Cohesive Zone Modeling

1 http://www.nenastran.com/Composites

NASTRAN COHESIVE ZONE MODELING OF

COMPOSITE BONDED JOINTS

Allan Hsu, Barton McPheeters

NEi Software

5555 Garden Grove Boulevard

Westminster, CA 92683, USA

ABSTRACT

A computationally efficient method for modeling delamination initiation and propagation

in composite bonded joints is presented to aid in aircraft structural analysis. The

modeling strategy incorporates the following state of the art finite element technologies

to accurately capture the complex damage mechanisms in bonded composite joints:

cohesive zone modeling (CZM), three-dimensional layered solid elements, and

micromechanics-based damage degradation. Simulation results are presented for a

Double Cantilevered Beam (DCB) Mode I Fracture Test and a Mixed Mode Skin

Stiffener Tension Test, and compared to experimental data from technical publications.

1. INTRODUCTION

Analytical assessment of composite bonded joints is a recurring issue in aircraft design

today. Various techniques have been applied to this problem with mixed results. Some

techniques involve a two-step approach where a conventional finite element stress

analysis predicts delamination initiation, and the initial debond results are passed to a

fracture mechanics approach for delamination propagation predictions. Other techniques

are not able to fully capture all of the critical components of complex damage

mechanisms: initial damage resulting from through thickness matrix cracking and

branching, matrix cracks driving delaminations to occur, stiffness degradations in fiber

and/or matrix directions, and complete delamination across the adhesive bondline. This

particular one-step modeling strategy is designed to capture the true failure mechanism of

the joint, and incorporates the following state of the art finite element technologies to

accurately capture the complex damage mechanisms in bonded composite joints:

cohesive zone modeling (CZM), three-dimensional layered solid elements, and

micromechanics-based damage degradation. As the micromechanics-based failure model,

MultiContinuum Theory (MCT) is utilized as part of the modeling strategy to extract

separate failure predictions for the fibers and epoxy matrices throughout the composite

joints. Using a Double Cantilevered Beam (DCB) Mode I Fracture Test and a Mixed

Mode Skin Stiffener Tension Test, the paper compares critical delamination propagation

failure loads predicted by NASTRAN with experimental data from previously published

technical publications.


1. FINITE ELEMENT METHODS

Cohesive Zone Modeling (CZM) is a generally applicable nonlinear methodology for

simulating crack initiation and crack growth in pure mode and mixed mode fractures. The

incremental growth of a crack due to environmental loading can be monitored from

analysis, and the damage tolerance of the composite structure can be assessed. Many

different delamination scenarios can be analyzed with CZM: delamination in the

adhesive/adherend interface of a bonded joint or delamination in the skin/core interface

of a large scale honeycomb sandwich structure, for example.

Cohesive Zone Elements are typically defined at bonded contact interfaces between

structural elements. The behavior of these Cohesive Zone Elements are governed by

Traction Separation laws shown in Figure 1: as the cohesive zones separate, the traction

first increases until a maximum interfacial allowable stress is reached, then the traction

decreases to zero at a maximum interfacial allowable displacement, which results in

complete local separation. Normal (Mode I) and shear (Mode II) behavior are both

controlled by this triangular Traction Separation law. The area under the triangular curve

is the Strain Energy Release Rate, and this energy dissipation rate can be experimentally

determined for various joints.

Figure 1. Traction Separation Laws of Cohesive Zone Elements

We will use NEi Nastran for the finite element solver, taking advantage of its three-

dimensional layered solid element capability. Like standard layered shell elements, the

layup/stackup for the laminate is defined by the materials, ply thicknesses, and fiber

orientation angles (if applicable) for each layer. A known limitation of layered shell

elements is the usage of beam shear theory to derive balancing transverse shear stresses


on a ply by ply basis. Layered solid elements do not have this limitation – transverse

shear and the through-thickness normal stresses are included as a direct result of the

triaxial stress state in the fully 3D orthotropic material. The through-thickness stresses are

taken into account to assess fiber failures, matrix cracking, and delamination between the

plies. These stress components are particularly important when the following conditions

are present: thick laminates and joints, out-of-plane loading, or the free edges of a

component are the main regions of concern. With the same number of elements, there is

no computational time penalty for using layered solid elements instead of layered shell

elements. Also, the layered solid elements have three integration points per layer in the

through-thickness direction to ensure accuracy of results.

