![A schematic diagram of the mixed-mode bilinear cohesive law (redrawn from [63]): (a) the tractionseparation relationship for any arbitrary mode-mix ratio is defined in terms of the pure mode I and mode II relationships; (b) the relationship between the static damage variable d s and the equivalent separation. The magnitude of the traction vector t c = t 2 n + t 2 τ .](https://www.researchgate.net/profile/Ravindra-Duddu/publication/331215559/figure/fig2/AS:753501512687616@1556660168914/A-schematic-diagram-of-the-mixed-mode-bilinear-cohesive-law-redrawn-from-63-a-the_Q320.jpg)
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A stabilized finite element method for enforcing stiff anisotropic
cohesive laws using interface elements
Gourab Ghosha, Ravindra Duddua,∗, Chandrasekhar Annavarapub,∗
aDepartment of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee.
bComputational Geosciences, Lawrence Livermore National Laboratory, Livermore, California.
Abstract
We present a stabilized finite element method that generalizes Nitsche’s method for enforcing stiff
anisotropic cohesive laws with different normal and tangential stiffness. For smaller values of
cohesive stiffness, the stabilized method resembles the standard method, wherein the traction on
the crack surface is enforced as a Neumann boundary condition. Conversely, for larger values of
cohesive stiffness, the stabilized method resembles Nitsche’s method, wherein the cohesive law is
enforced as a kinematic constraint. We present several numerical examples, in two-dimensions, to
compare the performance of the stabilized and standard methods. Our results illustrate that the sta-
bilized method enables accurate recovery of crack-face tractions for stiff isotropic and anisotropic
cohesive laws; whereas, the standard method is less accurate due to spurious traction oscillations.
Also, the stabilized method could mitigate spurious sensitivity of load-displacement results to dis-
placement increment in mixed-mode fracture simulation, owing to its stability and accuracy.
Keywords: Interface elements, Nitsche’s method, Traction oscillations, Numerical stability,
Cohesive zone models, Mixed-mode fracture
1. Introduction
Numerical simulation of fracture propagation in multi-dimensions is a challenging problem, in
part due to mixed-mode interactions and stiff anisotropic cohesive response at arbitrarily shaped
crack interfaces, particularly under compression. The cohesive zone modeling approach has been
widely used to analyze and predict mixed-mode fracture or delamination propagation at anisotropic
and/or dissimilar material interfaces, despite its limitations. Typically, cohesive zone models
(CZMs) are implemented in conjunction with the finite element method by introducing zero-
thickness interface elements along potential crack surfaces. The constitutive behavior of the in-
terface is defined by a cohesive law that relates the traction with the separation across the crack
surface. There are broadly two classes of CZMs: intrinsic with initially elastic cohesive laws and
extrinsic with initially rigid cohesive laws. Cohesive laws with large initial stiffness may be re-
quired for accurate fracture analysis and contact enforcement in the intrinsic approach; however,
∗Corresponding authors
Email addresses: ravindra.duddu@vanderbilt.edu (Ravindra Duddu), asc.sekhar@gmail.com
(Chandrasekhar Annavarapu)
stiff cohesive laws may also need to be enforced during cycle-by-cycle fatigue analysis even in
the extrinsic approach (see Fig. 1). It is well-known that the standard (penalty-like) method for
enforcing stiff cohesive laws using interface elements suffers from a distinct numerical instability
that is often manifested by spurious oscillations in crack-face tractions, which may cause inaccu-
racies and convergence issues. The purpose of this paper is to introduce a novel stabilized finite
element method by generalizing Nitsche’s method for enforcing stiff anisotropic cohesive laws that
alleviates the numerical instability issue afflicting the standard method.
(a) Intrinsic CZM (b) Extrinsic CZM
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Figure 1: Illustration of stiff (red line) cohesive laws encountered in fracture analysis under monotonic and cyclic
loading. (a) In intrinsic CZMs, stiff cohesive laws may be prescribed to define the linear elastic portion before damage
initiation or no-interpenetration (contact) condition; (b) In extrinsic CZMs, stiff cohesive laws may be encountered
during unloading/reloading immediately after damage initiation or contact enforcement during cyclic loading.
In an extrinsic CZM, it is assumed that the interface separates only after the cohesive traction
exceeds a finite cohesive strength and then the maximum cohesive traction decreases monotoni-
cally with the increase in separation until an ultimate separation value is reached (see Fig. 1b).
Since their introduction in the seminal paper by Camacho and Ortiz [1], the extrinsic CZM ap-
proach have been used to simulate dynamic fracture in a wide-range of engineering materials [e.g.,
2–7]. In practice, the extrinsic approach is implemented by adaptively inserting zero-thickness
interface elements in a finite element mesh adjacent to the crack tip when a specified criterion
for the onset of failure is met. Consequently, the numerical implementation of extrinsic CZMs is
computationally efficient, but it requires sophisticated topology change algorithms to modify the
associated finite element data structures consistently with the evolving fracture geometry, which
increases algorithmic complexity. Despite several algorithmic advances [8–11], numerical imple-
mentation of extrinsic CZMs in a legacy finite element framework remains non-trivial, compared
to intrinsic CZMs that are much more straightforward to implement.
In intrinsic CZMs, it is assumed that the cohesive traction increases gradually with separa-
tion till it reaches a finite cohesive strength in the linear elastic regime, and then it decreases
monotonically till the separation reaches an ultimate value, where complete de-cohesion occurs
(see Fig. 1a). Xu and Needleman [12] pioneered the intrinsic approach using a potential-based
traction-separation law to model dynamic fracture growth in brittle solids. Several phenomenolog-
ical and potential-based intrinsic CZMs with bilinear [13], trapezoidal [14, 15], and polynomial
shapes [16, 17] have been developed for specific applications. In practice, the intrinsic approach
is implemented by inserting zero-thickness interface elements in a finite element mesh along all
2
potential crack paths a priori. This approach has been extensively used to model both quasi-static
and dynamic failure in a variety of applications including particle-matrix interface debonding in
metal-based composites [18], delamination in laminated composite materials [19–21], and fiber-
metal laminate failure [22]. The intrinsic approach is relatively straightforward to implement in a
legacy finite element framework, but it has some shortcomings, including the well-known “artifical
complicance” [23, 24] and increased computational cost.
Despite their differences, the implementation of both extrinsic and intrinsic approaches for
stiff cohesive laws using the standard (penalty-like) method is prone to several numerical issues
during dynamic and quasi-static fracture analysis. The stiffness of the cohesive law is often defined
using a non-dimensional quantity M=α0h/E, where α0is the initial cohesive stiffness, Eis the
elastic modulus and his the mesh size parameter [25]. For laminated composites, Turon et al.
[26] suggested using the sub-laminate thickness t, instead of mesh size parameter h, to define
the non-dimensional quantity M. For a given size-scale parameter (i.e., hor t), choosing a large
cohesive stiffness α0relative to the elastic modulus Eleads to a stiff cohesive law with M > 1000.
In dynamic fracture analysis, to avoid the artificial compliance issue, it is required to assume an
adequately large value for the initial cohesive stiffness in intrinsic CZMs [23, 25, 27], but that
can lead to ill-conditioning of the tangent stiffness matrices. This ill-conditioning issue may also
arise in extrinsic CZMs under cyclic loading, if the interface is unloaded immediately after crack
initiation when the elastic unloading/reloading slope could be large [1]. This issue can be resolved
to some extent by restricting the time step in an explicit finite element scheme to an extremely
small value [4, 24], but this will result in an impractically high computation cost. In quasi-static
fracture analysis, intrinsic CZMs exhibit spurious traction oscillations along the cohesive interface,
especially near crack tips, if a large initial cohesive stiffness is specified [26, 28–30], or if the
cohesive interface has a curved geometry and its behavior under compressive loading is described
by an anisotropic cohesive law with different values for normal and tangential stiffness [31]. It
has been argued in [31] that the issue of spurious traction oscillations encountered in the standard
(penalty-like) method for intrinsic CZMs with stiff cohesive laws (using full or reduced integration)
arises due to the violation of the inf-sup or LBB condition.
