Applied Mechanics of Solids (a.F. Bower) Chapter 8_ Theory of FEA -8

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1. Objectives andApplications >

1.1 Defining aProblem >

1.1.1 Deciding what tocalculate1.1.2 Defining geometry1.1.3 Defining loading1.1.4 Choosing physics1.1.5 Defining materialbehavior1.1.6 A representativeproblem1.1.7 Choosing a methodof analysis

2. Governing Equations >2.1 Deformationmeasures >

2.1.1 Displacement andVelocity2.1.2 Deformation gradient2.1.3 Deformation gradientfrom two deformations2.1.4 Jacobian of deformationgradient2.1.5 Lagrange strain2.1.6 Eulerian strain2.1.7 Infinitesimal Strain2.1.8 Engineering ShearStrain2.1.9 Volumetric andDeviatoric strain2.1.10 Infinitesimal rotation2.1.11 Principal strains2.1.12 Cauchy-Greendeformation tensors2.1.13 Rotation tensor,Stretch tensors2.1.14 Principal stretches2.1.15 Generalized strainmeasures2.1.16 Velocity gradient2.1.17 Stretch rate and spin2.1.18 Infinitesimalstrain/rotation rate2.1.19 Other deformationrates2.1.20 Strain equations ofcompatibility

2.2 Internal forces >2.2.1 Surface traction/bodyforce2.2.2 Internal tractions2.2.3 Cauchy stress2.2.4 Kirchhoff, Nominal,Material stress2.2.5 Stress forinfinitesimal motions

2009-11-21 Applied Mechanics of Solids (A.F

solidmechanics.org//Chapter8_6 1/20

2.2.6 Principal stresses2.2.7 Hydrostatic,Deviatoric, Von Mises stress2.2.8 Stresses at a boundary

2.3 Equations ofmotion >

2.3.1 Linear momentumbalance2.3.2 Angular momentumbalance2.3.3 Equations usingother stresses

2.4 Work and VirtualWork >

2.4.1 Work done by Cauchystress2.4.2 Work done by otherstresses2.4.3 Work forinfinitesimal motions2.4.4 Principle ofvirtual work2.4.5 Virtual work withother stresses2.4.6 Virtual work forsmall strains

3. Constitutive Equations>

3.1 General requirements3.2 Linear elasticity >

3.2.1 Isotropic elasticbehavior3.2.2 Isotropic stress-strainlaws3.2.3 Plane stress & strain3.2.4 Isotropic material data3.2.5 Lame, Shear, & Bulkmodulus3.2.6 Interpreting elasticconstants3.2.7 Strain energy density(isotropic)3.2.8 Anisotropic stress-strain laws3.2.9 Interpretinganisotropic constants3.2.10 Anisotropic strainenergy density3.2.11 Basis change formulas3.2.12 Effect of materialsymmetry3.2.13 Orthotropic materials3.2.14 Transversely isotropicmaterials3.2.15 Transversely isotropicdata3.2.16 Cubic materials3.2.17 Cubic material data

3.3 Hypoelasticity3.4 Elasticity w/ largerotations3.5 Hyperelasticity >

3.5.1 Deformationmeasures3.5.2 Stress measures3.5.3 Strain energy



density3.5.4 Incompressiblematerials3.5.5 Energy densityfunctions3.5.6 Calibratingmaterial models3.5.7 Representativeproperties

3.6 Viscoelasticity >3.6.1 Polymer behavior3.6.2 Generalconstitutive equations3.6.3 Spring-damperapproximations3.6.4 Prony series3.6.5 Calibratingconstitutive laws3.6.6 Calibratingmaterial models3.6.7 Representativeproperties

3.7 Rate independentplasticity >

3.7.1 Plastic metal behavior3.7.2 Elastic/plastic straindecomposition3.7.3 Yield criteria3.7.4 Graphical yieldsurfaces3.7.5 Hardening laws3.7.6 Plastic flow law3.7.7 Unloading condition3.7.8 Summary of stress-strain relations3.7.9 Representativeproperties3.7.10 Principle of max.plastic resistance3.7.11 Drucker's postulate3.7.12 Microscopicperspectives

3.8 Viscoplasticity >3.8.1 Creep behavior3.8.2 High strain ratebehavior3.8.3 Constitutive equations3.8.4 Representative creepproperties3.8.5 Representative highrate properties