We will utilize NEi Nastran’s embedded MultiContinuum Technology (MCT) capability,

provided by Firehole Composites’ Helius product. The traditional FEM approach for

fiber reinforced composites is to smear the properties of the fiber and matrix together to

arrive at a homogeneous representation of each layer. The smeared properties can be

found from testing or a micro-mechanical approach. Failure in smeared-layer analyses is

determined using a failure index that relates the stresses and strains in this smeared layer

to experimentally-determined material strengths. The micromechanics-based MCT

separates the matrix and fiber properties, handling each composite constituent differently

and allowing failure to progress through a multi-step damage mechanism. For example,

compression of a composite often results in the matrix being squeezed around with

deformations due to the Poisson’s effect that drag the fibers with them. With the matrix

and fibers in different stress states, the composite laminate can be much closer to failure

than a smeared approach would predict. Effects like this can become significant

especially when thermal loads are added to structural ones. Failure theories like Puck [5]

and LARC02 [1] account for the different failure states of the fibers and matrix. They

predict a two-step failure, with matrix (or possibly fiber) failing first, followed later by

the other with a subsequent degradation of properties in the interim. MCT accounts for

multiple-step failure, as well as out-of-plane stresses. Out-of-plane stresses can

sometimes be a significant fraction of in-plane stresses, up to the point that the matrix can

fail in the out-of-plane direction before the in-plane directions have failed completely.

MCT also has the major advantage of being developed for weaves/fabrics and other

composites (metal matrix, long fiber, unaligned fiber, etc.), while LARC02 and PUCK

was developed strictly for unidirectional composite materials. Fertig provides the

equations used to implement MCT [3].

Progressive Ply Failure Analysis allows a composite to fail one ply at a time. Once an

iterative nonlinear model is setup (either nonlinear static or nonlinear transient dynamic),

a Progressive Ply Failure Analysis can be run to examine the progression of nonlinear

damage. The selected Failure Theory predicts when a particular ply will fail, and then the

stiffness of the failed directions will be reduced with the corresponding reduction scale

factor (10 percent is used for this investigation). Multiple papers have shown Progressive

Ply Failure Analysis results to not be very sensitive to the exact value of the stiffness

reduction scale factors – varying the reduction scale factors from 4-20 percent generally

affect the composite FEA results minimally. Equilibrium is reestablished after the

material properties are degraded, and then at the next time step/load increment, composite


stresses are again checked to see what plies and what direction the plies fail in. After the

nonlinear analysis is concluded, the analyst can postprocess output such as percentage of

plies failed and location of plies failed.

2. DOUBLE CANTILEVERED BEAM MODE I FRACTURE TEST

2.1 Lamina Properties

The laminate for the Double Cantilevered Beam (DCB) tests are prepared from a carbon

(T300)/toughened epoxy (977-2) prepreg supplied by Cytec Fiberite.

Table 1. Material Properties for T300/977-2 Carbon Fiber

2.2 Finite Element Model Details

A DCB FEM is constructed to explore the delamination propagation through a bonded

joint with a starter crack – see Figure 2. The carbon fiber laminate has a stackup of [0]24,

and has dimensions of 150mm long, 20 mm wide, and 4 mm thick. The starter crack is 55

mm long.

Enforced displacement in the +Z/-Z direction is applied to the top edge of the top beam

and the bottom edge of the bottom beam to enable a Mode I opening type fracture. Using

6 CHEXA (linear hexahedral solid elements) elements through the thickness, a total of

3600 CHEXA elements form the mesh. The elements are not connected in any fashion at

the initial precrack locations, and cohesive zone elements are specified at the bonded

contact interface.

For the triangular Cohesive Zone Law, the Strain Energy Release Rate is related by:

(1)


SDMAXT is the interfacial allowable stress, and UDMAXT is the interfacial allowable

displacement. Multiple combinations of SDMAXT and UDMAXT were iterated on to

fulfill the requirement of SERR = 0.3 N/mm [2]. For most combinations of SDMAXT

and UDMAXT, FEA results were similar and not very sensitive. SDMAXT of 20 MPa

and UDMAXT of 0.03 mm were found to provide good nonlinear convergence and

correlate to experimental data well. Lowering SDMAXT and increasing UDMAXT

usually helps with nonlinear convergence. Lowering SDMAXT excessively will result in

significantly altered results.

Progressive damage (PPFA) of the carbon fibers and epoxy matrices are ignored for this

simulation.

