Blade Out (FBO)

460

To model the composite glass canopy, a curved composite plate was built. The composite plate was modeled using PCOMP and MATD054 material.

A picture of the model is shown below.


• 2000 Quads

• 4200 Hexas

The glass plate was modeled with composite shell elements. The bird was modeled with solid elements. The relative speed of the bird to the airplane is modeled by defining an initial velocity of 7874 in/s (447 m.p.h.), applied on all the grid points of the bird. The simulation time is 0.002 seconds. The unit system is inch, lb-force, second.

Input

All nodes of the bird have an initial velocity specified by the TIC entry. Nodes on the straight edges of the curved plate have translational constrains in all directions. Contacts are defined between the plate (master) and the bird (slave).

Input File:





TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (2e-5 seconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.



BCTABLE is a SOL 700 bulk data entry that is also used by SOL 600 and it is meant to define Master-Slave as well as self-contact.

$ Direct Text Input for Executive ControlCENDTITLE = JOBNAME IS: BIRD_STRIKE2SUBCASE 1$ Subcase name : Default SUBTITLE=Default TSTEPNL = 1 BCONTACT = 1 SPC = 2 IC = 1 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL

$------- BULK DATA SECTION -------BEGIN BULKINCLUDE final_scale2.bdfTSTEPNL 20 100 2e-5 1 ADAPT 2 10

PARAM,DYDTOUT,4e-4

$ Define contactBCTABLE 1 1 SLAVE 3 0. 0. 0. 0. 0 0. 0 0 0

MASTERS 4


462

BCBODY is a bulk data entry that is used by SOL 600 and SOL 700 only and defines a flexible or rigid contact body in 2-D or 3-D. It could be specified with a BSURF, BCBOX, BCPROP or BCMATL entry.

BCPROP is a bulk data entry that is used by SOL 600 and SOL 700 only. It defines a 3-D contact region by element properties. All elements with the specified properties define a contact body.

BSURF is a SOL 700 bulk data entry that is used by SOL 600 and SOL 700 only and defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

Bulk data entry that defines material and element properties for composite glass plate:

PCOMP is a bulk data entry. It was used to define the properties of a 5-ply composite material laminate.

MATD054 is a SOL 700 bulk data entry that is used to model arbitrary orthotropic materials, e.g., unidirectional layers in composite shell structures. Various types of failure could be specified for the model. This model is only valid for thin shell elements.

$ Deform Body Contact LBC set: birdBCBODY 3 3D DEFORM 3 0BCPROP 3 2

$ Deform Body Contact LBC set: plateBCBODY 4 3D DEFORM 4 0BSURF 4 1 2 3 4 5 6 7....

PCOMP 1 70. 0. 1 .012 0. YES 2 .012 0. YES 1 .012 0. YES 2 .012 0. YES 1 .012 0. YES....


Bulk data entry that defines material and element properties for bird:

MATD010 is a SOL 700 bulk data entry and is used to model elastic-plastic hydrodynamic material.

PSOLIDD is a SOL 700 bulk data entry that defines properties for solid elements. It refers to entry MATD010 for material properties, and refers to entry EOSPOL for relation between the density and pressure of the hydrodynamic material.

EOSPOL is a SOL 700 bulk data entry that defines the properties of a polynomial equation of state.

$ Material Record : mat1.1MATD054 1 2.33-4 1.16E7 1.16E7 1.16E7 0.223 0.223 0.223 4.74E6 4.74E6 4.74E6 0,,,,,,,,,,,,,,,,,,,,,,,,,,0.02, 42000. 42000. 42000. 42000. 1035.$ Material Record : mat1.2$ Description of Material :MATD054 2 9.53-5 2812 2812 2812 0.48 0.48 0.48 950. 950. 950. 0,,,,,,,,,,,,,,,,,,,,,,,,,,2, 281.2 281.2 281.2 281.2 95....

MATD010 3 9E-5 145 2.9 0.145

$ Elements and Element Properties for region : bird_propPSOLIDD 2 3 1 1EOSPOL 1 0.0 0.0 4.25E6


464

Bulk data entry that defines geometric properties of the model:

Bulk data entry that defines boundary conditions and initial velocity of the bird:


End of input file.

$ Pset: "plate_prop" will be imported as: "pshell.1"CQUAD4 1 1 1 2 13 12 0.....$ Pset: "bird_prop" will be imported as: "psolid.2"CHEXA 2001 2 2115 2133 2092 2093 2256 225322542255 ....$ Nodes of the Entire ModelGRID 1 11.2583 -4.72577-6...

$ Loads for Load Case : DefaultSPCADD 2 1$ Initial Velocities of Load Set : bird_velTIC 1 2092 1 4609.88TIC 1 2092 3 -3601.6....$ Displacement Constraints of Load Set : fixed_endSPC1 1 123456 1 THRU 11....$ Referenced Coordinate FramesCORD2R* 1 -5.12207 0.* 4.0018 2.12867-5 0. 2.72458-5 -1.1203 0. 9.12389


Results

t = 0.0 seconds

t = 0.0008 seconds

t = 0.0016 seconds

t = 0.002 seconds


466

Bird Strike on Rotating Fan Blades with Prestress

Description

Aerospace companies perform bird strike test simulation to predict the impact-resistance behavior of the aircraft engines. This is an example of a bird made by solid elements impacting against rotating fan blades. The fan blades and the rotor are initially prestressed due to the high rotational velocity by the implicit solver prior to the transient run that is simulated by the explicit solver.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDEFAUL, DYNADYPARAM*, LSDYNAPRESTRSISTRSSHSPCD2TIC3BCTABLEBCBODYBCGRIDBCPROP EOSPOLHGSUPPRMATD010MATD024MATDEROPSOLIDDRFORCETICTLOAD1

Model

The models for implicit and explicit runs are basically the same except that in the implicit run the bird model is not included. The hourglass control and definition of rotational velocities between implicit and explicit runs also differ.

The rotor, hub and fan blades are modeled by shell elements while the bird is modeled by solid elements. The rotational speed of blades and rotor is 8000 rpm which is applied using RFORCE card (rotational static force) in the prestress run, and TIC3 card (rotational initial speed) and SPCD2 cards in bird strike run. The speed of the impacting bird to the fan blades is assigned by an initial velocity of 7692 inch/s (437 mph), applied on all the grid points of the bird. The simulation time is 0.004 seconds. The unit system is inch, lbf and second.


The thickness of fan blades varies from 0.02659 to 0.40227 inch depending on the location of blade element. The radius of fan blades is 13.6 inch and that of rotor is 0.9 inch. The length and radius of bird model are 6.24 and 2.36 inches respectively.

Two material models are used in the analysis. All blades and rotors are elastic-plastic material and modeled by MATD024. The material properties are:

Poisson’s ratio = 0.35

σy = 138000 psi Tangent modulus = 100000 psi

Young’s modulus = 1.60E+7 psiPlastic strain failure limit 0.2

To model the bird, a cylinder with 2.42 inch radius and 6 inch length was built. Material model MATD010 was used to simulate the elastic-plastic hydrodynamic properties of the bird with the following properties:

G = 145 psi

Tangent modulus = 0.145 psi

8000 rpm

Fully fixed

Fixed (x,y direction) (x,y rotation)

8000 rpm (initial speed)

Fixed (x,y,z direction)

Fixed (x,y direction)

437 mph

8000 rpm (enforced speed)

(a) Prestress Model (Implicit)

(b) Bird Strike Model (Explicit)

ρ 4.14e 4 lbf/inch3– s2 inch⁄–=

σy 138000 psi=

ρ 9E-5 lbf/inch3 s2 inch⁄–=

σy 2.9 psi=


468

The relation between density and pressure (P) of the bird material was specified through an equation of linear polynomial:

E = specific internal energy per unit mass

The model used for bird strike analysis (explicit) has a total of 20724 grid points and 19055 elements as follows:

Input

The simulation consists of two runs. The first run is a prestress analysis that computes the deformations and stresses due to rotation. This computation is essentially static using implicit solver. Conducting this computation in the impact run would require structural relaxation. To speed up the process, the static part of the computation is done by the double precision version of the implicit solver in the prestress run. Boundary conditions and initial conditions of the prestress run differ from the second run. In the prestress run the rear of the rotor is fixed and a force in circumferential direction is applied to the rotor and fan blades. In the impact run the rear of the rotor is given an angular velocity and the bird is given an initial velocity. Contacts are defined between the fan blades and the bird.

SOL 700 is a executive control entry that initiates the explicit nonlinear solver in the MD Nastran. PATH is set to 3 because a different solver must be assigned for the implicit run (double precision solver.)

First, consider the prestress input deck. In this input deck, only the rotor and fan blades are included.

NUMBER OF CHEXA ELEMENTS = 10752NUMBER OF CQUAD4 ELEMENTS = 8227NUMBER OF CTRIA3 ELEMENTS = 76

SOL 700,NLTRAN STOP=1 PATH=3

P a1μ a2μ2 a3μ3 b0 b1μ b2μ2 b3μ2+ + +( )ρoE+ + +=

μ ρ ρo 1–⁄= ρ overall material density=

ρo reference density=

a1 0.0= a2 0.0=

a3 4.25 E6 psi= b1 b2 b3 0.0= = =


The bulk entry section begins with

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (5) and Time Increment (1e-5 sec) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps are determined by MD Nastran SOL 700 solver during the analysis. The time step is a function of the smallest element dimension during the simulation.

By using the SOL 700 bulk data entry PRESTRS a prestress analysis is carried out. Here the implicit SOL 700 solver is used. This solver requires the analysis to be run with double precision executable. Final deformations and stresses of elements are written to a text file named imput_file_name.nastin to provide initial conditions for rotor and fan blades of the impact run.

Bulk data entries that define properties for shell elements.

These elements model the fan blades and rotor.

Bulk data entry that defines material properties for rotor and fan blade

MATD024 is a SOL 700 bulk data entry that represents MD Nastran Material #24. It is used to model elasto-plastic material.

LOADSET = 2TITLE = MD Nastran job created on 18-Jan-07 at 10:58:50$ Direct Text Input for Global Case Control DataSUBCASE 1$ Subcase name : Default SUBTITLE=Default TSTEPNL = 1 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL DLOAD = 3

TSTEPNL 1 5 1.-5 1 ADAPT 2 10 $

PSHELL1 135 3001 KeyHoff Lobatto 5 Mid ++ 0.05440

MATD024 1 4.14E-4 1.60E+7 0.35000 100000. 138000. 0.99000 + + 3001


470

Bulk data entry that defines geometric properties of the rotor and blades:

Bulk data entry that defines boundary conditions and loads:

RFORCE is a bulk data entry that defines a static loading condition due to angular velocity.

The following parameters are needed to get optimal results.

CQUAD4 6062 301 300425 300426 300446 300445..GRID 1 1.0762 4.6577 -1.945..

TLOAD1 5 6 321 LSEQ 2 6 1DLOAD 3 1. 1. 5RFORCE 1 299999 -133.3330.0 0.0 1. TABLED1 321 0. 1. .001 1. ENDTSPC1 1 123456 300425 THRU 300443SPC1 1 1245 400058SPC1 1 1245 400115

DYPARAM,LSDYNA,ACCURACY,OSU,1DYPARAM,lSDYNA,ACCURACY,INN,2$$ control_ENERGY$DYPARAM,LSDYNA,ENERGY,HGEN,2DYPARAM,LSDYNA,ENERGY,RWEN,1DYPARAM,LSDYNA,ENERGY,SLNTEN,1DYPARAM,LSDYNA,ENERGY,RYLEN,1$$ control_HOURGLASS$DYPARAM,LSDYNA,HOURGLASS,IHQ,1DYPARAM,LSDYNA,HOURGLASS,QH,0.100000001DYPARAM,LSDYNA,DATABASE,D3PLOT,0.1PARAM,DYDEFAUL,DYNA


Now consider the bird strike run.

The bulk entry starts with:

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (.4e-4 sec) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps are determined by MD Nastran solver during the analysis. The time step is a function of the smallest element dimension during the simulation.

The prestress results file is prestress_rotor.dytr.nastin. The name of this file was changed to rotor.dytr.nastin due to the long file name. It includes the information of grid points, elements and initial conditions. This makes the shell elements that are defined in the prestress run available to the impact run. Because the only properties of the rotor and fan blade that need to be defined in the impact run are material properties and boundary conditions, all other information like GRID, CQUAD, etc must be deleted in the explicit input deck.

The file rotor.dytr.nastin contains the entry ISTRSSH. This entry specifies the prestress condition of the shell element. The information in the upper box are that element number 2275 has (1) in-plane integration point and (5) through-thickness integration points followed by (5) additional history variables in the first row. The contents from next row to the end include the initial conditions of this element such as the initial

TITLE = MD Nastran job created on 18-Jan-07 at 10:58:50$ Direct Text Input for Global Case Control DataSUBCASE 1$ Subcase name : Default SUBTITLE=Default TSTEPNL = 1 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL IC = 1 SPC = 1 BCONTACT = 1

BEGIN BULKinclude rotor.dytr.nastinTSTEPNL 1 100 .4e-4 1 ADAPT 2 10

ISTRSSH* 2275 1 5 5* * * * 0.000E+00 6.916E+03 7.371E+03 1.480E+02* * 7.084E+03 -3.908E+01 1.150E+02 0.000E+00* * 0.000E+00 0.000E+00 0.000E+00 1.392E+05* * 1.000E+05 0.000E+00 0.000E+00 0.000E+00* * -1.000E+00 1.481E+04 1.149E+04 1.999E+03* * 1.096E+04 5.499E+00 2.964E+03 0.000E+00* * 0.000E+00 0.000E+00 0.000E+00 1.392E+05*


472

stress in xx direction, yy direction and so on. These result variables of the prestress run are to be carried over to the impact run. If other elements types instead of shell are used, ISTRSBE, ISTRSTS, and ISTRSSO cards must be included in nastin file.


BCTABLE is a SOL 700 bulk data entry and it is meant to define Master-Slave as well as self contact. In this analysis, adaptive contacts between the bird and the fan blades are used.

BCBODY is a bulk data entry that is used by SOL 600 and SOL 700 only, which defines a flexible or rigid contact body in 2-D or 3-D. It could be specified with a BSURF, BCBOX, BCPROP, or BCMATL entry. In this analysis, BCPROP cards are used because adaptive contact is applied.

Bulk data entries that define material and properties for the shell as same as the prestress run:

HGSUPPR card is for the definition of the hourglass suppression method, the corresponding hourglass damping coefficients and sets for the bulk viscosity method and coefficients.

Bulk data entry that defines boundary conditions and initial velocity:

BCTABLE 1 2 SLAVE 8001 0. 0. 0.1 0. 0 0 0 0 0.1 YES+++ MASTERS 1001 SLAVE 1001 0. 0. 0.1 0. 0

BCBODY 1001 3D DEFORM 1001 0BCPROP 1001 1 2 3 4 5 6 7

MATD024 3001 4.14E-4 1.60E+7 0.35000 100000. 138000. 0.20000 + + 3001 ...PSHELL1 1 3001 KeyHoff Lobatto 5 Mid ++ 0.22768 ...

HGSUPPR 1 SHELL 1 1...

BCGRID 1 300425 THRU 300443


BCGRID is also used to specify a set of nodes which are to be constrained with SPCD2 (angular velocity).

TIC3 defines the initial velocity as combination of translational part and a rotational part.

SPCD2 defines imposed nodal motion on a node or a set of nodes.

In bird1.dat, bulk data that defines the bird properties

PSOLIDD is a SOL 700 bulk data entry that defines properties for solid elements. It refers to entry MATD010, EOSPOL for material properties.

MATD010 is a SOL 700 bulk data entry that represents MD Nastran Material #10, which is used to model elastic-plastic hydrodynamic material.

EOSPOL is a SOL 700 bulk data entry that defines the properties of a polynomial equation of state.

MATERO is used to assign a failure criterion on material property.

Bulk data entry that defines geometric properties of the bird:

SPCD2 1 GRID 1 7 80 -1.TABLED1 80 ++ 0.0 837.758 1. 837.758 ENDT$ Displacement Constraints of Load Set : Disp1SPC1 1 3 21 THRU 31....$ Initial angular velocity for rotor +fan bladeTIC3 1 299999 1.

-837.758

1 THRU 6384 300000 THRU 300018 300020 THRU...

PSOLIDD 3004 3004 1 3004EOSPOL 3004 0.0 0.0 4.25E6MATD010 3004 9E-5 145 2.9 0.145

CHEXA 1000001 3004 1000001 1000002 1000003 1000004 1000005 1000006 1000007 1000008..GRID 1000001 2.35785 10.7693 -17.238..

TIC 1 1000001 3 7692.TIC 1 1000002 3 7692.....


474

TIC is a bulk data entry that defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees-of-freedom.

Results

Prestress Run

Using the TSTEPNL entry 5 results increments were archived. The results of all increments are essentially the same which indicates that the implicit calculations are stable. The results of the last increment were written to the file prestres_rotor.dytr.nastin.