3.9 Large strainplasticity >

3.9.1 Deformationmeasures3.9.2 Stress measures3.9.3 Elastic stress-strain relations3.5.4 Plastic stress-strain relations

3.10 Large strainviscoelasticity >

3.10.1 Deformationmeasures3.10.2 Stress measures3.10.3 Stress-strainenergy relations



3.10.4 Strain relaxation3.10.5 Representativeproperties

3.11 Critical statesoils >

3.11.1 Soil behavior3.11.2 Constitutive laws(Cam-clay)3.11.3 Response to 2Dloading3.11.4 Representativeproperties

3.12 Crystal plasticity>

3.12.1 Basiccrystallography3.12.2 Features ofcrystal plasticity3.12.3 Deformationmeasures3.12.4 Stress measures3.12.5 Elastic stress-strain relations3.12.6 Plastic stress-strain relations3.12.7 Representativeproperties

3.13 Surfaces andinterfaces >

3.13.1 Cohesive interfacemodels3.13.2 Contact and friction

4. Solutions to simpleproblems >

4.1 Axial/Spherical linearelasticity >

4.1.1 Elastic governingequations4.1.2 Spherically symmetricequations4.1.3 General sphericalsolution4.1.4 Pressurized sphere4.1.5 Gravitating sphere4.1.6 Heated sphericalshell4.1.7 Axially symmetricequations4.1.8 General axisymmetricsolution4.1.9 Pressurized cylinder4.1.10 Spinning circulardisk4.1.11 Interference fit

4.2 Axial/Sphericalelastoplasticity >

4.2.1 Plastic governingequations4.2.2 Spherically symmetricequations4.2.3 Pressurized sphere4.2.4 Cyclicallypressurized sphere4.2.5 Axisymmetricequations4.2.6 Pressurized cylinder



4.3 Sphericalhyperelasticity >

4.3.1 Governing equations4.3.2 Spherically symmetricequations4.3.3 Pressurized sphere

4.4 1D elastodynamics >4.4.1 Surface subjected topressure4.4.2 Surface undertangential loading4.4.3 1-D bar4.4.4 Plane waves4.4.5 Wave speeds inisotropic solid4.4.6 Reflection at asurface4.4.7 Reflection at aninterface4.4.8 Plate impactexperiment

5. Solutions for elasticsolids >

5.1 General Principles >5.1.1 Governingequations5.1.2 Navier equation5.1.3 Superposition &linearity5.1.4 Uniqueness &existence5.1.5 Saint-Venantsprinciple

5.2 2D Airy functionsolutions >

5.2.1 Airy solution inrectangular coords5.2.2 Demonstration of Airysolution5.2.3 Airy solution in polarcoords5.2.4 End loaded cantilever5.2.5 Line loadperpendicular to surface5.2.6 Line load parallel tosurface4.4.7 Pressure on a surface4.4.8 Uniform pressure on astrip4.4.8 Stress near a cracktip

5.3 2D Complex variablesolutions >

5.3.1 Complex variablesolution5.3.2 Demonstration of CVsolution5.3.3 Line force5.3.4 Edge dislocation5.3.5 Circular hole ininfinite solid5.3.6 Slit crack5.3.7 Bimaterialinterface crack5.3.8 Rigid flat punch ona surface5.3.9 Parabolic punch on



a surface5.3.10 General linecontact4.3.11 Frictional slidingcontact4.3.12 Dislocation near asurface

5.4 3D static problems >5.4.1 Papkovich-Neuberpotentials5.4.2 Demonstration of PNpotentials5.4.3 Point force ininfinite solid5.4.4 Point force normal tosurface5.4.5 Point force tangentto surface5.4.6 Eshelby inclusionproblem5.4.7 Inclusion in anelastic solid5.4.8 Spherical cavity ininfinite solid5.4.9 Flat cylindricalpunch on surface5.4.10 Contact betweenspheres4.4.11 Relations forgeneral contacts4.4.12 P-d relations foraxisymmetric contact

5.5 2D Anisotropicelasticity >

5.5.1 Governing equations5.5.2 Stroh solution5.5.3 Demonstration of Strohsolution5.5.4 Stroh matrices forcubic materials5.5.5 Degenerate materials5.5.6 Fundamental elasticitymatrix5.5.7 Orthogonality of Strohmatrices5.5.8 Barnett/Lothe &Impedance tensors5.5.9 Properties of matrices5.5.10 Basis change formulas5.5.11 Barnett-Lotheintegrals5.5.12 Uniform stress state5.5.13 Line load/dislocationin infinite solid5.5.14 Line load/dislocationnear a surface

