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Paper Number:

Title: A stabilized ﬁnite element method for modeling mixed-mode delamination of compos-

ites

Authors: Gourab Ghosh1

Chandrasekhar Annavarapu 2

Stephen Jim´

enez1

Ravindra Duddu1

1Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee.

2Atmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, Livermore, California.

ABSTRACT

Delamination of composite materials is commonly modeled using intrinsic cohesive zone models

(CZMs), which are generally incorporated into the standard ﬁnite element (FE) method through a

zero-thickness interface (cohesive) element; however, intrinsic CZMs exhibit numerical instabili-

ties when the cohesive stiffness parameters is assumed to be large relative to the elastic stiffness of

the composite material. To address this numerical instability issue, we propose a stabilized ﬁnite

element method by combining the traditional penalty method with the Nitsche’s method that is

equally effective for any speciﬁed initial stiffness of the cohesive (traction-separation) law. The

key advantage of the proposed method is that it generalizes the Nitsche’s method to any traction-

separation law with arbitrary large values of initial stiffness and provides a uniﬁed way to treat

cohesive fracture problems in a variationally consistent and stable manner. We implemented the

stabilized method in the commercial ﬁnite element software Abaqus via the user element subrou-

tine and simulated benchmark tests for mode I and mixed-mode delamination in isotropic materials

to establish the viability of the approach. Ongoing work is aimed at extending the method to model

delamination in transversely isotropic laminated composites.

INTRODUCTION

Cohesive zone models (CZMs) were ﬁrst proposed by [1],[2] and are widely used for modeling

fracture and fatigue crack growth. The key advantages of CZMs are that they do not require the

presence of a initial (or pre-existing) crack and account for the ﬁnite size of the fracture pro-

cess zone (FPZ) at the crack tip, unlike the linear elastic fracture mechanics approaches. There-

fore, CZMs are suited for modeling delamination of composite materials, where the FPZ may be

greater than the characteristic length scales. One of the ways to implement cohesive zone mod-

els in the ﬁnite element (FE) framework is using zero-thickness interface elements, wherein the

interface (or cohesive) elements are placed along the probable crack path and their constitutive

(traction-separation) behavior is deﬁned by so-called “cohesive laws”. There are two classes of

cohesive laws, namely intrinsic traction-separation laws or initially elastic cohesive laws, and ex-

trinsic traction-separation laws or initially rigid cohesive laws (see Figure 1). The major difference

between the intrinsic and the extrinsic traction separation laws is the presence of the initial elastic

curve [3]. In the case of intrinsic laws, the traction is assumed to gradually increase with the sep-

aration and after reaching a maximum value, it decreases monotonically till the separation reaches

the ultimate separation value (i.e., where complete de-cohesion is assumed to occur). Whereas, in

extrinsic laws, it is assumed that only after traction reaches a ﬁnite cohesive strength, the separation

starts and the traction decreases monotonically with the increase in separation.

(a) Intrinsic traction-separation law (b) Extrinsic traction-separation law

Figure 1: Intrinsic and extrinsic traction separation laws

The numerical implementation of extrinsic traction separation law is challenging [4] because

advanced data structures are required to store the ﬁnite element discretization. Furthermore, paral-

lelization of ﬁnite element codes in conjugation with extrinsic laws is not a trivial task due to the

change of the mesh topology with the advancement of cracks. Although researchers have proposed

several ways (e.g., topology-based data structures [5]; scalable parallel implementation [6]) to alle-

viate the above-mentioned challenges, the complexities in its implementation are key deterrent for

its widespread use. Intrinsic cohesive laws, on the other hand are easier to implement in a FE code;

however, they suffer from the artiﬁcial compliance [7] due to elasticity of the cohesive law. This

issue can be solved to some extent by restricting the time step to an extremely small value [8], but

this may result in an impractically high computation cost. Another approach to solve this problem

is to use very high initial elastic slope for the intrinsic law [7], but that causes an ill-conditioning of

the tangent stiffness matrices. Thus, following a conventional approach to treat above-mentioned

problems within a ﬁnite element framework poses numerical challenges, and has motivated re-

searchers to come up with novel approaches to overcome the disadvantages posed by the intrinsic

traction-separation laws. For example, a hybrid discontinuous Galerkin (dG) and extrinsic traction-

separation law was proposed by [9]; in this approach, interface elements are inserted between bulk

elements at the beginning of the analysis and continuity during the elastic regime is maintained

in a weak manner by a dG formulation, but upon onset of failure the extrinsic traction separation

law replaces the dG formulation. However, it has been recognized [4] that implementing extrinsic

traction-separation law in conjunction with dG method in commercial software (e.g., Abaqus) is

extremely difﬁcult. Thus, this hybrid approach is effective in overcoming some disadvantages of

the intrinsic CZMs, but it is not easy to implement them in commercial codes.

