(2009) Rapatunga Et Al. Modeling Delamnt in Compsts

American Institute of Aeronautics and Astronautics

1

Modeling Delamination in Composites via Continuum

Interfacial Displacement Discontinuities

Vipul Ranatunga1

Miami University, Middletown, OH, 45042

Brett A. Bednarcyk2 and Steven M. Arnold

3

NASA Glenn Research Center, Cleveland, OH

A method for performing delamination simulation within finite element analyses, based

on continuum level interfacial displacement discontinuities, is presented. The method is

based on an existing model that enables exponential evolution of the interfacial compliance,

resulting in unloading of the tractions at the interface after delamination occurs. Results

compare the proposed approach with the cohesive element approach and Virtual Crack

Closure Techniques (VCCT) within the ABAQUS®

finite element software. Selection of

optimal parameters for analysis with ABAQUS®/Standard is discussed with respect to

Mode-I delamination using a double cantilever beam model. Great care must be taken with

respect to selection of the control parameters and mesh size in the cohesive and VCCT

approaches in order to obtain a reliable solution. In contrast, the proposed continuum

approach exhibits minimal sensitivity to its parameters, thus simplifying delamination

problem solution.

I. Introduction

Simulating the phenomenon of delamination in composites has received a great deal of attention in recent years.

Mainly in the context of finite element analyses, the goal has been to capture not only the onset (initiation) of

delamination, but also the progression. Towards this end, the Virtual Crack Closure Technique1 (VCCT) enables the

extraction of mode-specific stain energy release rates at a crack tip within a finite element model in order to evaluate

if a delamination-type crack will extend. Cohesive elements2 also enable evaluation of delamination-type crack

extension as well as control of the delamination propagation. Thus, cohesive elements are capable of simulating

more ductile delamination. Both of these state-of-the-art methods have been incorporated into the ABAQUS® finite

element software3 for the simulation of initiation and extension of delamination. However, these methods can suffer

from issues related to stability, long execution times, viscous regularization parameter dependence, mesh

dependence, and small time increment requirements.

The present paper explores an alternative approach to the delamination problem within the finite element

method, using a displacement discontinuity within a continuum material to simulate the delamination in a finite

element problem. The displacement discontinuity model is based on the original work of Jones and Whittier4, which

has been extended and, in the past, utilized within micromechanics models to simulate fiber-matrix interfacial

debonding in composite materials. A version of this interfacial model, that enables exponential evolution of the

effective interfacial compliance, and thus the unloading of interfacial stress during debonding or delamination, is

available in the MAC/GMC composite micromechanics software released by NASA (Bednarcyk and Arnold5). The

MAC/GMC software is callable by the ABAQUS® finite element software (through user subroutines) such that the

MAC/GMC micromechanics analysis is used to simulate the material behavior at the integration points in the

ABAQUS® finite element model (Bednarcyk and Arnold

6,7). Therefore, the interfacial model is available for use

within ABAQUS® for both monolithic and materials and for interfaces within composite repeating unit cells.

Herein the above-described interfacial discontinuity model is applied to model delamination within the

ABAQUS® finite element software. Results compare this approach with the standard cohesive element approach as

well as Virtual Crack Closure Technique (VCCT) available within ABAQUS®.

1 Assistant Professor, 4200 E. University Blvd., Middletown, OH, 45042, [email protected], AIAA Member.

2 Materials Research Engineer, Mechanics and Life Prediction Branch, 21000 Brokpark Rd., AIAA Senior Member.

3 Chief, Mechanics and Life Prediction Branch, 21000 Brokpark Rd., AIAA Member.

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2561

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

II. Displacement Discontinuity Model

The displacement discontinuity approach employed herein follows the basic form proposed by Jones and

Whittier4, in which a debonded interface is modeled by imposing discontinuity (i.e., ―jump‖) in a given

displacement component across the interface. This basic form can be written,

I I

j j ju R (1)

where I

ju is the discontinuity in displacement component j at interface I, j is the corresponding stress

component at interface I, and jR is a proportionality constant that may be thought of as a flexibility for the interface