Result Increment 2


Result Increment 5: written to the .nastin file


476

Impact run

The prestress result variables have been initialized at the begin of the analysis (Time = 0)

t = 0 seconds

t = 0.001 seconds


t = 0.00152 seconds

t = 0.002 seconds


478

t = 0.003 seconds

t = 0.004 seconds


Multiple Bird-strikes on a Box Structure

Problem Description

Bird strike on a box structure is a typical problem in aircraft industries. The box structure simulates the leading edge of lifting surfaces, e.g. wing, vertical, and horizontal stabilizers. The box can be simplified to consist of a curve leading edge panel and a front spar. The acceptable design criteria for bird strike are that the leading edge panel may fail but the front spar strength may not degrade to a certain level.

In this example, two cylindrical panels are put in parallel. Two birds strike the upper panel. One bird strikes in horizontal direction and the second one vertically. The second bird will perforate the first panel and impact the second one. The birds are modeled as cylindrical slugs of jelly. The plate is constrained in such a way that the edges can only move in radial direction.

Figure 13-1 Initial Situations

The properties and initial conditions of the plate and birds are as follows:

Plate Ambience B Bird 1 Bird 2

Material Titanium Air Jelly Jelly

Density (kg/m3) 4527 1.1848 930 930

Bulk modulus (Pa) 1.03e11 2.2e9 2.2e9

Poisson’s ratio 0.314

Yield stress (Pa) 1.38e8

Gamma 1.4

Thickness (m) 0.0015


480

SOL 700 Model

Each curved plate is modeled using 33x16 BLT-shells. The boundary conditions applied at the edges of the plate are defined within a cylindrical coordinate system, where the local z-axis is aligned with the length axis of the plate. The cylindrical system is defined using a CORD2C entry. To create a closed surface, required by COUPLING option, the two plates are connected with dummy quad elements.

The two birds and air are modeled using Multi Material Eulerian (FV) elements, also known as MMHYDRO. The location of the bird in the Euler domain is defined using TICEUL option.

The material for the birds and air are modeled using EOSPOL and EOSGAM, respectively.

To allow the bird perforating the first plate and impact the second one, several modeling techniques can be used. One of them is using two Eulerian domains and two coupling surfaces. Both the Eulerian domains and the coupling surfaces have to be logically different. Each coupling surface associates with one Eulerian domain.

In this model, the two coupling surfaces share the same physical space. By specifying that one domain is covered outside and the other inside, the Eulerian domain represents the correct physical space. The two Eulerian domains cannot interact with each other except through coupling surfaces. When coupling surfaces share the same shell elements with some or all shells failing, then the material can flow from one Eulerian domain into another one. The interaction between the Eulerian domains is activated using COUP1INT option and PARAM, FASTCOUP, INPLANE, FAIL. The rest of the Euler domain is filled with air. Please notice that when the effect of air is neglected, then the rest of the Eulerian domain should be filled with void. It will speed up the analysis.

The first domain is associated with a coupling surface that is INSIDE covered. Therefore, it cannot be adaptive and is defined using MESH,, BOX option. The second domain is adaptive and defined using MESH,, ADAPT. The ADAPT option will let SOL 700 create and update the Eulerian domain to minimize memory allocation and consequently lowered CPU time. The default Eulerian boundary condition is set to that only outflow is allowed using FLOWDEF option. In this case, a bird that reaches the free face boundary will flow out of the domain. The initial velocity of the birds is defined using TICVAL option.

Radius (m) 0.25

Length (m) 0.25

Mass (kg) 0.36 0.285

Initial velocity (m/s) 150 200

Fail (equiv. Plas. Strain) 0.1

Plate Ambience B Bird 1 Bird 2


The finite element model of the upper and lower plates, the Eulerian domains and the initialization of the birds are shown in the figure below. The dummy quad elements used to create closed coupling surfaces are not shown in this figure.

Input File:

SOL 700 is an executive control that activates an explicit nonlinear transient analysis:

Case control cards for problem time, loads and initial conditions:


$ Direct Text Input for Executive ControlCENDTITLE = Multiple BIRD STRIKE on BOX StructureSUBCASE 1$ Subcase name: DefaultSUBTITLE=DefaultTSTEPNL = 1SPC = 1IC = 1


482

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (10) and Time Increment (0.0015 seconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

Define the Initial, the Minimum and the Safety factor of the time step:

Define coupling surface that can fail and Multi material overflow array to store material data. In a problem where more than 10% of the elements have more than one material, the default value of FMULTI(0.1) must be increased.

Define Output results request for every 0.00015 s and time history output request for coupling surfaces:

Euler domain 1:

Define an Euler mesh with 50x28x44 elements reference to PEULER1 (=1):

$------- BULK DATA SECTION -------BEGIN BULKTSTEPNL 1 10 0.0015 1

PARAM*, DYINISTEP, 1e-7PARAM*, DYMINSTEP, 1e-8DYPARAM, STEPFCTL, 0.9

DYPARAM, FASTCOUP, INPLANE, FAILDYPARAM, FMULTI, 0.2

DYPARAM, LSDYNA, BINARY, D3PLOT, .00015DYTIMHS,, .000001,,,,,,,++, CPLSOUT

$ domain 1$MESH, 1, BOX,,,,,,,++,-0.26,-0.015,-0.05,0.50,0.28,0.44,,,++, 50, 28, 44,,,, EULER, 1


Define FSI coupling surface from elements listed in the BSURF entry (covering inside):

Define Eulerian element properties with reference to TICEUL1 (=11).

The initial conditions of these elements are defined in geometric regions.

Define Regions with shapes, material, initial values and level indicators:

Define region shapes:

Define Initial values of the birds and the air:

$ COUPLING SURFACE 1$COUPLE , 1 , 1 , INSIDE , ON , ON , , , , ++ , , , , , , , , , ++ , , 1 $BSURF , 1 , 7393 , THRU , 8448 , 13729 , THRU , 14048 , 14577 , ++ , THRU , 15236

PEULER1 , 1 , , MMHYDRO , 11

$ Allocation of material to geometric regions.$ --------------------------------------------TICEUL1 , 11 , 11 TICREG , 1 , 11 , CYLINDER , 1 , 3 , 1 , 3TICREG , 2 , 11 , CYLINDER , 2 , 5 , 2 , 2TICREG , 3 , 11 , SPHERE , 4 , 4 , 5 , 1

CYLINDR , 1 , , .13 , .125 , .2252 , .17 , .125 , .2944 , ++ , .035CYLINDR , 2 , , -.1381 , .125 , .26 , -.2381 , .125 , .26 , ++ .035SPHERE , 4 , , -.1381 , .125 , .26 , 1000

TICVAL, 1 , , XVEL , -75 , ZVEL , -129.9 TICVAL, 2 , , XVEL , 200TICVAL, 5 , , SIE , 2.1388E5 , DENSITY , 1.1848


484

Define Eulerian materials for the birds and the environment (air):

Euler domain 2:

Define an adaptive Euler mesh reference to PEULER1 (=6):

Define FSI coupling surface from elements listed in the BSURF entry (covering outside):

Domain 2 has only 1 region with air.

Interaction between the coupling surfaces 1 and 2:

Define interaction between coupling surface 2 and 1:

$--------Material Bird ------------------------------------MATDEUL , 3 , 930 , 3 EOSPOL , 3 , 2.2e9MATDEUL , 5 , 930 , 5 EOSPOL , 5 , 2.2e9$ -------- Material Air id =4MATDEUL , 4 , 1.1848 , 4EOSG , 4 , 1.4

$-----------------------------Domain 2------------------------------$MESH , 2 , ADAPT , 0.01 , 0.01 , 0.01 , , , , ++ , -0.26 , -0.015 , -0.05 , , , , , , ++ , , , , , , , EULER , 6

$===Coupling Surface 2$COUPLE , 2 , 2 , OUTSIDE , , , , , , ++ , , , , , , , , , ++ , , 2$BSURF , 2 , 7393 , THRU , 8448 , 13729 , THRU , 14048 , 14577 , +

TICEUL1,12,12TICREG,11,12,SPHERE,7,4,5,1.0SPHERE,7,,0.0,0.0,0.0,500.0

$ coupling interaction$COUPINT,2,2,1


Define default Eulerian flow boundary condition:

Define cylindrical coordinate system:

Define properties of the panels:

Define properties of dummy elements to close the coupling surfaces

Results

In this simulation, the time history of total z-force on the coupling surface is requested as shown in Figure 13-2. This force is the sum of all z-forces on the nodes that belong to both the upper and the lower plate.

From this figure, it is obvious that there are three large impact forces occurring on the plate. The first one is when the first bird impacts the upper plate, which is subject to a significant damage. The second one is when the second bird impacts the upper plate. The last peak is caused by the first bird impacting the lower plate.

Snapshots of the motion of the two birds and the deformation of the plates are shown in Figure 13-3 at various time steps of the simulation. Figure 13-3a is the initial condition. Figure 13-3b is at the moment when the first bird penetrates the upper plate and second bird touches the plate.

This corresponds with the first peak in the time history plot shown in Figure 13-2. Figure 13-3c is at the moment when the second bird penetrates the upper plate. It corresponds with the second peak of the time history plot. Figure 13-3d is at the moment when the second bird has left the plate and the first bird penetrates the lower plate. This corresponds with the third peak in the time history plot.

$ Flow boundary$ -------------------------------------------------------------FLOWDEF , 1 , , MMHYDRO , , , , , , ++ , FLOW , OUT

$ --------------------CORD2C , 1 , , 0.0 , 0.0 , 0.0 , 0.0 , 0.25 , 0.0 , ++ , 0.0 , 0.125 , 0.25

PSHELL1 , 2 , 2 , Blt , Gauss , 3 , .83333 , Mid , , ++ , .0015 $MATD024 , 2 , 4527 , 1.150e11 , .314 , 1.38e8 , , 0.1

PSHELL,3,999,1.E-3PSHELL,4,999,1.E-3$MATD009,999,1.E-20


486

Figure 13-2 Time History of Totoal Z-force on Coupling Surface

Figure 13-3 Deformation of Plates


Files

Abbreviated SOL 700 Input File

SOL 700,NLTRAN STOP=1CENDTITLE = Multiple bird strike using Multi-Material-FVSurferIC = 1SPC = 1TSTEPNL=1$BEGIN BULKPARAM*,DYINISTEP,1e-7PARAM*,DYMINSTEP,1e-8DYPARAM,FASTCOUP,INPLANE,FAILDYPARAM,FMULTI,0.2Dyparam,stepfctl,0.9DYPARAM,LSDYNA,BINARY,D3PLOT,.00015DYTIMHS,,.000001,,,,,,,++,CPLSOUT$TSTEPNL, 1, 10, .00015, 1$$ Include model + SPCINCLUDE examp4_9_bs.bdf$$ domain 1$MESH,1,BOX,,,,,,,++,-0.26,-0.015,-0.05,0.50,0.28,0.44,,,++,50,28,44,,,,EULER,1$$ COUPLING SURFACE 1$COUPLE, 1, INSIDE, ON, ON, , , , , + +, , , , , , , , , ++, 1 $BSURF, 1, 7393, THRU, 8448, 13729, THRU, 14048, 14577, ++, THRU, 15236$$ Flow boundary, property, material and equation of state data.$ -------------------------------------------------------------FLOWDEF, 1, MMHYDRO, , , , , , +

examp4_9_bs.dat

examp4_9_bs.bdf SOL 700 input files

examp4_9_bs.dytr.d3plot SOL 700 structure result file

EXAMP4_9_BS.DYTR_EULER_FV1_*.ARC

EXAMP4_9_BS.DYTR_EULER_FV2_*.ARC SOL 700 Euler result file


488

+, FLOW, OUT$PEULER1, 1, , MMHYDRO, 11PEULER1, 6, ,MMHYDRO, 12EOSGAM,4,1.4$$--------Material Bird ------------------------------------MATDEUL, 3, 930, 3 EOSPOL, 3, 2.2e9MATDEUL, 5, 930, 5 EOSPOL, 5, 2.2e9$$=============================================================$$ Allocation of material to geometric regions.$ --------------------------------------------TICEUL1 11 11 TICREG 1 11 CYLINDER1 3 1 3TICREG 2 11 CYLINDER2 5 2 2TICREG 3 11 SPHERE 4 4 5 1$CYLINDR 1 .13 .125 .2252 .17 .125 .2944 ++ .035CYLINDR 2 -.1381 .125 .26 -.2381 .125 .26 ++ .035SPHERE,4,,-.1381, .125, .26, 1000$$ Initial material data.$ ----------------------TICVAL 1 XVEL -75 ZVEL -129.9TICVAL 2 XVEL 200$$ LAGRANGE$$ Property, material and yield model.$ -----------------------------------PSHELL1 2 2 Blt Gauss 3 .83333 Mid ++ .0015$MATD024,2,4527,1.150e11,.314,1.38e8,,0.1$PSHELL,3,999,1.E-3PSHELL,4,999,1.E-3$MATD009,999,1.E-20$$ Boundary constrain.$ --------------------CORD2C 1 0.0 0.0 0.0 0.0 0.25 0.0 ++ 0.0 0.125 0.25 $$ -------- Material Air id =4MATDEUL 4 1.1848 4


$ |$ -> density$$-----------------------------Domain 2------------------------------$TICEUL1,12,12TICREG,11,12,SPHERE,7,4,5,1.0SPHERE,7,,0.0,0.0,0.0,500.0TICVAL,5,,SIE,2.1388E5,DENSITY,1.1848$$===Coupling Surface 2$COUPLE,2,2,OUTSIDE,,,,,,++,,,,,,,,,++,,2$BSURF 2 7393 THRU 8448 13729 THRU 14048 14577++ THRU 15236MESH,2,ADAPT,0.01,0.01,0.01,,,,++,-0.26,-0.015,-0.05,,,,,,++,,,,,,,EULER,6$$ coupling interaction$COUPINT,2,2,1$ENDDATA


490

Chained Analysis - Fan Blade Out to Rotor Dynamics

Description

This example presents a multi-disciplinary, integrated implicit-explicit-implicit analysis process tailored for more accurate and efficient simulations of aero engine fan blade-out events using MD Nastran. A FBO event can be extremely nonlinear because of the heavy wide chord fan blades incorporated in the new generation of high by-pass ratio jet engines. These new wide chord blades are used to meet airframe manufacturers’ demand for higher thrust engines with improved performance and optimum weight. Airframe and engine manufacturers use computerized analysis procedures to support the design of both the propulsion system and adjacent wing structures.

However, the manufactures do not share their finite element models and traditionally resort to construction of a new model to suit their analysis objective. For example, typical FBO models are very detailed and can exceed two or three million elements whereby a rotordynamics models is much coarser and can be under 50,000 elements. So the challenge becomes as how to transfer the FBO loads computed by SOL 700 explicit solver based on a very fine meshed model to a coarse model for rotordynamics simulation, all within one common modeling environment.

This example demonstrates the automated, multi-disciplinary simulation capability in MD Nastran to streamline the FBO event simulation facilitated by SOL 700 and SOL 400 which normally consists of the following separate steps:

1. Pre-stress fan blades condition at the maximum rotating speed & static loads such as gravity with implicit solution (SOL 700 implicit solver or SOL 400).

2. Followed by an explicit solution for few cycles with release of a fan blade to simulate: damage to the trailing fan blade(s); fan rubs with the engine case; breakage or damage to the inlet or engine containment case; twisting and bending of the FAN shaft and/or other rotating shafts.

3. And converting to an implicit solution to continue the analysis more rapidly to reach the steady state windmilling speed. This is done by including realistic input forcing functions and damage incurred during the explicit solution. Techniques to reduce the loads for application to a coarser model are introduced to preserve the solution integrity.

4. This integrated SOL 700 explicit and MD Nastran SOL 400 implicit solution with the Nastran rotor dynamics capability is used to predict the engine unbalance and to extract the whirling diagrams and critical tolerances (Figure 13-4). This allows the engine manufacturers to share results of the explicit phase with other manufacturers of modern airframe/propulsion system components seamlessly and without compromising design secrets, thus achieving higher accuracy and improved productivity with fewer bottlenecks.


Figure 13-4 Chained FBO-RD Analysis

All these steps are integrated in one common modeling environment by using MD Nastran. The MD Nastran MASTER database is tailored to include only the FBO loads and other relevant information required for rotordynamic simulation without compromising the confidentiality of model geometry and modeling details. The MASTER database can be shared by MD Natran users among different companies and organizations for follow-up analysis. For example, after the FBO analysis simulated by SOL 700, the MASTER database can be sent to airframe manufacturers to use the loads in their rotordynamics analysis facilitated by SOL 400. The following steps are completely automated in SOL 400 rotordynamics simulation:

1. Read the MASTER database generated by SOL 700 to use the FBO loads as a pre-condition to RD analysis.

2. Map the loads on the coarse finite element model in RD simulation. An advanced search technology is implemented in SOL 400 to identify the closest element to a given load. The load is then distributed on the corner nodes of the element.