5.6 Dynamic problems >5.6.1 Love potentials5.6.2 Pressurizedspherical cavity5.6.3 Rayleigh waves5.6.4 Love waves5.6.5 Elastic waves inwaveguides

5.7 Energy methods >5.7.1 Definition of potential



energy5.7.2 Minimum energy theorem5.7.3 Simple example ofenergy minimization5.7.4 Variational approach tobeam theory5.7.5 Estimating stiffness

5.8 Reciprocal theorem >5.8.1 Statement and proofof theorem5.8.2 Simple example5.8.3 Boundary-internalvalue relations5.8.4 3D dislocation loops

5.9 Energetics ofdislocations >

5.9.1 Potential energy ofisolated loop5.9.2 Nonsingulardislocation theory5.9.3 Dislocation inbounded solid5.9.4 Energy of interactingloops5.9.5 Peach-Koehler formula

5.10 Rayleigh Ritz method>

5.10.1 Mode shapes, nat.frequencies, Rayleigh'sprinciple5.10.2 Natural frequency of abeam

6. Solutions for plasticsolids >

6.1 Slip-linefields >

6.1.1 Interpreting slip-line fields6.1.2 Derivation of slip-line fields6.1.3 Examples ofsolutions

6.2 Boundingtheorems >

6.2.1 Definition of plasticdissipation6.2.2 Principle of minplastic dissipation6.2.3 Upper bound collapsetheorem6.2.4 Lower bound collapsetheorem6.2.5 Examples of boundingtheorems6.2.6 Lower bound shakedowntheorem6.2.7 Examples of lower boundshakedown theorem6.2.8 Upper bound shakedowntheorem6.2.9 Examples of upper boundshakedown theorem

7. Introduction to FEA >7.1 Guide to FEA >

7.1.1 FE mesh7.1.2 Nodes and elements7.1.3 Special elements



7.1.4 Material behavior7.1.5 Boundary conditions7.1.6 Constraints7.1.7 Contactingsurface/interfaces7.1.8 Initialconditions/external fields7.1.9 Soln procedures / timeincrements7.1.10 Output7.1.11 Units in FEAcalculations7.1.12 Using dimensionalanalysis7.1.13 Scaling governingequations7.1.14 Remarks on dimensionalanalysis

7.2 Simple FEAprogram >

7.2.1 FE mesh andconnectivity7.2.2 Global displacementvector7.2.3 Interpolation functions7.2.4 Element strains &energy density7.2.5 Element stiffnessmatrix7.2.6 Global stiffness matrix7.2.7 Boundary loading7.2.8 Global force vector7.2.9 Minimizing potentialenergy7.2.10 Eliminating prescribeddisplacements7.2.11 Solution7.2.12 Post processing7.2.13 Example code

8. Theory & Implementationof FEA >

8.1 Static linearelasticity >

8.1.1 Review of virtualwork8.1.2 Weak form ofgoverning equns8.1.3 Interpolatingdisplacements8.1.4 Finite elementequations8.1.5 Simple 1Dimplementation8.1.6 Summary of 1Dprocedure8.1.7 Example 1D code8.1.8 Extension to 2D/3D8.1.9 2D interpolationfunctions8.1.10 3D interpolationfunctions8.1.11 Volume integrals8.1.12 2D/3D integrationschemes8.1.13 Summary of elementmatrices



8.1.14 Sample 2D/3D code8.2 Dynamicelasticity >

8.2.1 Governing equations8.2.2 Weak form of governingeqns8.2.3 Finite elementequations8.2.4 Newmark timeintegration8.2.5 Simple 1Dimplementation8.2.6 Example 1D code8.2.7 Lumped mass matrices8.2.8 Example 2D/3D code8.2.9 Modal time integration8.2.10 Naturalfrequencies/mode shapes8.2.11 Example 1D modaldynamic code8.2.12 Example 2D/3D modaldynamic code

8.3 Hypoelasticity >8.3.1 Governing equations8.3.2 Weak form of governingeqns8.3.3 Finite elementequations8.3.4 Newton-Raphsoniteration8.3.5 Tangent moduli forhypoelastic solid8.3.6 Summary of Newton-Raphson method8.3.7 Convergence problems8.3.8 Variations on Newton-Raphson8.3.9 Example code