(as it relates the interfacial displacement jump to the interfacial stress). This flexible interface model has been

applied extensively to model fiber-matrix interfacial debonding in composite materials. It was incorporated within

the method of cells micromechanics model by Aboudi8 and within the generalized method of cell by Sankurathri et

al.9. Achenbach and Zhu

10 added the condition to this flexible interface model that, when in compression, the

interface must behave as if well bonded (i.e., 0jR ). Wilt and Arnold11

added a finite bond strength to the

interface, I

jDB , such that the interface behaves as perfectly bonded until the interfacial stress reaches this

interfacial strength. Bednarcyk and Arnold12,13

further extended the Jones and Whittier4 flexible interface model to

admit time dependence in the jR parameter. By allowing this parameter to evolve with time, it is possible to enable

unloading and redistribution of the interfacial stress upon initiation of debonding or delamination. Considering the

components normal (n) and tangential (t) to a particular interface separately, the equations for the displacement

discontinuity components can be written,

;III I n

n n n n DBu R t (2)

;III I t

t t t t DBu R t (3)

where the time-dependence of the jR parameter, as well as the finite bond strength of the interface, has been

denoted. This form of the Jones and Whittier4 flexible interface model has been called the evolving compliant

interface (ECI) model in order to distinguish it from the previous form that incorporates a constant value for the jR

parameter. This ECI model is employed in the present investigation to examine delamination in the context of finite

element simulations.

Many forms for the time-dependence of the jR parameter, both implicit and explicit, were examined by

Bednarcyk and Arnold12

. It was determined that explicit exponential time-dependence provides the model with the

desired ability to unload interfacial stress, permitting redistribution of this stress. Further, the ECI model compares

well (in a qualitative sense) with several previous interfacial constitutive models that incorporate interfacial stress

unloading. The ECI model representation of the debonding parameter put forth by Bednarcyk and Arnold12,13

is,

exp 1DBj j DB

j

t tR t t t

(4)

where DBt is the time at which debonding occurs (for the particular interface), and j and j are parameters that

characterize the interfacial time-dependent behavior. The typical effective interfacial behavior represented by the

ECI model is plotted in Fig. 1. Until the interfacial stress reaches the bond strength DB , the interfacial

displacement is zero. Then, debonding occurs and the interfacial stress rises and then falls as the interface opens.


3

Also presented in Fig. 1 is a

comparison of the ECI model with two

previous interfacial constitutive models

that also admit unloading of the

interfacial stress. The well-known

model of Needleman14

(NI model) is one

such interfacial representation. While

this model does not incorporate a finite

bond strength, the effective interfacial

behavior is qualitatively similar to that of

the ECI model. Examples of studies that

have employed the NI model to simulate

the response debonding in composites

include McGee and Herakovich15

,

Eggleston16

, Lissenden and

Herakovich17

, and Lissenden et al.18

.

The fact that the NI model equations are

explicitly nonlinear in their relation

between interfacial tractions and

interfacial displacements can add

difficulty to the global solution when

incorporated within micromechanics

models.

The effective interfacial response of

the statistical interfacial failure (SIF)

model, developed by Robertson and

Mall19

, is plotted in Fig. 1 as well. Its qualitative similarity to the ECI model and NI model is obvious. Since the

SIF model, like the NI model, is nonlinear, Robertson and Mall19

incorporated a linear approximation of the SIF

model (also plotted in Fig. 1) within their employed micromechanics model (the method of cells).

III. Virtual Crack Closure Technique (VCCT)

The Virtual Crack Closure Technique uses the principle of Linear Elastic Fracture Mechanics (LEFM) to predict

the crack propagation taking place along a predefined surface. VCCT assumes that the strain energy released during

a crack extension is the same energy required to close the crack. The fracture criterion at any given node under

mixed-mode condition is expressed as3

(5)

where Geq is the equivalent strain energy release rate calculated at any given node, and GCeq is the critical equivalent

strain energy rate calculated based on a specific mode-mix criterion defined by the user. One such criterion is the

Benzeggagh-Kenane20

law (B-K law) given by3

(6)

where GIC and GIIC are the critical energy release rates associated with pure Mode I and Mode II fracture,

respectively. Power Law and Reeder Law are the two other formulae available in ABAQUS® to calculate equivalent

strain energy release rate, but only the B-K formula is used in this study.

IV. Cohesive Element Method

Cohesive elements in ABAQUS® are used to model bonded interfaces when the interface thinkness is negligibly

small. A traction-separation based constitutive response is defined by an adhesive material with number of material

parameters including interfacial penalty stiffness, interfacial strength, and the critical energy. Damage initiation

takes place as the degradation of the stiffness begins when the stresses and/or strains satisfy a specified damage

initiation criteria. In this study, quadratic nominal stress criterion, defined by Eq. (7) is used to initiate the damage.