Directed at achieving the same goal of removing the artiﬁcial compliance issue associated

with the intrinsic laws, a continuum approach was proposed by [10]. Two of the major types of

the continuum approaches for enforcing continuity weakly at the interface are: Lagrange multi-

plier methods and discontinuous Galerkin (dG) methods. The work of [10] was motivated by an

augmented Lagrange multiplier-based mixed interface element approach proposed by [11]. In La-

grange multiplier method, it is challenging to construct a stable Lagrange multiplier space. It has

been shown by [12] that for embedded ﬁnite element methods it is difﬁcult to ﬁnd a stable Lagrange

multiplier. A stable choice of Lagrange multipliers is important from the standpoint of removing

artiﬁcial oscillations in the interfacial traction. The second key alternative, that is, dG method

originated from the Nitsche’s method [13]. The Nitsche’s method can be seen as a variationally

consistent penalty method. In the penalty method (introduced in [14]), Dirichlet constraint at the

interface is enforced by introducing a spring at the interface, and a better approximation for the

Dirichlet constraint can be obtained with an increase in penalty parameter (i.e., the stiffness of the

spring/ slope of the traction separation law). The interfacial constraints are achieved exactly when

the stiffness approaches to inﬁnity, but this implies that the method is variationally inconsistent.

Also, a very high value of stiffness results in an ill-conditioned system of equations. Whereas,

Nitsche’s method not only eliminates the instability issues (evident from [15]) related with the

standard penalty methods by adding consistency terms ([16]), but also yields a well-conditioned

system of discrete equations if the method parameters are chosen appropriately. This method has

been used for solving a wide class of interface problems in an efﬁcient way ([17], [18]). A compre-

hensive review of the classical Nitsche’s method and it’s application to interface problems can be

found in [19]. Recently a precise deﬁnition of the weights and a closed form analytical expression

of the stabilization parameter was proposed by [18]; this weighted Nitsche method provides much

accurate results in comparison to the standard Nitsche method. Thus, it is evident that although

the Nitsche’s method-based consistent penalty and dG approaches are well established for treating

embedded interface problems, relatively little work has been done in extending these methods to

generalized intrinsic cohesive laws.

In this article, we will discuss a stabilized ﬁnite element method for alleviating the artiﬁcial

compliance issue inherent to the intrinsic cohesive law with very high value of cohesive stiff-

ness. The key advantage of the proposed method is that it generalizes the Nitsche’s method to any

traction-separation law with arbitrary large values of initial stiffness and provides a uniﬁed way to

treat cohesive fracture problems in a variationally consistent and stable manner. The rest of this

article is organized as follows: in the Section , we introduce the model problem and the associated

variational formulation. In section 3, we discuss the spatial discretization followed by numerical

examples demonstrating the accuracy and efﬁcacy of the approach in Section 4. Finally, the last

section provides a summary and some concluding remarks.

MODEL PROBLEM AND VARIATIONAL FORMULATION

In this section, we present details of the stabilized Nitsche formulation for treating general cohesive

laws. We ﬁrst explain the notation for variables and the problem domain, followed by a discus-

sion of the strong form and the weak form of the governing equations for linear elastostatics and

cohesive fracture. We present the model equations in indicial notation with Einstein’s summation

convention and reserve the right superscript for exponents (italicized) or descriptors (unitalicized).

Domain Description

We deﬁne a domain Ω⊂R2, which is partitioned into two non-overlapping bulk domains Ωm

(where m=1,2 and Ω=Ω1∪Ω2) separated by an embedded crack surface Γ∗(Fig. 2). Both the

bulk domains consist of homogeneous, linear, isotropic, and elastic material. Dirichlet and Neu-

mann boundary conditions are deﬁned on the parts of the domain boundary (Γ≡∂Ω) excluding

the embedded interface boundary (Γ∗). The parts of the boundary where Dirichlet and Neumann

conditions are deﬁned are denoted as Γm

dand Γm

n, respectively. The unit normal to the boundary of

each subdomain, nm, points outwards from the domain Ωm.