8.4 Hyperelasticity >8.4.1 Governing equations8.4.2 Weak form ofgoverning eqns8.4.3 Finite elementequations8.4.4 Newton-Raphsoniteration8.4.5 Neo-Hookean tangentmoduli8.4.6 Evaluating boundaryintegrals8.4.7 Convergenceproblems8.4.8 Example code

8.5 Viscoplasticity >8.5.1 Governing equations8.5.2 Weak form ofgoverning eqns8.5.3 Finite elementequations8.5.4 Integrating thestress-strain law8.5.5 Material tangent8.5.6 Newton-Raphsonsolution8.5.7 Example code

8.6 Advanced elements>



8.6.1 Shearlocking/incompatible modes8.6.2 Volumetriclocking/Reduced integration8.6.3 Incompressiblematerials/Hybrid elements

9. Modeling MaterialFailure >

9.1 Mechanisms of failure>

9.1.1 Monotonicloading9.1.2 Cyclicloading

9.2 Stress/strain basedcriteria >

9.2.1 Stress basedcriteria9.2.2 Probabilisticmethods9.2.3 Static fatiguecriterion9.2.4 Models ofcrushing failure9.2.5 Ductile failurecriteria9.2.6 Strainlocalization9.2.7 High cyclefatigue9.2.8 Low cycle fatigue9.2.9 Variableamplitude loading

9.3 Elastic fracturemechanics >

9.3.1 Crack tip fields9.3.2 Linear elasticfracture mechanics9.3.3 Calculating stressintensities9.3.4 Using FEA9.3.5 Measuring toughness9.3.6 Values of fracturetoughness9.3.7 Stable tearing9.3.8 Mixed mode fracture9.3.9 Static fatigue9.3.10 Cyclic fatigue9.3.11 Finding cracks

9.4 Energy methods infracture >

9.4.1 Definition of energyrelease rate9.4.2 Energy based fracturecriterion9.4.3 G-K relations9.4.4 G-compliance relation9.4.5 Calculating K withcompliance9.4.6 Integral expressionfor G9.4.7 The J integral9.4.8 Calculating K using J

9.5 Plastic fracturemechanics >

9.5.1 Dugdale-Barenblatt



model9.5.2 HRR crack tipfields9.5.3 J based fracturemechanics

9.6 Interface fracturemechanics >

9.6.1 Interface crack tipfields9.6.2 Interface fracturemechanics9.6.3 Stress intensityfactors9.6.4 Crack pathselection

10. Rods, Beams, Plates &Shells >

10.1 Dyadic notation10.2 Deformable rods -general >

10.2.1 Characterizing thex-section10.2.2 Coordinate systems10.2.3 Kinematic relations10.2.4 Displacement,velocity and acceleration10.2.5 Deformation gradient10.2.6 Strain measures10.2.7 Kinematics of bentrods10.2.8 Internal forces andmoments10.2.9 Equations of motion10.2.10 Constitutiveequations10.2.11 Strain energydensity

10.3 String / beamtheory >

10.3.1 Stretched string10.3.2 Straight beam (smalldeflections)10.3.3 Axially loaded beam

10.4 Solutions for rods>

10.4.1 Vibration of astraight beam10.4.2 Buckling undergravitational loading10.4.3 Post buckled shape ofa rod10.4.4 Rod bent into a helix10.4.5 Helical spring

10.5 Shells - general >10.5.1 Coordinate systems10.5.2 Using non-orthogonal bases10.5.3 Deformationmeasures10.5.4 Displacement andvelocity10.5.5 Deformationgradient10.5.6 Other strainmeasures10.5.7 Internal forcesand moments



10.5.8 Equations ofmotion10.5.9 Constitutiverelations10.5.10 Strain energy

10.6 Plates andmembranes >

10.6.1 Flat plates (smallstrain)10.6.2 Flat plates with in-plane loading10.6.3 Plates with largedisplacements10.6.4 Membranes10.6.5 Membranes in polarcoordinates

10.7 Solutions forshells >

10.7.1 Circular plate bent bypressure10.7.2 Vibrating circularmembrane10.7.3 Natural frequency ofrectangular plate10.7.4 Thin film on asubstrate (Stoney eqs)10.7.5 Buckling of heatedplate10.7.6 Cylindrical shellunder axial load10.7.7 Twisted open walledcylinder10.7.8 Gravity loadedspherical shell

A: Vectors & MatricesB: Intro to tensorsC: Index NotationD: Using polar coordinatesE: Misc derivations