In the traditional penalty method [32], a Dirichlet constraint at the interface is enforced by
introducing a spring-like tie constraint at the interface. A better approximation of the Dirichlet
constraint can be obtained by using a large value for the penalty parameter, which can be inter-
preted as the stiffness of the spring (i.e., initial slope of the traction separation law). Thus, the
standard finite element method for implementing an intrinsic CZM is equivalent to the penalty
method for stiff cohesive laws. Theoretically, the interfacial constraint of zero separation before
crack initiation (i.e., extrinsic CZM) can be achieved if the initial cohesive stiffness approaches
infinity, but using a very large cohesive stiffness leads to ill-conditioning and numerical instability
issues. Lagrange-multiplier-based mixed formulations can alleviate instability issues associated
with cohesive interface elements [33, 34] or with embedded contact interfaces in the extended fi-
nite element method (XFEM) [35, 36]. However, these approaches can be computationally costly
and complicated to implement, because it is difficult and non-trivial to find a stable Lagrange
multiplier space that alleviates traction oscillations [37]. Another alternative is Nitsche’s method,
which was originally introduced in [38] to weakly enforce Dirichlet boundary conditions. Later, it
3
was extended to weakly enforce the continuity of the displacement field at the interior boundaries
[39]. We note that the discontinuous Galerkin (DG) method essentially originated from Nitsche’s
method [40] and the latter has been referred to as the classical DG method [41]. A comprehensive
review of Nitsche’s method and its application to interface problems can be found in [42, 43].
The Nitsche’s method can be interpreted as a variationally consistent penalty method for weakly
enforcing interfacial constraints. This method can eliminate the instability issues associated with
the penalty method by adding consistency terms [44, 45], and can yield oscillation-free traction
profiles at embedded interfaces [46]. In the recent decades, Nitsche’s method has been utilized for
solving a wide range of interface problems in an efficient way [46–55]. More recently, Nitsche-
based methods have been developed for frictional-sliding on embedded interfaces [56, 57] and
small-sliding contact on frictional surfaces, including stick-slip behavior [58]. In this article, we
propose a stabilized finite element method for cohesive fracture problems, which is inspired by
the Nitsche’s method for general boundary conditions developed by Juntunen and Stenberg [59].
The proposed method ensures accurate recovery of crack-surface traction even for large values of
cohesive stiffness (e.g., 8–16 orders of magnitude more than bulk stiffness). We further demon-
strate the ability of the stabilized method to alleviate numerical instability associated with the
implementation of stiff, anisotropic cohesive laws with different interface properties in the nor-
mal and tangential directions. The main novelty of this paper is that it extends Nitsche’s method
for cohesive fracture so that it is applicable to both intrinsic and extrinsic approaches, including
stiff elastic loading and unloading conditions. The rest of this paper is organized as follows: in
Section 2, we introduce the governing equations of the cohesive fracture problem and the weak
forms corresponding to the standard and stabilized methods; in Section 3, we discuss the numer-
ical implementation of the stabilized method in the commercial finite element software Abaqus;
in Section 4, we present several numerical examples to compare the standard and stabilized meth-
ods with a particular emphasis on the accuracy of evaluating crack-face tractions for stiff cohesive
laws in quasi-static simulation, including mixed-mode fracture; in Section 5, we conclude with a
summary and closing remarks.
2. Model formulation
In this section, we present details of the stabilized finite element method for enforcing stiff
cohesive laws. We first present the strong form of the governing equations followed by a brief
description of the interface cohesive law for mixed-mode loading. We next derive the weak form
for the standard (penalty-like) and stabilized (Nitsche-inspired) methods.
2.1. Strong Form
We consider the initial domain Ω⊂Rnd containing a linearly elastic solid with nd = 2 in two
dimensions, as shown in Fig. 2. The domain boundary Γ≡∂Ωis partitioned into two disjoint
parts such that ∂Ω = ΓD∪ΓNwith ΓD∩ΓN=∅, where the Dirichlet and Neumann boundary
conditions are enforced. The domain Ωcontains an internal cohesive interface Γ∗, which divides Ω
into two non-overlapping sub-domains Ω1and Ω2. The outward unit normal to the boundary ∂Ωis
denoted by ne, and the unit normal vector associated with the interface Γ∗denoted by npoints from
4
Ω2to Ω1. We use a rectangular Cartesian coordinate system and the total Lagrangian description
for variables with Xdenoting the position of the material points.
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Figure 2: A schematic of the domain for the quasi-static cohesive fracture problem.
The displacement field u(X)is discontinuous across Γ∗, but continuous in Ω1and Ω2; there-
fore, it can be represented by two continuous functions u1(X)and u2(X)in the respective sub-
domains. For brevity, henceforth we will suppress the spatial dependence of variables. Assuming
small displacements, the Cauchy stress tensor can be defined in Ω1and Ω2as
σm=Dm:m, m ={1,2},(1)
where Ddenotes the fourth-order elasticity tensor and the small strain tensor =1
2(∇u+ (∇u)T)
is defined by the symmetric part of the displacement gradient tensor and ∇is the spatial gradient
operator with respect to the material coordinates X.
The strong form of the quasi-static boundary value problem in the absence of body force is:
∇·σm=0in Ωm, m ={1,2},(2)
u=¯
uon ΓD,(3)
σ·ne=¯
ton ΓN,(4)
tc(δ) = σ·n,on Γ∗,(5)
where ¯
tis the prescribed traction or stress vector on the Neumann boundary ΓN,¯
uis the prescribed
displacement vector on the Dirichlet boundary ΓD, and the traction on the cohesive interface tcis
given by a function of the interface separation or displacement jump as
δ= [[u]] = u2−u1(6)
Note that the traction tcis continuous across the cohesive interface Γ∗and is related to the Cauchy
stress tensor evaluated in the sub-domains Ω1and Ω2as
tc=σ·n=−σ1·n1=σ2·n2.(7)
In the above two equations, we followed the notation convention used in [52] for defining the
displacement jump and outward normal, and use it to establish the weak form in Section 2.3.
5
2.2. Intrinsic cohesive law
For simplicity, we consider the bilinear intrinsic cohesive law with an initial (increasing) linear
elastic portion followed by a (decreasing) linear softening response. The corresponding relation
between crack-face traction and interface separation can be defined using the damage mechanics
framework as [20, 21, 60]
tc=−α δ,(8)
where αis the cohesive stiffness matrix including the effect of damage. Note that the crack-
face traction tcis the Newton’s third law pair to the cohesive traction. Thus, the negative sign in
the above equation indicates that tccauses a restoring force at the interface, which is equal and
opposite to the deforming force. To represent the mixed mode-I and mode-II fracture behavior in
two dimensions, we use the normal and tangential coordinate system. Accordingly, the tangential
tτand normal tncomponents of the traction vector tcare related to the tangential δτand normal δn
components of the interface separation δas
tτ
tn=−(1 −ds)α0
τ0
0 (1 −ds)α0
nδτ
δn,(9)
where α0
nand α0
τrepresent the initial cohesive stiffness in the normal and the tangential directions,
respectively. The scalar damage variable dsdescribing interface degradation under quasi-static
mixed-mode loading in two dimensions is given by
ds=
0if δe< δc
e,
δu
e(δe−δc
e)
δe(δu
e−δc
e)if δc
e≤δe< δu
e,
1if δu
e≤δe,
(10)
where δe=pδ2
n+δ2
τis the equivalent separation, δc
eand δu
eare interface parameters correspond-
ing to critical and ultimate separations, respectively, defined as [61]
1
δc
e
=sα0
ncos I
σmax 2
+α0
τcos II
τmax 2
(11)
1
δu
e
=α0
nδc
e(cos I)2
2GIC +α0
τδc
e(cos II)2
2GIIC (12)
where the direction cosines cos I=δn/δeand cos II =δτ/δe,σmax and τmax are the pure mode I
and mode II cohesive strengths, and GIC and GIIC are the pure mode I and mode II critical fracture
energies (see Fig. 3a). For monotonic loading, when the equivalent interface separation δeis less
than the critical separation δc
e, there is no damage in the cohesive interface elements. After the
critical separation is exceeded, damage starts to accumulate till the separation reaches the ultimate
value δu
e, when the cohesive elements are completely damaged (see Fig. 3b). The mixed-mode
cohesive law described above was previously proposed by Jiang et al. [61], wherein quadratic
damage initiation and mixed-mode failure criteria were used to obtain the equivalent critical and
6
ultimate separations. If the parameter values of cohesive stiffness, cohesive strength and fracture
energy are chosen to be the same for both normal and shear modes, then we get an isotropic
cohesive law, else we get an anisotropic cohesive law. For non-monotonic (cyclic) loading, we
can enforce irreversibility of damage evolution by ensuring that damage does not change during
unloading and reloading cycles, until the previous maximum damage is exceeded [60, 62].