Figure 2: Domains Ω1and Ω2separated by a shared boundary Γ∗. The Dirichlet boundaries (Γ1

d,Γ2

d) and the Neumann

boundaries (Γ1

n,Γ2

n) are as shown. The complementary part of the boundary is traction free. The normal to the boundary

of each subdomain, nm, points outwards from the domain Ωmas shown

Strong Form

The governing equations of equilibrium without body force in each of the domains are given by:

σm

i j,j=0 in Ωm,(1)

um

i=¯um

ion Γm

d,(2)

σm

i j nm

j=hm

ion Γm

n,(3)

where σm

i j and um

idenote the components of the stress and displacement ﬁelds in the domain Ωm,

respectively, and nm

jdenotes the components of the unit outward normal. On the Dirichlet portion

of the boundary, the displacement is ﬁxed to the prescribed ﬁeld ¯um

iand on the Neumann portion

of the boundary, prescribed traction is denoted by hm

i. The traction ﬁeld tm

ion the embedded crack

interface can be obtained by projecting the stress from each domain and is related to the interface

separation [[ui]] = u2

i−u1

iaccording to an assumed traction-separation law, that is,

tm

i=σm

i j nm

j=f([[ui]]) on Γ∗,(4)

To represent the mode I and mode II cohesive fracture behavior in two dimensions, we use the

normal and tangential coordinate system (nm,τm). Accordingly, the tangential (tτ) and normal (tn)

components of the traction vector are deﬁned by:

tm

i=tnnm

i+tττm

i,(5)

um

i=unnm

i+uττm

i,(6)

tn=−αn[[un]],(7)

tτ=−ατ[[uτ]],(8)

where αnand ατrepresents the cohesive stiffness in the normal and the tangential direction, re-

spectively. The force balance on the interface is given by:

t1

i+t2

i=0onΓ∗.(9)

Equations (7) and (8) can used to represent general cohesive laws by using the damage mechanics

framework described in [20, 21, 22, 23]; herein, we consider a bilinear intrinsic traction separation

law. The constitutive equations for the linear, elastic, isotropic bulk domains are given by:

σm

i j =Cm

i jkl εm

kl =Cm

i jkl um

(k,l)in Ωm,(10)

where Cm

i jkl ,εm

kl and um

(k,l)denote the fourth-order elasticity tensor, the second-order strain tensor,

and the symmetric gradient of the displacement ﬁeld, respectively.

Weak Form

In this section, we follow the weighted residual approach [18] and deﬁne the solution spaces U=

U1×U2and trial spaces W=W1×W2respectively, such that:

Um={um∈H1(Ωm),um=¯um

ion Γm

d},(11)

Wm={wm∈H1(Ωm),wm=0onΓm

d}.(12)

Following the standard ﬁnite element approach, we get

ZΩ

wm

(i,j)σm

i j dΩ−∑

mZΓ∗

(wm

ntm

n+wm

τtm

τ)dΓ=ZΓn

wm

ihm

idΓ.(13)

By substituting expressions for tnand tτfrom Equations (7) and (8) in the above equation, we get

ZΩ

wm

(i,j)σm

i j dΩ+ZΓ∗

([[wn]]αn[[un]] + [[wτ]]ατ[[uτ]])dΓ=ZΓn

wm

ihm

idΓ.(14)

Equation (14) is the standard weak form for the penalty method. It is evident that as αn→∞

and ατ→∞, solving Equation (14) is an ill-posed problem. To alleviate the ill-posedness of the

weak form, we adopt a Nitsche’s method-based stabilized FE approach. We begin by re-scaling the

normal and tangential traction components and followed by some algebraic manipulations arrive

at the ﬁnal stabilized weak form as given by

ZΩ

w(i,j)σi jdΩ−ZΓ∗

(( αn

αn+βn

)[[wn]]pγ+ ( ατ

ατ+βτ

)[[wτ]] fγ)dΓ

+ZΓ∗

(αnβn

αn+βn

[[wn]][[un]] + ατβτ

ατ+βτ

[[wτ]][[uτ]])dΓ=ZΓn

wihidΓ,

(15)

where pγand fγare the weighted interfacial pressure and shear, respectively, as deﬁned by

pγ=n2

j<σi j >γn2

j;fγ=τ2

j<σi j >γn2

j; (16)

and <σi j >γ=γ1σ1

i j +γ2σ2

i j represents a weighted average of stress across the interface. The

weights γ1and γ2are positive and satisfy the condition γ1+γ2=1. Note that Equation (15) is well-

deﬁned even as αnor ατ→∞. On the other hand, if βnand βτ→∞, we get a standard penalty

weak-form deﬁned in Equation (14).

