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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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