et al. eco-hydrological modeling: a case study in upstream ...
(2009) Rapatunga Et Al. Modeling Delamnt in Compsts ...
description
Transcript of (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.
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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,
American Institute of Aeronautics and Astronautics
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
American Institute of Aeronautics and Astronautics
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
American Institute of Aeronautics and Astronautics
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
Damping Factor = 0.01
No Adaptive Stabilizations
Maximum Time Inc. = 0.01 s
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
Damping Factor = 0.01
No Adaptive Stabilizations
Maximum Time Inc. = 0.01 s
Even mesh over the entire area with
0.01‖x0.01‖ elements 0.68‖
Case 4
No Viscosity
Damping Factor = 0.01
No Adaptive Stabilizations
Maximum Time Inc. = 0.01 s
Even mesh over the entire area with
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
American Institute of Aeronautics and Astronautics
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‖
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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
American Institute of Aeronautics and Astronautics
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.
References
1 Krueger, R., ―The Virtual Crack Closure Technique: History, Approach and Applications‖ NASA/CR-2002-
211628, 2002.
2 Comanho, P.P. and Davila, C.G., ―Mixed-Mode Decohesion Finite Elements for the Simulation of
Delamination in Composite Materials‖ NASA/TM-2002-211737, 2002.
3 Dassault Systemes Simulia Corp., Abaqus Analysis User’s Manual, Providence, RI, 2008.
4 Jones, J.P., and Whittier, J.S., ―Waves at Flexibly Bonded Interfaces‖ Journal of Applied Mechanics 34, 1967,
pp. 905-909.
5 Bednarcyk, B. A., and Arnold, S. M., MAC/GMC 4.0 User’s Manual, NASA/TM—2002-212077, 2002.
6 Bednarcyk, B.A., and Arnold, S.M., ―A Framework for Performing Multiscale Stochastic Progressive Failure
Analysis of Composite Structures‖ NASA/TM-2007-214694, 2007.
7 Bednarcyk, B.A. and Arnold, S.M., ―Coupling Analytical Micromechanics with FEA for Progressive Failure
Modeling of Composite Structures‖ ASME Applied Mechanics and Materials Conference, June, Austin, Texas,
2007.
8 Aboudi, J., ―Damage in Composites – Modeling of Imperfect Bonding‖ Composites Science and Technology
28, 1987, pp. 103-128.
9 Sankurathri, A., Baxter, S. and Pindera, M. J., ―The Effect of Fiber Architecture on the Inelastic Response of
Metal Matrix Composites with Interfacial and Fiber Damage‖ in Damage and Interfacial Debonding in Composites,
G.Z. Voyiadjis and D.H. Allen (eds.), Elsevier, New York, 1996, pp. 235-257.
10 Achenbach, J.D. and Zhu, H., ―Effect of Interfacial Zone on Mechanical Behavior and Failure of Fiber-
Reinforced Composites,‖ Journal of the Mechanics and Physics of Solids, 37 (3), 1989, pp. 381-393.
11 Wilt, T.E. and Arnold, S.M., ―Micromechanics Analysis Code (MAC) User Guide: Version 2.0,‖ NASA TM
107290, 1996.
American Institute of Aeronautics and Astronautics
15
12
Bednarcyk, B.A. and Arnold, S.M., ―A New Local Debonding Model with Application to the Transverse
Tensile and Creep Behavior of Continuously Reinforced Titanium Composites‖ NASA/TM–2000-210029, 2000.
13 Bednarcyk, B.A. and Arnold, S.M., ―Transverse Tensile and Creep Modeling of Continuously Reinforced
Titanium Composites with Local Debonding‖ International Journal of Solids and Structures 39 (7), , 2002, pp.
1987-2017.
14 Needleman, A., ―A Continuum Model for Void Nucleation by Inclusion Debonding,‖ Journal of Applied
Mechanics 54, 1987, pp. 525-531.
15 McGee, J.D. and Herakovich, C.T., ―Micromechanics of Fiber/Matrix Debonding‖ Report AM-92-01, Applied
Mechanics Program, University of Virginia, Charlottesville, Va, 1992.
16 Eggleston, M.R., ―Testing, Modeling, and Analysis of Transverse Creep in SCS-6/Ti-6Al-4V Metal Matrix
Composites at 482 C‖ GE Research and Development Center Report 93CRD163, 1993.
17 Lissenden, C.J. and Herakovich, C.T., ―Interfacial Debonding in Titanium Matrix Composites‖ Mechanics of
Materials 22, 1996, pp. 279-290.
18 Lissenden, C.J., Herakovich, C.T., and Pindera, M. J., ―Inelastic Deformation of Titanium Matrix Composites
Under Multiaxial Loading‖ in Life Prediction Methodology for Titanium Matrix Composites, ASTM STP 1253, W.S.
Johnson, J.M Larson, and B.N. Cox (eds.), American Soc. for Testing and Mat., Philadelphia, 1996, pp. 257-277.
19 Robertson, D.D. and Mall, S., ―Micromechanical Analysis of Metal Matrix Composite Laminates with
Fiber/Matrix Interfacial Debonding‖ Composites Engineering 4 (12), 1994, pp. 1257-1274.
20 Benzeggagh, M. L. and Kenane, M., ―Measurement of Mixed-mode Delamination Fracture Toughness of
Unidirectional Glass/epoxy Composites with Mixed-mode Bending Apparatus,‖ Compos. Sci. Technol., Vol. 56,
No. 4, pp. 439-449, 1996.
21 Song, K., Davila, C. G., and Rose, C. A., ―Guidelines and Parameter Selection for the Simulation of
Progressive Delamination‖, Abaqus Users‘ Conference, Newport, RI, 2008.
22 Turon, A., Dávila, C. G., Camanho, P. P., and Costa, J., ―An Engineering Solution for Mesh Size Effects in the
Simulation of Delamination Using Cohesive Zone Models,‖ Engineering Fracture Mechanics, Vol. 74, No. 10, 2007,
pp. 1665-1682.
23 Turon, A., Costa, J., Camanho, P., and Maimi, P., Computational Methods in Applied Sciences: Mechanical
Response of Composite Materials, Springer, Berlin, 2008, Chapter 4.
24 Cox, B.N., Marshall, D.B., ―Concepts for Bridge Cracks in Fracture and Fatigue‖, Acta Metallurgica
Materialia, Vol. 42, 1994, pp. 341-363.