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A Stabilized Finite Element Method for Modeling Mixed-mode Delamination of Composites


Abstract and Figures

Delamination of composite materials is commonly modeled using intrinsic cohesive zone models (CZMs), which are generally incorporated into the standard finite element (FE) method through a zero-thickness interface (cohesive) element; however, intrinsic CZMs exhibit numerical instabili-ties when the cohesive stiffness parameters is assumed to be large relative to the elastic stiffness of the composite material. To address this numerical instability issue, we propose a stabilized finite element method by combining the traditional penalty method with the Nitsche's method that is equally effective for any specified initial stiffness of the cohesive (traction-separation) law. The key advantage of the proposed method is that it generalizes the Nitsche's method to any traction-separation law with arbitrary large values of initial stiffness and provides a unified way to treat cohesive fracture problems in a variationally consistent and stable manner. We implemented the stabilized method in the commercial finite element software Abaqus via the user element subrou-tine and simulated benchmark tests for mode I and mixed-mode delamination in isotropic materials to establish the viability of the approach. Ongoing work is aimed at extending the method to model delamination in transversely isotropic laminated composites.
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Title: A stabilized finite element method for modeling mixed-mode delamination of compos-
Authors: Gourab Ghosh1
Chandrasekhar Annavarapu 2
Stephen Jim´
Ravindra Duddu1
1Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee.
2Atmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, Livermore, California.
Delamination of composite materials is commonly modeled using intrinsic cohesive zone models
(CZMs), which are generally incorporated into the standard finite element (FE) method through a
zero-thickness interface (cohesive) element; however, intrinsic CZMs exhibit numerical instabili-
ties when the cohesive stiffness parameters is assumed to be large relative to the elastic stiffness of
the composite material. To address this numerical instability issue, we propose a stabilized finite
element method by combining the traditional penalty method with the Nitsche’s method that is
equally effective for any specified initial stiffness of the cohesive (traction-separation) law. The
key advantage of the proposed method is that it generalizes the Nitsche’s method to any traction-
separation law with arbitrary large values of initial stiffness and provides a unified way to treat
cohesive fracture problems in a variationally consistent and stable manner. We implemented the
stabilized method in the commercial finite element software Abaqus via the user element subrou-
tine and simulated benchmark tests for mode I and mixed-mode delamination in isotropic materials
to establish the viability of the approach. Ongoing work is aimed at extending the method to model
delamination in transversely isotropic laminated composites.
Cohesive zone models (CZMs) were first proposed by [1],[2] and are widely used for modeling
fracture and fatigue crack growth. The key advantages of CZMs are that they do not require the
presence of a initial (or pre-existing) crack and account for the finite size of the fracture pro-
cess zone (FPZ) at the crack tip, unlike the linear elastic fracture mechanics approaches. There-
fore, CZMs are suited for modeling delamination of composite materials, where the FPZ may be
greater than the characteristic length scales. One of the ways to implement cohesive zone mod-
els in the finite element (FE) framework is using zero-thickness interface elements, wherein the
interface (or cohesive) elements are placed along the probable crack path and their constitutive
(traction-separation) behavior is defined by so-called “cohesive laws”. There are two classes of
cohesive laws, namely intrinsic traction-separation laws or initially elastic cohesive laws, and ex-
trinsic traction-separation laws or initially rigid cohesive laws (see Figure 1). The major difference
between the intrinsic and the extrinsic traction separation laws is the presence of the initial elastic
curve [3]. In the case of intrinsic laws, the traction is assumed to gradually increase with the sep-
aration and after reaching a maximum value, it decreases monotonically till the separation reaches
the ultimate separation value (i.e., where complete de-cohesion is assumed to occur). Whereas, in
extrinsic laws, it is assumed that only after traction reaches a finite cohesive strength, the separation
starts and the traction decreases monotonically with the increase in separation.