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3. Synchronization of the explicit and implicit timesteps, The explicit timestep is much smaller than the implicit timestep, so the timesteps need to be “synched up” when the analysis is switched from FBO analysis to RD simulation. The synchronization is based on Fast Fourier Transformation (see Timestep Control on SOL 400 for details)

It is believed that this process can result in much higher levels of accuracy and dramatically reduce the cost of analysis and design of the propulsion system and wing while still protecting any proprietary information. The example problem that is used in this paper is a representative finite element model of an engine mounted on a wing.

SOL 700 Entries IncludedSOL 700TSTEPNLDYPARAM,LSDYNA,DATABASE,SSSTATMDBEXSSSDYPARAM,LSDYNA,DATABASE,NCFORCCSPOTBLDOUTSOL400ANALYSIS=NLTRANROTORGRSPINTCONM2UNBALNCCBUSHPBUSHPBUSHT

Loadings Types

The dynamic loads on the engine after the FBO can be classified under two categories:

a. Large amplitude transient impact loads generated inside the engine due to the released blade hitting the containment and the impact loads generated as a result of contact with the trailing blade(s).

b. The so-called “Seizure Torque” being applied on the fan rotor due to unbalance caused by the missing blade. The seizure torque is as a result of rubbing loads caused by contact between the tip of the blades and the fan case. If the torque is large enough it could stall the rotor and the engine causing a “seizure” (see below for more details).

The impact type transient loadings are calculated and stored by MD Nastran SOL 700. In this release, only three types of loadings are taken into account.

1. Impact loads between the broken blade and the case

2. Rub loads on fan case

3. Rub loads on blade tips


Additionally, this release is limited to the analysis of only one released blade and assumes that there are no other failed trailing blades. In other words, only the released blade is considered for unbalance. The other types of loads and unbalances, such as impact loads between the broken blade and remaining blades unbalance generated by breaking some of the remaining blades due to the impact between the broken and remaining blades and so on, will be considered in future releases of MD Natran.

The impact loads are generated due to released blade(s) “impacting” the containment structure or other components. These forces contain both a normal component (to the fan case) and tangential components which change with time as the blade hits various parts of the containment ring. The released blade, pre-determined in the analysis and in testing, is the only blade which is actually released at the hub and impacts the fan case. In many cases, the trailing blade will impact the root of the released blade, causing the trailing blade to fail and break at a different location. As a result, one or more trailing blades will behave like shrapnel and will contribute significantly to the impact loads. These forces and their contact locations are stored in SOL 700 “binout” as well as the MD Nastran database MASTER file in the Nastran basic coordinate system. The new entry BLDOUT in MD Nastran defines blade out force output information and mapping criteria for a combined SOL 700 – SOL 400 Blade-out analysis (used both in the SOL 700 and subsequent SOL 400 analyses).

Rub loads occur as a result of fan blade tips coming into contact with the inner surface of the containment

ring and/or its trench-filler materials. During the Fan blade-out event, as the unbalance forces

on the rotor make it to go off-center and the running tip-clearance between the rotating blades and the stator structure gets consumed, the tips of the blades will rub against the enclosure. The rubbing loads are distinguished between those that are applied on the fan case and those that are located on the blade tips. There are equal and opposite sets of forces on the containment ring and on the blade tips. The primary difference is that the rubbing loads on the containment ring can be stationary whereas the rubbing loads on the blade tips are varying as the blades rotate and at any given instant different blades on the rotor continue to contact the stator structure in the same location. The rub loads have radial and

tangential components, with radial component acting along the span of the blade and tangential

component opposite to the direction of motion of the rotor. Using the relationships for tip Coulomb

damping with the coefficient of friction , the magnitude of the tangential component is computed

as: . Since the torque on the rotor produced by the frictional force always opposes the

motion, its tendency would invariably be to slow-down the spin RPM of the rotor-shaft. These loads, if severe enough, do not only slow down the rotor rather may even stop it, a phenomenon called “seizure torque”. Thus, rub forces have normal and tangential components and , respectively at the

points of contact. Similar to impact forces, MD R3 SOL 700 will compute the contact location and magnitude of the rubbing loads and store them into “binout” as well as the “MASTER” file. The loads that have zero magnitude are filtered out and are not written to the database to save time and disk space.

MurΩ2( )

Frub( )

Fr( )

Ft( )

μ

Ft μFr= Ft( )

Fr( ) Ft( )


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The unbalance force , which results from the mass of the missing blade material, occurs

whether the remaining unbroken blades contact the containment ring or not. This force is output by SOL 700 in the Nastran basic coordinate system and saved in the SOL 700 “binout” file. In addition, the mass of the broken blade will be saved for use in the subsequent creation of UNBALNC entries for the SOL 400 rotor dynamics analysis.

Time Step Control in SOL 400

The contact forces computed by SOL 700 are stored and transferred to SO 400. These forcing functions have very small time intervals and they may increase the analysis time. In order to increase the timestep and synchronize the explicit and implicit timesteps, a technique based on Fast Fourier Transformation (FFT) and Inverse Fast Fourier Transformation (IFFT) is used to eliminate the high frequencies of the data.

First, the time histories from SOL 700 are changed by FFT from time domain to frequency domain. Next, the frequency domain histories are passed through a low pass filter where the low pass frequency can be selected by the user. Finally, the histories are changed by Inverse FFT from frequency domain back to time domain. Because the high frequencies are removed by the low pass filter, the final time histories do not include the high frequency vibration.

Load Mapping Scheme

MurΩ2( ) Mu( )


Model

Figure 13-5 Implicit Prestress Blade and Rotor Model and Location of Bearings on the Rotor

Figure 13-6 Explicit FBO Engine Model and Location of Bearings on the Rotor


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Figure 13-7 Implicit Rotor Dynamics Model

A simplified generic engine model was provided by Boeing for the purpose of this study. The engine model was modified and enhanced by MSC to include realistic fan blades, rotor, three bearings and other components. Typically, full FBO models can easily have millions of elements and degrees of freedom to represent a realistic jet engine. However, for the purpose of this study, even though the FBO model was constructed with a much finer mesh density than the rotor dynamics model, it is not as elaborate as the full engine models that are used by manufacturers in their explicit simulation. The FBO model has 8864 nodes and 8256 shell elements and is deemed to be sufficiently detailed to capture the physics of the problem and compute the impact and rub loads. The fan blades were constructed by shell elements with various thicknesses across their width and length. The rotor was made of a hollow rod with varied cross sections across its length and a rotational velocity of 4500 rpm. The material for both rotor and fan blades is titanium grade with the following properties:

Material: (Titanium)

Blade and Rotor (SI Unit)

The bearings were modeled by constructing two concentric rings with pre-determined stiffness properties that can contact each other. The flange on the bearings prevents the axial movement of the rotor during the fan-blade-out. The bearing models and their properties are important design considerations to simulate the “fusing” during the FBO and rotor dynamics analysis. Fusing is an event where a bearing or other support structure fails as a result of high loads beyond the design strength of the fusing structure, and its stiffness is reduced to zero.

ρ = 4.466 g/cm3 Poisson’s ratio = 0.35

σy = 1009 MPa Tangent modulus = 731 MPa

Young’s modulus = 117 GPa Plastic strain failure limit 0.2


Input

The simulation consists of three runs. The first run is a prestress analysis that computes the deformations and stresses due to rotational velocities. This computation is essentially linear static and an implicit solver is selected for the purpose of computational efficiency.. Boundary conditions and initial conditions of the prestress run differ from the FBO run. In the prestress run the three bearing points are fixed and a force in circumferential direction is applied to the rotor and fan blades.

Implicit Prestress Run

Since the entries and details of the prestress input file are quite similar to that of “Bird Strike on rotating fan blades with prestress” example, explanation of the prestress input will be skipped.

Explicit FBO Run

Since the explicit FBO input is also similar to of the explicit input of “Bird Strike on rotating fan blades with prestress” example, only additional or different entries will be explained.

BLADEOUT option activates the chaining simulation. All FBO forces assigned in BLDOUT entry will be stored in “MASTER” file after the simulation.

TSTEPNL entry describes the number of Time Steps (300) and Time Increment (1.e-4 seconds) of the simulation. End time is the product of the two entries (30 ms).

DBEXSSS entry requests the statistics of subsystems. The subsystems are defined by BCPROP entries.

DYPARAM, LSDYNA,DATABASE,SSSTATM parameter requests to store the mass, mass center and mass inertia tensor of the subsystems which are assigned by DBEXSSS entry. All information will be stored to jid.dytr.ssstat ascii file at every 0.000008 seconds and will be used for unbalance input in the rotor dynamic simulation.

SOL 700,NLTRAN STOP=1 PATH=3 BLADEOUT

TSTEPNL 1 300 .1e-3 1 ADAPT 2 10

DBEXSSS 111 21 2 3 4 5 6 7 ++ 8 9 10 11 12 13 14 15 ++ 16 17 18 19 20 101DYPARAM LSDYNA DATABASESSSTATM .00008

$$ ALL BLADES$BCPROP,101,1011106,1011107,1011108,1011109,1011110,1011111,1011112,+


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BCPROP 101 includes the properties of all blades and BCPROP 21 includes only the broken blade properties. These two subsystem information will be used in UNBALANC and CONM2 entries of SOL 400 rotor dynamics simulation.

In the example, 9 contacts are defined. In order to reduce the size of binout file which includes the contact forces, only the fan case is considered to capture the FBO loads.. Only the contact forces between the remaining blades and the fan case (contact 01) and the broken blade and the fan case (contact 02) are stored in the binout file. To store contact forces in binout files, two options in BCTABLE and one parameter are required. SPR and MPR options can store the contact forces on SLAVE and MASTER parts respectively. DYPARAM*,LSDYNA,DATABASE,NCFORC parameter controls the timestep of contact forces output which are defined in BCTABLE.

To define the release mechanism, breakable joints (CSPOT) are used. These are elements that have coincident nodes on the hub and the blade roots but are distinct.

Figure 13-8 Adding Breakable Joints

$$ CONTACT ID SLAVE BODY MASTER BODY DESCRIPTION$ 01 1 22 remaining all blades to direct contact case (recording)$ 02 21 22 broken blade to direct contact case (recording)$ 03 21 1 broken blade to remaining all blades$ 04 1 remaining all blades (self contact)$ 05 21 23 broken blade to non-direct contact case$ 06 22 direct contact case (self contact)$ 07 1001 1004 bearing point 1 : fuse at 6.0E-3 seconds$ 08 1002 1005 bearing point 2$ 09 1003 1006 bearing point 3$BCTABLE 1 9 SLAVE 1 0. 0. 0.1 0. 0 0 0 0 0.1 SS1WAY+++ 1 1


The breakable joints between the hub and the release blade are added using CSPOT. The joints will be released at 0.00001 seconds after the start of FBO simulation.

CSPOT entry defines the complex or combined welds. This is used to connect two nodes which are defined by BCGRID entry and are released (broken) at 0.00001 seconds.

BLDOUT entry defines the contact force output information and mapping criteria for a sequential SOL 700 FBO and SOL 400 RD analyses. Using this entry, the all forces can be categorized and stored to MASTER file in the SOL 700 run. All slaves and maters in the BCTABLE must be assigned to BLDOUT entry using six different types of flags in ISLVi’s and IMASTi’s. In the example, nine ISLVi’s and nine IMASTi’s are required because there are nine contact definitions in BCTABLE. See MD Nastran Quick Reference Guide for other fields.

The spin down event after the blade out can be defined by using a time-dependent pre-determined rotational speed of the turbine using SPCD2, BCGRID, and TABLED1 entries.

CSPOT 1111 101 10 1111 0.00001..CSPOT 1126 116 10 1126

BLDOUT 1 0 1.0E-6 0.0 0.09204 2.90E+1 0.244 12 0 0. 1. 0. 2 4 1 3 1 2 2 2 1 99 3 3 99 99 99 99

SPCD2 1 GRID 123 5 80 -1.$BCGRID 123 20003787THRU 2000386220003867THRU 2000394220003947++ THRU 20004022


500

Rotor Dynamics Run (SOL 400)

The FBO loads computed in SOL 700 are read by SOL 400 by assigning the _FBO.MASTER file to DBSET in the File Management Section (FMS) of SOL 400 run.

SOL 400 executive control entry activates nonlinear static and transient analysis.

Case control commands are defined in the following box.

ANALYSIS=NLTRAN actives “nonlinear transient analysis”.

Bulk data starts with BEGIN BULK.

In order to use equivalent material properties in SOL 400, all MATD024 materials models used in SOL 700 are translated to MAT1 and MATEP with slope option.

nastran buffsize=65536nastran dbcfact=4nastran system(151)=1init scratch logi=(scratch(9999000))assign dbloc1='impact_FBO.MASTER'dblocate datablk=(GEOM3K) logical=dbloc1 , where(projno>0 and version=* and wildcard)

SOL 400

analysis=nltranrigid=linearRGYRO= 100DISPLACEMENT(print,plot,SORT1,REAL)=ALLSTRESS(plot,SORT1,REAL)=ALLSTRAIN(plot,SORT1,REAL)=ALL

BEGIN BULK..$MATD024 403153 4.14E-4 1.60E+7 0.35000 2.5E5 1.38E5 0.25000$MAT1 101 1.60E+7 0.35000 4.14E-4MATEP 101 SLOPE 2.5E5 1.38E5

TSTEPNL 100 4000 1.0E-4 2 1.0E-2 1.0E-2 0


TSTEPNL entry of SOL 400 controls the convergence criteria and data for nonlinear transient analysis.

ROTORG entry defines the rotor which consists of GRID IDs from 10 to 21. RSPINT entry indicates the rotational direction which is assigned to the rotational axis from GRID 11 to GRID 10. The rotational speed is defined in TABLED1, 10 for describing the speed down at various time steps. Note that the magnitude of the rotational velocities defined in SOL 400 differ from SOL 700. This is because the unit of rotational velocity used in SOL 400 is RPM and is different to that used by SOL 700 (radian/seconds)

ROTORG 10 10 THRU 21$RSPINT 10 11 10 RPM 1000$TABLED1 1000 ++ 0.0 4500.0 0.012 4255.0 0.016 4096.6 0.028 3834.2 ++ 0.042 3689.1 0.055 3605.1 0.25 2915.7 0.5 2250.0 ++ 100. 2250.0 ENDT

$ impact_FBO.dytr.ssstatsubsystem: 1

total mass of subsystem = 0.91899477E-01 x-coordinate of mass center = 0.16037865E+03 y-coordinate of mass center =-0.28772884E+02 z-coordinate of mass center = 0.10338639E+03.... subsystem: 21

total mass of subsystem = 0.59591613E+01 x-coordinate of mass center = 0.16148860E+03 y-coordinate of mass center = 0.41383951E-05 z-coordinate of mass center = 0.10000020E+03

inertia tensor in global coordinates row1= 0.2385E+04 -0.5329E-03 -0.2139E+01 row2= -0.5329E-03 0.1748E+04 -0.1921E-02 row3= -0.2139E+01 -0.1921E-02 0.1748E+04

-> translate

UNBALNC,100,0.0919,12,0.,1.,0.,,++,29.00,180.0,1.10995$$ blade + hub$CONM2, 2001,12, ,5.959,,,,,++, 0.2385E+04,0,0.1748E+04,0,0,0.1748E+04$GRID 12 161.488 0. 100.


502

The mass, mass center and mass inertia tensor computed in SOL 700 are stored in the impact_FBO.dytr.ssstat file. These values are then used in SOL 400 to define mass unbalance by UNBALNC and CONM2 entries. As shown in the box above, the order of the subsystem id numbers in ssstat file is determined by the order of DBEXSSS as defined in SOL 700. For example, subsystem 1 represents the released blade while subsystem 21 represents all blades and hub information. The unbalance mass in the UNBALNC entry is the same value of total mass as defined in subsystem 1. ROFFSET and ZOFFSET of UNBALNC entry are calculated by the difference of the mass locations between subsystem 1 and 21. In the example, the x-direction in SOL 700 FBO simulation is coincident with the z-direction of the rotor in SOL 400 RD simulation. In addition, the mass inertia tensor of subsystem 21 is recorded to Iij fields of CONM2 entry. GRID 12 which describes the mass location of hub and blades is also set to the same center location of subsystem 21.

In the gyroscopic nonlinear transient analysis, only the additional unbalance mass is considered as opposed to FBO simulation, where the unbalance mass results from losing mass due to blade out. Therefore, the additional mass must be added to the opposite side of the location where blade-out occured. To add the mass to the opposite side of the blade out, the unbalance is assigned at the location which is measured 180 degrees in the positive direction of the local unbalance coordinate system.

BLDOUT entry is also used in SOL400. BLDOUT entry in SOL 400 can control and apply the FBO forces to the nonlinear transient analysis using different time steps.

Bearings in SOL 400 are modeled using CBUSH elements. PBUSHT controls the failure criteria. The CBUSH element is defined to fail at 1.65E5 lbf in radial (y-z) direction.

BLDOUT,1, 1, 1.0E-6, 0.0, 0, 0, 1.0E-3, 1+,0.0919,2.9000E+1, 1.10995, 12, 0, 0.,1.,0.