Problems1. Objectives andApplications >

1.1 Defining aProblem

2. Governing Equations >2.1 Deformationmeasures2.2 Internal forces2.3 Equations ofmotion2.4 Work and VirtualWork

3. Constitutive Equations>

3.1 General requirements3.2 Linear elasticity3.3 Hypoelasticity3.4 Elasticity w/ largerotations3.5 Hyperelasticity3.6 Viscoelasticity3.7 Rate independentplasticity3.8 Viscoplasticity3.9 Large strainplasticity



3.10 Large strainviscoelasticity3.11 Critical statesoils3.12 Crystal plasticity3.13 Surfaces andinterfaces

4. Solutions to simpleproblems >

4.1 Axial/Spherical linearelasticity4.2 Axial/Sphericalelastoplasticity4.3 Sphericalhyperelasticity4.4 1D elastodynamics

5. Solutions for elasticsolids >

5.1 General Principles5.2 2D Airy functionsolutions5.3 2D Complex variablesolutions5.4 3D static problems5.5 2D Anisotropicelasticity5.6 Dynamic problems5.7 Energy methods5.8 Reciprocal theorem5.9 Energetics ofdislocations5.10 Rayleigh Ritz method

6. Solutions for plasticsolids >

6.1 Slip-linefields6.2 Boundingtheorems

7. Introduction to FEA >7.1 Guide to FEA7.2 Simple FEAprogram

8. Theory & Implementationof FEA >

8.1 Static linearelasticity8.2 Dynamicelasticity8.3 Hypoelasticity8.4 Hyperelasticity8.5 Viscoplasticity8.6 Advanced elements

9. Modeling MaterialFailure >

9.1 Mechanisms of failure9.2 Stress/strain basedcriteria9.3 Elastic fracturemechanics9.4 Energy methods infracture9.5 Plastic fracturemechanics9.6 Interface fracturemechanics

10. Rods, Beams, Plates &Shells >



10.1 Dyadic notation10.2 Deformable rods -general10.3 String / beamtheory10.4 Solutions for rods10.5 Shells - general10.6 Plates andmembranes10.7 Solutions forshells

A: Vectors & MatricesB: Intro to tensorsC: Index NotationD: Using polar coordinatesE: Misc derivations

FEA codesMapleMatlab

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Chapter 8

Theory and Implementation of the Finite Element Method 8.6 Advanced element formulations Incompatible modes; reducedintegration; and hybrid elements Techniques for interpolating the displacement field within 2D and 3D finite elements were discussed in Section8.1.9 and 8.1.10. In addition, methods for evaluating the volume or area integrals in the principle of virtual workwere discussed in Section 8.1.11. These procedures work well for most applications, but there are situationswhere the simple element formulations can give very inaccurate results. In this section

1. We illustrate some of the unexpected difficulties that can arise in apparently perfectly well designed finiteelement solutions to boundary value problems;

2. We describe a few more sophisticated elements that have been developed to solve these problems. We focus in particular on `Locking phenomena. Finite elements are said to l`ock if they exhibit an unphysicallystiff response to deformation. Locking can occur for many different reasons. The most common causes are (i)the governing equations you are trying to solve are poorly conditioned, which leads to an ill conditioned system offinite element equations; (ii) the element interpolation functions are unable to approximate accurately the straindistribution in the solid, so the solution converges very slowly as the mesh size is reduced; (iii) in certain elementformulations (especially beam, plate and shell elements) displacements and their derivatives are interpolatedseparately. Locking can occur in these elements if the interpolation functions for displacements and theirderivatives are not consistent. 8.6.1 Shear locking and incompatible mode elements Shear locking can be illustrated by attempting to find a finiteelement solution to the simple boundary value problem illustratedin the picture. Consider a cantilever beam, with length L, height2a and out-of-plane thickness b, as shown in the figure. The topand bottom of the beam are traction free, the left handend is subjected to a resultant force P, and the right hand end isclamped. Assume that b

quadrilateral plane stress elements. Results are shown for two different ratios of , with

for both cases.