When the nominal stress ratios reaches a value of one, the damage initiates and the criteria is expressed as

Figure 1. Comparison of the ECI model with the Needleman14

(1987) interface (NI) model and the statistical interfacial failure

(SIF) model developed by Robertson and Mall19

.

0

40

80

120

160

200

0.000 0.002 0.004 0.006 0.008 0.010

Inte

rfacia

l S

tress (

MP

a)

Normalized Interfacial Displacement

NI Model

SIF Model

SIF Model Linear Approximation

ECI Model

DB (SIF Approximation)

σDB (SIF,


4

(7)

where, indicates that a pure compressive stress state will not initiate damege3 above3. Damage evolution

represents the rate of stiffness degradation after a damage initiation criteria is reached. Damage evolution is defined

by the fracture energy, which is the area under traction-separation curve of the cohesive model. Components of

fracture energies associated with each mode of fracture (pure Mode I/Mode II/Mode III) must be specified as a

material property similar to VCCT criteria, and the B-K formulation is used for damage evolution. Damage

evolution in thecohesive model is captured by a scalar damage variable D. After the damage initiation, the damage

variable D monotonically evolves from 0 to 1, and the cohesive elements with a damage variable that has reached

this maximum are removed. This element removal process is more appropriate for modeling fracture and separation

of components similar to a DCB test. If the elimination process is not needed, cohesive elements can be left in the

model after completely damaged, which may be appropriate in cases where it is necessary to resist the

interpenetration of the surrounding components. Element removal has been enabled during the current DCB

simulations to indicate the crack propagation and the final crack tip.

V. Results and Discussion

To begin the comparison between the interfacial displacement discontinuity approach, VCCT approach, and the

cohesive element approach, a transversely isotropic double cantilever beam (DCB) specimen geometry is

considered. As shown in Fig. 2, the selected beam has a length (L) of 4‖, width (2W) of 0.12‖ and thickness of 0.3‖

as used by Song et al.21

. The beam includes two sub-laminates, each with a thickness of 0.06‖ and an initial crack

length (CL) of 1.15‖ between the two sub-laminates. Material properties of AS4/3501-621

given in Table 1 are used

in the plane-strain finite element models, considering unidirectional fibers aligned with the length direction (Y) of

the specimen.

Figure 2. Geometry of the DCB specimen with applied boundary conditions.

The middle strip of elements shown in blue in Fig. 2 in the case of the interfacial displacement discontinuity

approach, are represented with a MAC/GMC micromechanics repeating unit cell (RUC) through an ABAQUS® user

material. Usually such an RUC consists of a number of subcells and is used to represent a composite material6,7

.

However, in the present case, the material is treated as homogenized and transversely isotropic, so the RUC consists

of a single graphite/epoxy subcell with X-Z plane of isotropy, and interfacial debonding activated at the RUC

interface as indicated in Fig. 2. The same elastic transversely isotropic epoxy material properties are used within the

one subcell MAC/GMC RUC. It should be noted that MAC/GMC enforces periodicity conditions at the RUC

boundaries, thus the shown RUC represents a material that is continuous in the Y-direction, but allows an interfacial

displacement discontinuity, according to the ECI model described above, in the X-direction. Remainder of the

geometry (yellow) is meshed using reduced integration plane strain (CPE4R) elements with elastic orthotropic

material model with transverse isotropy as given in Table 1.

In the case of simulations with ABAQUS® cohesive elements, the geometry shown in Fig. 2 remains the same

with the exception of zero-thickness cohesive elements are been employed by meshing the blue region with cohesive

(COH2D4) elements and then editing the cohesive element nodes such that they all lie on the centerline of the blue

region. The remainder of the model is meshed using reduced integration plane strain (CPE4R) elements with the

L

CL

2W

Applied displacements +/- 0.05‖

Fixed BC Debond

interface

MAC/GMC RUC

Y

X


5

standard ABAQUS® elastic orthotropic model with transverse isotropy, representing the same graphite/epoxy

material discussed above.

Table 1. Mechanical properties of AS4/3501-6 including parameters used with traction-separation based

cohesive element simulations21

.