CBUSH 101 101 1002 1012 0PBUSH 101 K 1.0E7 1.0E7


Results of Prestress Implicit Simulation

Figure 13-9 Displacement Contours on Fan Blades and Rotor – SOL 400 vs SOL 700

SOL 400 SOL 700


504

Figure 13-10 Stress Contours on Fan Blades and Rotor – SOL 400 vs SOL 700

The stress and deformation results between SOL 400 and SOL 700 were within 2% of each other, which is quite acceptable (see Table 13-1) . However, for this particular analysis, which took a few minutes to complete, SOL 400 ran the same model three times faster than the SOL 700 implicit solver. One reason for the speed up difference is because SOL 700 implicit solver is based on a double precision version.

Table 13-1 Comparison of SOL 400 vs SOL 700 Pre-stress Results

SOL400 SOL 700Difference

(refer to SOL 400 results)

Analysis Time 135 seconds 398 seconds 300%

Maximum Resultant Displacement 24.66 mm(0.971 inch)

24.13 mm(0.950 inch)

2.2 %

Maximum Equivalent Stress 710.2 MPa(103 ksi)

696.4 MPa(101 ksi)

2.0 %

SOL 400 SOL 700


Results of Fan Blade Out (FBO) Explicit Simulation

Figure 13-11 Effective Stress Contour

0.0015 seconds 0.0037 seconds

0.0037 seconds at a different angle


506

Figure 13-12 Snapshot of Contact Between Released Blade and Containment

Figure 13-13 Fan Case t = 3ms

Plastic Strain Effective Stresses


Figure 13-14 Loads

Node 3175 of Case

Impact Loads

Rubbing Load (Radial) Rubbing Load (Tangential)

Node 1405440 of Blade Tip

Rubbing Load (Radial) Rubbing Load (Tangential)


508

Figure 13-15 Node 31795 and 1405440 Locations on Case and Blade Tip

Results of Rotor Dynamics Implicit Simulation

In this analysis, the orbit diagrams between SOL 400 and SOL 700 (see Figure 13-16) demonstrate that the whirling characteristics are similar and deformation magnitudes of the whirl are very close in the z-direction but differ in the y-direction. The difference could be due to two contributing factors. One factor is that the contact forces between the trailing blades and the released blade were not taken into consideration in the rotordynamics simulation and only the impact and rubbing loads were included. The contact forces between the blades will have some contribution to the magnitude of the whirling deformation in the y-direction. The second reason is because the sectional properties of the rotor at center where hub is located are approximated in the stick model for the rotor dynamics simulation.


Figure 13-16 Comparison of Orbit at the Tip of Rotor

In the implicit rotordynamic analysis, the failure load for bearing 1 is set to 734 kN (1.6E5 lbf). A radial dependence is specified for the fuse option. Figure 13-17 shows the time history for the force in this bearing. The bearing is found to fuse in less than a revolution after the FBO event. The time-to-fuse is then used to modify the explicit FBO analysis. In the FBO analysis, fusing is modeled by deactivating contact between the two rings of the bearing at the analysis time recorded in the implicit rotordynamic analysis.

Figure 13-17 Loadings on the First Bearing and Fusing After 0.004 Seconds

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideDrop Test

510

Drop Test

Drop Test Simulation of a Computer Package

Description

Many companies perform drop test simulation to predict the impact-resistance properties of the package for their products. This is an example of a computer package which includes the CPU box, foam and the clipboard box which is dropped at a velocity of 3.81 m/s on a rigid floor. To model the simulation, contact relations were defined between the ground, the clipboard box, the foam and the CPU box itself. A picture of the model without the clipboard box is shown below.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDTOUTPARAM, DYSHELLFORMPARAM, DYSHINPBCTABLEBCBODYBSURF BCPROPMATD020MATD014

Model


• 13024 Trias

• 1859 Quads

• 112514 Tetras

511Chapter 13: ExamplesDrop Test

The ground, clipboard box, and CPU box were modeled with shell elements. The foam was modeled with Tet elements. All shell elements are Belytschko-Tsay formulation. The gravity is defined to take into account the mass of the CPU box and the bouncing effect after the drop. The drop speed of the CPU box is modeled by defining an initial velocity of 3.81m/s, applied on all the grid points of the CPU in a vertical direction towards the ground. The simulation time is 0.025 seconds. The unit system is Newton, meters, and seconds.

Input

All nodes of the CPU box have an initial velocity specified by the TIC entry. All nodes of the rigid ground have been constrained in all the degrees of freedom. Contacts are defined between:

1. The ground (master) and bottom of the clipboard box (slave).

2. The CPU box (master) and the foam (slave).

3. The clipboard box (master) and the foam (slave).

Input File


The bulk entry section begins...

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (2.5e-4 seconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.


TIME 10000CEND LOADSET = 1 SPC = 2 DLOAD = 2 IC=1 TSTEPNL = 20 BCONTACT = 1 weightcheck=yes

$------- BULK DATA SECTION -------BEGIN BULKINCLUDE final_scale2.bdfTSTEPNL 20 100 2.5e-4 1 ADAPT 2 10


512


PARAM*, DYSHELLFORM is a SOL 700 bulk data entry to define the shell formulation. If DYSHELLFORM = 2, the default shell formulation is Belytschko-Tsay.

PARAM,DYSHINP is a SOL 700 bulk data entry that specifies the number of integration points for SOL 700 shell elements.

Bulk data entry that defines gravity load effects:

GRAV is a bulk data entry that defines acceleration vectors for gravity or other acceleration loading.


BCTABLE is a SOL 700 bulk data entry that is also used by SOL 600 and it is meant to define Master-Slave as well as self-contact.

PARAM,DYDTOUT,2.5e-3

PARAM*,DYSHELLFORM,2

PARAM,DYSHNIP,2

$ ------- GRAVITATION ----- $TLOAD1 4 5 100LSEQ 1 5 3DLOAD 2 1. 1. 4GRAV 3 0 9.806 0. 0. -1.TABLED1 100

$ Define contactBCTABLE 1 3 SLAVE 1 MASTERS 2 SLAVE 3 MASTERS 4 SLAVE 3 MASTERS 5



BSURF is a bulk data entry that is used by MD Nastran Implicit Nonlinear (SOL 600) and SOL 700 only, which defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.


$ bottom of cardboard as slaveBCBODY 1 3 DEFORM 1 0$$ ground as masterBCBODY 2 3 DEFORM 2 0$$ foam as slaveBCBODY 3 3 DEFORM 3 0$$ CPU as masterBCBODY 4 3 DEFORM 4 0$$ cardbox box as masterBCBODY 5 3 DEFORM 5 0

BSURF 1 231364 THRU 231420....

$ define ground using property 1BCPROP 2 1$$ define foam using property 4BCPROP 3 4$$ define CPU using property 3BCPROP 4 3$$ define cardboard box using property 2BCPROP 5 2


514

Bulk data entry that defines properties for shell and solid elements:



MATD014 is a SOL 700 bulk data entry that is used to model foam materials.

Bulk data entry that defines boundary conditions and initial velocity of the package:

$ ========== PROPERTY SETS ========== $ * groung_prop *PSHELL 1 1 .005$ * cardboard_prop *PSHELL 2 2 .005$ * cpu_prop *PSHELL 3 3 .005$ * foam_prop *PSOLID 4 4

$ -------- Material ground id =1$MATRIG 1 1e+20 1e+11 .3$Define rigid material and apply constraint in all dof'sMATD020 1 7850 1e11 0.3 1 7 7

$ -------- Material cardboard id =2MAT1,2,2.5e9,,0.3,1.e2$$ -------- Material cpu id =3MATD020 3 2.16e3 2.1e11 0.3$ -------- Material foam id =4MATD014,4,20,3e6,2e6,0.,0.,3.,-1.e20,++,,,,,,,,,++,-0.0513,-0.1054,-0.1984,-0.5108,-0.7985,-1.0217,-1.347,,++,,,,,,,,,++,60000,78000,90000,111000,150000,180000,240000,,++,,,,,,,,,....

$ ------- Initial Velocity BC nodal_vel ----- $ Loads for Load Case : DefaultTIC 1 1 1 -3.81TIC 1 2 1 -3.81....TIC 1 8228 1 -3.81TIC 1 8237 1 -3.81



End of input file.


516

Results

t = 0.0 seconds

t = 0.008 seconds

t = 0.0175 seconds

t = 0.0175 seconds


Drop Test Simulation of a Computer Package


518

Wheel Drop Test

Description

This is an example of a wheel drop test as required in automotive industry to comply with government regulations. In this test a rigid block of 540 Kg is dropped at 13 degrees on a wheel. The drop velocity is 2052.8 mm/s. Several contacts are defined to predict the interaction between wheel, tire and the rigid block.

SOL 700 Entries IncludedSOL700TSTEPNLBCTABLEBCBODYBSURFTICTLOAD1MATD027PCOMPPLOAD4HGSUPPR

Model

230 mm

13°



• 11040 Quads

• 40077 Pentas

• 27036 Hexas

The impact velocity of the rigid block modeled by 48 hexa elements is 2052.8 mm/s in the x-direction (vertical). The velocity is applied on all nodes of the block by using the TIC entry. Since the tire is inflated, PLOAD4 and Table entries are used to apply a pressure load of 0.2 MPa to internal surface of the tire as shown in the figure shown above. Gravity is also taken into account. The wheel is constrained in all directions at surface located around the bolt locations as marked in color.

Four Contacts are defined between:

1. Rigid block and tire

2. Rigid block and wheel

3. Tire and wheel

4. Self contact of tire

Total simulation time is 0.04 s.

Input

SOL 700 is the command for the initiation of an explicit nonlinear transient analysis.


This part is fixed


520

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (0.4 ms) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of time increments and the exact value of the Time Steps are determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

BCTABLE is a SOL 700 bulk data entry to define Master-Slave contact as well as self contact.

BCBODY and BSURF are bulk data entries that are used by MD Nastran Implicit and Explicit Nonlinear only. BCBODY defines a flexible or rigid contact body in 2-D or 3-D and BSURF assigns a contact body or surface defined by Element IDs.


BEGIN BULKTSTEPNL 1 100 4.-4 1 ADAPT 2 10

BCTABLE 1 7 SLAVE 3 0. 0. .3 0. 0 0. 0 0 0

MASTERS 3

BCBODY 3 3D DEFORM 3 0 .3BSURF 3 200001 200002 200003 200004 200005 200006 ...

GRID 1 82.3 144.119 -138.485...$CHEXA 200001 200 203800 203813 203713 203700 213800 213813 213713 213700...CPENTA 232641 230 201803 201802 201809 211803 211802 211809...CQUAD4 310001 310 201204 211204 211703 201703...


Bulk data entry that defines boundary conditions:

Bulk data entry that defines initial velocities for the rigid block model:

TIC is a bulk data entry that defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees-of-freedom.

TLOAD1 is a bulk data entry that defines a time-dependent dynamic load or enforced motion for use in transient response analysis.

Five material models are used in this analysis. MATD027 is used to model the rubber of the tire, with the following properties:

Density = 9.6E-10 ton/mm3

Poisson’s ratio = 0.49 and Coefficients of strain energy equation are 0.6933 and 0.3224 (see MD Nastran Quick Reference Guide.)

SPC1 1 123456 864 874 875 876 882 883...SPC1 3 23 60001 THRU 60108...

TIC 2 60001 1 -2052.8...

SPCADD 2 1 3TLOAD1 6 7 1

MATD027 200 9.6-10 .49 0.6933 .03224 0.

PSOLID 200 200 0...PCOMP 360 0. 0. 301 .5 90. YES 301 .5 90. YES 303 .6 25. YES 305 .5 0. YES...


522

The other material models are used to model the alloy wheel and the composite tire. PCOMP is used for the definition of a composite material laminate shell. The composite layers are shown in schematic above. It consists of a total of 4 material layers 301, 301, 303 and 305. The thicknesses and orientation angles are shown in the definition of PCOMP entry above.

PLOAD4 is a bulk data entry which describes a pressure load on a face of 2D or 3D elements.

Several tables and coordinates are defined in the input file.

HGSUPPR card is for the definition of the hourglass suppression method, the corresponding hourglass damping coefficients and sets for the bulk viscosity method and coefficients.

PLOAD4 4 232401 1. 200105 210101...GRAV 5 2 9800. -1. 0. 0.

TABLED1 2 0. 1. 1000. 1. ENDT...

HGSUPPR, 200, SOLID, 200, 1, , , , 0.040 , , , 0...


Results

t = 0.000 second

t = 0.013 second

t = 0.027 second

t = 0.040 second

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideDefense

524

Defense

Rod Penetration

Description

This is an example of a rod penetrating through a plate. The initial velocity of the rod is 124.6 m/s in horizontal direction, and 33.39 m/s in the vertical direction. To model the simulation, Adaptive Contact was defined between the rod and the plate to predict the stress and deformations of both parts after the penetration

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDTOUTPARAM*,DYCONSLSFACBCTABLEBCBODYBCPROPMATD024TICD

Model

Because of symmetry, only half of the actual structures (rod and plate) were modeled in this example. Boundary conditions were applied along the center line of the structures to ensure symmetric behavior. The model has a total of 7668 grid points and 5664 solid elements. All the elements were hexahedrals (CHEXA). The initial velocity of the rod is 124.6 m/s in horizontal direction, and 33.39 m/s in the vertical direction toward the plate, applied on all the grid points. The simulation time is 0.11 seconds. The unit system is Kilonewton, meters, and milliseconds.

525Chapter 13: ExamplesDefense

Input

All nodes of the rod have an initial velocity specified by the TICD entry. All nodes along the center line have a boundary condition that ensures symmetric behavior of the structures. All nodes of the plate have been constrained in all the degrees of freedom. Adaptive Contact is defined between the rod and the plate.

Input File:

SOL 700 is a executive control entry that activates an explicit nonlinear transient analysis.



TIME 10000CENDparam,marcbug,0 ECHO = NONE DISPLACEMENT(SORT1,print,PLOT) = ALL Stress(SORT1,PLOT) = ALL Strain(SORT1,PLOT) = ALL accel(print,plot)= ALL velocity(print,plot)= ALL echo=both$ LOADSET = 1 SPC = 2$ DLOAD = 2 IC=1 TSTEPNL = 20 BCONTACT = 1 weightcheck=yes page

BEGIN BULK$$$TSTEPNL 20 10 11. 1 5 10 ++ ++ 0


526

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (10) and Time Increment (11 milliseconds) of the simulation. The total time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.


PARAM*,DYCONSLSFAC is a SOL 700 bulk data entry to define the default scale factor for contact forces

Bulk data entry that defines Adaptive Contact relations and Contact bodies:




PARAM,DYDTOUT,5

PARAM*,DYCONSLSFAC,1.0

BCTABLE 1 4 SLAVE 3+ YES MASTER 4

BCBODY 3 3 DEFORM 3 0BCBODY 4 3 DEFORM 4 0

BCPROP 3 2BCPROP 4 1


Bulk data entry that defines properties for solid elements:


MATD024 is a SOL 700 bulk data entry. It is used to model an elasto-plastic material with an yield stress versus strain curve and arbitrary strain rate dependency. Failure can also be defined based on the plastic strain or a minimum time step size.

Bulk data entry that defines initial velocity of the rod:

TICD is a bulk a SOL 700 bulk data entry that defines transient analysis initial conditions with increment options.

$ ========== PROPERTY SETS ========== $$ * projectile *$PSOLID 1 1$$ * plate *$PSOLID 2 2

$ ========= MATERIAL DEFINITIONS ==========$$$$ -------- Material MAT_PLASTIC_KINE.2 id =2MATD024 1 18.62 1.17 .22 0.0179 0.8$ -------- Material MAT_PLASTIC_KINE.1 id =1MATD024 2 7.896 2.1 .284 0.01 0.8

$ ------- Initial Velocity BC ini ----- $TICD 1 1 1 0.1246 2586 1TICD 1 1 3 -0.03339 2586 1


528


Bulk data entry that defines boundary conditions along the center line, and the fixed boundary conditions of the plate:

End of input file.

$ --- Define 7668 grid points --- $GRID 1 9.24175-1.5e-05.0513793....GRID 7668 23.0000 4.80000 .00000$$ --- Define 5664 elements$CHEXA 1 1 1 2 5 4 10 11+A000001+A000001 14 13....CHEXA 5664 2 7619 7620 7628 7627 7659 7660+A005664+A005664 7668 7667

SPC1 1 246 1 2 3 10 11 12.... 2559 2560 2563 2564 2567 2568 2571 2572 2575 2576 2579 2580 2583 2584SPC1 3 123456 2587 2596 2605 2614 2623 2632....SPC1 3 123456 7660 THRU 7668SPCADD 2 1 3 $ENDDATA


Results

time = 0.0 seconds

time = 0.025 seconds

time = 0.055 seconds

time = 0.11 seconds


530

Shaped Charge, using IG Model, Penetrating through Two Thick Plates

Problem Description

When a metal cone is explosively collapsed onto its axis, a high-velocity rod of molten metal, the jet, is ejected out of the open end of the cone. The cone is called a liner and is typically made of copper. The jet has a mass approximately 20 percent of the cone mass, and elongates rapidly due to its high velocity gradient. This molten rod is followed by the rest of the mass of the collapsed cone, the slug. Typical shaped charges have liner slope angles of less than 42 degrees ensuring the development of a jet; with jet velocities ranging from 3000 to 8000 m/s. A typical construction of a shaped charge is shown in Figure 13-18.