(a)
Mode I Mode II
(b)
(1 )
Figure 3: A schematic diagram of the mixed-mode bilinear cohesive law (redrawn from [63]): (a) the traction-
separation relationship for any arbitrary mode-mix ratio is defined in terms of the pure mode I and mode II rela-
tionships; (b) the relationship between the static damage variable dsand the equivalent separation. The magnitude of
the traction vector ktck=pt2
n+t2
τ.
2.3. Weak Form
We follow the Galerkin method of weighted residuals to derive the weak forms corresponding to
the standard and stabilized methods. We define the space of trial functions Uand the space of test
functions W, such that:
U={u∈H1(Ω),u=¯
uon ΓD},(13)
W={w∈H1(Ω),w=0on ΓD}.(14)
By weighting Eq. (2) with the test function w, integrating by parts, applying the divergence the-
orem, and using the traction continuity condition at the interface in Eq. (7), and the constitutive
relation in Eq. (1), we can derive the weak form as follows:
ZΩ
∇sw:D:∇sudΩ−ZΓ∗
(w2·σ2·n2+w1·σ1·n1)dΓ−ZΓN
w·¯
tdΓ = 0,(15)
ZΩ
∇sw:D:∇sudΩ−ZΓ∗
[[w]] ·tcdΓ = ZΓN
w·¯
tdΓ,(16)
Note that in Eq. (15) we considered the integrals on the two sides of the cohesive interface, sepa-
rately, and in Eq. (16) we defined the jump in the test function as [[w]] = w2−w1.
7
2.3.1. Standard method
Substituting the traction-separation relation in Eq. (8) into the weak form in Eq. (16) we get
ZΩ
∇sw:D:∇sudΩ + ZΓ∗
[[w]] ·α δ dΓ = ZΓN
w·¯
tdΓ.(17)
Thus, in the standard method the cohesive tractions are enforced as a Neumann boundary condition
on the interface. Because the cohesive tractions and separations are defined in the normal and
tangential directions, the weak form is implemented as,
ZΩ
∇sw:D:∇sudΩ + ZΓ∗
(1 −ds)[[wn]]α0
nδn+ [[wτ]]α0
τδτdΓ = ZΓN
w·¯
tdΓ.(18)
If the initial cohesive stiffness parameters α0
nand α0
τare taken to be sufficiently large the standard
method resembles the penalty method for enforcing displacement continuity across the interface.
However, for stiff cohesive laws, that is, if cohesive stiffness is several orders of magnitude greater
than the elastic modulus, the standard method becomes ill-conditioned leading to instability and/or
convergence issues. In the limiting case where α0
n→ ∞ and/or α0
τ→ ∞, that is, for a non-
interpenetration (contact) constraint or an extrinsic cohesive law, the standard method is not well
defined. To circumvent the above issues, discontinuous Galerkin or Nitsche-based methods have
been proposed in [55, 64–67] based on the extrinsic approach, wherein the interface is perfectly
bonded until a certain stress threshold (i.e., interface separation tends to zero and cohesive stiffness
tends to infinity). While in [64] interface bonding prior to separation was enforced using the
interior penalty method, in [66] Riemann solutions were used to enforce interface conditions. In
the following section, we will present an alternative stabilized finite element method for cohesive
fracture that is applicable for the whole range of values cohesive stiffness αn, ατ>0, so that it is
applicable to both intrinsic and extrinsic approaches.
2.3.2. Stabilized method
The proposed method adopts the approach developed in [52, 59] and generalizes it to cohesive
fracture problems. By multiplying both sides of Eq. (8) with a stabilization matrix Swe obtain
Stc=−Sα δ.(19)
After multiplying the above equation by the weighting function wand integrating over the cohesive
interface Γ∗we get
ZΓ∗
[[w]] ·StcdΓ = −ZΓ∗
[[w]] ·Sα δ dΓ.(20)
By adding the above equation to the weak form in Eq. (16), we obtain
ZΩ
∇sw:D:∇sudΩ−ZΓ∗
[[w]] ·(I−S)tcdΓ + ZΓ∗
[[w]] ·Sα δ dΓ = ZΓN
w·¯
tdΓ,(21)
8
where Iis the identity matrix. The interface traction can be defined as
tc=hσiγ·non Γ∗,(22)
where the weighted average of the stress tensors on both sides of the interface is given by
hσiγ= (γ1σ1+γ2σ2)∀γ1+γ2= 1, γ1>0, γ2>0.(23)
Choosing the weights γ1=γ2= 0.5, yields the mean of the stress tensors evaluated on both sides
of the interface. To complete the formulation, we define the stabilization matrix Sas
S=
βτ
α0
τ(1 −ds) + βτ
0
0βn
α0
n(1 −ds) + βn
,(24)
where βτ, βnare the stabilization parameters. The stabilization parameters βτ, βnand the weights
γ1, γ2play a key role in the numerical performance of the method. This so-called weighted Nitsche
method [58] is particularly advantageous for dissimilar material interfaces with large contrast in
material properties or for unstructured meshes with significant variations in mesh size. For constant
strain triangular (CST) and tetrahedral elements, Annavarapu et al. [46] provided estimates for the
stabilization parameters using a local coercivity analysis as given by
βn=βτ= 2 |D1|(γ1)2
meas(Ω1)+|D2|(γ2)2
meas(Ω2)!meas(Γ∗)(25)
where |D|denotes the two-norm of the elasticity tensor, meas(Ω) denotes the area of neighboring
bulk element in 2D, and meas(Γ∗)is the length of the interface element. With a judicious choice
of the weights γ1, γ2, the stabilization parameters βτ, βnscale as 1/h, where h≈meas(Γ∗)is
the mesh/element size parameter. For all h∈(0,∞), both the initial cohesive stiffness αand
the stabilization parameter βscale as 1/h; thus, the stabilized method provides a well-conditioned
discrete system, irrespective of the mesh size, by ensuring that the cohesive stiffness terms and the
bulk stiffness terms have the same scaling. For an elaborate discussion on the appropriate choice
of weights, we refer the reader to Ref. [46]. Note that, in this study, we used the estimates given in
(25) to calculate the stabilization parameters for bilinear quadrilateral elements; precise estimates
can be derived as in [46], but such analysis is beyond the scope of this paper.
Finally, the weak form for the stabilized method can be written as
ZΩ
∇sw:D:∇sudΩ−ZΓ∗
[[w]] ·(I−S)hσiγ·ndΓ + ZΓ∗
[[w]] ·Sα δ dΓ = ZΓN
w·¯
tdΓ.(26)
In the above equation, the second and third terms on the left hand side ensure consistency and
stability of the proposed method, respectively. The stabilized method presented here is unsym-
metric and resembles the incomplete interior penalty method [43, 68]. It can be proved that the
9
displacement solution uof the strong form equations (2) – (5) is satisfied by the solution to the
weak form equation (26), which establishes consistency for any value of cohesive stiffness; the
mathematical procedure for proving this is similar to that described in [59, Lemma 2.1]. As
(1−ds)α0
n,(1−ds)α0
τ→ ∞ (refer to Eq. (24)), we recover the Nitsche-based method for frictional
contact as [56]
ZΩ
∇sw:D:∇sudΩ−ZΓ∗
[[w]] · hσiγ·ndΓ + ZΓ∗
[[w]] ·(βτ[[uτ]] + βn[[un]]) dΓ = ZΓN
w·¯
tdΓ.
(27)
As (1 −ds)α0
n,(1 −ds)α0
τ→0, we recover the weak form for a traction-free crack surface as
ZΩ
∇sw:D:∇sudΩ = ZΓN
δu·¯
tdΓ.(28)
Thus, the stabilized method remains well-defined for any arbitrarily values of the cohesive stiffness
terms, that is, for (1 −ds)α0
n,(1 −ds)α0
τ∈[0,∞). Comparing the weak forms in equations (16)
and (26), we can obtain an alternative definition for crack-surface traction as
tc= (I−S)hσiγ·n−Sα δ.(29)
Thus, the key idea of the Nitsche-inspired stabilized method for cohesive fracture is to evaluate
the crack surface traction in terms of the weighted average stress in the bulk material across the
interface and the traction in the cohesive interface.
3. Numerical implementation
In this section, we discuss the finite element approximation using matrix notation along with
the expression for the residual and tangent matrices for the bulk and interface elements. We also
present algorithms for implementing this stabilized method into the commercial software Abaqus
using user defined subroutines for two-dimensional plane strain analysis.