SPATIAL DISCRETIZATION

We discretized the domain Ωinto bilinear plane strain quadrilateral bulk or continuum elements

and introduce zero-thickness cohesive elements at the interface boundary. The approximated dis-

placement ﬁeld is deﬁned as

um=Nmam,m=1,2,(17)

where Nmis the element shape function matrix and amdenotes the element vector containing the

displacement degrees of freedom (DOFs). The displacement jump at the interface [[u]] is given by

[[u]] = u1−u2=N1a1−N2a2,(18)

where am(m=1,2) denotes the nodal displacement vector of the continuum subdomain Ωm(m=1,2)

adjacent to Γ∗. The shape function matrix Nmcan be represented as

Nm=Nm

10Nm

20Nm

30Nm

40

0Nm

10Nm

20Nm

30Nm

4,(19)

where Nm

J(J=1,2,3,4) are the shape functions of four-noded quadrilateral bulk elements. Af-

ter we introduce the discretized form for the approximation spaces into the variational form in

Equation (13), it leads us to the following discrete equation of equilibrium in the residual form

R(u) =fext −(fb

int +fc

int).(20)

Note that the residual vector contributes to the RHS term in the Abaqus UEL subroutine. Neglect-

ing body forces, fext is obtained by assembling the element contributions from traction boundary

conditions on the Neumann boundary

fext =∑

eZΓm

ne

NmThmdΓefor m=1,2.(21)

where ∑

e

indicates the matrix (or vector) assembly of the global system from the element matrices

(or vectors) in entire computational domain. The internal bulk force vector fb

int is assembled as

fb

int =∑

eZΩm

e

BmTCmBmum

edΩefor m=1,2,(22)

where Bmis the strain-displacement relationship matrix, and Cmis the elasticity matrix in Voigt

notation. The strain-displacement relationship matrix is given by

Bm=

Nm

1,10Nm

2,10Nm

3,10Nm

4,10

0Nm

1,20Nm

2,20Nm

3,20Nm

4,2

Nm

1,2Nm

1,1Nm

2,2Nm

2,1Nm

3,2Nm

3,1Nm

4,2Nm

4,1

,(23)

where Nm

J,1denotes the derivative of the shape function Nm

Jwith respect to x1in Ωm. The cohesive

element contribution to the internal force vector fc

int is assembled as

fc

int =fstabilized +fconsistency =ZΓ∗e

NTSt dΓe+ZΓ∗e

NT(I−S)TσγdΓe(24)

where Sis the 2 ×2 stabilization matrix, Iis the 2 ×2 identity matrix, t= [tτ,tn]Tis the cohesive

traction vector that is a function of the displacement jump vector [[u]],Tis the stress transformation

matrix and σγis the weighted stress vector in Voigt notation for in-plane stress components.

Let us now deﬁne the element tangent matrix Kthat contributes to the AMATRX term in the

Abaqus UEL subroutine. The element tangent matrix is obtained by assembling the contributions

of the bulk and cohesive tangent matrices, that is,

K=−∂R

∂u=∂fb

int

∂u+∂fc

int

∂u=Kb+Kc,(25)

where,

Kb=∑

eZΩm

e

BTCmBdΩefor m=1,2 (26)

and

Kc=Kstabilized +Kconsistency =∑

eZΓ∗e

[[N]]TSM[[N]] dΓe+∑

eZΓ∗e

[[N]]T(I−S)TCBγdΓe.(27)

In the above equation: the jump in shape function matrix [[N]] is represented as follows

[[N]] = −N10−N20N20N10

0−N10−N20N20N1,(28)

where Nm(m=1,2) represents the linear shape functions evaluated at the interface Gauss points

(GPs); the weighted shape function gradient matrix Bγis given by

Bγ=γB+(1−γ)B−,(29)

where B+and B−are matrices containing gradient of the shape functions calculated at the interface

GPs from the neighboring bulk domains of the cohesive element; the stress transformation matrix

Tis deﬁned as

T=−CS CS C2−S2

S2C2−2CS ,(30)

where Cand Srepresent cosθand sin θ, respectively, and θis the inclination of the cohesive

element with the x1coordinate direction; the cohesive stiffness matrix Mand the stabilization

matrix Sare deﬁned as

M="∂tτ

∂uτ

∂tτ

∂un

∂tn

∂uτ

∂tn

∂un#;S="βt

M11+βt

βt

M12+βt

βn

M21+βn

βn

M22+βn#.(31)