Figure 13-18 Typical Construction of Shaped Charge


An example simulation of shaped charge formation is carried out to demonstrate the ability of SOL 700 to perform such a simulation. A simplified axisymmetric model of explosives and a copper liner is created in a finite volume Euler mesh. Explosive are detonated starting from a point on the axis of symmetry at the end of the explosives. The simulation is carried out for 60 μs after detonation of the explosives. The jet is formed and penetrates two thick plates. See Figure 13-19 for the model layout.

Figure 13-19 SOL 700 Model Setup

Typical shaped charges are axisymmetric. However, aiming at higher velocity, 3-D designs are targeted. 3-D simulation of shaped charge formation would be necessary to avoid excessive experimental work. SOL 700 has full abilities to perform such a 3-D simulation.

SOL 700 Model

The model is simplified as shown in Figure 13-19. The aluminum casting is replaced with a rigid body.

Detonation is assumed to start at a point on the axis at the rear end of the explosives. The liner shape is slightly simplified as shown in the figure. The retaining ring is assumed rigid and is modeled as a wall boundary for the Euler Mesh (BARRIER). SI units are used in this example.


532

A. Euler Mesh and Liner:

A triangular prismatic Finite Volume Euler mesh is used with head angle of 5 degrees as shown in Figure 13-20. A very fine mesh is used to accurately simulate the behavior of the extremely thin liner. The liner is placed in this Euler mesh. Symmetry conditions (closed boundary, default Euler boundary condition) are imposed on the two rectangular faces of the prism to create an axisymmetric behavior.

Figure 13-20 Euler Mesh

The liner material pressure – density relationship is modeled with EOSPOL model. The liner is made of copper and the constants are taken as follows:

Material yield strength is modeled with a Johnson-Cook yield model. The constants are taken as follows:

a1 1.43E11 N/m2

a2 0.839E11 N/m2

a3 2.16E9 N/m2

b1 0.0

b2 0.0

b3 0.0

A 1.2E8 N/m2

B 1.43E9 N/m2

C 0.0

n 0.5

m 1.0

ε0 1.0


Other liner material properties of liner are as follows:

In the input deck:

MATDEUL 701 8960. 711 712 713 714EOSPOL, 711, 1.43+11, 0.839+11, 2.16+9SHREL,712,0.477E11$ Johnson-Cook$ A B n C m EPS0 CvYLDJC,713, 1.2E8, 1.43E9, 0.5, 0.0, 1.0, 1.0, 399.0,+$ TMELT TROOM+, 1356.0, 293.0$PMINC,714,-2.5E10

It is very easy to define the shape and position of the liner by using the method of geometrical regions when creating the initial conditions of the liner material.

CYLINDR, 1,, -0.5391, -0.56, 0., 2.0, 0.4147, 0.,++,0.2958CYLINDR, 2,, -0.5391, -0.56, 0., 2.0, 0.4147, 0.,++,0.2939CYLINDR, 3,, 0.2, 2.0406, 0., 0.2047, 2.0406, 0.,++,2.0019TICVAL,2,,DENSITY,8960.

B. Casting and Retaining Ring:

The casting is assumed to be rigid. It is modeled by the default Eulerian boundary condition (closed boundary). The retaining ring is also assumed to be rigid and is modeled by a barrier.

C. Plates:

Two thick plates are placed in this Euler mesh. Plate material is defined as steel:

MATDEUL 801 7830. 811 812 813 814EOSPOL, 811, 1.64E+11SHREL,812,0.818E11YLDVM,813,1.4E9PMINC,814,-3.8E9

The shapes and positions of the plates are defined by using the method of geometrical regions.

Tmelt 1356.0 K

Troom 293.0 K

Cv 399.0 J/kg

Density 8960 Kg/m3

Constant shear model 0.477E11 N/m2

Constant spallation model -2.5E10 N/m2


534

CYLINDR, 4,, 0.22, 2.0406, 0., 0.223, 2.0406, 0.,++,2.05CYLINDR, 5,, 0.27, 2.0406, 0., 0.273, 2.0406, 0.,++,2.05TICVAL,3,,DENSITY,7830.

D. Explosive:

The explosive is modeled by ignition and growth equation of state. The explosive is placed in this Euler mesh.

EOSIG,100,,,,,,,,++,,,,,,,,,++,,,,,99,,MCOMPB,SI

The explosive material is taken from the database that is build into SOL 700.

To initialize the whole Euler mesh, a TICEUL card will be defined.

TICEUL1 1 1TICREG 1 1 ELEM 1 100 1 1.TICREG 2 1 CYLINDER1 701 2 2.TICREG 3 1 CYLINDER2 3.TICREG 4 1 CYLINDER3 701 2 4.TICREG 5 1 CYLINDER4 801 3 5.TICREG 6 1 CYLINDER5 801 3 6.$SET1 1 1 THRU 15342TICVAL,1,,DENSITY,1630.,SIE,4.29E6

ResultsFigure 13-21 shows the initial position of the copper liner and two thick plates at 0μs, snap shots of

liner collapse, jet formation and plates penetrated at 10 μs, 20 μs, 30 μs, 40 μs, 50 μs and 60 μs.


Figure 13-21 Initial Position of the Copper Liner and Two Thick Plates, Snap Shots of Liner Collapse, Jet Formation and Plates Penetrated (Courtesy – Postprocessing by CEI Ensight)

Figure 13-22 shows the velocity field of explosive gases, liner, and jet at 20 μs. A jet velocity of about 6000 m/s is achieved

Figure 13-22 Velocity Field of Explosive Gases, Liner, and Jet

Abbreviated SOL Input File

SOL 700,NLTRAN STOP=1CENDTITLE = SHAPED CHARGES TEST$ for QA purpose, run shorter time$ENDTIME = 1.E-5 IC = 1TSTEPNL=1$$$BEGIN BULK


536

TSTEPNL 1 10 1.E-06 1PARAM*,DYINISTEP,1.E-11PARAM*,DYMINSTEP,1.E-13DYPARAM,VELMAX,20.0E+03DYPARAM,LSDYNA,BINARY,D3PLOT,1.E-5$INCLUDE model.bdfINCLUDE wall.dat$ EXPLOSIVE$MATDEUL 100 1630. 100 101 102$EOSIG,100,,MCOMPB,SI,,,,,++,,,,,,,,,++,,,,,,,,,++,,,,,99$SHREL,101,3.E9$YLDVM,102,2.E8$$ COPPER$MATDEUL 701 8960. 711 712 713 714EOSPOL, 711, 1.43+11, 0.839+11, 2.16+9SHREL,712,0.477E11$ Johnson-Cook $ A B n C m EPS0 CvYLDJC,713, 1.2E8, 1.43E9, 0.5, 0.0, 1.0, 1.0, 399.0,+$ TMELT TROOM+, 1356.0, 293.0 $PMINC,714,-2.5E10$$ STEEL$MATDEUL 801 7830. 811 812 813 814EOSPOL, 811, 1.64E+11SHREL,812,0.818E11YLDVM,813,1.4E9PMINC,814,-3.8E9$TICEUL1 1 1TICREG 1 1 ELEM 1 100 1 1.TICREG 2 1 CYLINDER1 701 2 2.TICREG 3 1 CYLINDER2 3.TICREG 4 1 CYLINDER3 701 2 4.TICREG 5 1 CYLINDER4 801 3 5.TICREG 6 1 CYLINDER5 801 3 6.$PEULER1, 1 ,, MMSTREN, 1SET1 1 1 THRU 15342CYLINDR, 1,, -0.5391, -0.56, 0., 2.0, 0.4147, 0.,++,0.2958


CYLINDR, 2,, -0.5391, -0.56, 0., 2.0, 0.4147, 0.,++,0.2939CYLINDR, 3,, 0.2, 2.0406, 0., 0.2047, 2.0406, 0.,++,2.0019CYLINDR, 4,, 0.22, 2.0406, 0., 0.223, 2.0406, 0.,++,2.05CYLINDR, 5,, 0.27, 2.0406, 0., 0.273, 2.0406, 0.,++,2.05$TICVAL,1,,DENSITY,1630.,SIE,4.29E6TICVAL,2,,DENSITY,8960.TICVAL,3,,DENSITY,7830.$BARRIER,1,2$ENDDATA


538

Mine Blast

Problem Description

This is a simulation of an explosion under a vehicle. The vehicle has triggered a mine that is exploding underneath the bottom shield. In this example, the actual explosion of the mine is not modeled. Instead, the simulation is started moments after the mine explodes. This is called the blast wave approach. At the location of the mine, a high density and high specific energy is assumed in the shape of a small sphere. During the simulation, this region of high density, energy, and high pressure, expands rapidly. The blast wave interacts with the bottom shield and causes an acceleration of parts of the flexible body. The intent of this simulation is to find the location and the value of the maximum acceleration.

SOL 700 Model

An outline of the basic numerical model is shown in Figure 13-23 below. It is composed of the following main components:

a. Vehicle Structure

b. Euler Domain 1 - air outside vehicle and compressed air (explosive)

c. Euler Domain 2 - air inside vehicle

d. Ground

e. Fluid Structural Coupling

Figure 13-23 Outline of Basic Numerical Model


A. The Vehicle:

Vehicle structure is modeled by QUAD, TRIA shell elements and some BAR elements.

Figure 13-24 Vehicle Structure

Material properties are taken as follows:

Assumed that there will be no failure of the structure. In a part of the structure, there is a hole through which air and pressure waves can freely flow. This hole will be modeled with dummy shell elements.

B. Euler Domain 1:

The first Euler domain is the air on the outside of the vehicle. The properties of air at rest are:

In the input deck:

MATDEUL,230,1.29e-12,203,,,,,,++,,1.01TICVAL,5,,DENSITY,1.29E-12,SIE,1.938e11

Density 7.85E-9 tonne/mm3

Modulus of elasticity 210000. tonne/mm/s2

Poison ratio 0.3

Yield stress 250. tonne/mm/s2

Density 1.29E-12 tonne/mm3

Gamma 1.4

Specific internal energy 1.9385E8 tonne-mm2/s2


540

At the location of the mine, a small region will be modeled with high density and specific internal energy equivalent to TNT of 7kg when the sphere has a radius of .25 meter:

The input deck will show:

TICVAL,4,,DENSITY,107E-12,SIE,3.9e12SPHERE,400,,1797.5,0.,-450.,250.

The Euler region will be modeled by using the MESH card. The region will have to be large enough to contain the entire vehicle, including when the vehicle is in motion:

MESH,1,BOX,,,,,,,++,-2623.,-1403.,-903.,6100.,2800.,2150.,,,++,30,10,10,,,,EULER,201

For the most accurate blastwave simulations, it is advised to use the Second-order Euler solver of SOL 700. This is activated by specifying the second-order option on the Euler property card and specifying the parameter to use the second-order Range Kutta integration method:

PARAM,RKSCHEME,3PEULER1,201,,2ndOrder,101

To initialize the whole first Euler mesh, a TICEUL card will be defined. To initialize the Euler domain, other than within the sphere of the explosion, a second large sphere is used. Because it has lower priority, the Euler elements within the mine blast are will still initialized with high density and energy:

TICEUL1,101,11TICREG,1,11,SPHERE,400,230,4,20.TICREG,2,11,SPHERE,501,230,5,1.SPHERE,400,,1797.5,0.,-450.,250.SPHERE,501,,0.,0.,-5000.,10000.

The Euler domain has infinite boundaries. This can be achieved by defining a zero gradient flow boundary on the outside of the Euler mesh. Use an empty FLOWDEF card:

FLOWDEF,202,,HYDRO,,,,,,++,FLOW,BOTH

C. Euler Domain 2:

The second Euler region represents the air inside the vehicle. Also for the second Euler region, a MESH card is used. The air is at rest again, so the same properties apply:

PEULER1,202,,2ndOrder,102TICEUL1,102,12TICREG,3,12,SPHERE,502,230,5,5.SPHERE,502,,0.,0.,-5000.,10000.

Density 107E-12 tonne/mm3

Specific Internal Energy 4.9E12 tonne-mm2/s2


Many of the previous cards will be used to initialize the density and energy (TICVAL) and material (DMAT/EOSGAM) in this Euler region:

TICVAL,4,,DENSITY,107E-12,SIE,3.9e12TICVAL,5,,DENSITY,1.29E-12,SIE,1.938e11

MATDEUL,230,1.29e-12,203,,,,,,++,,1.01EOSGAM,203,1.4

D. The Ground:

The ground is modeled as rigid body using dummy QUAD elements. It is used to close the Euler boundary under the vehicle so the blast wave will reflect on this boundary:

PSHELL,999,999,1.MATRIG,999,,,,1.0E10,0.00,0.00,-800.,++,1.E10,0.0,0.0,1.E10,0.0,1.E10,,,++,,,,,,,,,++,,,,1,7,7

E. Fluid Structure Interaction:

In order to make fluid structure interaction possible, a closed volume needs to be defined. The car model itself is not closed, so a dummy boundary will be defined to close the volume. This extra surface consists of three parts:

Part 1 resides on the back,

Part 2 is the top cover, and

Part 3 is the vent on the bottom of the vehicle.

For all these parts, dummy shell elements are defined and hole definitions will be defined.

Figure 13-25 Dummy Shell Elements Defined to Close the Volume

The input for dummy shell elements

PSHELL,900,901,1.PSHELL,910,901,1.PSHELL,920,901,1.MATD009,901,1.E-20


542

With this closed volume, the coupling surface can be defined. For each Euler domain, a separate surface is required. However, in this model, the interaction surface consists of the same elements, except for the extra ground elements (pid=999) for the outer Euler domain region 1. The surface definition will make use of the properties of the elements.

The outer surface:

BCPROP,97,60,61,62,110,135,150,900,++,910,920,999

The inner surface:

BCPROP,98,60,61,62,110,135,150,900,++,910,920

Now the coupling surfaces can be defined. For the outer region, all elements inside the volume are not active. The covered option will, therefore, be set to INSIDE. Attached to this surface will be the first Euler MESH:

COUPLE,1,97,INSIDE,ON,ON,11,,,++,,,,,,,,,++,,1

The inner Euler domain is constrained by surface 2. For this volume, the outer Euler elements will be covered:

COUPLE,2,98,OUTSIDE,ON,ON,,,,++,,,,,,,,,++,,2

As discussed before, there are holes in the coupling surface. To this end, a flow definition is required for one of the coupling surfaces. In this example, the flow cards are referenced from the first coupling surface. The input to define flow between the regions is:

LEAKAGE,1,11,1,PORFCPL,84,CONSTANT,1.0BCPROP,1,900

Also, for each of the other two flow surfaces, these set of cards are repeated

$LEAKAGE,2,11,2,PORFCPL,84,CONSTANT,1.0BCPROP,2,910$LEAKAGE,3,11,3,PORFCPL,84,CONSTANT,1.0BCPROP,3,920$

Finally, the flow definition itself prescribes that the Euler region from coupling surface 1 is interacting with the Euler region from coupling surface 2:

PORFCPL,84,LARGE,,BOTH,2


F. Miscellaneous:a. Because this model uses the coupling surface interface, the time step safety factor for Eulerian

elements has to be .6. However, the Lagrangian elements (the quadratic and triangular elements) determine the time-step, and it is beneficial to use a higher time step safety factor for the Lagrangian elements:

PARAM,STEPFCTL,0.9

b. To show results every .0002 seconds the following output request was added:

DYPARAM, LSDYNA, BINARY, D3PLOT,.0002

PARAM, CPLSARC,.0002

Results

The Figure 13-26 below shows the location, value, and time of the maximum acceleration. The stress distribution at this time is also in Figure 13-27.

