A Stabilized Finite Element Formulation
Remedying Traction Oscillations in Cohesive
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
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
we will verify the accuracy using simple patch tests considering uniaxial tension, compression and
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,
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., ). 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 .
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  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 , 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
. 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 . 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.
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.
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
" # "$% "&
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Figure 1: A schematic diagram of the domain for the linear elastostatic cohesive fracture problem
boundary value problem in absence of body force is :
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
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.
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
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]
0if δe< δc
e< δe< δu
τis the equivalent separation, δc
eare interface parameters correspond-
ing to critical and ultimate separations, respectively, deﬁned as [15, 17]
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
e, when the cohesive elements are completely damaged.
Figure 2: A schematic diagram of the mixed-mode bilinear cohesive law (redrawn from ): (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
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:
[[w]] ·tcdΓ = ZΓn
Upon substituting the cohesive traction-separation relations in normal and tangential directions,
we write the ﬁnal expression for the standard weak form as
n[[un]] + [[wτ]]α0
τ[[uτ]]dΓ = ZΓn
When the initial cohesive stiffnesses 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
. 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
[[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
[[w]] ·Stc([[u]]) dΓ = ZΓ
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
τwithout making the stabilized weak form variationally
inconsistent or ill-posed.
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 . 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.
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).
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
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
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
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.
Figure 4: Rectangular plate loaded in shear: (a) schematic diagram; (b) ﬁnite element mesh.
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
τ= 1014 N/mm3
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
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/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.
Figure 7: Rectangular notched beam: (a) schematic diagram; (b) ﬁnite element mesh.
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
In this work, we illustrated the ability of the stabilized formulation for cohesive zone models,
originally proposed in , 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
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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