3.1. Finite element approximation
The sub-domains Ω1and Ω2are discretized by four-noded plane strain quadrilateral bulk ele-
ments and zero-thickness four-noded interface elements are introduced at the cohesive interface Γ∗
(Fig. 4). The displacement field um
2×1at any point Xcan be approximated as
um(X) = N(X)¯
um, m = 1,2,(30)
where ¯
um= [u1
1, u1
2, ..., u4
1, u4
2]8×1is the nodal displacement vector for a bulk element in the sub-
domain Ωm,Nis the element shape function matrix given by
N=N10N20N30N40
0N10N20N30N42×8
,(31)
10
30
20
31
21
32
22 23
33
3
12
4
Cohesive
element
12
43
Bulk element
Local node numbering
Global node numbering
Figure 4: Finite element discretization with bulk and zero-thickness cohesive elements
and NJ(J= 1,2,3,4) are the standard finite element shape functions for the four-noded quadri-
lateral element. Using Voigt notation, the small-strain strain tensor m= [11 , 22, 12 ]|
3×1in the
bulk element can be approximated as
m=B¯
um, m = 1,2,(32)
where the strain-displacement relationship matrix is defined as
B=
N1
,10N2
,10N3
,10N4
,10
0N1
,20N2
,20N3
,20N4
,2
N1
,2N1
,1N2
,2N2
,1N3
,2N3
,1N4
,2N4
,1
3×8
,(33)
and NJ
,i denotes the derivative of the shape function NJwith respect to material coordinate Xi
(i= 1,2) in two dimensions. The normal and tangential components of the displacement jump or
separation across the interface element can be approximated as
[[u]]2×1=δ2×1=δτ
δn= [[N]]ˆ
u,(34)
where ˆ
u= [ˆu1
1,ˆu1
2, ..., ˆu4
1,ˆu4
2]8×1is the nodal displacement vector for the four-noded interface
element in Γ∗and [[ˆ
N]] is the jump in the interfacial shape function matrix given by [69]
[[N]] = −Cˆ
N1−Sˆ
N1−Cˆ
N2−Sˆ
N2Cˆ
N2Sˆ
N2Cˆ
N1Sˆ
N1
Sˆ
N1−Cˆ
N1Sˆ
N2−Cˆ
N2−Sˆ
N2Cˆ
N2−Sˆ
N1Cˆ
N12×8
,(35)
and ˆ
NJ(J= 1,2) are the standard finite element shape functions for the one-dimensional linear
element, and Cand Srepresent the cosine and sine, respectively, of the angle θthat defines the
orientation of the interface element with the global x1coordinate axis (see Fig. 5). All the finite
element shape functions are evaluated using the standard isoparametric concept.
11
x1
x2
x'2
x'1
θ
x'1
x'2
1K(𝑢M$
$, 𝑢M &
$)
2K(𝑢M$
&, 𝑢M &
&)
3K(𝑢M$
O, 𝑢M &
O)
𝛿𝜏
4K(𝑢M$
Q, 𝑢M &
Q)
𝛿𝑛
(a) (b)
Before crack opening After crack opening
Figure 5: Four-noded linear cohesive element: (a) the orientation θof the local (normal-tangential) coordinates with
respect to the global Cartesian coordinates before crack opening; (b) the relation between nodal displacements in
global coordinates and interface separations in local coordinates after crack opening.
3.2. Discretization and Linearization
By introducing the finite element approximation into the variational form in Eq. (26), we write
the discretized form of the equilibrium equations as
R(U) = fext −(fb
int(U) + fc
int(U)) = 0,(36)
where Ris the global residual vector, Uis the global displacement vector, fext and fb
int are the
global external and internal force vectors, respectively, assembled from all the bulk elements in the
domain Ω, and fc
int is the internal force vector from all the cohesive elements on the interface Γ∗.
The solution to Eq. (36) in the generalized case can be obtained an iterative solution procedure.
Let k+1
j+1 Ube the global (nodal) displacement vector at an applied load/displacement step j+ 1
and iteration k+ 1. Using a Taylor’s expansion we can linearize the global displacement and the
residual vectors as
Rk+1
j+1 U=0=Rk
j+1U+k
j+1∆U=Rk
j+1U+"∂Rk
j+1U
∂k
j+1U#k
j+1∆U.(37)
Rewriting the above equation into a fully discretized and linearized system of equation, we obtain
k
j+1Kk
j+1∆U=Rk
j+1U(38)
where Kis the algorithmically consistent tangent matrix obtained by assembling the contributions
of the bulk (Kb) and cohesive (Kc) tangent matrices as
K=−∂R
∂U=∂fb
int
∂U+∂fc
int
∂U=Kb+Kc,(39)
12
3.3. Bulk elements
The internal bulk force vector fb
int is assembled as
fb
int =X
eZΩm
e
B|σm=X
eZΩm
e
B|DmB¯
umdΩefor m={1,2},(40)
where X
e
indicates the matrix (or vector) assembly of the global system from the element matrices
(or vectors) in the entire computational domain, Bis the strain-displacement relationship matrix
defined in Eq. (33), σm= [σ11, σ22 , σ12]|
3×1is the stress tensor in Voigt notation containing only
the in-plane components, and Dmis the 3×3plane strain elasticity matrix in Voigt notation. The
corresponding bulk tangent matrix is
Kb=X
eZΩm
e
B|DmBdΩefor m={1,2},(41)
Neglecting body forces, fext is obtained by assembling the element contributions from any applied
traction on the Neumann boundary
fext =X
eZΓne
N|¯
tdΓe(42)
Note that Abaqus automatically handles the evaluation of fext, so simple traction boundary condi-
tions need not be defined using user defined subroutines.
3.4. Interface elements
The cohesive internal force vector fc
int has contributions from both the consistency and stabi-
lization terms in the variational form in Eq. (26) and can be assembled as
fc
int =fconsistency +fstabilized =X
eZΓ∗e
[[N]]|(I−S)TσγdΓe+X
eZΓ∗e
[[N]]|Sα δ dΓe,(43)
where [[N]] is the jump in shape function matrix in Eq. (35); S,α,Iare the 2×2stabilization,
cohesive stiffness and identity matrices, σγ= [hσ11iγ,hσ22iγ,hσ12iγ]|
3×1is the weighted Cauchy
stress for in-plane components in Voigt notation, and the 2×3stress transformation matrix is
T=−CS CS C2−S2
S2C2−2CS (44)
Thus, the matrix Tdefines the relation between the Cauchy stress tensor at any point on a bulk
element edge and the traction vector at that point on the cohesive interface based on its orientation.
The interface element’s tangent stiffness matrix consists of both the consistency and stabilized
13
terms and can be assembled as
Kc=Kconsistency +Kstabilized =X
eZΓ∗e
[[N]]|(I−S)TDBγdΓe+X
eZΓ∗e
[[N]]|Sα[[N]] dΓe.
(45)
where Bγdenotes the weighted shape function gradient matrix defined as,
Bγ=γ2B2γ1B13×16 .(46)
In the above equation, B1and B2are matrices containing the gradient of the shape functions cal-
culated from the adjacent bulk elements at the position of an interface Gauss point of the cohesive
element. Note, by setting the matrix S=Iin Eq. (45) we can revert to the standard method.
3.5. Abaqus implementation
The proposed method is implemented in the commercial software Abaqus, as illustrated in
Figure 6. All the element force and stiffness matrices described in Section 3.3 and Section 3.4 have
been evaluated via the user-element-material (UELMAT) and user-element (UEL) subroutines for
4-noded bulk and 4-noded interface elements, respectively. The bulk element force vector fb
int and
the tangent matrix Kbare computed using the UELMAT subroutine, because it allows for the usage
of in-built constitutive models via the material lib mech function. The cohesive element
force vector fc
int and the corresponding tangent matrix Kbare computed using the UEL subroutine,
according to Eqs. (43) and (45). Recall that fc
int and Kccontain the weighted average of stress
σγand shape function derivative matrix Bγ, which are calculated using the nodal displacement
vectors and shape function matrices of the two neighboring bulk elements. To avoid repetition
of computations in our implementation, we calculate σγand Bγat the cohesive element Gauss
points lying on the bulk element edges in the UELMAT subroutine and pass them to the UEL
subroutine using global modules. The tangent matrix Kcis unsymmetric owing to the consistency
term Kconsistency in our formulation, whereas the stabilization term Kstabilized is symmetric. The
matrix Bγdefined in Eq. (46) has the dimension of 8×16, where 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 (see Fig. 7). However, the UEL subroutine for the cohesive
interface element allows access to only its four nodes (i.e., eight DoFs), so we can only assemble
an 8×8element stiffness matrix. To assemble the 8×16 element stiffness matrix computed
in the UEL subroutine into the global stiffness matrix, we create “dummy” elements in the mesh
(elements IV-VII in Fig. 7). We partition the 8×16 element stiffness matrix as,
Kconsistency,e8×16 =
KIV4×4
KVI4×4KIII 8×8KV4×4
KVII4×4
.(47)
14
and assemble the 8×8matrix through the cohesive element and the four 4×4matrices through
the dummy elements using global modules into the global stiffness matrix. Note that the dummy
elements are only used for matrix assembly.