E1 / ksi E2 / ksi E3 / ksi G12 / ksi G13 / ksi G23 / ksi

21,500 1640 1640 870 870 522

12 = 13 23 GIC/ in-lb/in2 GIC/ in-lb/in2 GIC/ in-lb/in2

0.27 0.45 0.468 3.17 3.17 1.8

/ psi / psi / psi K1 / psi K2 / psi K3 / psi

3576 12600 12600 1.37e9 1.37e9 1.37e9

For simulations with VCCT, same geometry is used without the middle zone (in blue) and the two sub-laminates

have been created with a set of initially bonded nodes along the middle-seam. The same crack length of CL is

maintained and the entire geometry is meshed with reduced integration plane strain (CPE4R) elements and the

orthotropic material model with transverse isotropy is used.

A. Identification of Control Parameters for MAC/GMC Simulations

The parameters that control the interfacial behavior in the MAC/GMC approach are the normal and tangential

bond strengths, , , and the parameters , , , and appearing in Eq.(4). In the present case of a DCB

simulation, there is no shear at the interface, thus only the three normal parameters are relevant. Change of force-

displacement curve for various and values for two different values are shown in Fig. 3. These parameters

characterize an interface within a generalized method of cells repeating unit cell, and thus, their effects are

homogenized in to the effective nonlinear behavior of that repeating unit cell.

Change of while is held constant at 0.001 and

interfacial strength is at 28608 psi.

Change of while is held constant at 3.0 and

interfacial strength is at 44700 psi.

Figure 3. Change of force-response curve due to changing parameter values of and

The interfacial elements call MAC/GMC to determine this effective nonlinear behavior and the interfacial

response is completely encapsulated in the provided constitutive response. Therefore, no special effort is required

with respect to the finite element model control parameters and mesh size in order to obtain a reliable solution. As


6

will be shown below, the same cannot be said for the VCCT and cohesive approaches. Note, however, that like the

VCCT and cohesive element approaches, the MAC/GMC approach does exhibit mesh dependence in that a denser

mesh will lead to higher stress concentrations and earlier onset of delamination. In the case of MAC/GMC, though,

a converged solution for a given mesh size is much easier to achieve.

B. Identification of Control Parameters for VCCT Simulations

In order to simulate the progressive delamination growth using the VCCT approach, it is necessary to identify

the optimal values of the control parameters associated with convergence. Simulation of crack propagation is a

challenging problem and stabilization techniques are required to overcome the convergence difficulties.

Viscous regularization is one such parameter that introduces localized damping to overcome convergence

difficulties. Viscosity is added as a factor when specifying the debonding master and slave surfaces, and is often

considered as an iterative procedure to set a correct viscosity value. Once an appropriate value of viscosity is

selected to overcome any convergence issues, it is necessary to compare the energy consumed as the ‗damping

energy‘ with total strain energy of the model to ensure that the former is not too high in relation to the latter.

Automatic stabilization of an unstable quasi-static problem can also be achieved by the addition of volume-

proportional damping factor to the model. An automatic stabilization with a constant damping factor can be included

in any nonlinear analysis step in ABAQUS®. The damping factor can be calculated based on the dissipated energy

fraction or it can be directly specified as a reasonable estimate. If automatic stabilization is applied to a problem, it is

again necessary to ensure that the viscous damping energy is reasonably low compared to the total strain energy of

the model. Obtaining an optimal value for the damping factor may require a number of trials until the issues with

convergence are solved and the dissipated energy due to stabilization is sufficiently small compared to total strain

energy of the model. An adaptive stabilization scheme is also available in ABAQUS®, where the damping factor is

controlled by convergence history and the ratio of energy dissipated by viscous damping to the total strain energy. In

order to characterize the effects of these parameters, the following preliminary cases listed in Table 2 have been

studied with and without viscous regularization, while maintaining a constant damping factor without adaptive

stabilization. In these simulations, a constant displacement of 0.05‖ as shown in Fig. 2 is applied and the length of

the crack beyond the initial crack opening (CL in Fig. 2) is observed.