(a) Intrinsic traction-separation law (b) Extrinsic traction-separation law
Figure 1: Intrinsic and extrinsic traction separation laws
The numerical implementation of extrinsic traction separation law is challenging [4] because
advanced data structures are required to store the finite element discretization. Furthermore, paral-
lelization of finite element codes in conjugation with extrinsic laws is not a trivial task due to the
change of the mesh topology with the advancement of cracks. Although researchers have proposed
several ways (e.g., topology-based data structures [5]; scalable parallel implementation [6]) to alle-
viate the above-mentioned challenges, the complexities in its implementation are key deterrent for
its widespread use. Intrinsic cohesive laws, on the other hand are easier to implement in a FE code;
however, they suffer from the artificial compliance [7] due to elasticity of the cohesive law. This
issue can be solved to some extent by restricting the time step to an extremely small value [8], but
this may result in an impractically high computation cost. Another approach to solve this problem
is to use very high initial elastic slope for the intrinsic law [7], but that causes an ill-conditioning of
the tangent stiffness matrices. Thus, following a conventional approach to treat above-mentioned
problems within a finite element framework poses numerical challenges, and has motivated re-
searchers to come up with novel approaches to overcome the disadvantages posed by the intrinsic
traction-separation laws. For example, a hybrid discontinuous Galerkin (dG) and extrinsic traction-
separation law was proposed by [9]; in this approach, interface elements are inserted between bulk
elements at the beginning of the analysis and continuity during the elastic regime is maintained
in a weak manner by a dG formulation, but upon onset of failure the extrinsic traction separation
law replaces the dG formulation. However, it has been recognized [4] that implementing extrinsic
traction-separation law in conjunction with dG method in commercial software (e.g., Abaqus) is
extremely difficult. Thus, this hybrid approach is effective in overcoming some disadvantages of
the intrinsic CZMs, but it is not easy to implement them in commercial codes.
Directed at achieving the same goal of removing the artificial compliance issue associated
with the intrinsic laws, a continuum approach was proposed by [10]. Two of the major types of
the continuum approaches for enforcing continuity weakly at the interface are: Lagrange multi-
plier methods and discontinuous Galerkin (dG) methods. The work of [10] was motivated by an
augmented Lagrange multiplier-based mixed interface element approach proposed by [11]. In La-
grange multiplier method, it is challenging to construct a stable Lagrange multiplier space. It has
been shown by [12] that for embedded finite element methods it is difficult to find a stable Lagrange
multiplier. A stable choice of Lagrange multipliers is important from the standpoint of removing
artificial oscillations in the interfacial traction. The second key alternative, that is, dG method
originated from the Nitsche’s method [13]. The Nitsche’s method can be seen as a variationally
consistent penalty method. In the penalty method (introduced in [14]), Dirichlet constraint at the
interface is enforced by introducing a spring at the interface, and a better approximation for the
Dirichlet constraint can be obtained with an increase in penalty parameter (i.e., the stiffness of the
spring/ slope of the traction separation law). The interfacial constraints are achieved exactly when
the stiffness approaches to infinity, but this implies that the method is variationally inconsistent.
Also, a very high value of stiffness results in an ill-conditioned system of equations. Whereas,
Nitsche’s method not only eliminates the instability issues (evident from [15]) related with the
standard penalty methods by adding consistency terms ([16]), but also yields a well-conditioned
system of discrete equations if the method parameters are chosen appropriately. This method has
been used for solving a wide class of interface problems in an efficient way ([17], [18]). A compre-
hensive review of the classical Nitsche’s method and it’s application to interface problems can be
found in [19]. Recently a precise definition of the weights and a closed form analytical expression
of the stabilization parameter was proposed by [18]; this weighted Nitsche method provides much
accurate results in comparison to the standard Nitsche method. Thus, it is evident that although
the Nitsche’s method-based consistent penalty and dG approaches are well established for treating
embedded interface problems, relatively little work has been done in extending these methods to
generalized intrinsic cohesive laws.
In this article, we will discuss a stabilized finite element method for alleviating the artificial
compliance issue inherent to the intrinsic cohesive law with very high value of cohesive stiff-
ness. The key advantage of the proposed method is that it generalizes the Nitsche’s method to any
traction-separation law with arbitrary large values of initial stiffness and provides a unified way to
treat cohesive fracture problems in a variationally consistent and stable manner. The rest of this
article is organized as follows: in the Section , we introduce the model problem and the associated
variational formulation. In section 3, we discuss the spatial discretization followed by numerical
examples demonstrating the accuracy and efficacy of the approach in Section 4. Finally, the last
section provides a summary and some concluding remarks.
In this section, we present details of the stabilized Nitsche formulation for treating general cohesive
laws. We first explain the notation for variables and the problem domain, followed by a discus-
sion of the strong form and the weak form of the governing equations for linear elastostatics and
cohesive fracture. We present the model equations in indicial notation with Einstein’s summation
convention and reserve the right superscript for exponents (italicized) or descriptors (unitalicized).