Figure 2. DCB Finite Element Model with Cohesive Zone Elements

2.3 DCB Mode I Fracture Results

Nonlinear static analysis (Nastran SOL 106) is run with large displacement effects that

include follower forces and differential stiffness.

The laminate opens up in a pretty stable manner – see Figure 3. The red regions indicate

locations where the cohesive zones have completely failed and the adhesive is no longer

functional.

Figure 3. Deformed Configuration of DCB (1:1 Actual Def)


Figure 4. Results Comparison – Simulation vs. Experiment

Delamination initiates at a certain critical load (P Crit), and as the delaminations

propagate through the adhesive bondline, the stiffness of the DCB decreases.

Looking at Figure 4, the simulation results have good correlation with the experimental

and analytical results. The critical delamination failure load from experiment is 58 N, and

the failure load from Node 153 in the simulation is 53.6 N. The experimental results are

taken from De Morais, et a. [2], and is a typical load-displacement curve from the

testing. A critical load (P Crit) was also calculated to be 63.7 N through analytical

relations – these calculations are not presented here.

Nodes 151-155 are the 5 loaded nodes on the edge of the top beam (153 is in the center of

the edge, 152 and 154 are next to 153, 151 and 155 are on the ends of the edge). Nodes

151 and 155 have the same load-displacement response, while nodes 152 and 154 have

the same load-displacement response.

Load-displacement response is plotted for all 5 nodes to show the variance of behavior

through the width of the specimen. Experimental testing has shown that at the crack tip in

DCB specimens, the Strain Energy Release Rate decreases close to the edges. The

maximum load for the ends of the edge (Nodes 151 and 155) is 39.7 N.


3. MIXED MODE SKIN STIFFENER TENSION TEST

3.1 Lamina Properties

Both skin and flange are made from IM6/3501-6 graphite/epoxy prepreg tape with a

nominal thickness of 0.188 mm. Along with lamina material properties, constituent (fiber

and matrix) material properties are also input into the finite element simulation for use

with the MCT micromechanical failure model.

Table 2. Material Properties for IM6/3501-6 Carbon Fiber [4]

3.2 Finite Element Model Details

A bonded skin and flange assembly FEM is constructed to explore the complex damage

mechanism of a complete delamination of a composite flange – see Figure 5. The skin

layup, consisting of 14 plies, is [0/45/90/-45/45/-45/0]S, and the flange layup, consisting

of 10 plies, is [45/90/-45/0/90]S. The skin width is 25.4 mm, and the skin length is 127.0

mm. Maximum flange length at the adhesive bond is 50 mm, and the minimum flange

length is 42 mm. The angle at the flange tip is 27 degrees.


One side of the composite skin is constrained in X, Y, and Z translations. The other side

is constrained transversely (Y and Z directions) and enforced to displace in the +X

direction. Using 2 CHEXA (linear hexahedral solid elements) elements through the

thickness for both the skin and the flange, a total of 828 layered solid elements form the

mesh. Cohesive Zone Elements are specified for the bonded contact interface between the

flange and the skin. The two tapered flange ends are meshed differently to prevent

simultaneous delamination from occurring at both ends.

For Mode I opening mode, SDMAXT is set to 11.3 MPa and UDMAXT is set to 0.0162

mm to satisfy SERR = 0.0931 N/mm [4]. For Mode II shearing mode, SDMAXS is set to

11.3 MPa and UDMAXS is set to 0.084 mm to satisfy SERR = 0.467 N/mm. For these

types of CZM simulations, experience suggests we use a realistic value for interfacial

allowable stresses and then adjust, if needed, to improve convergence.

Micromechanics based damage degradation is utilized in this simulation.

Figure 5. Bonded Skin and Flange FEM

3.3 Skin Stiffener Tension Test Results

3.3.1 Displacement-Load Behavior

Nonlinear static analysis (Nastran SOL 106) is run with large displacement effects that

include follower forces and differential stiffness. NEi Nastran’s layered solid element and

MultiContinuum Theory (MCT) capabilities add virtually zero computational penalty to a

typical nonlinear finite element run – the biggest factor for run time is model size

(number of degrees of freedom) and number of cohesive zone elements. The system is

setup for large complex multiscale composite models, with simultaneous in-plane and

out-of-plane damage progression, to run within hours. This particular analysis ran

through 392 load increments, 9 load bisections, and 1625 equilibrium iterations in about

half an hour (depending on hardware used).