Abaqus Standard Solver
UELMAT Subroutine
Bulk Element
UEL Subroutine
Cohesive Element
Kb, fbint Kc, fcint
σm, Bm
Figure 6: Flow chart showing the interaction between UELMAT and UEL subroutines and the Abaqus standard solver
for implementing the stabilized method through user-defined bulk and cohesive/interface elements.
12
3
4
I
3
4
56
III
56
7
8
II
3
4
7
8
V
56
7
8
VII
12
3
4
IV
12
56
VI
Bulk
12
3
4
56
7
8
Bulk
Bulk
Cohesive
Figure 7: Assembly of the cohesive element matrix Kconsistency,edefined in Eq. (47) into the global tangent matrix in
Abaqus requires the creation of four dummy elements (IV-VII) for each cohesive element (III).
Although the above implementation with dummy elements for the cohesive stiffness matrix
assembly may appear convoluted, it is efficient and may even be advantageous when interface
elements are inserted along all element edges. This is because all bulk stress and shape function
matrices at interface Gauss points are computed once in the UELMAT subroutine and stored in
global modules, instead of recomputing it wherever needed in the UEL subroutine. We found that
our implementation of the stabilized method did not increase the wall clock time of computation
(compared to standard method) in all the numerical examples presented in Section 4. However, we
note that alternative implementations of the stabilized method in existing finite element codes are
possible. For example, Versino et al. [65] used Abaqus UEL subroutine for a 8-noded interface
element in 2D to implement a discontinuous Galerkin based extrinsic cohesive zone model. This
implementation may be more advantageous when interface elements are inserted along simple
(straight) interfaces between laminate plies in composite materials.
The UELMAT and UEL subroutines are detailed in Algorithms 1 and 2, respectively. As
discussed in [60], we formulate the bilinear cohesive zone model within the damage mechanics
framework that allows us to automatically handle the unloading/reloading conditions based on a
15
previous maximum damage (history variable). Despite the bilinear shape of the cohesive law, the
damage variable is a nonlinear function of interface separation and this nonlinearity is handled by
Abaqus/Standard outside of the user subroutines. As detailed in the Abaqus manual [70, Chapter
7: Analysis Solution and Control], Abaqus/Standard combines incremental and iterative (Newton-
Raphson) procedures for solving nonlinear problems. The total load/displacement is applied incre-
mentally as smaller increments (pseudo-time steps) and the user typically suggests the size of the
first increment and Abaqus/Standard automatically chooses the size of the subsequent increments.
Within each increment, Abaqus/Standard automatically performs iteration to find an equilibrium
solution based on a user-defined criteria for residual force and displacement correction. We note
that, in cohesive fracture simulations, the Abaqus/Standard default criteria may be too small that
numerical convergence may not be attainable. Therefore, we increase these tolerances appropri-
ately so as to attain convergence and maintain adequate accuracy, to obtain the results in Section
4.4 and Section 4.5.
Algorithm 1 : Abaqus UELMAT subroutine for the bulk element
Given all the variables at the previous iteration of the current increment, at the next iteration:
1. Compute the shape function derivative matrix Baccording to Eq. (33)
2. Determine the 3×3plane strain elasticity tensor Dvia the material lib mech function
3. Compute and assemble the bulk element contributions to the tangent matrix and the internal
force vector using four-point Gauss integration
4. Determine σmand shape function derivatives Bmat the two interface Gauss integration
points and store them in global modules
4. Numerical examples
In this section, we present four examples to demonstrate the numerical stability and accuracy
of the proposed stabilized method in two-dimensions. For all the simulations we assumed bilinear
quadrilateral plane strain elements with four-point Gauss integration scheme and four-noded linear
cohesive elements with two-point Gauss integration scheme. Currently, the user element subrou-
tines are written only for 2D plane-strain and plane-stress elements in Abaqus software. Additional
patch tests and benchmark problems have been presented in [71, 72].
4.1. Square plate with horizontal interface
In this example, we assess the accuracy of the stabilized method in recovering normal traction
on a straight, horizontal interface with isotropic CZMs using a constant strain patch test [73]. We
consider a square plate of side length L= 1 mm with a horizontal interface at mid-height (see Fig.
8a). Both vertical and horizontal displacements are constrained at the bottom edge of the plate,
16
Algorithm 2 : Abaqus UEL subroutine for the cohesive element
Given all the variables at the previous iteration of the current increment, at the next iteration:
1. Compute the jump in the interfacial shape function matrix [[N]] according to Eq. (35) and the
interface separation vector [[u]] using Eq. (34)
2. Calculate equivalent separation δe, and equivalent critical δc
eand maximum δuseparations
according to Eq. (12), and static damage dsusing Eq. (10)
3. Calculate the stabilization matrix Saccording to Eq. (24)
4. Calculate the weighted stress hσiγand shape function derivative matrix Bγ, according to
Eqs. (46) and using the information passed from the UELMAT subroutine
5. Compute and assemble the cohesive element contributions corresponding to the stabilized
part of the tangent matrix and the internal force vector using two-point Gauss integration
6. Define the 3×3plane strain elasticity tensor Dand the stress transformation matrix T.
7. Compute the cohesive element contributions corresponding to the consistency part of the
tangent matrix and the internal force vector using two-point Gauss integration
8. Partition Kconsistency,e8×16 into one 8×8matrix KIIIand four 4×4matrices
KIV,KV,KVI ,KVII, as described in Eq. (47).
9. Assemble partitioned matrices into the global stiffness matrix using the cohesive element
(III) and four dummy elements (IV–VII) (see Fig. 7).
17
whereas a uniform vertical displacement ∆ = 0.1mm is applied at the top edge of the plate, and
traction-free conditions are specified on the left and right edges of the square plate. The Young’s
modulus and Poisson’s ratio of the isotropic linearly elastic material in the bulk elements are as-
sumed as E= 1 N/mm2and ν= 0.2, respectively. We use a 10 ×10 structured square mesh with
an element size of 0.1 mm and the stabilization parameters βn=βτ= 14 N/mm3. The analysis
is conducted under the assumption of small deformations and no interface damage (i.e., ds= 0),
although the bulk applied strain is 10%. The normal traction profile along the horizontal interface
obtained from the standard and stabilized methods for different cohesive stiffness values is shown
in Fig. 9. For the smaller cohesive stiffness value of 100 N/mm3both methods yield oscillation-
free traction profile at the cohesive interface. For the larger stiffness value of 1016 N/mm3, the
standard method exhibits instability resulting in spurious traction oscillations; whereas, the stabi-
lized method does not exhibit any instability.
∆∆
(a) (b)
X1
X2
xx x x x x x x x xx
Figure 8: Square plate with horizontal interface: (a) schematic diagram; (b) finite element mesh
(a) (b)
Figure 9: Square plate with a horizontal interface: normal traction profiles obtained from the standard and stabilized
methods with (a) α0
n=α0
τ= 102N/mm3and (b) α0
n=α0
τ= 1016 N/mm3.
To demonstrate the accuracy of stabilized method for stiff cohesive laws, we report the relative
l2-error (vector norm) in normal traction and separation for different values of initial cohesive
18
Table 1: Square plate with a horizontal cohesive interface: relative l2-error in normal traction and separations from
the standard and stabilized finite element methods for different cohesive stiffness. The bilinear isotropic CZM is used,
wherein the tangential and normal cohesive stiffness are taken to be equal.