Table 2. Parameters Used for VCCT Simulations with Viscous Regularization and Automatic Stabilization

Damping Other Details

Extension of

the Crack

Length

Case 1

Viscosity = 0.2

Damping Factor = 0.01

No Adaptive Stabilizations

Maximum Time Inc. = 0.05 s

Even mesh over the entire area with

0.02‖x0.02‖ elements 0.73‖

Case 2

Viscosity = 0.2




Selected area near the crack meshed with

0.01‖x0.01‖ elements, remainder with

0.01‖x0.02‖ mesh

0.73‖

Case 3

Viscosity = 0.2





0.01‖x0.01‖ elements 0.68‖

Case 4

No Viscosity





0.01‖x0.01‖ elements 0.70‖

The different element sizes and mesh patterns used in these four cases are shown in Fig. 4. The first three cases have

used relatively high damping as evident from the comparison of viscous energy to total strain energy as shown in

Fig. 5 (only Case 3 is shown), while a much lower energy is consumed as the automatic stabilization energy (static


7

dissipation). For these four DCB simulations under displacement control, reaction force verses time is shown in Fig.

6 and the response displays considerable oscillations during delamination propagation.

a) Case 1

b) Case 2

c) Cases 3&4

Figure 4. Different mesh sizes used in VCCT simulation cases 1-4.

Figure 5. Comparison of strain energy, viscous

energy, and stabilization energy for VCCT Case 3

Figure 6. Effect of damping, stabilization, and

viscosity on reaction force for VCCT cases 1-4

In order to reduce the oscillations observed in the reaction force, a finer mesh has been utilized and another set of

simulations have been completed. When an analysis with small mesh is utilized with VCCT, it is necessary to

maintain a small time increment. In order to understand the optimal mesh size and maximum allowable time

increment, five cases have been studied with two different mesh sizes of 0.01‖x0.01‖ and 0.005‖x0.005‖ with four

different maximum allowable time increments as given in the Table 3.

Table 3. VCCT simulation studies with mesh size, maximum allowable time increment, and the resultant

crack length with and without viscosity

Mesh size Maximum allowable time

increment DTmax /(s)

Extension of crack length

without viscosity

Extension of crack length

with viscosity

Case 5 0.01‖x0.01‖ 0.005 0.27‖ Not available

Case 6 0.005‖x0.005‖ 0.01 0.185‖ 0.24‖

Case 7 0.005‖x0.005‖ 0.005 0.27‖ 0.20‖

Case 8 0.005‖x0.005‖ 0.001 0.27‖ 0.27‖

Case 9 0.005‖x0.005‖ 0.0005 Not available 0.27‖


8

Reaction forces predicted by the set of

simulations given in Table 3 are shown in Fig. 7.

Based on the simulations with an element size of

0.005‖, it is understood that if a small mesh size is

needed for VCCT, the time increment has to be

reduced in order to receive a smooth load-

displacement curve. Time increments of 0.001

seconds and 0.0005 seconds produced identical

results indicating a maximum time increment of

0.001 seconds is adequate for the selected mesh

size of 0.005 inches. Additionally, if a finer mesh

is not needed, a time increment of 0.005 could be

utilized as proven by 0.01‖x0.01‖ mesh size while

producing almost identical results according to

Fig. 7. The crack extension observed in cases 5, 7,

and 8 has been predicted consistently at 0.27‖

while the case 6 with small element size but

relatively high time increment produced a much

lower crack extension.

Also, these crack extension values have been

smaller than the values reported under Table 2

with large element size and high viscosity.

Therefore, mesh size as well as maximum

allowable time increment must be considered in

order to obtain smooth and accurate crack

propagation with VCCT.

VCCT simulation Case 5 reported in Table 3

maintained a viscosity of 0.1 compared to a a

higher viscosity value of 0.2 used with Case 3 in

Table 2. The comparison of viscous dissipation

energies with total strain energy for these two

cases is shown in Fig. 8. A relatively low ratio of

viscous dissipation energy to total strain energy is

reported in Case 5, while maintaining a lower

oscillation during crack propagation. In contrast,

Case 3 displays a higher viscous dissipation and

an oscillating response as reported in Fig. 6.

C. Identification of Control Parameters for Cohesive Element Simulations

For the cohesive element approach, selection of simulation parameters such as viscous regularization and

automatic stabilization must be done by trial and error. Comparison of the stabilization energy and viscous

dissipation energy to the overall strain energy of the model for validation of these parameters is also necessary.