Figure 13-26 Acceleration Plot


544

Figure 13-27 Stress Distribution Plot

Abbreviated SOL 700 Input FileSOL 700,NLTRAN STOP=1CENDTITLE= Job name: mine blast (mm/tonne/s/K)IC=1SPC=1$TSTEPNL=1$------- BULK DATA SECTION -------BEGIN BULK$------- Parameter Section ------$DYPARAM,RKSCHEME,3DYPARAM,FASTCOUPDYPARAM,STEPFCTL,0.9PARAM*,DYINISTEP,.5E-7PARAM*,DYMINSTEP,1.E-13$$DYPARAM,LSDYNA,BINARY,D3PLOT,.0002PARAM,CPLSARC,.0002$MESH,1,BOX,,,,,,,++,-2623.,-1403.,-903.,6100.,2800.,2150.,,,++,30,10,10,,,,EULER,201$MESH,2,BOX,,,,,,,++,-2621.,-1201.,-251.,5900.,2400.,1250.,,,++,30,10,10,,,,EULER,202$PEULER1,201,,2ndOrder,101$TICEUL1,101,11


$TICREG,1,11,SPHERE,400,230,4,20.TICREG,2,11,SPHERE,501,230,5,1.$SPHERE,400,,1797.5,0.,-450.,250.SPHERE,501,,0.,0.,-5000.,10000.$PEULER1,202,,2ndOrder,102$TICEUL1,102,12$TICREG,3,12,SPHERE,502,230,5,5.$SPHERE,502,,0.,0.,-5000.,10000.$TICVAL,4,,DENSITY,107E-12,SIE,3.9e12TICVAL,5,,DENSITY,1.29E-12,SIE,1.938e11$MATDEUL,230,1.29e-12,203,,,,,,++,,1.01$EOSGAM,203,1.4$FLOWDEF,202,,HYDRO,,,,,,++,FLOW,BOTH$COUPLE,1,97,INSIDE,ON,ON,11,,,++,,,,,,,,,++,,1$$ Define flow thru the holes$LEAKAGE,1,11,1,PORFCPL,84,CONSTANT,1.0BCPROP,1,900$LEAKAGE,2,11,2,PORFCPL,84,CONSTANT,1.0BCPROP,2,910$LEAKAGE,3,11,3,PORFCPL,84,CONSTANT,1.0BCPROP,3,920$PORFCPL,84,LARGE,,BOTH,2$COUPLE,2,98,OUTSIDE,ON,ON,,,,++,,,,,,,,,++,,2$BCPROP,97,60,61,62,110,135,150,900,++,910,920,999$BCPROP,98,60,61,62,110,135,150,900,++,910,920$$ ========== PROPERTY SETS ==========


546

$$ * pbar.9988 *$PBAR 9988 222 3600.1000000.1000000.2000000.$$ * pbar.9989 *$PBAR 9989 222 100000. 3.E+8 3.E+8 6.E+8$$ * pbar.9990 *$PBAR 9990 222 3000. 200000.2500000.3000000.$$ * pbar.9993 *$PBAR,9993,111,459.96,25066.,55282.,16543.$$ * pbar.9996 *$PBAR,9996,111,895.52,309450.,55349.,48782.$$ * pbar.9999 *$PBAR,9999,111,736.,490275.,827555.,2095137.$$ * pshell.30 *$PSHELL 30 111 3 $$ * pshell.40 *$PSHELL 40 111 4 $$ * pshell.50 *$PSHELL 50 111 5 $$ * pshell.60 *$PSHELL 60 111 6PSHELL 61 111 6 PSHELL 62 111 6 $ * pshell.80 *$PSHELL 80 111 8 $$ * pshell.110 *$PSHELL 110 111 11 $$ * pshell.120 *$PSHELL 120 111 12 $


$ * pshell.135 *$PSHELL 135 111 13.5 $$ * pshell.150 *$PSHELL 150 111 15 PSHELL 151 111 15 $$ * pshell.200 *$PSHELL 200 111 20 $$ * pshell.450 *$PSHELL 450 111 45 $$ dummy elements for coupling surface$ holePSHELL,900,901,1.$ top coverPSHELL,910,901,1.$ side coverPSHELL,920,901,1.$MATD009,901,1.E-20$$ groundPSHELL,999,999,1.$MATRIG,999,,,,1.0E10,0.00,0.00,-800.,++,1.E10,0.0,0.0,1.E10,0.0,1.E10,,,++,,,,,,,,,++,,,,1,7,7$$ * conm2 *$CONM2,5000,1145,,1.5 CONM2,5001,1146,,1.7$$ ========= MATERIAL DEFINITIONS ==========$MATD024,111,7.85e-09,210000.,.3,250E10$MAT1,222,210000.,,.3,7.85e-09$INCLUDE model.bdfINCLUDE ground.dat$ENDDATA


548

Blastwave Hitting a Bunker

Problem DescriptionThe purpose is to demonstrate application of multi-Euler domains to failing coupling surfaces. The problem simulates a bunker, located on the ground that is open at the sides and is surrounded by air. Gas can flow freely through the sides of the bunker. A blast wave is ignited close to the bunker and expands into the air. When by the impact of the blast wave, the bunker surface fails gas will flow trough the bunker surface.

SOL 700 ModelingThe bunker and the ground consist of cquad4 shell elements. The elements of the bunker are Lagrangian deformable shells and the ground is modeled as rigid, using a MATRIG. The explosive/air region is modeled by two Euler meshes. The first domain models the inside of the bunker, and the second one models the outside of the bunker. For the interaction between the bunker and an Euler domain, a unique coupling surface has to be used, therefore, two coupling surfaces are needed.

The first coupling surface, for modeling the inside of the bunker, consists of the following facets:

• The 180 degrees cylindrical surface and the two open sides of the bunker. The two open sides are represented by dummy shell elements. These are elements 1 to 2240.

• The top of the ground that lies within the bunker. This is a square and is formed by elements 2241 to 3280.

These facets make up a closed coupling surface, as shown in Figure 13-28.


This coupling surface contains gas inside, and therefore Euler elements outside the coupling surface should not be processed and so the COVER is OUTSIDE.

Figure 13-28 Coupling Surface 1

The second coupling surface consists of the following facets:

• The 180 degrees cylindrical surface and the two open sides of the bunker. These are elements 1 to 2240. The top of the ground inside the bunker is not part of the second COUPLE.

• The top of the ground that is outside the bunker and 5 dummy surfaces of the ground that are used to close the coupling surfaces. These are formed by the elements 3413 to 4012, 4095 to 4340, 4505 to 4709, 4894 to 7904.

These facets make up a closed coupling surface, as shown in Figure 13-29.

Figure 13-29 Coupling Surface 2


550

This coupling surface is used for simulating the gas outside the coupling surface. So Euler elements inside the coupling surface should not be processed and the COVER has to be set to INSIDE. The second coupling surface uses the second Euler mesh and serves as inner boundary surface for this Euler mesh. The outside boundary of this mesh is where the Euler domains ends and boundary conditions for this boundaries are provided by a FLOWDEF. The FLOWDEF is chosen as non-reflecting. Waves exit the Euler domain with only little reflection.

To get an accurate expansion of the blast wave, the diffusion should be kept at a minimum, and therefore the Roe solver with second-order is used. Interactive failure will be used for the bunker structure, while porosity will be used for the open sides:

• The bunker elements can fail and gas flows through the failed elements from outside the bunker into the bunker. All elements of the bunker are assigned to a BSURF, and occur in both coupling surfaces. They are able to fail interactively, using the COUP1FL entry. These parts are formed by elements 1 to 1600. The nodes of the failed elements are constrained in space by using PARAM, NZEROVEL, YES, to preserve the geometry of the coupling surfaces from severe distortion.

• Since gas can flow through the two sides without any obstruction, these two areas are modeled with BSURF entries, and are opened by using a PORFLCPL entry. These sides are modeled with dummy shell elements and consist of elements 1601 to 2400.

The couple cards refer to mesh-number. The first mesh for the Euler elements inside the bunker is created and initialized by:

PEULER1,301,,2ndOrder,111MESH,2,BOX,,,,,,,++,-430.,0.,-1287.,837.,480.,1296.,,,++,24,16,30,,,,EULER,301

The value "2ndOrder" activates the Roe solver with second-order accuracy. The property id is the link between the TICEUL1 card 101 and the mesh card. The second Euler mesh for the Euler elements outside the bunker is created and initialized by:

PEULER1,201,,2ndOrder,101MESH,1,BOX,,,,,,,++,-647.,0.,-1293.,1057.,447.,1293.,,,++,33,23,37,,,,EULER,201


Results

Figures 13-30 and 13-31 show a fringe plot and an isosurface. Figure 13-31 has been created by Ensight.

Figure 13-30 Deformed Effective Stress Plot of the Bunker

Figure 13-31 Isosurfaces Created using SIE Variable for the Two Euler Domains

Abbreviated SOL 700 Input FileSOL 700,NLTRAN STOP=1CENDTITLE= Job name is: bunkerIC=1SPC=1TSTEPNL=1$BEGIN BULK$------- BULK DATA SECTION -------$


552

$INCLUDE mesh.datINCLUDE model.dat$TSTEPNL 1 20 0.0005 1$------- Parameter Section ------DYPARAM,FASTCOUP,,FAILPARAM*,DYINISTEP,1E-7PARAM*,DYMINSTEP,1E-8DYPARAM,LIMITER,ROEDYPARAM,RKSCHEME,3DYPARAM,LSDYNA,BINARY,D3PLOT,0.002$$ ========== PROPERTY SETS ========== $$ * steel prop *$PSHELL 1 1 .150$$ * dummy_shell *$PSHELL,2,2,1E-3MATD009,2,1E-20$$PSHELL 3 4 .100$$ ========= MATERIAL DEFINITIONS ==========$$$ -------- Material steel id =1MATD024,1,.000734,2.9e+07,.3,50000,,.21$$ -------- Material AIR id =3MATDEUL 3 1.2e-07 3EOSGAM 3 1.4 $$ -------- ground MATRIG 4 .000734 2.9e+07 .3 $$ ======== Load Cases ========================$$ ------- General Coupling: GENERAL ----- $$COUPLE 7 1 INSIDE ON ON 16 ++ ++ 1 1$BSURF 1 1 THRU 2240 3413 THRU 4012 4095++ THRU 4340 4505 THRU 4709 4894 THRU 7904$$COUP1FL,1,1.2e-07,3e+08$


COUPLE 8 11 OUTSIDE ON ON ++ ++ 2 2$BSURF 11 1 THRU 3280$$COUP1FL,2,1.2e-07,3e+08$$COUPINT,1,7,8$PORFCPL,81,LARGE,,BOTH,8LEAKAGE,1,16,32,PORFCPL,81,,1.0BSURF 32 1601 THRU 2240 $$ ------- Rigid Body Constraints ----- $SPCD2 1 RIGID MR4 1 0 1 0.SPCD2 1 RIGID MR4 2 0 1 0.SPCD2 1 RIGID MR4 3 0 1 0.SPCD2 1 RIGID MR4 5 0 1 0.SPCD2 1 RIGID MR4 6 0 1 0.SPCD2 1 RIGID MR4 7 0 1 0.TABLED1 1 ++ 0. 1. 1. 1. ENDT$$-----Mesh.dat---------------$MESH,1,BOX,,,,,,,++,-647.,0.,-1293.,1057.,447.,1293.,,,++,33,23,37,,,,EULER,201$$ Inner Euler $MESH,2,BOX,,,,,,,++,-430.,0.,-1287.,837.,480.,1296.,,,++,24,16,30,,,,EULER,301$PEULER1,201,,2ndOrder,101PEULER1,301,,2ndOrder,111$$TICEUL1 101 101 TICREG 1 101 SPHERE 8 3 5 2TICREG 2 101 SPHERE 5 3 4 1 $SPHERE 8 -536.4 165 -453.6 85SPHERE 5 -536.4 165 -453.6 10000$$TICEUL1 111 111 TICREG 3 111 SPHERE 9 3 4 1 $


554

SPHERE 9 -53.4 100 -673.6 10000$$$ ------- TICVAL BC AIR-INI ----- TICVAL 4 DENSITY 1.2e-07 SIE 3e+08$$ ------- TICVAL BC EXP-INI ----- TICVAL 5 DENSITY3.84e-06 SIE 3e+09$$FLOWDEF,202,,HYDRO,,,,,,++,FLOW,BOTH$ENDDATA

555Chapter 13: ExamplesTime Domain NVH

Time Domain NVH

Chassis

Description

This is an example of a Time Domain NHV simulation on a pickup truck chassis. A chassis of a car was modeled and an impulse loading was applied at one of front corner points. Time histories were obtained at 12 points and they were translated to a frequency domain by applying Fast Fourier Transform (FFT) to obtain dynamic properties such as mode shapes and frequencies.

SOL 700 Entries IncludedSOL 700TSTEPNLTIMNVHPARAM,S700NVH1TIMNAT


556

Model

The chassis consists of 1986 grid points and 1996 quadratic shell elements. An impact loading is applied at the end point of upper picture. The impact loading which is defined by FORCE and TABLE entries starts 0 at 0 seconds, goes up to 0.01 at 0.001 second and down to 0 at 0.002 second like lower figure. There is no boundary condition. The units are g, mm, and second.

Figure 13-32 Impluse Load applied at One Corner of the Plate

Time Domain NVH Scheme

MD Nastran SOL 700 (impluse loading)

Time history

- Displacement- Velocity- Acceleration (default)

Time domain -> Frequency domain using FFT

Extract dynamic properties:

Natural frequencies and Mode shapes


Input

On node 902517, the impact loading is applied. This analysis is done by the following three steps.

Step 1: Find modal properties using TIMNVH entryStep 2: Check the obtained modal properties and select required natural frequenciesStep 3: Re-run with selected natural frequencies

Input file timnvh.bdf

SOL 700 is the initiating command of an explicit nonlinear transient analysis.

Case control section is below.

Bulk entry section starts.

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (10 ms) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by MD Nastran during the analysis. The time step is a function of the smallest element dimension during the simulation.

TIMNVH entry is for Time NVH analysis.

Detail for TIMNVH entry:


CENDLOADSET = 1SUBCASE 1

TSTEPNL = 1 SPC = 2 DLOAD = 2

BEGIN BULKTSTEPNL 1 100 .01 1 ADAPT 2 10TIMNVH,1, , , 1.0, 1000., 3,.0005, 2,++, 0, 3, 1, 0.015, 0, 3, 13, .0030,++, 901581, 901641, 901697, 901865, 902061, 902097, , ,++, 902580, 902595, 902609, 902797, 902996, 903063

TIMNVH, 1, , , 1.0, 500., 3, 0.00005, 2,+


558

The range of natural frequencies to obtain is from 1.0 Hz to 500 Hz and only z-translational degrees of freedom is considered (3). The sampling rate is 0.00005 seconds. The peaking criterion is two, which means a peak is selected if the number of increasing and decreasing amplitude around a peak is over 2.

Acceleration is selected for the response (0) and translational eigenvectors are only requested as ASCII format (3). Eignevalues are normalized by 1.0 (1) and 0.015 is selected as a tolerance value which means if there are two modes which distance is smaller than 0.015 Hz, it is assumed to be the same mode. ACII file format of natural frequencies and eigenvalues are requested (0) as well as translational time histories of z-direction (3). Frequency-amplitude data of z-direction are requested (13) and a peak which amplitude is less than 0.0030×the maximum amplitude is ignored (.0030)

The grid points (901581, 901641, 901697, 901865, 902061, 902097, 902580, 902595, 902609, 902797, 902996 and 903063) are only considered for Time NVH analysis.


Bulk data entry that defines properties for shell elements with 3.5mm thickness:.

Bulk data entry that defines applying forces.

+, 0, 3, 1, 0.015, 0, 3, 13, .0030,+

+, 901581, 901641, 901697, 901865, 902061, 902097, , ,+

+, 902580, 902595, 902609, 902797, 902996, 903063

CQUAD4 901092 5 901572 901573 901705 901704 0. 0...GRID 64011 1711.76 -595.649577.221..

PSHELL 5 63 3.5 63 63..

TLOAD1 4 5 1LSEQ 1 5 3DLOAD 2 1. 1. 4FORCE 3 902517 0 .01 0. 0. -1.


Bulk data entry that defines tables.

$ Nodes of the Entire Model$ Loads for Load Case : DefaultSPCADD 2 1$ Displacement Constraints of Load Set : fixSPC1 1 2 903448 903909 904202 904558

Input file chassis.bdf:

This input is for the refinement of the selection of modal frequencies and mode shapes.

Only different part is shown

PARAM, S700NVH is for the re-run of Time Domain NVH analysis. Using 1 as option Time Domain NVH analysis is carried out without re-running SOL 700.

The PEAK value (original: 2) in TIMNVH entry is changed to -2 to use TIMNAT entry.

TIMNAT entry is for the control of the natural frequency selection. In this job, 35, 43, 49, 101, 108 Hz are selected to get the results.

Results

In this analysis, frequency resolution is just 1 Hz because end time of the analysis is defined by 1.0 second. As increasing an end time, the accuracy may increases.

There are three types of new results file from Time NVH analysis.

1. mode.out: the natural frequencies and eigenvalues selected are restored.

2. ampl-freq-00901865-3.txt : amplitude-frequency output of DOF =3 at grid point 901865.

3. time-hist-00901865-3.txt: time history output of DOF =3 at grid point 901865.

TABLED1 1 -10. 0. 0. 0. .001 1. .002 0. 10. 0. ENDT..ENDDATA

PARAM,S700NVH1,1TSTEPNL 1 100 .01024 1 ADAPT 2 10TIMNVH,1, , , 1.0, 1000., 3,.0005, -2,++, 0, 3, 1, 0.015, 0, 3, 13, .0030,++, 901581, 901641, 901697, 901865, 902061, 902097, , ,++, 902580, 902595, 902609, 902797, 902996, 903063 TIMNAT,1,35.,43.,49.,101.,108.


