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A Stabilized Finite Element Formulation

Remedying Traction Oscillations in Cohesive

Interface Elements

GOURAB GHOSH

1

,

CHANDRASEKHAR ANNAVARAPU

2

and

RAVINDR

A

DUDDU

1

ABSTRACT

The standard finite element implementation of intrinsic cohesive zone models (CZMs) based on the

penalty method exhibits a distinct lack of numerical stability and/or convergence for stiff cohesive

laws

.

This lack of stability is typically observed in the form of spurious oscillations in the normal

and tangential tractions recovered at the cohesive interface

.

In this paper, we will present a robust,

stabilized finite element formulation for CZMs that remedies traction oscillations, thus ensuring

stability and convergence for any value of initial cohesive stiffness

.

A key advantage of the pro-

posed formulation is that it generalizes the Nitsche’s method for modeling cohesive fracture with a

large initial cohesive stiffness, thus enabling the implementation of intrinsic and extrinsic CZMs in

a unified and variationally consistent manner

.

We present several numerical examples to demon-

strate the stability, convergence and accuracy of the proposed formulation in two-dimensions

.

First,

we will verify the accuracy using simple patch tests considering uniaxial tension, compression and

shear loadings

.

Second, we will demonstrate the lack of spurious traction oscillations at cohesive

interfaces of rectangular beams loaded under shear and three-point bending

.

To demonstrate the

stability issues related with the spurious traction oscillation, we consider both isotropic as well

as anisotropic CZMs, wherein the normal and tangential cohesive stiffness values are different.

Our numerical results for high stiffness cases clearly show that the proposed formulation yields

a smooth oscillation-free traction profile and ensures stability, whereas the standard formulation

suffers from instability and/or convergence issues.

___________________

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

Atmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, Livermore,

California.

1

2

INTRODUCTION

Numerical simulation of fracture initiation and propagation in laminated composite materials is

complex due to several plausible damage mechanisms, including ﬁber debonding, ﬁber breakage,

and matrix cracking. Cohesive zone models (CZMs) are often used for modeling fracture propa-

gation involving delamination or debonding at laminate interfaces. The advantage of CZMs is that

they can be implemented within the ﬁnite element method by introducing zero-thickness interface

elements along crack interfaces, whose constitutive behavior is deﬁned by a traction-separation law

or a cohesive law. In general, there are two classes of cohesive laws, namely intrinsic or initially

elastic cohesive laws, and extrinsic or initially rigid cohesive laws. On the one hand, the extrinsic

cohesive zone models may consistently describe fracture initiation at stiff cohesive interfaces, but

are difﬁcult to implement in a legacy ﬁnite element software (e.g., [1]). On the other hand, intrin-

sic CZMs are fairly straightforward to incorporate in a legacy ﬁnite element framework, but are

plagued by numerical issues for stiff cohesive law. Speciﬁcally, the issue is that the ﬁnite element

implementation can suffer from ill-conditioned discrete systems for “stiff” cohesive laws (i.e., co-

hesive stiffness is much greater than the value of bulk elastic modulus) or “anisotropic” cohesive

laws (i.e., different normal and tangential cohesive stiffness), resulting in poor convergence and

spurious oscillations in secondary variables such as interfacial tractions [2–4]. It bears emphasis

that these issues can also arise with extrinsic CZMs during the analysis of stiff behavior under

compressive loading or low-cycle fatigue loading, when unloaded and reloaded in the early stages

of the softening regime of the traction-separation law [5].

In this work, we address the numerical issues associated with the ﬁnite element implementa-

tion of CZMs by generalizing the Nitsche’s method to formulate a robust, stabilized formulation

for treating stiff and/or anisotropic cohesive laws. The Nitsche’s method [6] can be viewed as a

variationally consistent penalty method, with the advantage that the discrete system of equations

are better conditioned provided the stabilization parameters are chosen appropriately. As a result

of the pioneering work of Hansbo [7], the method has become popular for a wide class of interface

(contact) problems [8–10]. A more detailed account of the Nitsche’s method and its application

to various interface problems in computational mechanics can be found in the review article by

[11]. While the Nitsche’s method has been used in the context of contact problems before, to

our knowledge, the extension to cohesive fracture problems is entirely new [12]. The rest of this

paper is organized as follows: ﬁrst, we will brieﬂy introduce the model problem and present the

generalized Nitsche formulation for cohesive laws; second, we will summarize the numerical im-

plementation of the proposed formulation in the commercial software ABAQUS using user-deﬁned

subroutines; third, we will present numerical examples demonstrating the ability of the formula-

tion to remedy spurious traction oscillations; and, ﬁnally, we conclude with a few closing remarks.

