Content uploaded by Gourab Ghosh
Author content
All content in this area was uploaded by Gourab Ghosh on Dec 21, 2017
Content may be subject to copyright.
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)
References
[1] D.S. Dugdale. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8(2):100–
104, 1960.
[2] G.I. Barenblatt. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechan-
ics, 7:55–129, 1962.
[3] G.H. Paulino, Z. Zhangb, and W. Celesc. Dynamic failure branching and fragmentation using coesive zone
modeling. In Convegno IGF XVIII Cetraro 2006, 2008.
[4] V.P. Nguyen. Discontinuous Galerkin/extrinsic cohesive zone modeling: Implementation caveats and applica-
tions in computational fracture mechanics. Engineering Fracture Mechanics, 128:37–68, 2014.
[5] K. Park, G.H. Paulino, W. Celes, and R. Espinha. Adaptive mesh refinement and coarsening for cohesive zone
modeling of dynamic fracture. International Journal for Numerical Methods in Engineering, 92(1):1–35, 2012.
[6] R. Espinha, K. Park, G.H. Paulino, and W. Celes. Scalable parallel dynamic fracture simulation using an extrinsic
cohesive zone model. Computer Methods in Applied Mechanics and Engineering, 266:144–161, 2013.
[7] N. Blal, L. Daridon, Y. Monerie, and S. Pagano. Artificial compliance inherent to the intrinsic cohesive zone
models: criteria and application to planar meshes. International Journal of Fracture, 178(1-2):71–83, 2012.
[8] P.D. Zavattieri and H.D. Espinosa. Grain level analysis of crack initiation and propagation in brittle materials.
Acta Materialia, 49(20):4291–4311, 2001.
[9] J. Mergheim, E. Kuhl, and P. Steinmann. A hybrid discontinuous Galerkin/interface method for the computa-
tional modelling of failure. International Journal for Numerical Methods in Biomedical Engineering, 20(7):511–
519, 2004.
[10] T.J. Truster and A. Masud. A discontinuous/continuous Galerkin method for modeling of interphase damage in
fibrous composite systems. Computational Mechanics, 52(3):499–514, 2013.
[11] E. Lorentz. A mixed interface finite element for cohesive zone models. Computer Methods in Applied Mechanics
and Engineering, 198(2):302–317, 2008.
[12] H. Ji and J.E. Dolbow. On strategies for enforcing interfacial constraints and evaluating jump conditions with
the extended finite element method. International Journal for Numerical Methods in Engineering, 61(14):2508–
2535, 2004.
[13] J. Nitsche. ¨
Uber ein Variationsprinzip zur L¨
osung von Dirichlet-Problemen bei Verwendung von Teilr¨
aumen, die
keinen Randbedingungen unterworfen sind. In Abhandlungen aus dem mathematischen Seminar der Universit ¨
at
Hamburg, volume 36, pages 9–15. Springer, 1971.
[14] I. Babuˇ
ska. Numerical solution of boundary value problems by the perturbed variational principle. Institute for
Fluid Dynamics and Appl. Math., Technical Note BN-624, University of Maryland, College Park, Md., 1969.
[15] F. Liu and R.I. Borja. A contact algorithm for frictional crack propagation with the extended finite element
method. International Journal for Numerical Methods in Engineering, 76(10):1489–1512, 2008.
[16] P. Wriggers and G. Zavarise. A formulation for frictionless contact problems using a weak form introduced by
Nitsche. Computational Mechanics, 41(3):407–420, 2008.
[17] A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsches method, for elliptic interface
problems. Computer Methods in Applied Mechanics and Engineering, 191(47):5537–5552, 2002.
[18] C. Annavarapu, M. Hautefeuille, and J.E. Dolbow. A robust Nitsche’s formulation for interface problems. Com-
puter Methods in Applied Mechanics and Engineering, 225:44–54, 2012.
[19] P. Hansbo. Nitsche’s method for interface problems in computational mechanics. GAMM-Mitteilungen,
28(2):183–206, 2005.
[20] X. Liu, R. Duddu, and H. Waisman. Delamination analysis of composites using a finite element based discrete
damage zone model. Conference Proceedings, Society for the Advancement of Materials and Process Engineer-
ing (SAMPE), Baltimore, MD, pages 1–15, 2012.
[21] X. Liu, R. Duddu, and H. Waisman. Discrete damage zone model for fracture initiation and propagation. Engi-
neering Fracture Mechanics, 92:1–18, 2012.
[22] Stephen Jimenez, Xia Liu, Ravindra Duddu, and Haim Waisman. A discrete damage zone model for mixed-
mode delamination of composites under high-cycle fatigue. International Journal of Fracture, 190(1-2):53–74,
2014.
[23] S. Jim´
enez and R. Duddu. On the parametric sensitivity of cohesive zone models for high-cycle fatigue delami-
nation of composites. International Journal of Solids and Structures, 82:111–124, 2016.
[24] Y. Mi, M.A. Crisfield, G.A.O. Davies, and H.B. Hellweg. Progressive delamination using interface elements.
Journal of Composite Materials, 32(14):1246–1272, 1998.
[25] S. Jim´
enez, X. Liu, R. Duddu, and H. Waisman. A discrete damage zone model for mixed-mode delamination
of composites under high-cycle fatigue. International Journal of Fracture, 190(1-2):53–74, 2014.
[26] J.R. Reeder and J.H. Crews. Mixed-mode bending method for delamination testing. AIAA Journal, 28(7):1270–
1276, 1990.