Furthermore, selection of the interfacial penalty stiffness and interfacial strength values along the normal and shear

directions are challenging and discussed in detail by Turon et al.22,23

, and further extended by Song et al.21

for

simulations with ABAQUS®. During the present study, a number of finite element simulations have been conducted

to identify the appropriate element size and necessary minimal damping to avoid convergence issues while

accelerating the solution process. Element sizes and viscous damping values used in these simulations are listed in

Table 4. Figure 9 shows the energy dissipation due to viscous effects and automatic stabilization for a selected set of

simulations with an element size of 0.01‖. With an element size of 0.01 inches, the crack propagation was unable to

initiate without the use of viscous damping coefficient due to numerical convergence difficulties, as shown by Case

4 on the Table 4. But the same simulation with a smaller mesh size of 0.005 inches was completed without viscosity,

but took a longer time to complete the simulation.

Figure 7. Effect of mesh size and corresponding small time

increment required by VCCT cases 5-9

Figure 8. Comparison of viscous dissipation energy and

total strain energy with different viscosity values used in

VCCT simulations.


9

Table 4. Simulations with cohesive elements at different element sizes and damping.

Case Element

Size

Viscous

Damping

Crack

Extension Case

Element

Size

Viscous

Damping

Crack

Extension

Case 1 0.01‖ 1e-5 0.77‖ Case 6 0.005‖ 1e-5 0.77‖

Case 2 0.01‖ 1e-4 0.72‖ Case 7 0.005‖ 5e-4 0.57‖

Case 3 0.01‖ 5e-4 0.58‖ Case 8 0.005‖ 0 N/A

Case 4 0.01‖ 0 Failed Case 9 0.002‖ 5e-4 0.57‖

Case 5

COH3D8

by Song et

al.21

1e-6 0.77‖ Case 10 0.001‖ 5e-4 0.57‖

Simulations with a higher viscosity value tends to overpredict the response while producing a smooth load-

displacement (time) curve avoiding oscillations, and the solution process is faster. Even though the response reflects

small oscillations, a mesh size of 0.01 elements is sufficient enough to reach the mesh convergence, as verified by

the simulations with finer mesh sizes of 0.005‖, 0.001‖, and 0.002‖ in Fig. 10.

a) Dissipation of energy due to static stabilization

b) Dissipation of energy due to viscous damping

Figure 9. Dissipation of energy due to viscosity and stabilization in cohesive element simulation cases 1-3.

a). Reaction force for element size 0.01‖

b). Reaction force for element size 0.005‖

Figure 10. Reaction force with different element sizes and viscosity in selected cohesive element simulations.

According to the values reported on Table 4, the length of the crack extension has been consistantly predicted at

0.77‖ by all the models with low viscosity. When a significant portion of the energy is consumed as viscous energy,

length of the crack has been predicted at a lower value, irrespective of the element size used in these simulations.


10

According to these tabulated values

in Table 4, both the 0.01‖ mesh and

0.005‖ mesh with a viscosity of 1e-5

predicted the crack length at 0.77‖ while

a higher viscosity of 5e-4 predicted the

crack length at 0.57‖ and 0.58‖ for the

same element sizes. Therefore, low-

viscous models predicted the crack

length more accurately and consistantly

even though the time to reach a solution

is significantly longer.

Reaction forces predicted by models

with different element sizes with the

same viscosity of 5e-4 are compared

against a case with zero viscosity and a

case reported by Song et al.21

, as shown

in Fig. 11. According to these

comparisons, an element size of 0.01‖

with viscosity of 5e-5 could produce

results almost identical to much finer element sizes. Simulations reported by Song et al.21

have utilized a viscosity of

1e-6 and a three-dimansional DCB model with COH3D8 elements. Smooth reaction force as observed in the cases

with two-dimansional geometry can be achived by increasing the viscosity in Song‘s three-dimensional model.

Additionally, time taken for two dimensional simulations with higher viscosity has been considerably lower

compared to a three-dimensional model with lower viscosity. Therefore, based on these observations it is concluded

that, for the selected set of material data on Table 1, the element size of 0.01‖ is adequate for simulations with

cohesive elements.

According to Turon et al.22

, the length of the cohesive zone under Mode-I loading can be expressed in terms of

material properties as

(8)

where M is a factor that depends on the constitutive laws of the material, and various authors have proposed

different values23

. E2 is the modulus of elasticity for the material in transverse direction and is the nominal

strength of the material when the deformation is purely normal to the interface. The cohesive zone, as observed with

both 0.01‖ mesh and 0.005‖ mesh in Fig. 12, has a length of about 0.04 inches. Based on the material data used, the

quantity (M·E2) is found as 1,052,492 psi. Since the transverse modulus E2 is 1640 ksi, a value for M is found as

0.64, which is reasonably close to as predicted by Cox and Marshall24

.