For the thick beam, the finite element and exact solutions agree nearly perfectly. For the thin beam, however,the finite element solution is very poor even though the mesh resolution is unchanged. The error in the finiteelement solution occurs because the standard 4 noded quadrilateral elements cannot accurately approximate thestrain distribution associated with bending. The phenomenon is known as s`hear locking because the elementinterpolation functions give rise to large, unphysical shear strains in bent elements. The solution would eventuallyconverge if a large number of elements were added along the length of the beam. The elements would have tobe roughly square, which would require about 133 elements along the length of the beam, giving a total mesh sizeof about 500 elements. Shear locking is therefore relatively benign, since it can be detected by refining the mesh, and can be avoided byusing a sufficiently fine mesh. However, finite element analysts sometimes cannot resist the temptation toreduce computational cost by using elongated elements, which can introduce errors. Shear locking can also be avoided by using more sophisticated element interpolation functions that can accuratelyapproximate bending. I`ncompatible Mode elements do this by adding an additional strain distribution to theelement. The elements are called i`ncompatible because the strain is not required to be compatible with thedisplacement interpolation functions. The approach is conceptually straightforward:

1. The displacement fields in the element are interpolated using the standard scheme, by setting

where are the shape functions listed in Sections 8.1.9 or 8.1.10, are a

set of local coordinates in the element, denote the displacement values andcoordinates of the nodes on the element, and is the number of nodes on theelement.

2. The Jacobian matrix for the interpolation functions, its determinant, and its inverse are defined in theusual way

3. The usual expression for displacement gradient in the element is replaced by

where p=2 for a 2D problem and p=3 for a 3D problem, are a set of unknown displacementgradients in the element, which must be determined as part of the solution.

4. Similarly, the virtual displacement gradient is written as

where is a variation in the internal displacement gradient field for the element. 5. These expressions are then substituted into the virtual work equation, which must now be satisfied for

all possible values of virtual nodal displacements and virtual displacement gradients . At firstsight this procedure appears to greatly increase the size of the global stiffness matrix, because a set ofunknown displacement gradient components must be calculated for each element. However, the

unknown are local to each element, and can be eliminated while computing the element stiffnessmatrix. The procedure to do this can be shown most clearly in a sample code.



A sample small-strain, linear elastic code withincompatible mode elements is provided inFemlinelast_incompatible_modes.mws. When run withthe input file shear_locking_demo.txt it produces theresults shown in the figure. The incompatible modesclearly give a spectacular improvement in theperformance of the element. HEALTH WARNINGS: (i) The procedure outlined here only works for small-strain problems. Finite strainversions exist but are somewhat more complicated. (ii) Adding strain variables to elements can dramaticallyimprove their performance, but this procedure must be used with great care to ensure that the strain anddisplacement degrees of freedom are independent variables. For further details see Simo, J. C., and M. S. Rifai,Int J. Num. Meth in Eng. 29, pp. 1595 1638, 1990. Simo, J. C., and F. Armero, Int J. Num. Meth in Eng., 33,pp. 1413 1449, 1992. 8.6.2 Volumetric locking and reduced integration elements Volumetric locking can be illustrated using a simple boundary value problem.Consider a long hollow cylinder with internal radius a and external radius b asshown in the figure. The solid is made from a linear elastic material with Youngsmodulus and Poissons ratio . The cylinder is loaded by an internal pressure

and deforms in plane strain. The analytical solution to this problem is given in Section 4.1.9. The figures below compare the analytical solution to a finite element solution withstandard 4 noded plane strain quadrilateral elements. Results are shown for twovalues of Poissons ratio . The dashed lines show the analytical solution, whilethe solid line shows the FEA solution.

The two solutions agree well for , but the finite element solution grossly underestimates the displacementsas Poissons ratio is increased towards 0.5 (recall that the material is incompressible in the limit ). In thislimit, the finite element displacements tend to zero this is known as `volumetric locking The error in the finite element solution occurs because the finite element interpolation functions are unable toproperly approximate a volume preserving strain field. In the incompressible limit, a nonzero volumetric strain atany of the integration points gives rise to a very large contribution to the virtual power. The interpolationfunctions can make the volumetric strain vanish at some, but not all, the integration points in the element. Volumetric locking is a much more serious problem than shear locking, because it cannot be avoided by refiningthe mesh. In addition, all the standard fully integrated finite elements will lock in the incompressible limit; andsome elements show very poor performance even for Poissons ratios as small as . Fortunately, mostmaterials have Poissons ratios around 0.3 or less, so the standard elements can be used for most linear elasticityand small-strain plasticity problems. To model rubbers, or to solve problems involving large plastic strains, theelements must be redesigned to avoid locking. Reduced Integration is the simplest way to avoid locking. The basic idea is simple: since the fully integratedelements cannot make the strain field volume preserving at all the integration points, it is tempting to reduce thenumber of integration points so that the constraint can be met. `Reduced integration usually means that theelement stiffness is integrated using an integration scheme that is one order less accurate than the standardscheme. The number of reduced integration points for various element types is listed in the table below. Thecoordinates of the integration points are listed in the tables in Section 8.1.12.