Figure 3 gives an overview of the implementation of the proposed approach in commercial FE

software Abaqus. The presence of weighted average of stress and shape function derivatives across

the interface implies that the computation of cohesive element tangent matrices and the residual

vectors depends on the displacement shape functions associated with their nodes as well as those

in the two neighboring bulk elements. In our scheme, we calculate these quantities at the interface

GPs in the UELMAT subroutine for the bulk elements, and then pass them to the UEL subroutine

for the cohesive elements using global modules. These set of calculations are done separate from

the standard loop over bulk GPs for assembling the bulk stiffness matrix and right hand side (RHS)

vector. The element tangent matrix is unsymmetric and Kconsistency has the dimension of 8×16

(the number of rows correspond to the interfacial degrees of freedoms (DoFs) and the number of

columns correspond to the interfacial and adjacent bulk element DoFs) for the setup (Fig. 4). As

a cohesive interface element has only four nodes associated with it and two DoFs deﬁned at each

of them, it can only assemble a stiffness matrix of size 8×8. To resolve this implementation issue,

we use “dummy” elements (elements IV-VII in Fig. 4) in the UEL subroutine that facilitate the

partition and assembly of the stiffness matrix in Abaqus.

NUMERICAL EXAMPLES

In this section, we ﬁrst verify the proposed stabilized ﬁnite element formulation, by performing

uniaxial tension test and comparing numerically obtained solution with the analytical solution.

Next, we simulate mode I and mixed mode bending tests using the proposed methodology to

demonstrate the its applicability to large scale complex problems. All simulations are performed in

two dimensions assuming plane strain conditions. We assume linear, elastic, and isotropic material

Figure 3: Abaqus ﬂowchart showing interaction between UELMAT and UEL

1 2

3 4

5 6

7 8

3 4

7 8

IV

5 6

7 8

V

1 2

3 4

VI

1 2

5 6

VII

1 2

3 4

I

5 6

7 8

II

3 4

5 6

III

Figure 4: Bulk (I and II), cohesive (III), and dummy (IV-VII) elements

behavior for the simulations. Table I shows a summary of the material properties considered for

each of the test cases.

Uniaxial Tension Test

To perform the uniaxial tension test, a vertical displacement (δ) is applied at the upper two nodes

(i.e., nodes 7 and 8) of the model (leftmost diagram in Fig. 4) , and the bottom nodes (i.e., nodes

1 and 2) are constrained using pinned/roller boundary conditions. The computational domain con-

sists of two bilinear quadrilateral plane strain elements (CPE4), and a user deﬁned zero thickness

cohesive spring element at the interface. To establish the accuracy of the formulation, the differ-

ence between the computed values of the vertical displacements at the middle nodes (i.e., nodes

3, 4, 5, 6) and the corresponding theoretical values for the inﬁnitely stiff case is evaluated. Our

numerical results reported in Table II indicate that the proposed formulation guarantees stability

and accuracy (close to machine precision) for large values of cohesive stiffness (i.e., for cohesive

Table I: SUMMARY OF MATERIAL PROPERTIES IN EACH NUMERICAL EXAMPLE

Material parameter E νGIC GIIC σmax τmax

(units) (N/mm2) (N/mm) (N/mm) (N/mm2) (N/mm2)

Patch test 1.00E+05 0.35 0.28 - 5.7 -

Mode I 1.00E+05 0.35 0.28 - 5.7 -

Mode II 1.00E+05 0.35 - 4 - 57

Mixed mode 1.00E+05 0.35 4 4 57 57

Table II: ACCURACY OF THE FORMULATION FOR THE UNIAXIAL TENSION TEST

Cohesive Stiffness (αn) (N/mm) % Error

1.00E+03 96.61

1.00E+05 22.17

1.00E+08 2.84E-02

1.00E+12 2.85E-06

1.00E+15 2.85E-09

1.00E+20 1.39E-14

1.00E+100 9.71E-14

stiffness up to 10100 N/mm). As expected the error between the computed and theoretical values

decreases with an increase in the cohesive stiffness.

Mode I: Double Cantilever Beam (DCB) Test

Figure 5: Geometry and boundary conditions for the mode I double cantilever beam (DCB) test. The dimensions are:

L=100mm,H=4mmanda0= 25 mm

Fig. 5 shows the set up of the double cantilever beam test. Fixed boundary condition is ap-

plied at the right end of the beam. A displacement is applied at the upper and lower nodes at the

left end to initiate mode I delamination and the corresponding load is recorded. The simulation

is displacement controlled so as to capture the softening portion of the load-displacement curve.