MODEL FORMULATION

In this section, we brieﬂy present the details of the stabilized ﬁnite element formulation for co-

hesive fracture. We ﬁrst present the strong form of the governing equations along with the brief

description of the interface cohesive law. We next derive the weak form for the standard (penalty-

like) and stabilized (Nitsche-based) using the standard Galerkin method of weighted residuals.

Strong Form

We deﬁne a domain Ω⊂R2, which is partitioned into two non-overlapping bulk domains Ω1and

Ω2(such that Ω=Ω1∪Ω2) separated by a pre-deﬁned cohesive interface Γ∗(Fig. 1). Both the

bulk domains consist of a homogeneous isotropic linearly elastic material. Boundary conditions

are deﬁned on the parts of the domain boundary Γ≡∂Ωexcluding the interface Γ∗. The parts of

the domain boundary where Dirichlet and Neumann conditions are prescribed are denoted as Γd

and Γn, respectively and ∂Ω=Γd∪Γnwith Γd∩Γn=∅. The outward unit normal to the boundary

∂Ωis denoted by neand unit normal vector associated with the interface boundary Γ∗is denoted

by nand points from Ω2to Ω1(thus n=−n1=n2). The strong form of the linear elastostatic

!

" # "$% "&

'" # ()% (*

()

(*

+,

+-./0 12

(3

"$"&

x1

x2

X

+4

Figure 1: A schematic diagram of the domain for the linear elastostatic cohesive fracture problem

boundary value problem in absence of body force is [13]:

∇·σ=0in Ω,(1)

u=udon Γd,(2)

σ·ne=ton Γn,(3)

tc([[u]]) = σ2·n2=−σ1·n1=f([[u]]) on Γ∗,(4)

where uis the unknown displacement ﬁeld vector, σis the Cauchy stress tensor, tis the prescribed

traction or stress vector on the Neumann boundary Γn,udis the prescribed displacement vector

on the Dirichlet boundary Γd, and tcis the traction on the crack interface. Because the traction

is continuous across the interface, it can be related to the stress tensor on each domain as tc=

σ2·n2=−σ1·n1. The traction tccan also be deﬁned as a function of the interface separation

δ=[[u]] = u2−u1based on an assumed cohesive law. Assuming small displacements, the Cauchy

stress tensor can be deﬁned as

σ=D:,(5)

where Dis the fourth order elasticity tensor and =1

2(∇u+ (∇u)T)is the small strain tensor

deﬁned as the symmetric gradient of the displacement vector.

Cohesive law

The intrinsic cohesive law deﬁning a traction-separation relationship can be introduced by using the

damage mechanics framework, which is detailed in [14, 15]. For simplicity, we consider a bilinear

intrinsic traction separation law that has an initial (increasing) linear elastic portion followed by a

(decreasing) linear softening response (see Fig. 2). To represent the mode I and mode II cohesive

fracture behavior in two dimensions, we employ the normal and tangential coordinate system.

Accordingly, the tangential (tτ) and normal (tn) components of the traction vector tcare related to

the normal (δn) and tangential (δτ) components of the interface separation as

tc([[u]]) = tτ

tn= (1 −ds)α0

τ0

0α0

nδτ

δn,(6)

where α0

nand α0

τrepresent the initial cohesive stiffness in the normal and the tangential directions,

respectively; δτand δnrepresent the components of the interface separation vector in normal and

tangential directions, respectively; dsis the isotropic damage variable describing interface degra-

dation under general mixed-mode loading as given by [16, 17]

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

(7)

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 [15, 17]

1

δc

e

=sα0

ncos I

σmax 2

+α0

τcos II

τmax 2

(8)

1

δu

e

=α0

nδc

e(cos I)2

2GIC +α0

τδc

e(cos II)2

2GIIC (9)

In the above equation, 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. 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.

(a)

Mode I

Mode II

(b)

(1 )

Figure 2: A schematic diagram of the mixed-mode bilinear cohesive law (redrawn from [18]): (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

Weak Form

We follow the standard Galerkin ﬁnite element approximation to obtain the traditional weak form.

The equilibrium equation Eq. (1) is weighted with a test function w, and after applying integration

by parts we get the following expression:

ZΩ

∇sw:D:∇sudΩ−ZΓ∗

[[w]] ·tcdΓ = ZΓn

w·tdΓ,(10)

Upon substituting the cohesive traction-separation relations in normal and tangential directions,

we write the ﬁnal expression for the standard weak form as

ZΩ

∇sw:D:∇sudΩ−ZΓ∗

(1 −ds)[[wn]]α0

n[[un]] + [[wτ]]α0

τ[[uτ]]dΓ = ZΓn

δu·tdΓ.(11)

When the initial cohesive stiffnesses α0

nand α0

τare taken to be large (i.e., stiff intrinsic cohesive

law), ill-conditioning may occur leading to instability and lack of convergence [2, 19].