Cohesive Stiffness Standard FEM Stabilized FEM
(N/mm3)||εtn||2
||t∗
n||2
||δn||2
||u∗
n||2
||εtn||2
||t∗
n||2
||δn||2
||u∗
n||2
1021.4% 2.1×10−21% 2.1×10−2
1082.5% 2.2×10−81×10−82×10−8
1015 4.6% 2.2×10−15 3.7×10−15 2×10−15
1016 25.7% 1.3×10−16 1.8×10−17 2.4×10−16
stiffness in Table 1. The relative l2-errors are calculated as
||εtn||2
||t∗
n||2
=qPNGP
i=1 (ti
n−t∗i
n)2
qPNGP
i=1 (t∗i
n)2;||δn||2
||u∗
n||2
=qPNGP
i=1 (δi
n)2
qPNGP
i=1 (u∗i
n)2
(48)
where NGP is the total number of Gauss (integration) points on the cohesive interface, interface
separation δnand traction tnare evaluated using equations (6) and (29) respectively, and the refer-
ence values of the traction t∗
nand displacement u∗
nat the horizontal interface are calculated for the
perfectly bonded interface case under the plane strain linear elastic assumption as
t∗
n=∆
LE
(1 −ν2);u∗
n=∆
2.(49)
In the above equation, the normal traction is calculated based on the engineering strain (defined
with respect to the initial length) under the assumption of small deformations and E/(1 −ν2)is
the plane strain elastic modulus [74]. From Table 1 it is evident that the stabilized method ensures
accurate recovery of interface traction compared to the standard method. As the cohesive stiffness
is increased to a large value the interface separation tends to zero (to machine precision) in both
the standard and stabilized methods, but the error in interface traction tends to zero (to machine
precision) only in the stabilized method. We also notice that the computational time with the
standard and stabilized method is comparable. We next investigate the effect of mesh refinement
on the accuracy of traction evaluation for structured meshes. For an initial cohesive stiffness of
108N/mm3, we see that the traction error does not change with mesh refinement in both methods,
as given in Table 2. This further illustrates that the improvement in accuracy with the stabilized
method ensues from the consistent weak formulation in Eq. (26).
4.2. Square plate with inclined interface
In this example, we assess the accuracy of the stabilized method in recovering normal and
tangential tractions on an straight, inclined interface with isotropic and anisotropic CZMs using
the constant strain patch test [73]. We consider a square plate of side length L= 1 mm with a
straight interface inclined at an initial angle of 140.4◦with the global x1(i.e., horizontal) axis, as
19
Table 2: Square plate with a horizontal cohesive interface: relative l2-error in normal traction and separations from
the standard and stabilized finite element methods for different mesh resolutions. The bilinear isotropic CZM is used,
wherein the tangential and normal cohesive stiffness are taken to be 10 8N/mm3.
Mesh size Standard FEM Stabilized FEM
(mm) ||εtn||2
||t∗
n||2
||δn||2
||u∗
n||2
||εtn||2
||t∗
n||2
||δn||2
||u∗
n||2
0.1 2.5% 2.2×10−81×10−82×10−8
0.04 2.5% 2.2×10−81×10−82×10−8
0.02 2.5% 2.2×10−81×10−82×10−8
shown in Fig. 10. We use a 13 ×18 semi-structured mesh so that the interface is divided into 13
∆∆
(a) (b)
X1
X2
xxxxxxxxxxxxxx
Figure 10: Square plate with inclined interface: (a) schematic diagram; (b) finite element mesh
elements. The support conditions, loading and material properties are identical to those discussed
in Section 4.1, and the stabilization parameters βn=βτ= 30 N/mm3. The analysis is conducted
under the assumption of small deformations and no interface damage (i.e., ds= 0). The normal and
tangential tractions along the inclined interface obtained from the standard and stabilized methods
for different cohesive stiffness values are shown in Fig. 11. For the smaller cohesive stiffness
value of 100 N/mm3both methods yield oscillation-free traction profiles at the cohesive interface.
For the larger stiffness value of 1016 N/mm3, the standard method exhibits instability resulting in
spurious traction oscillations; whereas, the stabilized method does not exhibit such an instability.
To illustrate the accuracy of the stabilized method for stiff isotropic CZMs, we report the rela-
tive l2-error in normal and tangential tractions for different values of initial cohesive stiffness using
the isotropic bilinear CZM in Table 3. The reference value of the normal and tangential tractions
t∗
nand t∗
τunder the plane strain and perfectly bonded interface assumptions can be evaluated as
t∗
n=∆
LE
(1 −ν2)sin2θ;t∗
τ=∆
LE
(1 −ν2)sin θcos θ, (50)
where θ= 140.4◦is the angle that the inclined interface makes with the global x1axis in the
undeformed (initial) configuration. From Table 3, it is evident that the stabilized method ensures
20
Figure 11: Square plate with an inclined interface: traction profiles obtained from the standard and stabilized methods
for isotropic cohesive zone models for two different cohesive stiffness values.
accurate recovery of interface traction for large values of cohesive stiffness; whereas, in the stan-
dard method the interface traction error increases with the cohesive stiffness.
We next examine the performance of the standard and stabilized methods for anisotropic CZMs,
wherein the normal and tangential cohesive stiffness values are not equal. It can be seen from Fig.
12, that the standard method yields an oscillation-free traction profile for smaller stiffness values,
but it suffers from instability for larger values with spurious oscillations in the normal traction
profile. In contrast, the stabilized method yields an oscillation-free traction profile regardless of
the choice of cohesive stiffness. In Table 4 we report the accuracy of the stabilized and standard
methods for the anisotropic CZM. We observe that for the assumed normal and tangential stiffness
values, the stabilized method recovers crack-surface traction more accurately compared to the
standard method; the error in traction decreases close to machine precision in the stabilized method
as the stiffness is increased to a very large value.
4.3. Square plate with semicircular interface
In the previous examples, we considered constant strain patch tests with straight interfaces
described by isotropic and anistropic CZMs; in these cases traction oscillations do not appear if
full Gauss integration is used with the standard method, except if the initial cohesive stiffness
values is assumed to be very large (i.e., α0
n≥1015 N/mm3). However, Svenning [31] has shown
21
Table 3: Square plate with inclined interface: relative l2-error in normal and tangential tractions from the standard
and stabilized finite element methods for different cohesive stiffness. The bilinear isotropic CZM is used, wherein the
tangential and normal cohesive stiffness are taken to be equal.
Cohesive Stiffness Standard FEM Stabilized FEM
(N/mm3)||εtn||2
||t∗
n||2
||εtτ||2
||t∗
τ||2
||εtn||2
||t∗
n||2
||εtτ||2
||t∗
τ||2
1021.7% 1.8% 0.7% 0.8%
1082.6% 2.6% 1.1×10−73.4×10−8
1014 3.0% 2.6% 8.3×10−10 1.1×10−10
1016 64.9% 80.2% 9.3×10−11 9.1×10−11
Table 4: Square plate with inclined interface: relative l2-error in normal and tangential tractions from the standard and
stabilized finite element methods for different cohesive stiffness. The bilinear anisotropic CZM is used, wherein the
tangential and normal cohesive stiffness are taken to be different from each other.
Cohesive Stiffness Standard FEM Stabilized FEM
(N/mm3)||εtn||2
||t∗
n||2
||εtτ||2
||t∗
τ||2
||εtn||2
||t∗
n||2
||εtτ||2
||t∗
τ||2
α0
n= 105,α0
τ= 1022.2% 2.2% 0.3% 0.3%
α0
n= 1011,α0
τ= 1072.6% 2.6% 1.1×10−74.9×10−8
α0
n= 1015,α0
τ= 1011 10.5% 2.6% 8.3×10−10 1.1×10−10
that spurious tractions oscillations can appear with curved interfaces described by an anisotropic
CZM under compression, even for moderately large values of cohesive stiffness (e.g., 2–4 orders
of magnitude more than the Young’s modulus). Therefore, in this example, we demonstrate the
ability of the stabilized method to alleviate traction oscillations on a semicircular interface with an
anisotropic CZM. We consider a square plate of side length L= 100 mm with a cohesive interface
of diameter 60 mm, as shown in Fig. 13. A horizontal compressive displacement of ∆=1mm
is applied on the right edge of the plate, the displacement at the left edge is constrained in both
directions, and traction-free condition is specified for the top and bottom edges of the plate. We
assume the plane strain condition and a linear elastic bulk material with a modulus of elasticity
E= 20 GPa, and a Poisson’s ratio ν= 0.2, following the example in [31]. The analysis is
conducted under the assumption of small deformations and no interface damage (i.e., ds= 0).