The computational mesh consists of bilinear quadrilateral elements (CPE4) and user-deﬁned zero

thickness cohesive elements. The load vs upper left node displacement curves for the isotropic ma-

terial from the traditional CZM approach, proposed stabilized approach, and the analytical LEFM

solution ([24]) are shown in Fig. 6. It can be seen in Fig. 6 that:

1. In traditional cohesive zone formulations, if the initial cohesive stiffness is very high, traction

oscillations are commonly observed as a result of numerical instability in the formulation.

2. The proposed formulation alleviates numerical instability, consequently, the post-peak load-

displacement curve is free of oscillations for any choice of initial cohesive stiffness.

3. The initial portion of the load-displacement curve obtained from proposed method shows an

excellent match with the analytical curve.

4. The peak load obtained from the numerical simulation approaches the analytical (i.e., LEFM)

peak load value as the cohesive strength is increased.

(a) Traditional CZM formulation (b) Stabilized ﬁnite element formulation

Figure 6: Load vs. displacement curves for an isotropic material from the mode-I delamination:

(a) traditional CZM formulation shows instability for large stiffness values; (b) the stabilized for-

mulation alleviates instabilities for large stiffness values.

Mixed Mode: Mixed Mode Bending (MMB) Test

The MMB test setup is shown in Fig. 7 (proposed by [25] as an alternative to the original conﬁgu-

ration proposed in [26]). The beam of length 2L is simply supported at the lower left and right end

nodes. The forces applied on the upper and lower arm are obtained from superposition of mode I

and mode II are given as:

P

u=Pc+L

4L+P3c−L

4L,(32)

P

d=Pc+L

4L−P3c−L

4L,(33)

where c is the length parameter that decides the ratio between the two forces, and thus deﬁnes the

mixed mode ratio. The computational mesh for the beam is similar to that used for mode I. The

load (Pu) vs displacement (δ) curves for the isotropic material from the traditional CZM, proposed

stabilized method and the analytical LEFM solution [24] are shown in Fig. 8. Displacement control

is used for the simulation in this case as well. The results reafﬁrm the applicability of the proposed

stabilized approach.

Figure 7: Geometry and boundary conditions for the mixed mode bending (MMB) test. The

dimensions are: L = 100 mm, H = 4 mm and a0= 25 mm

Conclusion

In this work, we propose a uniﬁed formulation based on the Nitsche method that is equally ef-

fective for any speciﬁed stiffness of a cohesive law. We showed that when the cohesive stiffness

approaches zero, the formulation collapses to a traditional ﬁnite element formulation with the cohe-

sive law enforced as a Neumann boundary condition; on the other hand, as the stiffness approaches

a large value, the proposed approach becomes identical to that of a standard Nitsche method and the

cohesive law is enforced as a kinematic constraint. Thus, the key advantage is that it generalizes the

Nitsche approach to a traction-separation law of any arbitrary initial stiffness and provides a uniﬁed

way to treat such problems in a variationally consistent and stable manner. The stabilized ﬁnite

element formulation would naturally extend to general nonlinear forms of traction-separation rela-

tionships, although we only use the bilinear cohesive law. We performed several numerical studies

to demonstrate the advantages of the proposed approach over the standard CZM approach through

(a) Traditional CZM formulation (b) Stabilized ﬁnite element formulation

Figure 8: Load vs. displacement curves for the mixed mode bending test. Similar to the results

of mode-I test, traditional CZM approach shows instability for large stiffness but the proposed

stabilized approach is free of that

two benchmark problems: mode-I and mixed-mode delamination tests. The load-displacement

curves obtained from these tests for traditional CZM formulation show instability and oscillation

for large stiffness values, whereas, the stabilized formulation results are free of any numerical in-

stabilities or oscillations. Moreover, results obtained from the stabilized formulation show much

better agreement with the analytical solution in comparison to the traditional CZM formulation.

Currently, the formulation is only implemented assuming linear isotropic elastic behavior in the

bulk material domain and our future work will focus on extending the proposed method to model

static delamination in transversely isotropic laminated composites. Another direction of future

work is to model high cycle fatigue delamination of composites.

Acknowledgements

GG and RD gratefully acknowledge the funding support from the Ofﬁce of Naval Research award

#N0014-17-12040 (Program Ofﬁcer: Mr. William Nickerson)

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