To alleviate the ill-posedness of the weak form and ill-conditioning of the discretized system

for stiff cohesive laws, we developed a Nitsche-method-based stabilized ﬁnite element formulation

[12]. First, by pre-multiplying both the sides of Eq. (4) with a stabilization matrix Swe obtain

Stc([[u]]) = Sσ2·n2=−Sσ1·n1on Γ∗.(12)

Taking n=n2=−n1, we can write the above equation as

Stc([[u]]) = Shσiγ·non Γ∗,(13)

where the weighted average of stress tensors on either side of the interface deﬁned as

hσiγ= (γ1σ1+γ2σ2)∀γ1+γ2= 1, γ1>0, γ2>0.(14)

Next, by multiplying both the sides of Eq. (13) with the jump in the test function [[w]] and integrat-

ing over the cohesive interface, we get

ZΓ∗

[[w]] ·Shσiγ·ndΓ = ZΓ∗

[[w]] ·Stc([[u]]) dΓ.(15)

Finally, by adding the above Eq. (15) to the standard weak form in Eq. (10), we obtain the

stabilized weak form for the exact same cohesive fracture problem as

ZΩ

∇sw:D:∇sudΩ−ZΓ∗

[[w]] ·(I−S)hσiγ·ndΓ−ZΓ∗

[[w]] ·Stc([[u]]) dΓ = ZΓ

δu·tdΓ

(16)

where Iis the identity matrix, and tc([[u]]) denotes the interface cohesive law. In the above equation,

the second term on the left hand side ensures consistency of the method and the third term on the

left hand side ensures the stability of the method. Evidently, if we select the stabilization matrix

Sto be an identity matrix, the consistency term vanishes and the stabilized weak form in Eq. (16)

reduces to the standard weak form represented by Eq. (10); whereas, if we choose S=0, the stability

term vanishes and the stabilized weak form represents that corresponding to the perfectly bonded

interface case. Proper choice of S(based on the criteria established in [8, 20, 21]) allows us to

take extremely large values of α0

nand α0

τwithout making the stabilized weak form variationally

inconsistent or ill-posed.

NUMERICAL IMPLEMENTATION

We have implemented the proposed method in the commercial FE software ABAQUS through two

user-deﬁned subroutines: UELMAT to complete the bulk/continuum element computations, and

UEL to complete the cohesive/interface element computations. The FORTRAN codes of these sub-

routines will soon be made available through our project website [22]. These subroutines are con-

ﬁgured for bilinear quadrilateral plane strain elements using four-point Gauss integration scheme

and four-noded linear cohesive elements using two-point Gauss integration scheme. Currently, we

have the user element subroutines written for 2D plane strain and stress elements in Abaqus, but

the stabilized method can be extended to 3D elements. An overview of the numerical implementa-

tion is provided below:

The algorithm for UELMAT subroutine is summarized as follows:

1. Compute the stiffness matrix and internal force vector for the bulk element

2. Assemble the stiffness matrix and the right hand side vector for the bulk element

3. Compute stress and shape function derivatives at interface Gauss integration points

4. Store computed stress and shape function derivatives in global modules

The algorithm for UEL subroutine is summarized as follows:

1. Compute the nodal cohesive separation

2. Calculate equivalent, equivalent critical and equivalent maximum separation

3. Compute static damage at interface element Gauss integration points

4. Calculate the stabilization matrix based on the stability criteria [8, 20, 21]

5. Compute the cohesive traction

6. Calculate the averaged stress using the information passed from the bulk element subroutine

7. Compute the weighted shape function derivative matrix using the information passed from

the UELMAT subroutine through global modules

8. Compute the stabilized and consistency terms of the stiffness matrix

9. Partition and assemble consistency part of the stiffness matrix terms using dummy elements

10. Update the stiffness matrix of the user element

11. Compute the internal force of the cohesive element and update the right hand side vector of

the user element

The presence of weighted average of stress and shape function derivatives across the interface

implies that the computation of cohesive element tangent matrices and the residual vectors depends

on the displacement shape functions associated with the nodes of the cohesive element, as well as

those in the two neighboring bulk elements. In our numerical implementation, we calculate these

quantities at the interface Gauss integration points in the UELMAT subroutine for the bulk ele-

ments, and then pass them to the UEL subroutine for the cohesive elements using global modules.