560

From the ampl-freq-*** files, the frequency-amplitude plots are obtained.

Comparison of natural frequencies between SOL 103 and SOL 700 (Hz)

Since an impact is applied to vertical direction and the peaks are detected by the vertical direction acceleration, the horizontal dominant modes are hard to catch.

Mode SOL103 SOL 700 Diff(%) Comparison

1 3.601695E+01 3.500018E+01 2.82% Vertical motion dominant


3 5.250646E+01 4.900025E+01 6.68% Lateral motion dominant

4 6.742810E+01 Small peak Lateral motion dominant

5 8.472197E+01 Small peak Lateral motion dominant



1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

0 20 40 60 80 100 120 140

node 901581

node 902996

node 902797

node 902609

node 902595

node 902580

node 902097

node 902061

node 901865

node 901697

node 901641

node 903063

1st mode ≈ 35. 2nd mode ≈ 43.

3rd mode ≈ 49.

4th mode ≈ 101.

5th mode ≈ 108.


In mode.out file

Comparison of mode shapes between SOL 103 and SOL 700


1

2

EIGV 1 3.500018E+01MODES 1 5

902595-4.37831381E-02 2.30181446E-04-9.77437006E-02

901697-4.15069142E-02 2.55256359E-04-6.31611930E-01 901641-4.29914555E-02 7.70991520E-05-1.08571907E-01 901581-3.32998498E-02-2.49243337E-04 7.08997618E-01

901865 4.37855265E-02-1.51550001E-04-4.18557096E-01 902061 7.97601410E-02 4.34427876E-04 5.67705213E-01 902097 8.68013598E-02 8.02417982E-03 1.00000000E+00 902580-3.38588683E-02 2.97715028E-04 7.28400224E-01

902609-4.24521220E-02-1.61168521E-04-6.35288211E-01 902797 4.11242103E-02-3.00773060E-04-4.29582120E-01 902996 7.69986448E-02 7.40153667E-04 5.51699503E-01 903063 8.41026922E-02-3.47784987E-03 9.82653769E-01..

1st Mode

1st Mode Natural Frequency

Output Node Number x-eigenvector z-eigenvectory-eigenvector


562

3

4

5



Because the vertical mode shapes of mode 2 and 3 are very similar although the lateral mode shapes and the amplitudes of vertical mode are quite different, the mode shapes 2 and 3 obtained from SOL 700 are nearly similar.

6

7


Vertical Mode Shape of Mode 3Vertical Mode Shape of Mode 2

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuidePrestress

564

Prestress

Simulation of Prestress and Impact on Rotating Fan Blades

Description

Aerospace companies perform bird strike test simulation to predict the impact-resistance properties of the aircraft structure. This is an example of a solid object, impacting against rotating fan blades. The radius of the rotor and the fan blades are respectively 0.040 m and 0.195 m. The rotational speed of the rotor is 10 rev/s.

The impactor is a cube of 0.0 4 x 0.0215 x 0.05 m and its velocity is 50 m/s (112 mph). Material model MATD024 was used to simulate the elastic-plastic properties of the impactor with parameters as follows:

To model the fan blades and the rotor MATD024 material was used as:

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM, DYDEFAUL, DYNADYPARAM*, LSDYNAPRESTRSISTRSSHSPCD2TIC3BCTABLEBCBODYBCGRIDBSURF DLOADMATD024RFORCETICTLOAD1

Plastic strain failure limit 0.15

Plastic strain failure limit 0.15

ρ 1000kg m3⁄= PR 0.3=

σy 2.E 7Pa+= Eh 2.1E 10Pa+=

ETAN 5E 7+=

ρ 7800kg m3⁄= PR 0.3=

σy 2.E 8Pa+= Eh 2.1E 11Pa+=

ETAN 5E 8+=

565Chapter 13: ExamplesPrestress

Model


• 2800 Quads

• 150 Hexas

The rotor and fan blades are modeled by shell elements. The impactor is modeled with solid elements. The relative speed of the impactor to the rotor is modeled by defining an initial velocity of 50 m /s (112 mph), applied on all the grid points of the impactor. The simulation time is 0.01 seconds. The unit system is m, N, sec.

Input

The simulation consists of two runs. The first run is a presstress analysis that computes the deformations and stresses due to rotation. This computation is essentially static. Conducting this computation in the impact run would require structural relaxation. Since this would interfere with the impact process the static part of the computation is done by an implicit solver in a presstress run. Boundary conditions and initial conditions of the prestress run differ from the second run. In the prestress run the rear of the rotor is fixed and a force in circumferential direction is applied to the rotor and fan blades. In the impact run the rear of the rotor is given an angular velocity and the impactor is given an initial velocity. Contacts are defined between the fan blades and the impactor.

Input file



566

SOL 700 is a executive control entry similar to SOL 600. It activates an explicit nonlinear transient analysis using SOL 700 solver.

First consider the prestress input deck. In this input deck only the rotor and fan blades are included.

The bulk entry section begins with

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (5) and Time Increment (1e-5 sec) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

By using the SOL 700 bulk data entry PRESTRS, a prestress analysis is carried out. Here the implicit MD Nastran solver is used. This solver requires the analysis to be run with double precision executable. Final deformations and stresses of elements are written to a text file named imput_file_name.nastin to provide initial conditions for rotor and fan blades of the impact run.

Bulk data entries that define properties for shell elements

These elements model the fan blades and rotor.

Bulk data entry that defines material properties for rotor and fan blade

MATD024 is a SOL 700 bulk data entry that models arbitrary elasto-plastic material.

LOADSET = 2TITLE = MD Nastran job created on 07-Dec-06 at 16:50:17$ Direct Text Input for Global Case Control DataSUBCASE 1 TITLE=This is a default subcase. TSTEPNL = 1 SPC = 1 DLOAD = 3 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL STRESS(SORT1,REAL,VONMISES,BILIN)=ALL

TSTEPNL 1 5 1.-5 1 ADAPT 2 10

PSHELL1 1 1 BLT GAUSS 2+ 0.0025 0.0025 0.0025 0.0025

$ Material Record : mat1.1MATD024 1 7800. 2.1+11 .3 2.+8 5.+8 .15


Bulk data entry that defines geometric properties of the rotor and blades:

Bulk data entry that defines boundary conditions and loads

RFORCE is a bulk data entry that defines a static loading condition due to angular velocity.

The following parameters are needed to get optimal results.

$ Pset: "p1" will be imported as: "pshell.1"CQUAD4 1 1 77 78 6 5....$ Nodes of the Entire Model

$ Loads for Load Case : Default$ Displacement Constraints at rear of rotorSPC1 1 123456 21 THRU 31....$RFORCE applied to rotor + fan bladesTLOAD1 5 6 1LSEQ 2 6 1DLOAD 3 1. 1. 5TABLED1 1 ++00003D++00003D 0. 1. .001 1. ENDT +00003ERFORCE 1 7000 10.00 0. 0. -1.

DYPARAM,LSDYNA,ACCURACY,OSU,1DYPARAM,lSDYNA,ACCURACY,INN,2$$ control_ENERGY$DYPARAM,LSDYNA,ENERGY,HGEN,2DYPARAM,LSDYNA,ENERGY,RWEN,1DYPARAM,LSDYNA,ENERGY,SLNTEN,1DYPARAM,LSDYNA,ENERGY,RYLEN,1$$ control_HOURGLASS$DYPARAM,LSDYNA,HOURGLASS,IHQ,1DYPARAM,LSDYNA,HOURGLASS,QH,0.100000001DYPARAM,LSDYNA,DATABASE,D3PLOT,0.1

PARAM,DYDEFAUL,DYNA


568

Now consider the impact run.

The bulk entry starts with

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (1e-4 sec) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps is determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

The inputdeck of the impact run includes the nastran file prestr10.dytr.nastin. This makes the shell elements that are defined in the prestress run available to the impact run. The only properties of the rotor and fan blade that need to be defined in the impact run are material properties and boundary conditions.

The file prestr10.dytr.nastin contains the entry ISTRSSH. This entry specifies what result variables of the prestress run are to be carried over to the impact run.


$ Direct Text Input for Executive ControlCENDTITLE = JOBNAME IS: BIRD_STRIKE2SUBCASE 1$ Subcase name : Default SUBTITLE=Default TSTEPNL = 1 BCONTACT = 1 SPC = 2 IC = 1 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL

$------- BULK DATA SECTION -------BEGIN BULKTSTEPNL 1 100 1e-4 1 ADAPT 2 10

INCLUDE prestr10.dytr.nastin

$ Define contactBCTABLE 1 4 SLAVE 3 0. 0. 0. 0. 0 0. 0 0 0

MASTERS 3..


BCTABLE is a SOL 700 bulk data entry that is also used by SOL 600 and it is meant to define Master-Slave as well as self contact. In addition to contact between impactor and fan blade there is also self contact defined.

BCBODY is a bulk data entry that is used by SOL 600 and SOL 700 only, which defines a flexible or rigid contact body in 2-D or 3-D. It could be specified with a BSURF, BCBOX, BCPROP, or BCMATL entry.

BSURF is a SOL 700 bulk data entry that is used by SOL 600 and SOL 700 only, which defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

Bulk data that defines the properties of elements:

PSOLID is a SOL 700 bulk data entry that defines properties for solid elements. It refers to entry MATD024 for material properties.

Bulk data entries that define material properties for the shell and solid elements:

Bulk data entry that defines geometric properties of the impactor:

$ Deform Body Contact LBC set: impactorBCBODY 3 3D DEFORM 3 0BSURF 3 2116 2117 2118 2119 2120 2121 2122

$ Deform Body Contact LBC set: bladeBCBODY 4 3D DEFORM 4 0BSURF 4 2801 2802 2803 2804 2805 2806 2807....

$ Elements and Element Properties for region : p2PSOLID 2 2 0

$ shell elements $ Material Record : mat1.1MATD024 1 7800. 2.1+11 .3 2.+8 5.+8 .15

$ solid elements$ Material Record : mat1.2MATD024 2 1000. 2.1+10 .3 2.+7 5.+7 .15

$ Pset: "p2" will be imported as: "psolid.2"CHEXA 2801 2 2921 2922 2928 2927 2957 2958 2964 2963....$ Nodes of the Entire ModelGRID 2921 .146905 .0092747-.023994..


570

The shell elements are defined through the Nastran initialization file prestr10.nastin.

Bulk data entry that defines boundary conditions and initial velocity:

BCGRID is also used to specify a set of nodes which are to be constrained with SPCD2 (angular velocity).

TIC is a bulk data entry that defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees of freedom. TIC3 defines the initial velocity as combination of translational part and a rotational part.

SPCD2 defines imposed nodal motion on a node or a set of nodes.

INCLUDE inserts an external file into the input file.

End of input file.

BCGRID 1 21 THRU 31 1346 THRU 1355 2064 ++ THRU 2073 2741 THRU 2749

$ Loads for Load Case : Default$ Initial velocvity of impactor$ Initial Velocities of Load Set : inivel_solidTIC 1 2921 3 -50TIC 1 2922 3 -50....$ Displacement Constraints of Load Set : Disp1SPC1 1 3 21 THRU 31....$ Initial angular velocity for rotor +fan bladeTIC3 1 70000 1. ++ -62.832+ 1 THRU 2920 $$ Imposed angular velocity at the rear of the rotorSPCD2 1 GRID 1 7 80 -1....and the referenced tableTABLED1 80 ++ 0.0 62.832 1. 62.832 ENDT$


Results

Prestress Run

In accordance to the TSTEPNL entry 5 results increments were archived. The results of all increments are essentially the same which indicates that the implicit calculations are stable. The results of the last increment were written to the file prestres10.dytr.nastin.

Result Increment 2


572

Result Increment 5: written to the .nastin file.

Impact Run:

The prestress result variables have been initialized at the begin of the analysis (Time = 0)

t = 0 sec


t = 0.0001 sec

t = 0.001 sec


574

t = 0.002 sec

t = 0.005 sec


t = 0.01 sec

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideSmooth Particle Hydrodynamics (SPH)

576

Smooth Particle Hydrodynamics (SPH)

Ball Penetration using SPH Method

Description

This is an example of a ball penetrating through a plate using SPH method. The initial velocity of the ball is 6.18 km/s (0.618 cm/µs) in vertical direction. In the simulation, the center part of the plate and the ball projectile are modeled by SPH elements.

SOL 700 Entries IncludedSOL 700TSTEPNLDYPARAM,LSDYNA,BINARY,D3PLOTCSPHPSPHEOSGRUNSPHDEFTICMATD010PSOLIDDMATD003

577Chapter 13: ExamplesSmooth Particle Hydrodynamics (SPH)

Model

The circular section located at the center of plate was modeled using SPH elements and the remainder of the plate was modeled using structural material property (MATD003) in the example. The ball is modeled by SPH elements and is impacting the the center of plate at 6.18 km/s . The model has a total of 19479 grid points, 300 solid elements and 18759 SPH particle elements. All the structural elements were Hexas. The simulation time is 20.0 seconds.

Input file

SOL 700 is an executive control card similar to SOL 600. It activates an explicit nonlinear transient analysis.

Case control section is below:

The bulk entry section starts:

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (6000) and Time Increment (3.33 ms) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps are determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

LSDYNA,BINARY,D3PLOT option of DYPARAM entry controls the output time steps of d3plot binary file. The result plots at every 0.5 seconds are stored in d3plot binary file.


DLOAD = 1IC = 1SPC = 1BCONTACT = 1TSTEPNL = 1

BEGIN BULK

$

TSTEPNL*1 6000 3.3333334E-03 *

*

$

$ DATABASE_BINARY

$$ Number of SPH = 18759

CSPH* 10000001 11 8.2517613E-04 **


578

CSPH entry defines a SPH particle. The SPH element number (10000001) must correspond to the grid number which describes the SPH element location. The lumped mass (8.25E-4) is applied on the SPH element.

PSPH entry defines the property of SPH particle. Both of the material property and equation of state are set to 2. The smoothing length of the particles is set to 1.2. The scale factors for the minimum and maximum smoothing length are set to 0.2 and 4.0, respectively.

EOSGRUN entry defines a Gruneisen Equation of State.

The Gruneisen equation of state defines pressure for compression as

And for tension as

All fields are set for the coefficients of two equations above. Please see MD Nastran Quick Reference Guide for details.

$$ Part = material type #10$PSPH* 10 2 2 ** ** 1.200000 0.2000000 4.000000 0.0 ** 0.0 0.0

$$ Number of EOS = 2$EOSGRUN*2 0.5328000 1.339000 0.0 ** 0.0 2.000000 0.4800000 0.0 ** 0.0 **

SPHDEF* 1 0 0.0 ** 0 0 0.0 0.0 ** **

pρ0C2μ 1 1

γ0

2---- μ–

a2---μ2–⎝ ⎠

⎛ ⎞+

1 S1 1–( )μ– S2μ2

μ 1+-------------– S3

μ3

μ 1+( )2---------------------–

2------------------------------------------------------------------------------------------------------ γ0 aμ+( )E+=

p ρ0C2μ γ0 aμ+( )E+=


SPHDEF entry defines and controls the physics of SPH particles. All values of this example are assigned using default values. See MD Nastran Quick Reference Guide for details.

MATD003 entry defines an isotropic and kinematic hardening plastic material including rate effects. This

material is used to model the boundary structural plate in the example. The density is 2.785 kg/cm3 and Young’s modulus is 0.7 GPa. The Poisson’s ratio, the yield stress and the tangential modulus are set to 0.269, 0.0029 GPa and 0.07 GPa, respectively. The hardening parameter is set to 1, which describes the isotropic hardening only.

MATD024 is a SOL 700 bulk data entry. It was used to model an elasto-plastic material with an arbitrary stress versus strain curve and arbitrary strain rate dependency. Failure can also be defined based on the plastic strain or a minimum time step size.

Bulk data entries that defines initial velocity of the rod:

TIC entry defines a nodal initial condition. In the example, all SPH grids have initial velocity conditions. Grid point 10000001 is located at the center of the plate and has zero velocity to all directions. Other nodes on the center of the plate have zero velocities similar to grid point 10000001. Grid point 100001 is located at the ball which has a velocity of 0.618 cm/s in y direction. All nodes on the ball have the same velocity as grid point 100001.

$MATD003*1 2.785000 0.7000000 0.2690000 ** 2.9000000E-03 7.0000000E-02 1.000000 ** 0.0 0.0 0.0 0 **

$$ *INITIAL_VELOCITY_NODE$TIC* 1 10000001 1 ** 0.0…$$ *INITIAL_VELOCITY_NODE$TIC* 1 100001 1 ** 0.0TIC* 1 100001 2 ** 0.6180000…

$MATD010 2 2.785 0.269 2.9E-03 0.0 -2.0E-2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


580

MATD010 entry defines an elastic-plastic hydrodynamic material. This material is used to model SPH

elements. The density, shear modulus, yield stress and cutoff pressure are set to 2.785 kg/cm3, 0.269 GPa, 0.0029 GPa and -0.02 GPa, respectively.