Number of integration points for reduced integration schemes



Linear triangle (3 nodes) 1 pointQuadratic triangle (6 nodes): 3 pointsLinear quadrilateral (4 nodes): 1 pointQuadratic quadrilateral (8 nodes): 4 points

Linear tetrahedron (4 nodes): 1 pointQuadratic tetrahedron (10 nodes): 4 pointsLinear brick (8 nodes): 1 pointQuadratic brick (20 nodes): 8 points

HEALTH WARNING: Notice that the integration order cannot be reduced for the linear triangular andtetrahedral elements these elements should not be used to model near incompressible materials, although indesperation you can a few such elements in regions where the solid cannot be meshed using quadrilaterals. Remarkably, reduced integration completely resolves locking in some elements (the quadratic quadrilateral andbrick), and even improves the accuracy of the element. As an example, the figure below shows the solution tothe pressurized cylinder problem, using both full and reduced integration for 8 noded quadrilaterals. Withreduced integration, the analytical and finite element results are indistinguishable.

Reduced integration does not work in 4 noded quadrilateral elements or 8noded brick elements. For example, the figure on the right shows thesolution to the pressure vessel problem with linear (4 noded) quadrilateralelements with reduced integration. The solution is clearly a disaster. Theerror occurs because the stiffness matrix is nearly singular the system ofequations includes a weakly constrained deformation mode. This phenomenon is known as `hourglassing because of the characteristic shapeof the spurious deformation mode. Selectively Reduced Integration can be used to cure hourglassing. Theprocedure is illustrated most clearly by modifying the formulation for static linear elasticity. To implement themethod:

1. The volume integral in the virtual work principle is separated into a deviatoric and volumetric part bywriting

Here, the first integral on the right hand side vanishes for a hydrostatic stress.2. Substituting the linear elastic constitutive equation and the finite element interpolation functions into

the virtual work principle, we find that the element stiffness matrix can be reduced to

3. When selectively reduced integration is used, the first volume integral is evaluated using the fullintegration scheme; the second integral is evaluated using reduced integration points.

Selective reduced integration has been implemented in the sample programfem_selective_reduced_integration.mws. When this code is run with theinput file volumetric_locking_demo.txt it produces the results shown in thefigure. The analytical and finite element solutions agree, and there are nosigns of hourglassing. In many commercial codes, the f`ully integrated elements actually useselective reduced integration. The B-bar method: Like selective reduced integration, the B-bar method works by treating the volumetricand deviatoric parts of the stiffness matrix separately. Instead of separating the volume integral into two parts,however, the B-bar method modifies the definition of the strain in the element. Its main advantage is that theconcept can easily be generalized to finite strain problems. Here, we will illustrate the method by applying it tosmall-strain linear elasticity. The procedure starts with the usual virtual work principle



Hourglass base vectorsLinear quadrilateral

Linear brick

In the B-bar method1. We introduce a new variable to characterize the volumetric strain in the elements. Define

where the integral is taken over the volume of the element.2. The strain variation in each element is replaced by the approximation

3. Similarly, the virtual strain in each element is replaced by This means that the volumetric strain in the element is everywhere equal to its mean value. The virtual workprinciple is then written in terms of and as

Finally, introducing the finite element interpolation and using the constitutive equation yields the usual system oflinear finite element equations, but with a modified element stiffness matrix given by

This expression can be integrated using a standard, full integration scheme. The B-bar method has been implemented in the sample code FEM_Bbar.mws. The code can be run with theinput file volumetric_locking_demo.txt. Run the code for yourself to verify that the analytical and finite elementsolutions agree, and there are no signs of hourglassing. Reduced integration with hourglass control:Hourglassing in 4 noded quadrilateral and 8 nodedbrick elements can also be cured by adding anartificial stiffness to the element that acts toconstrain the hourglass mode. The stiffness mustbe carefully chosen so as to influence only thehourglass mode. Only the final result will be givenhere for details see D.P. Flanagan and T.Belytschko, International J. Num Meth in Engineering, 17, pp. 679 706, (1981). To compute the corrective term:1. Define a series of `hourglass base vectors which specify the displacements of the ath node in the ith

hourglass mode. The 4 noded quadrilateral element has only one hourglass mode; the 8 noded brick has 4modes, listed in the table.