These set of calculations are done separately from the standard loop over bulk Gauss integration

points for assembling the bulk stiffness matrix and right hand side vector. The element tangent

matrix is unsymmetric and the consistency part of the stiffness matrix has the dimension of 8×16,

because the number of rows correspond to the 8 interfacial degrees of freedoms (DoFs) and the

number of columns correspond to the 16 interfacial and adjacent bulk element DoFs. As the four-

noded cohesive interface element has knowledge of only its 8 DoFs in the UEL subroutine, we can

only assemble a stiffness matrix of size 8×8, which is main issue with implementing the stabilized

formulation in Abaqus. However, we resolved this implementation issue by partitioning the 8×16

matrix into one 8×8 matrix and four 4×4matrices and assembling them into the global stiffness

matrix using “dummy” elements, as illustrated in Fig. 3.

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 3: Assembly of the cohesive element matrix into the global tangent matrix in Abaqus. This requires the creation

of four dummy elements (IV-VII) for each cohesive element (III) along with its two adjacent bulk elements (I and II).

NUMERICAL EXAMPLES

In this section, we ﬁrst perform patch tests to assess the ability of the proposed stabilized ﬁnite

element formulation to enforce perfect bonding at the cohesive interface before damage initiation,

as usually deﬁned in an extrinsic traction-separation law. Next, we simulate two benchmark tests

for a rectangular plate loaded in shear and a rectangular notched beam with straight cohesive in-

terfaces to illustrate the ability of the formulation to alleviate spurious traction oscillations. In all

Table I: THE ACCURACY OF THE STABILIZED METHOD TO ENFORCE PERFECTLY BONDED INTERFACE

IN THE UNIAXIAL TENSION PATCH TEST. AS THE COHESIVE STIFFNESS TENDS TO INFINITY THE

INTERFACE SEPARATION GOES TO MACHINE PRECISION.

Cohesive Stiffness 1031051081015 1020 10100

(N/mm3)

Displacement 96.61 22.17 2.84×10−22.85×10−91.39×10−14 9.71×10−14

error (%) at node 6

these simulation studies, we assume linearly elastic isotropic bulk material behavior.

Patch Test for Veriﬁcation

We conducted 2D patch tests by considering a computational domain deﬁned by a unit square (1

mm ×1 mm) consisting of two bulk elements and one cohesive element as shown in Fig. 3. We

ﬁrst performed a uniaxial tension test by applying a vertical displacement ∆ = 0.05 mm at the

upper two nodes (i.e., nodes 7 and 8), while constraining the bottom two nodes (i.e., nodes 1 and

2) using pinned and roller boundary conditions. The Young’s modulus and Poisson’s ratio of the

isotropic elastic material in the bulk elements are assumed to be 105N/mm2and 0.35, respectively.

We calculated the percentage error between the computed vertical displacements at the middle

nodes (i.e., nodes 3, 4, 5, 6) for different values of initial cohesive stiffness and the corresponding

theoretical vertical displacement for inﬁnite cohesive stiffness. The error values reported in Table

I illustrate the accuracy of the proposed formulation in enforcing a perfectly bonded interface

tends to machine epsilon (double precision) as the cohesive stiffness value is increased to large

value. Even for an extremely high value of cohesive stiffness (i.e., 10100 N/mm3) the stabilized

formulation guarantees convergence; whereas, the standard formulation fails to converge beyond a

moderately high value of cohesive stiffness (i.e., 1015 N/mm3).

We also conducted uniaxial compression and simple shear tests using the same set up as before,

with the only difference being that the direction of applied vertical displacement was reversed at

the upper two nodes (i.e., nodes 7 and 8) for compression test, and horizontal displacements were

applied at these nodes for shear test. We calculated the percentage error between the computed ver-

tical and horizontal displacements at the middle nodes (i.e., nodes 3, 4, 5, 6) for different values of

initial cohesive stiffness and the corresponding theoretical vertical displacement for inﬁnite cohe-

sive stiffness. The obtained error values are almost identical to those reported in Table I. We have

also observed that for high values (e.g., 1020 N/mm3or 1030 N/mm3) the standard formulation fails

to converge, whereas the proposed formulation still yields an accurate solution. In summary, these

patch tests clearly demonstrate that the stabilized formulation ensures convergence even for very

large cohesive stiffness values, thus enabling us to enforce stiff cohesive laws without encountering

any numerical instability.

Rectangular Plate Loaded in Shear

We considered a 16 mm ×5 mm rectangular plate with a vertical interface, as shown in Fig. 4.