We first examine the performance of the standard method for isotropic and anisotropic CZMs
using unstructured meshes. We consider a coarse mesh with 240 elements and a fine mesh with
1780 elements with bilinear quadrilateral elements. For the isotropic CZM we take normal and
tangential cohesive stiffnesses α0
n=α0
τ= 1011 N/mm3and for the anisotropic CZM we take α0
n=
1011 N/mm3and α0
τ= 109N/mm3. Fig. 14 shows the normal traction profiles along the cohesive
interface obtained from the standard method. Fig. 14(a) indicates that the standard method yields
a smooth traction profile with the isotropic CZM if the mesh is adequately refined. However, mesh
refinement cannot alleviate spurious traction oscillations with the anisotropic CZM, as evident
from Fig. 14(b). This issue with the standard (penalty-like) method for anisotropic CZMs was
22
Figure 12: Square plate with an inclined interface: traction profiles obtained from the standard and stabilized methods
for anisotropic cohesive zone models for different cohesive stiffnesses.
previously reported in [31].
In Figure 15, we compare the performance of the standard and stabilized methods for isotropic
and anisotropic CZMs using the fine mesh (with 1780 elements). According to Eq. (25), we take
the stabilization parameters βn=βτ= 2 ×105N/mm3. Figure 15(a) and (b) show that for
isotropic CZMs the standard method suffers from numerical instability only for very large values
of cohesive stiffness (i.e., 1018 N/mm3); whereas, the stabilized method yields an oscillation-free
traction profile. Figure 15(c) shows that the stabilized method is able to alleviate the spurious
oscillations in normal traction observed with the standard method for the anisotropic CZM with
α0
n= 1011 N/mm3and α0
τ= 109N/mm3. These results illustrate that the standard method is not
robust when dealing with stiff anisotropic cohesive laws, unlike the stabilized method.
We next evaluate the relative l2-error in interface traction from the standard and stabilized
methods. Assuming that the interface is perfectly bonded for large values of cohesive stiffness
α0
n>108N/mm3, α0
τ>106N/mm3, the exact value of the normal and tangential tractions t∗
nand
t∗
τat interface Gauss points can be calculated using Eq. (50), wherein θis the local orientation
of the interface element that varies along semicircular interface. Table 5 shows that the stabilized
method ensures the accurate recovery of normal traction at the interface for both isotropic and
anisotropic CZMs, unlike the standard method. Even if the cohesive stiffness α0
n= 108N/mm3
and α0
τ= 106N/mm3are 2–4 orders of magnitude larger than the Young’s modulus E= 20 ×103
23
∆
L
d
∆
(a) (b)
x
x
x
xxxxxxxxxxxxxxxxxxxx
x
x
x
X1
X2
Figure 13: Square plate with semicircular interface: (a) schematic diagram; (b) finite element mesh.
Table 5: Square plate with semicircular interface: accuracy of the standard and stabilized finite element methods. We
consider both isotropic and anisotropic CZMs and relative l2-error in normal tractions is reported.
Cohesive Stiffness Standard FEM Stabilized FEM
(N/mm3)||εtn||2
||t∗
n||2
||εtn||2
||t∗
n||2
α0
n=α0
τ= 1082.9% 4.3×10−8
α0
n=α0
τ= 1011 2.9% 4.3×10−8
α0
n= 108,α0
τ= 106117.1% 4.2×10−8
α0
n= 1011,α0
τ= 109117.6% 1.3×10−7
N/mm3, we see that the standard method performs quite poorly with 117 % error in normal traction
and exhibits spurious oscillations (results are identical to Fig. 14b). Noting that in mode II fracture
analysis, an anisotropic cohesive laws with α0
n> α0
tis typically used to enforce the no inter-
penetration condition at the crack surface, the stabilized method can offer a significant advantage
owing to its stability and accuracy.
4.4. Asymmetric double cantilever beam
In all the previous examples, we conducted linear elastic analysis under the assumption of
small deformations and no interface damage (i.e., ds= 0) to illustrate the accuracy of the stabi-
lized method in recovering crack-surface traction. In this example, we will examine the accuracy
of the stabilized method for analyzing mixed-mode delamination crack growth in composite mate-
rials using the asymmetric double cantilever beam (DCB). While the specimen geometry and test
set-up shown in Fig. 16 resembles that in [75, 76], we altered the applied load configuration. The
fixed boundary condition is applied at the right end of the beam, and vertical displacements ∆1
and ∆2are applied at the upper and lower nodes at the left end (with ∆2/∆1= 0.095), to initiate
24
(a) (b)
Figure 14: Square plate with semicircular interface: effect of mesh refinement on normal traction profiles predicted by
the standard method. (a) α0
n=α0
τ= 1011 N/mm3and (b) α0
n= 1011 N/mm3and α0
τ= 109N/mm3.
Table 6: Material properties and cohesive parameters for the asymmetric double cantilever beam. The cohesive pa-
rameters are assumed from [78] for mixed-mode loading conditions.
EνGIC GIIC σmax τmax
(N/mm2) (N/mm) (N/mm) (N/mm2) (N/mm2)
1050.35 4.0 4.0 57 57
the delamination process. Thus, using a displacement-controlled simulation, we capture the soft-
ening portion of the load-displacement curve due to the evolution of damage dsin the interface
elements; however, the mode-mix ratio between mode I and mode II fracture is not a constant,
because the applied loads are not constant. We now perform numerical convergence studies using
a structured square mesh with an element size of 0.125 mm, so that the cohesive process zone
is adequately resolved according to the guidelines described in [77]. The material properties and
cohesive parameters assumed for this test are listed in Table 6, and the stabilization parameters
βn=βτ= 2 ×106N/mm3.
To solve this nonlinear quasi-static fracture problem, we need to linearize the discretized equi-
librium equation using the Taylor series expansion (see Eq. (37)) at a given iteration and applied
load/displacement step; in order to ensure accuracy it is necessary to take the applied displacement
increment small enough within a load/displacement step [79]. In Abaqus, this can be achieved by
prescribing a small displacement increment ˙
∆within the default quasi-static pseudo-time stepping
algorithm. To avoid nonlinear convergence we choose the force residual tolerance Rα
n= 1.0[70,
Chapter 7: Analysis Solution and Control]. Because the weak form and the corresponding dis-
cretized equilibrium equation from standard and stabilized methods are different, we investigate
their accuracy in relation to the choice of displacement increment along with the cohesive stiff-
25
(a) (b) (c)
Figure 15: Square plate with semicircular interface: Normal traction profiles obtained from the standard and stabilized
method for different cohesive stiffness. (a) isotropic CZM with α0
n=α0
τ= 1011 N/mm3, (b) isotropic CZM α0
n=
α0
τ= 1018 N/mm3and (c) anisotropic CZM α0
n= 1011 N/mm3and α0
τ= 109N/mm3
L
H
a0
P1,Δ1
P2,Δ2
Figure 16: Geometry and boundary conditions for the asymmetric double cantilever beam test. The dimensions are: L
= 100 mm, H = 4 mm and a0= 25 mm
ness. In Fig. 17, we plot the load-displacement responses for two different displacement rates
(0.005 mm/step and 0.001 mm/step) and cohesive stiffness values (α0
n=α0
τ= 108N/mm3and
α0
n=α0
τ= 1012 N/mm3). For the smaller stiffness of 108N/mm3(see Fig. 17a), the standard and
stabilized methods predict the same load-displacement curve for ˙
∆=0.001 mm/step; however,
both methods are slightly inaccurate for ˙
∆=0.005 mm/step. We also used a smaller displace-
ment increment ˙
∆ = 0.0005 mm/step (results not shown) and observed that the load-displacement
curves match exactly with those obtained with ˙
∆ = 0.001 mm/step. For the larger stiffness of 1012
N/mm3(see Fig. 17b), the standard and stabilized methods predict the same load-displacement
curve for ˙
∆=0.001 mm/step, but the standard method is significantly inaccurate for ˙
∆=0.005
mm/step and shows oscillations in the softening portion of the load-displacement curve. In con-
trast, the load-displacement curve obtained from the stabilized method for ˙
∆=0.005 mm/step is
reasonably accurate without any oscillations in the softening portion.
To further explore the reason behind the inaccuracy in load-predictions for larger displacement
26
rates, we plot the crack versus displacement curves in Fig. 18 for two cohesive stiffness values
considered above. For the smaller stiffness of 108N/mm3(see Fig. 18a), the standard and stabilized
methods predict the same crack growth behavior for the two displacement rates. However, for
the larger stiffness of 1012 N/mm3(see Fig. 18b), the standard method predicts slower crack
growth for ˙
∆ = 0.005 mm/step, which leads to the inaccurate prediction of softening portion
of the load-displacement response. In contrast, the stabilized method predicts reasonably similar
crack growth behavior for the two displacement rates. We next examine the normal and tangential
traction profiles along the cohesive interface predicted by the standard and stabilized methods for
the larger cohesive stiffness of 1012 N/mm3. In Fig. 19, we plot the traction versus interface length
at an applied displacement ∆=6.4mm for ˙
∆=0.005 mm/step and 0.001 mm/step. In Fig.