According to these simulations with material data reported in Table 1 (and shown in Fig. 12a), an element size of

0.01‖ could produce more than four elements along the cohesive zone. Therefore, an element size of 0.01‖ satisfies

the recommendations of Turon et al.22

of having at least three elements in the cohesive zone to represent the fracture

energy accurately.

D. Comparison of VCCT, Cohesive, and MAC/GMC predictions

In order to compare the reaction forces predicted by VCCT, cohesive elements, and MAC/GMC, following set of

cases has been selected based on the resullts reported in the previous sections. Simulations with VCCT used a mesh

size of 0.01 inches, a maximum allowable time increment of 0.005 seconds, and zero viscosity, as verified by

simulations reported in Fig. 7. In the case of cohesive elements, same element size of 0.01 inches, a maximum

allowable time increment of 0.01 seconds, and a viscosity of 5e-5 have been used for optimal results (see Fig. 11).

Figure 11. Comparison of reaction forces with changing element

size and viscosity for selected cohesive element simulations.


11

a) Mesh with element size of 0.01‖ , Ne=4 b) Mesh with element size of 0.005‖, Ne=8

Figure 12. Deformed mesh with cohesive zone before initiating the crack propagation (deformation scale

factor of 20 is used to magnify the deformed cohesive zone for better visualization).

All the other material parameters have been kept the same as used in previous cases. Comparison of reaction

forces predicted by VCCT and cohesive elements are shown in Fig. 13. With the same material data used in both

models, cohesive model displays a significantly lower reaction force compared to VCCT predictions. Even though

the mesh size, lowest possible viscosity, element size, and interface strengths have been chosen based on previous

simulation results to represent cohesive zone accurately, the cohesive model predictions are considerably lower to

VCCT values.

In order to find a more

representative value for , stresses at

the integration points next to debonding

front on the VCCT model have been

considered as shown in Fig. 14(a), and a

value of 44.7 ksi has been selected as a

starting value. With this modified ,

the calculated value for the length of the

cohesive zone using Eq. (8) is 0.0003‖,

which would require at most a 0.0001‖

element length along the cohesive zone

to represent the fracture energy

accurately. With such a small element

size along the cohesive zone, the

cohesive model may become highly

time-consuming and computationally

expensive. Therefore, a number of

simulations have been conducted to find

the appropriate material parameters that

could lower the overall reaction force

predicted by VCCT, while increasing the force response predicted by cohesive model. For both VCCT and cohesive

models, GIC has been reduced to 0.185 lb-in/in2 from the original 0.486 lb-in/in

2 to reduce the overall force needed.

The most influencial parameter that can be changed for cohesive model is the nominal strength . But according to

Eq. (8), the mesh has to be refined if the values of GIC and are changed in order to allow at least three elements

along the cohesive zone.

With the modified critical strain energy value of 0.185 lb-in/in2, a new simulation with VCCT model has been

performed and the stress distribution at the integration points along the delamination front for this model is shown in

Fig. 14 (b). Based on these stresses, a possible interfacial strength value of 28.6 ksi has been chosen as the nominal

strength ( ) for Eq. (8), and found the required element size of 0.002‖ in order to satisfy the minimum number of

elements according to Eq. (8).

Figure 13. Comparison of reaction forces predicted by VCCT and

cohesive models.


12

a) Stress distribution along the delamination front in the

VCCT model with Gc=0.468 lb-in/in2

b) Stress distribution along the delamination front in the

VCCT model with adjusted Gc=0.185 lb-in/in2

Figure 14. Distribution of stress along the crack-front observed with VCCT model.

Based on these adjusted material parameters (critical strain energy value of 0.185 lb-in/in2 and interfacial

strength value of 28.6 ksi), comparison of reaction forces predicted by the three models are shown in Fig. 15. For

the cohesive model, both critical strain energy rate as well as interfacial strength have been changed while the

VCCT model used only the modified critical strain energy. For the MAC/GMC model, the same interfacial strength

value of 28.6 ksi has been used as the bond strength as expressed in Eq. (2). In addition to the , two other

parameters critical to interfacial time-dependent behavior of the ECI model are the and as reported in Eq. (4).