A vertical displacement of 1 mm is applied on the right edge of the plate, and both the horizontal

and vertical displacements are constrained on the left edge. The Young’s modulus and Poisson’s

ratio of the isotropic linearly elastic material in the bulk elements are assumed to be E= 1 N/mm2

and ν= 0.2, respectively, for illustration purposes. In Fig. 5, we plot the normal and tangential

tractions at the cohesive interface for large values cohesive stiffness (α0

n=α0

τ= 1014 N/mm3)

obtained from the standard and stabilized formulations. Clearly, the traction proﬁles obtained from

the stabilized formulation are free of any spurious oscillations, unlike the standard formulation.

We note that the standard formulation fails to converge for extremely large values of cohesive

stiffness (e.g., α0

n=α0

τ= 1025 N/mm3), showing the lack of robustness. In contrast, the stabilized

formulation guarantees convergence and stability for any value of cohesive stiffness. Fig. 6 shows

the normal and tangential traction obtained from the standard and the proposed formulation for the

anisotropic cohesive zone model (α0

n= 1014 N/mm3and α0

τ= 1013 N/mm3). Clearly, the standard

formulation yields oscillations in normal and tangential traction proﬁles; whereas the stabilized

formulation yields an oscillation-free traction proﬁle, illustrating its stability.

∆

x

x

x

x

x

x

x

x

x

x

x

x

∆

(a) (b)

Figure 4: Rectangular plate loaded in shear: (a) schematic diagram; (b) ﬁnite element mesh.

(a) (b)

Figure 5: Simulation results for the rectangular plate in shear obtained from the standard and stabilized formulations

for an isotropic cohesive zone model: (a) Normal and (b) tangential traction proﬁle for α0

n=α0

τ= 1014 N/mm3

(a) (b)

Figure 6: Simulation results for the rectangular plate in shear obtained from the standard and stabilized formulations

for an anisotropic cohesive zone model: (a) Normal and (b) tangential traction proﬁle for α0

n= 1014 N/mm3and α0

τ=

1013 N/mm3

Rectangular Notched Beam

We considered a 450 mm ×100 mm rectangular beam with a vertical interface, as shown in Fig.

7. A vertical displacement of 10 mm is applied at the midpoint of the top edge of the beam, and

the bottom two corner nodes are constrained using pinned and roller boundary conditions. The

Young’s modulus and Poisson’s ratio of the isotropic linearly elastic material in the bulk elements

are assumed to be E= 2 ×104N/mm2and ν= 0.2, respectively. In Fig. 8, we plot the normal

and tangential tractions at the cohesive interface for cohesive stiffness values of (α0

n=α0

τ= 108

N/mm3) obtained from the standard and stabilized formulations. We note that even for moderately

large values of cohesive stiffness (i.e., 3–4 orders of magnitude larger than Young’s modulus) the

standard formulation yields an oscillations in traction proﬁle at the ends of the cohesive interface,

illustrating the instability. Clearly, the traction proﬁles obtained from the stabilized formulation

are free of any spurious oscillations, unlike the standard formulation.

∆∆

(a) (b)

x

x

x

x

x

x

x

x

x

Figure 7: Rectangular notched beam: (a) schematic diagram; (b) ﬁnite element mesh.

(a) (b)

Figure 8: Simulation results for the notched beam obtained from the standard and stabilized formulations for an

isotropic cohesive zone model: (a) Normal and (b) tangential traction proﬁle for α0

n=α0

τ= 108N/mm3

Conclusion

In this work, we illustrated the ability of the stabilized formulation for cohesive zone models,

originally proposed in [12], for remedying traction oscillation in interface elements described by

stiff cohesive laws. The proposed formulation generalizes the Nitsche method for cohesive fracture

problems and allows us to use extremely large values of initial stiffness, thus providing a uniﬁed

way to treat intrinsic and extrinsic cohesive zone models in a variationally consistent and stable

manner. We performed numerical simulation studies that demonstrate the superior convergence

behavior of the proposed formulation over the standard formulation based on the standard penalty

method. The simulation studies also highlighted the lack of robustness and numerical instability

issues associated with the traditional formulation and the efﬁcacy of the proposed formulation in

eliminating them. Our future work is focused on developing user subroutines for 3-D fracture

propagation in composites subjected to monotonic and cyclic loading, including high-cycle fatigue

delamination.

Acknowledgements

We gratefully acknowledge the funding from our sponsors: GG and RD were supported by the

Ofﬁce of Naval Research – award #N0014-17-12040 (Program Ofﬁcer: Mr. William Nickerson).

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