19(a), the normal traction profiles from both methods match well for ˙
∆ = 0.001 mm/step; but the
profile predicted by the standard method for ˙
∆ = 0.005 mm/step shows traction oscillations near
the tension peak and plateau region near the compression peak. Because the crack length predicted
by the standard method is smaller, the corresponding traction profile lags behind the other profiles.
The tangential traction profiles in Fig. 19(b) also show similar behavior. Notably, oscillations
in tangential traction can be observed even for ˙
∆=0.001 mm/step in the standard method. In
summary, this study illustrates that the accurate recovery of crack-surface traction by the stabilized
method can mitigate spurious sensitivity of load-displacement curves to displacement increment,
and thus enable the reliable prediction of delamination crack propagation.
(a) (b)
Figure 17: Load versus displacement curves for the asymmetric double cantilever beam test: (a) α0
n=α0
τ= 108N/mm3
and (b) α0
n=α0
τ= 1012 N/mm3
4.5. Double cantilever beam
In this example, we investigate the accuracy of the stabilized method in recovering crack-
face traction for mode-I delamination crack growth using the double cantilever beam (DCB).
We also examine the sensitivity of load-displacement curves to interface cohesive strength and
27
(a) (b)
Figure 18: Crack versus displacement curves for the asymmetric double cantilever beam test: (a) α0
n=α0
τ= 108N/mm3
and (b) α0
n=α0
τ= 1012 N/mm3
Table 7: Material properties and model parameters for the double cantilever beam. The cohesive parameters are
assumed from [78] for mode I loading conditions.
EνGIC σmax
(N/mm2) (N/mm) (N/mm2)
1050.35 0.28 57
mesh/element size. The specimen geometry and test set-up are identical to that in the previous ex-
ample shown in Fig. 16, except for the applied load configuration. Specifically, at the left end of the
beam we apply vertical displacements ∆1and ∆2on the upper and lower nodes with ∆1/∆2=−1
to initiate the delamination process. Thus, using a displacement-controlled simulation, we capture
the softening portion of the load-displacement curve due to the evolution of damage dsin the inter-
face elements. The material properties and cohesive parameters assumed for this test are listed in
Table 7. We choose the displacement increment ˙
∆=0.0001 mm/step, cohesive stiffness α0
n= 108
N/mm3, and and the stabilization parameters βn=βτ= 2 ×106N/mm3for all the simulations.
In Fig. 20, we plot the load-displacement responses for different interface strengths and mesh
sizes along with the linear elastic analytical solution given in [78]. For the fine mesh with 0.125
mm ×1 mm rectangular elements, Fig. 20a shows that the cohesive strength effects the peak
load prediction, and it is important to take the cohesive strength adequately large to ensure a better
match with the analytical solution. For smaller cohesive strengths, the reduced peak load prediction
is a consequence of crack initiation and propagation at smaller applied displacements. Beyond a
certain value of the interface strength (σmax = 57 N/mm2) there is little effect from increasing the
cohesive strength, as the load-displacement curve converges to the analytical solution. However,
increasing the cohesive strength decreases the cohesive process zone size, so a smaller element size
28
(a) (b)
Figure 19: Traction versus interface length curves for the asymmetric double cantilever beam test for α0
n=α0
τ= 1012
N/mm3: (a) normal traction and (b) tangential traction.
is required to accurately recover the crack-face traction. Fig. 20b depicts the effect of mesh size on
the load-displacement response for cohesive strength σmax = 57 N/mm2. Choosing a coarse mesh
(e.g., h= 1 mm) yields a noisy load-displacement curve due to inaccuracies in crack-face traction
and episodic crack growth. Our study suggests that the interface element size h=meas(Γ∗)should
be chosen smaller than 0.25 mm to better capture the softening portion of the load-displacement
curve. This is consistent with the guidelines described in [77] that the cohesive process zone needs
to be resolved with at least three interface elements to ensure sufficient accuracy.
We next compare the normal traction profile along the cohesive interface obtained from the
standard and the stabilized method at an applied displacement ∆=0.67 mm for coarse and fine
meshes. In Fig. 21, the traction profile obtained from the standard method shows spurious oscil-
lations, whereas that obtained from the stabilized method is free of oscillations. We also observe
that the peak traction in the coarse mesh (h= 1 mm) is less that the cohesive strength σmax = 57
N/mm2, whereas that in the fine mesh (h= 0.125 mm) it is equal to the cohesive strength, which
illustrates that a finer mesh is required to accurately capture the crack-face traction. Thus, this
study demonstrates the superior stability of the Nitsche-based, stabilized method compared to the
standard method.
5. Conclusion
In this paper, we proposed a stabilized finite element method for enforcing stiff istropic and
anistropic cohesive laws using zero-thickness interface elements. The stabilized method gener-
alizes Nitsche’s method to cohesive fracture problems and the key advantage is that our method
remains well defined for any arbitrarily large value of cohesive stiffness. We presented several
29
(a) (b)
Figure 20: Load versus displacement curves for the symmetric double cantilever beam test: (a) different cohesive
interface strength and (b) different mesh size
numerical examples demonstrating the stability and accuracy of the proposed method over the
standard (penalty-like) method in two-dimensions. We first determined the numerical accuracy of
the stabilized method in recovering crack surface traction at straight and semi-circular interfaces
using constant strain patch tests. We demonstrated that the traction error in the stabilized method
(measured with respect to the analytical solution for perfectly-bonded interface) approaches ma-
chine precision for large values of cohesive stiffness; whereas, the error increased in the standard
method as the cohesive stiffness was increased. We next evaluated the numerical stability of the
proposed method in alleviating spurious traction oscillations along the interface for stiff isotropic
and anisotropic cohesive laws (i.e, with equal and unequal normal and tangential stiffnesses).
Our numerical results clearly showed the presence of spurious traction oscillations in the stan-
dard method when enforcing anisotropic cohesive laws on curved interfaces under compression
and sliding fracture; whereas, the stabilized method yielded oscillation-free traction profiles and
ensured accurate recovery of crack surface traction, regardless of the choice of cohesive stiffness.
We next simulated mixed-mode delamination crack growth in an isotropic material using the
asymmetric double cantilever beam test configuration. For stiff isotropic cohesive laws, we investi-
gated the sensitivity of load-displacement curves predicted by the standard and stabilized methods
to applied displacement increment (or load step size) and cohesive stiffness. If the displacement
increment is taken small enough ( ˙
∆=0.001 mm/step), then both the standard and stabilized
methods predict the same load-displacement curve. However, for the larger cohesive stiffness and
displacement increment of ˙
∆ = 0.005 mm/step, the standard method is less accurate compared
to the stabilized method (see Fig. 17). Our results indicate that this discrepancy in the standard
method potentially arises from the inaccurate prediction of crack growth behavior (see Fig. 18)
and crack surface traction (see Fig. 19). In contrast, the stabilized method is sufficiently accurate
in predicting peak load and crack growth even for the larger displacement increment ˙
∆ = 0.005
30
(a) (b)
Figure 21: Normal traction versus interface length curves from the standard and stabilized methods for the double
cantilever beam test for α0
n=α0
τ= 108N/mm3: (a) fine mesh (h= 0.125 mm) and (b) coarse mesh (h= 1 mm).
mm/step. Thus, this study illustrated that stabilized method could mitigate the spurious sensi-
tivity of load-displacement results to displacement increment in mixed-mode fracture simulation.
Finally, we investigated the effect of cohesive interface strength and mesh refinement on the load-
displacement response using the mode I double cantilever beam test. We illustrated that the cohe-
sive strength parameter affects the peak-load prediction and choosing a larger value (σmax = 57
N/mm2) ensures a better match with the linear elastic analytical solution obtained from beam the-
ory. We also show that it is necessary to choose the interface element size small enough (according
to the criteria in [26, 77]), in order to obtain accurate load-displacement curve and crack-face trac-
tion profiles. These cohesive fracture simulation studies clearly illustrate the superior stability and
accuracy of the proposed Nitsche-based, stabilized finite element method compared to the standard
finite element method.
Acknowledgements
GG and RD gratefully acknowledge the financial support of the Office of Naval Research –
award #N0014-17-12040 (Program Officer: Mr. William Nickerson). We also thank Prof. Caglar
Oskay at Vanderbilt University for his helpful comments on composite delamination anaylsis.
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