The displacement-load curve stays linear until matrix cracking occurs at the flange tip,

which quickly propagates into debond failures along the adhesive bondline. A load drop

occurs as the flange completely and unstably separates from the composite skin. In Figure

6, the x-axis is the displacements in the X direction extracted at the flange tip, and the y-

axis is the total load applied in the X direction at the enforced motion boundary.


Figure 6. Displacement-Load Behavior of Skin Stiffener FEM

Five tension tests were performed by Krueger [4]; the FE simulation results are tabulated

with the experimental data in Figure 7. The correlation is good - the damage initiation

load and flange debond load is slightly lower for the FE simulation, but for both

simulation and test, damage initiation occurs at or above 90 % of flange debond load. It is

interesting to note that for one test specimen, no visible damage could be detected prior to

flange debond at 21,100 N.

Figure 7. Results Comparison – Simulation vs. Experiment


3.3.2 Bond Failures and Matrix Failures

From Figure 8, it is observed that the specimen is elongated by the tensile load, but also

bends due to the local effects of the flange. The locations of matrix cracking and

branching, after the complete separation of flange from skin, are shown in Figure 10. The

delaminated bondline regions are shown in Figure 9.

Figure 8. Deformed Configuration at Damage Initiation (10 to 1 Actual Def)

Figure 9. Delamination Failure Regions

Figure 10. Matrix Failure Regions


3.3.3 Assessing Probability of Fiber Damage

From using the micromechanics-based failure model MCT for damage degradation, we

can extract matrix-specific and fiber-specific failure indices for the laminated structure at

any load level. For example, at damage initiation load, matrix failure indices and fiber

failure indices for all the plies are shown in Figures 11 and 12. Using this type of

constituent-based information, the analyst can conclude that at damage initiation load, the

fibers are not close to being damaged with a max Fiber FI of 0.22. Also, with the flange

tips cracking, most of the flange exhibits matrix failure indices less than 0.1. The matrix

cracking that occurs is in the through-thickness direction, and the middle of the

composite flange does not experience much load.

Figure 11. Matrix Failure Indices at Damage Initiation


Figure 12. Fiber Failure Indices at Damage Initiation

4. CONCLUSIONS

Composite bonded joints historically have been difficult to analyze for structural integrity

and damage tolerance, and current aerospace programs have needed to depend heavily on

expensive experimental testing to catch design flaws.

An integrated Finite Element Analysis solution in the NASTRAN environment has been

proposed for aerospace engineers to efficiently and accurately optimize complex

composite structures. The combination of Cohesive Zone Modeling with 3D

micromechanics based damage degradation offers invaluable design insight into the

complex damage mechanism of composite bonded joints.

For the test cases described in this paper, DCB Mode I Fracture Test and Skin Stiffener

Tension Test, NASTRAN has proved to be fairly accurate in predicting critical

delamination propagation failure loads. The failure load for DCB Mode I Fracture Test is

predicted within 10 % of experiment, and the failure load for Skin Stiffener Tension Test

is predicted within 20 % of experiment.

Future research will focus on using this same coupling of in-plane and delamination

simulation to model advanced dynamic load events – Barely Visible Impact Damage

(BVID) and high speed ballistic impacts.


5. REFERENCES

1. Dávila, C.G., Jaunky, N., and Goswami, S. “Failure Criteria for FRP Laminates in

Plane Stress.” 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and

Materials Conference, Norfolk, Virginia, 7-10 April 2003.

2. De Morais, A.B., De Moura, M.F., Marques, A.T., De Castro, P.T., “Mode-I

Interlaminar Fracture of Carbon/Epoxy Cross-Ply Composites”. Composites Science

and Technology 2002;62:679-686.

3. Fertig, Ray S. III, “Bridging the Gap Between Physics And Large-Scale Structural

Analysis: A Novel Method For Fatigue Life Prediction Of Composites” 2009 SAMPE

Fall Technical Conference, Wichita, Kansas, 19-21 October 2009.

4. Krueger, R., M. K. Cvitkovich, T. K. O’Brien, and P. J. Minguet, “Testing and

Analysis of Composite Skin/Stringer Debonding under Multi-Axial Loading,”

Journal of Composite Materials, vol. 34, no. 15, pp. 1263-1300, 2000.

5. Puck, A. and Schürmann, H. “Failure analysis of FRP laminates by means of

physically based phenomenological models.” Composites Science and Technology,”

62:12-13 (2002) 1633-1662.

Cohesive Zone Modeling

Documents

Transcript of Cohesive Zone Modeling