PSOLIDD entry defines a solid element property using element formulation and equation of state. In the example, this entry can be changed to PSOLID because only material property is assigned.

Bulk data entries that define grid points and solid type elements.

Bulk data entries that define parameters.

End of input file.

$$ Part = material type # 3$PSOLIDD*1 1 0 **

$$ Number of Nodes = 19479$GRID 1 -0.3E+1 0.404 -0.30E+1..GRID 101791 0.667E-1-0.467 0.467$$ Number of Solid elements = 300$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678$2345678CHEXA 1 1 13 14 20 19 1 2 8 7 ..CHEXA 300 1 714 715 720 719 704 705 710 709

$$ control_OUTPUT$DYPARAM*ELDLTH 100 **..DYPARAM*LSDYNA ENERGY RYLEN 1 **$ENDDATA


Results

Figure 13-33 Ball Penetration at Various Increments

t = 0.00 s t = 1.98 s

t = 3.98 s t = 5.98 s

t = 8.99 s t = 20.0 s

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideSheet Metal Forming

582

Sheet Metal Forming

Square Cup Deep Drawing using Forming Limit Diagram

Description

This is a sheet metal forming example of a plate with anisotropic behavior that is drawn through a square hole by means of a punch. This particular example has experimental results as it was provided as a verification problem for participants of the 1993 NUMISHEET Conference held in Japan. The analysis involved results obtained at single punch depth (20 mm punch travel) for an aluminum alloy plate. The material is seen to be anisotropic in its planar directions; i.e., the material behavior is different in all directions in the plane of the sheet metal as well as in the out of plane direction. The data obtained from the NUMISHEET Conference were as follows:

Aluminum Alloy

Thickness = 0.81 mmYoung’s modulus = 71 GPaPoisson’s ratio = 0.33Density = 2700 kg/m3

Yield stress = 135.3 MPaStress = 576.79 * (0.01658 + p)0.3593 MPaLankford parameters: R0 = 0.71, R45 = 0.58, R90 = 0.70Friction coefficient = 0.162

Size of the plate modeled was 0.15 x 0.15 (in meters). No strain-rate dependency effects were included in the material data, so the metal sheet was analyzed without these effects. The dimensions of the plate, die, punch, and clamp are all given in Figure 13-34.

SOL 700 Entries IncludedSOL 700TSTEPNLDYPARAM,LSDYNA,BINARY,D3PLOTCSPHPSPHEOSGRUNSPHDEFTICMATD010PSOLIDDMATD003

583Chapter 13: ExamplesSheet Metal Forming

Figure 13-34 Dimensions of Plate, Die, Punch, and Clamp (in Millimeters)


584

Model

Figure 13-35 SOL 700 Model (Exploded View)

The SOL 700 model is shown in Figure 13-35. The main parts in the finite element model are:

• sheet metal

• punch

• die

• clamp

Sheet Metal

The SOL 700 material model for sheet metals is a highly sophisticated model and includes full anisotropic behavior, strain-rate effects, and customized output options that are dependent on material choice. Since not all of the materials can be derived from the simplified set given by the NUMISHEET organization, most participants in the conference used an isotropic material model. In reality, the process is definitely anisotropic and effect due to these differences can be seen in the transverse direction. For materials displaying in-plane anisotropic behavior, the effect would be even more noticeable. The parameters on the MAT190 (refer to the MD Nastran Quick Reference Guide) specify planar anisotropic behavior and are as follows (for the aluminum sheet):

• MATD190 elastic material properties.

• Isotropic behavior was assumed in the elastic range:

Exx = 71.0 GPa = 0.33


• Planar anisotropic yielding and isotropic hardening were assumed in the plastic range:

A = Stress constant = 0.0 MPa

B = Hardening modulus = 576.79 MPa

C = Strain offset = 0.01658

n = Exponent for power-law hardening = 0.3593

• Lankford parameters:

R0 = 0.71

R45 = 0.58

R90 = 0.70

Punch, Die, and Clamp

These three components provide the constraints and driving displacement for the analysis and are modeled as rigid bodies. Contact is then specified with the metal sheet using the friction coefficient values provided. The three contact types are specified as following:

• Contact between the punch and the sheet

• Contact between the die and sheet

• Contact between the clamp and sheet

Finally, the punch is given a scaled downward velocity providing the driving displacement for the analysis.

Input File

SOL 700 is an executive control card and activates an explicit nonlinear transient analysis.

Case control section is below:

The bulk entry section starts:


DLOAD = 1IC = 1SPC = 1BCONTACT = 1TSTEPNL = 1

BEGIN BULK$TSTEPNL 1 20 2.0E-3$DYPARAM LSDYNA BINARY D3PLOT 0.002


586

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (20) and Time Increment (2.00 ms) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of Time Increments and the exact value of the Time Steps are determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

LSDYNA,BINARY,D3PLOT option of DYPARAM entry controls the output time steps of d3plot binary file. The result plots at every 0.002 seconds are stored in d3plot binary file.

Bulk data entries that define properties for shell elements

MATD020 entry defines the rigid material property. In the example, the clamp, die, and punch are modeled by the rigid materials.

MATD190 entry defines an anisotropic material developed by Barlat and Lian (1989) for modeling sheets under plane stress conditions and failure criteria of Forming Limit Diagram. This material allows the use of the Lankford parameters for the definition of the anisotropy.

In the model, Gosh’s hardening rule is used.

Forming limit diagram is defined in by TABLED1 as shown above.

All fields are set for the coefficients of equations. See MD Nastran Quick Reference Guide for details.

PSHELL1 1 1 BLT Gauss ++ .81

MATD020 2 1.0 210.E9 0.3 1 4 7

MATD190 1 2.7E-4 7.1E7 0.33 2.0 576.79E3.3593 0 ++ 6.0 .71 .58 .70 .01658 ++ 2.0 77 ++ 1.0 0.0 0.0 ++ 0.0 1.0 0.0TABLED1,77,,,,,,,,++,-100.0,196.67,0.0,30.,30.,45.,40.,47.,++,50.,45.,ENDT

SPCD2,1,RIGID,MR2,3,0,100,1.0,,++TABLED1,100,,,,,,,,++,0.0,-1000.,0.02,-1000.,ENDT

σY εp( ) k ε0 εp+( )np–=


SPCD2 entry defines imposed nodal motion on a node, a set of nodes or nodes of a rigid body. The rigid punch is moving downward at 1000 mm/s from 0 to 0.02 seconds.

FORCE entry defines a force on the grid point as well as rigids. Since the forces on the rigid body are not yet supported by Nastran input processor, TODYNA and ENDDYNA entries are used in conjunction with FORCE entry to by-pass the IFP (Input File Processor) and directly access SOL 700.

BCBODY entry defines a flexible or rigid contact body in 2-D or 3-D. Although SOL 700 only supports a flexible contact in BCTABLE, the rigid contact can be applied using the rigid material of contact bodies. In the example, all contacts have 0.162 static and kinetic friction coefficients. The surface to surface one way contact method is used on the all contact definition.

BCBODY entry defines a flexible or rigid contact body in 2-D and 3-D.

BSURF entry defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

FORCE 9999 MR3 -19.6E6 1.

BCTABLE 1 3 SLAVE 1 0. 0. 0.162 0. 0 0 0 0 0.162 SS1WAY++

BCBODY 1 DEFORM 1..$BSURF 1 1 THRU 1600..

$GRID 1 -75. 75. 0.0..GRID 4528 -8.33333-37.0067-75.405$CQUAD4 1 1 1 2 43 42..CQUAD4 4468 63 4527 4273 4274 4528


588

Bulk data entries that define parameters.

End of input file.

Results

Figure 13-36 Comparison of Major Principal Strain Along Line OB(Numisheet and Dytran Results vs MD Nastran SOL 700)

PARAM,DYDEFAUL,DYNADYPARAM,LSDYNA,OUTPUT,NPOPT,1..$ENDDATA


Figure 13-37 Comparison of Minor Principal Strain Along Line OB(Numisheet and Dytran Results vs MD Nastran SOL 700)

Figure 13-38 Forming Limit Diagram Along Line OB at 0.019 Seconds


590

Figure 13-39 Maximum Principal Strain Contour Plots at Mid Surface at Various Times

Note that the FLD diagram correctly predicts the failure of elements at t = 0.019 as shown in the stress fringe plots.

t = 0.000 s t = 0.004 s

t = 0.008 s t = 0.012 s

t = 0.016 s t = 0.020 s


PART 2. Square Cup Deep Drawing using Implicit Spring Back

SOL 700 Entries IncludedSOL 700MATD036SEQROUTSPRBCK

Description

The springback is refered to an event when the sheet metal starts vibrating after the punch is removed and might alter its final desired shape. Since it takes some time before the workpiece comes to a rest, the springback simulation is performed by the implicit solver to speed up the analysis. Using explicit-implicit switching available in SOL 700, the residual deformations after sheet metal forming are computed and used as a pre-condition for springback analysis. Because in this example, there was a failure at around 0.019 seconds in the sheet metal as shown in Part 1, the explicit simulation was terminated at 0.018 seconds. The initial condition including the final stresses and deformation and the element connectivity of the explicit run is transferred to the implicit run. The analysis scheme is described below.

Figure 13-40 Analysis Scheme

Model

The model of explicit run is the same as Part 1. In the implicit run, only the sheet metal is used.

Input File

Explicit Input File

BEGIN BULK$TSTEPNL 1 10 1.8E-3

SOL 700 Explicit

(Use SEQROUT Entry)

Generate jid.dytr.nastin

SOL 700 Implicit

(Include jid.dytr.nastin)

(Use SPRBCK Entry)


592

As mentioned above, the end time of simulation is assigned to 0.018 seconds.

SEQROUT entry generates the jid.dytr.nastin file at the end of simulation. The nastin file includes the final deformations and stresses of the assigned part. The nastin file can be used for a subsequent explicit or implicit SOL 700 run. In the example, only the result for Part 10 which includes the sheet metal is written out to the nastin file.

Implicit Input File

As mentioned above, the end time of simulation is assigned to 0.018 seconds.

SEQROUT entry generates the jid.dytr.nastin file at the end of simulation. The nastin file includes the final deformations and stresses of the assigned part. The nastin file can be used for a subsequent explicit or implicit SOL 700 run. In the example, only the result for Part 10 which includes the sheet metal is written out to nastin file.

Because all information of nodes and element connectivity is in jid.dytr.nastin file, Grid and CQUAD entries are removed in the implicit input. Only one point boundary condition at the center and SPRBCK entry are added in the input file.

Since MATD190 is not available in the implicit analysis, MATD036 is used instead of MATD190. MATD036 and MATD190 are identical material models except that FLD is supported only in MATD190.

MATD036 is only different in the failure criteria using FLD. Others are the same as MATD190 in the explicit simulations of Part 1 and 2.

SEQROUT 10BCPROP 10 1

BEGIN BULK$TSTEPNL 1 10 1.8E-3

SEQROUT 10BCPROP 10 1

MATD036 1 2.7E-4 7.1E7 0.33 2.0 576.79E3.3593 0 ++ 6.0 .71 .58 .70 .01658 ++ 2.0 ++ 1.0 0.0 0.0 ++ 0.0 1.0 0.0

SPRBCK 1 0.005 ++ 200 0.0 1.00E-3 ++ 2 1 100 1.0E-2 0.10 ++ 1 1


SPRBCK activates the implicit spring back analysis. Nonlinear with BFGS updates solver type is used in the example. See MD Nastran Quick Reference Guide for other fields.

Only one point at the center of the sheet metal is fixed to prevent singular condition in the implicit simulation.

SPC1 1 123456 841


594

Results

Vertical (Z-direction) Displacement Contour Plot:

Explicit Simulation

t = 0.000 s t = 0.018 s (end of explicit run)

Because the final results are applied as the initial condition for implicit simulation, the initial deformation of implicit simulation is set to zero.

Initial Condition of Implicit Run Final Residual Stress of Implicit Run

Implicit Simulation


Figure 13-41 Comparison of Vertical Displacements (z-direction) After Explicit and Springback Simulations Along Diagonal Line of Plate

MD Natran R3 Explicit Nonlinear (SOL 700) User’s GuideMiscellaneous

596

Miscellaneous

Paper Feed

Description

The objective of the paper feeding analysis is to predict the paper jamming and the capacity of the printer. In this example, angular velocity is applied on five rollers to feed the paper in the printer and 31are defined to simulate the paper feeding process.

SOL 700 Entries IncludedSOL 700TSTEPNLPARAM*,DYMINSTEPPARAM*,LDTSTEOFCTLBCTABLECELASIDCDAMPIDTLOAD1SPCDFORCEPELASPDAMPCONM2RBE2

Model

597Chapter 13: ExamplesMiscellaneous

The model has a total of 6946 grid points, 6170 elements, and 20 rigid body connections as follows:

• 4972 Quads

• 1188 Hexas

• 10 Concentrated masses

The paper is modeled by 1830 quad elements. All shell elements are Belytschko-Tsay formulation. The circumferential velocite is set to 1500 mm/s using different angular velocities of drives and pinches. Gravity is also taken into account. Total time of simulation is 0.4 seconds.

Loading and Boundary Conditions

The angular velocity of each drive and pinch is defined such that a 1500 mm/s circumferential velocity is created. The rotational velocities are applied sequentially at center node of the drive starting from drive 1 through drive 5 by defining Tables and SPCD. To close the gap between all the drives and the pinches, two opposite direction vertical forces are applied by using a combination of FORCE and Table entries. The magnitude of the load is predefined at each drive location.

paper

drive_2

drive_3

drive_4

drive_5

drive_1

lower guide_1

upper guide_1

entrance

guide_2 lower guide_3

upper guide_3

lower guide_4

upper guide_4

upper guide_5

lower guide_5

pinch_1 pinch_2

pinch_5

pinch_3

pinch_4


598

Thirty-one contacts are defined between:

1. Self contact of a paper

2. Paper and drive_1

3. Drive_1 and pinch_1 (two ways)









12. Paper and entrance

13. Paper and lower guide_1

14. Paper and upper guide_1

15. Paper and guide_2







RBE2

Damper Spring

Adding angular velocity to get 1500 mm/s circumferential velocity


22. Paper and pinch_1





SOL 700 initiates the explicit nonlinear transient analysis.

TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (100) and Time Increment (0.004 s) of the simulation. End time is the product of the two entries. Notice here the Time Increment is only for the first step. The actual number of time increments and the exact value of the Time Steps are determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.



BSURF is a bulk data entry that is used by MD Nastran Implicit Nonlinear (SOL 600) and SOL 700 only, which defines a contact body or surface defined by Element IDs


BEGIN BULKTSTEPNL 1 100 .004 1 ADAPT 2 10

BCTABLE 1 31 SLAVE 98 0. 0. .3 0. 0 0. 0 0 0

MASTERS 98...

BCBODY 98 3D DEFORM 98 0BSURF 98 3001 3002 3003 3004 3005 3006 3007...


600


CELAS1D and CDAMP1D are for the scalar spring connection and scalar damper connection, respectively.

TLOAD1 is a bulk data entry that defines a time-dependent dynamic load or enforced motion for use in transient response analysis. SPCD is for the enforced motion value, velocity in this example, in dynamic analysis. Using FORCE card, a concentrated force at a grid point by specifying a vector is applied.

All materials are defined by isotropic elastic materials.

PELAS and PDAMP are bulk data entries which describe properties of scalar spring and damper, respectively. CONM2 is for the definition of mass on a grid point.

GRID 1 12.5 20.05 0....$CQUAD4 1 1 1 2 12 11...CHEXA 325 2 361 362 366 365 401 402 406 405...CELAS1D 21001 18 21001 2 21002 2...CDAMP1D 21002 19 21001 2 21002 2...

TLOAD1 19 20 VELO 1LSEQ 1 20 21SPCD 21 21001 6 -120....DLOAD 2 1. 1. 19 1. 22 1. 25...FORCE 4 21001 0 9800. 0. 1. 0....GRAV 3 0 9800. 0. -1. 0.

MAT1 6 3.+6 .3 8.4-7...

PELAS 18 4.9...PDAMP 19 196....CONM2 51001 21001 .001...


Bulk data entries for element properties:

RBE2 is a bulk data entry which defines a rigid body with independent degrees of- freedom that are specified at a single grid point and with dependent degrees of freedom that are specified at an arbitrary number of grid points.

Several tables and coordinates are defined in the input file.

RBE2 55003 21001 123456 1001 1002 1003 1004 1005...

SPCADD 2 1 3 4 5 6 7 8...SPC1 1 123456 21002...

TABLED1 1 -.12 0. 0. 0. 1.-4 .5 2.-4 .8 4.-4 1. .1201 1. .1205 0. 1.2 0.


602

Results

t = 0.00 second t = 0.20 second

t = 0.40 second

MD Nastran R3 Explicit Nonlinear (SOL 700) User’s Guide

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Transcript of MD Nastran R3 Explicit Nonlinear (SOL 700) User’s Guide