In order to create an interfacial debonding behavior similar to cohesive and VCCT approaches, a value of 0.00005 is

selected for while 3.0 is selected for the parameter .

External work and strain energy of cohesive, VCCT and MAC/GMC models are shown in Fig. 16. The stress

distributions for these three models along the crack front are shown in Fig. 17. As expected with cohesive modeling,

the local stress distribution near the crack-tip for the cohesive model is relatively high compared to VCCT and

MAC/GMC models. Both VCCT and MAC/GMC models use the same element size of 0.01‖ over the entire body of

the model. Regardless of the disagreement of local stresses near the crack-tip, overall stress distribution shows

similarity while the overall response of these three models remain consistant as evidenced from reaction forces

shown in Fig. 15.

Figure 15. Reaction forces predicted by VCCT,

cohesive model, and MAC/GMC using adjusted Gc and

.

Figure 16. Comparison of work and energy for

VCCT, MAC/GMC and cohesive model.


13

a) Cohesive Model b) VCCT Model c) MAC/GMC Model

Figure 17. Comparison of stress component along the fiber (vertical) direction for cohesive, VCCT, and

MAC/GMC models at the beginning of crack propagation for a similar width of the geometry.

The time taken to complete the execution of each model

is shown in Table 5. A significant advantage with execution

time is displayed by MAC/GMC model over the VCCT and

cohesive models. The cohesive model with maximum

allowable time of 0.01 seconds took ten times more CPU

time compared to MAC/GMC simulation with 0.075 second

constant time increment. If a large maximum allowable time

increment is used with the cohesive models, the solution

process becomes unstable due to severe nonlinearities at the verge of a crack initiation. Abaqus/Standard offers a set

of stabilization mechanisms to handle nonlinear problems by allowing the user to specify number of solution control

parameters such as number of equilibrium iterations (I0) and number of equilibrium iterations after which the

logarithmic rate of convergence check begins (IR). Even with current increment of 0.01 seconds for the cohesive

model, these convergence parameters have to be modified in order to stabilize the solution process.

E. VCCT and MAC/GMC Models

Comparison of reaction force for

the original material data with GIC as

0.486 lb-in/in2 and a debond strength of

46108 psi, which is slightly less than

the values observed from VCCT model

at the integration points (44.7 ksi), have

been shown for VCCT and MAC/GMC

models in Fig. 18. An excellent

agreement between two models is

observed for this slightly adjusted

debonding strength value in the

MAC/GMC model. The extended crack

length predicted by VCCT is 0.28‖

while the crack length predicted by

MAC/GMC is 0.30‖, indicating a close

agreement between the values predicted

by the two models.

Table 5. Total CPU time take for completing a

simulation with MAC/GMC, VCCT, and

cohesive models

Model Total CPU Time/s

MAC/GMC Model 1423

VCCT Model 3241

Cohesive Model 14, 806

Figure 18. Comparison of reaction forces predicted by VCCT and

MAC/GMC with original material data.

MAC/GMC Model


14

VI. Conclusions and Future Work

An approach to finite element simulation of composite delamination based on a continuum level interfacial

displacement discontinuities has been compared to the VCCT and cohesive element approaches available in the

ABAQUS®

finite element software. It has been demonstrated that the proposed continuum approach, available

within NASA‘s MAC/GMC software, is considerably more robust and simpler to apply than VCCT and cohesive

elements, while producing similar results. A simple plane strain DCB delamination simulation was used for the

comparison, yet this still required a great deal of operator effort and execution time to obtain reliable results in the

case of VCCT and cohesive elements. In contrast, because the continuum interfacial approach embeds the

delaminating interface within the effective material constitutive model operating within the interfacial elements, the

approach is far less sensitive to the finite element model control parameters.

Future work will seek to expand the application of the proposed interfacial displacement discontinuity approach

to mixed mode delamination problems and problems in which the path of damage, delamination, or cracking is not

known a priori. The proposed approach is ideal for such problems as interfaces can be placed within the material of

any element with very low overhead and effort. Delamination problems can also easily be combined with micro

scale continuum damage effects (applicable to the fiber/matrix constituents) in MAC/GMC to simulate more

realistic, multiple mode failure in composites.

Acknowledgments

First authors would like to acknowledge the support received through Miami University Shoupp award for

initiating research collaborations with NASA-Glenn Research Center. Also, the software and hardware support

received from Research Computing Support Group at Miami University is very much appreciated.

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Documents

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