2. Calculate the `hourglass shape vectors for each mode as follows

where denotes the number of nodes on the element.3. The modified stiffness matrix for the element is written as

where denotes the volume of the element, and is a numerical parameter that controls the stiffness of

the hourglass resistance. Taking where is the elastic shear modulus

works well for most applications. If is too large, it will seriously over-stiffen the solid. Sample code: Reduced integration with hourglass control has beenimplemented in the sample code Fem_hourglasscontrol.mws. When runwith the input file volumetric_locking_demo.txt it produces the resultsshown in the picture. Hourglassing has clearly been satisfactorilyeliminated. HEALTH WARNING: Hourglass control is not completely effective: itcan fail for finite strain problems and can also cause problems in adynamic analysis, where the low stiffness of the hourglass modes canintroduce spurious low frequency vibration modes and low wave speeds.



8.6.3 Hybrid elements for modeling near-incompressible materials The bulk elastic modulus is infinite for a fully incompressible material, which leads to an infinite stiffness matrixin the standard finite element formulation (even if reduced integration is used to avoid locking). This behaviorcan cause the stiffness matrix for a nearly incompressible material to become ill-conditioned, so that smallrounding errors during the computation result in large errors in the solution. Hybrid elements are designed to avoid this problem. They work by including the hydrostatic stress distribution asan additional unknown variable, which must be computed at the same time as the displacement field. This allowsthe stiff terms to be removed from the system of finite element equations. The procedure is illustrated mosteasily using isotropic linear elasticity, but in practice hybrid elements are usually used to simulate rubbers ormetals subjected to large plastic strains. Hybrid elements are based on a modified version of the principle of virtual work, as follows. The virtual workequation (for small strains) is re-written as

Here1. is the deviatoric stress, determined from the displacement field;2. is the hydrostatic stress, again determined from the displacement field;3. p is an additional degree of freedom that represents the (unknown) hydrostatic stress in the solid;4. is an arbitrary variation in the hydrostatic stress;5. is the bulk modulus of the solid.

The modified virtual work principle states that, if the virtual work equation is satisfied for all kinematicallyadmissible variations in displacement and strain and all possible variations in

pressure , the stress field will satisfy the equilibrium equations and traction boundary conditions, and thepressure variable p will be equal to the hydrostatic stress in the solid. The finite element equations are derived in the usual way

1. The displacement field, virtual displacement field and position in the each element are interpolatedusing the standard interpolation functions defined in Sections 8.1.9 and 8.1.10 as

2. Since the pressure is now an independent variable, it must also be interpolated. We write

where are a discrete set of pressure variables, is an arbitrary change in these pressure variables, are a set of interpolation functions for the pressure, and is the number of pressure variables

associated with the element. The pressure need not be continuous across neighboring elements, so thatindependent pressure variables can be added to each element. The following schemes are usually used

a. In linear elements (the 3 noded triangle, 4 noded quadrilateral, 5 noded tetrahedron or 8 nodedbrick) the pressure is constant. The pressure is defined by its value at the centroid of eachelement, and the interpolation functions are constant.

b. In quadratic elements (6 noded triangle, 8 noded quadrilateral, 10 noded tetrahedron or 20noded brick) the pressure varies linearly in the element. Its value can be defined by thepressure at the corners of each element, and interpolated using the standard linear interpolationfunctions.

3. For an isotropic, linear elastic solid with shear modulus and Poisson ratio the deviatoric stress isrelated to the displacement field by , while the hydrostatic

stress is , where is the bulk modulus.4. Substituting the linear elastic equations and the finite element interpolation functions into the virtual

work principle leads to a system of equations for the unknown displacements and pressures ofthe form

where the global stiffness matrices are obtained by summing the following element stiffnessmatrices



and the force F is defined in the usual way. The integrals defining may be evaluated using the fullintegration scheme or reduced integration (hourglass control may be required in this case). Theremaining integrals must be evaluated using reduced integration to avoid element locking.

5. Note that, although the pressure variables are local to the elements, they cannot be eliminated from theelement stiffness matrix, since doing so would reduce the element stiffness matrix to the usual, non-hybrid form. Consequently, hybrid elements increase the cost of storing and solving the system ofequations.

(c) A.F. Bower, 2008This site is made freely available for educational purposes.

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Applied Mechanics of Solids (a.F. Bower) Chapter 8_ Theory of FEA -8

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Transcript of Applied Mechanics of Solids (a.F. Bower) Chapter 8_ Theory of FEA -8