Domain Description
We define a domain R2, which is partitioned into two non-overlapping bulk domains m
(where m=1,2 and =12) separated by an embedded crack surface Γ(Fig. 2). Both the
bulk domains consist of homogeneous, linear, isotropic, and elastic material. Dirichlet and Neu-
mann boundary conditions are defined on the parts of the domain boundary (Γ) excluding
the embedded interface boundary (Γ). The parts of the boundary where Dirichlet and Neumann
conditions are defined are denoted as Γm
dand Γm
n, respectively. The unit normal to the boundary of
each subdomain, nm, points outwards from the domain m.
Figure 2: Domains 1and 2separated by a shared boundary Γ. The Dirichlet boundaries (Γ1
d) and the Neumann
boundaries (Γ1
n) are as shown. The complementary part of the boundary is traction free. The normal to the boundary
of each subdomain, nm, points outwards from the domain mas shown
Strong Form
The governing equations of equilibrium without body force in each of the domains are given by:
i j,j=0 in m,(1)
ion Γm
i j nm
ion Γm
where σm
i j and um
idenote the components of the stress and displacement fields in the domain m,
respectively, and nm
jdenotes the components of the unit outward normal. On the Dirichlet portion
of the boundary, the displacement is fixed to the prescribed field ¯um
iand on the Neumann portion
of the boundary, prescribed traction is denoted by hm
i. The traction field tm
ion the embedded crack
interface can be obtained by projecting the stress from each domain and is related to the interface
separation [[ui]] = u2
iaccording to an assumed traction-separation law, that is,
i j nm
j=f([[ui]]) on Γ,(4)
To represent the mode I and mode II cohesive fracture behavior in two dimensions, we use the
normal and tangential coordinate system (nm,τm). Accordingly, the tangential (tτ) and normal (tn)
components of the traction vector are defined by:
where αnand ατrepresents the cohesive stiffness in the normal and the tangential direction, re-
spectively. The force balance on the interface is given by:
Equations (7) and (8) can used to represent general cohesive laws by using the damage mechanics
framework described in [20, 21, 22, 23]; herein, we consider a bilinear intrinsic traction separation
law. The constitutive equations for the linear, elastic, isotropic bulk domains are given by:
i j =Cm
i jkl εm
kl =Cm
i jkl um
(k,l)in m,(10)
where Cm
i jkl ,εm
kl and um
(k,l)denote the fourth-order elasticity tensor, the second-order strain tensor,
and the symmetric gradient of the displacement field, respectively.
Weak Form
In this section, we follow the weighted residual approach [18] and define the solution spaces U=
U1×U2and trial spaces W=W1×W2respectively, such that:
ion Γm
Following the standard finite element approach, we get
i j d
By substituting expressions for tnand tτfrom Equations (7) and (8) in the above equation, we get
i j d+ZΓ
([[wn]]αn[[un]] + [[wτ]]ατ[[uτ]])dΓ=ZΓn
Equation (14) is the standard weak form for the penalty method. It is evident that as αn
and ατ, solving Equation (14) is an ill-posed problem. To alleviate the ill-posedness of the
weak form, we adopt a Nitsche’s method-based stabilized FE approach. We begin by re-scaling the
normal and tangential traction components and followed by some algebraic manipulations arrive
at the final stabilized weak form as given by
w(i,j)σi jdZΓ
(( αn
)[[wn]]pγ+ ( ατ
)[[wτ]] fγ)dΓ
[[wn]][[un]] + ατβτ
where pγand fγare the weighted interfacial pressure and shear, respectively, as defined by
j<σi j >γn2
j<σi j >γn2
j; (16)
and <σi j >γ=γ1σ1
i j +γ2σ2
i j represents a weighted average of stress across the interface. The
weights γ1and γ2are positive and satisfy the condition γ1+γ2=1. Note that Equation (15) is well-
defined even as αnor ατ. On the other hand, if βnand βτ, we get a standard penalty
weak-form defined in Equation (14).
We discretized the domain into bilinear plane strain quadrilateral bulk or continuum elements
and introduce zero-thickness cohesive elements at the interface boundary. The approximated dis-
placement field is defined as
where Nmis the element shape function matrix and amdenotes the element vector containing the
displacement degrees of freedom (DOFs). The displacement jump at the interface [[u]] is given by
[[u]] = u1u2=N1a1N2a2,(18)
where am(m=1,2) denotes the nodal displacement vector of the continuum subdomain m(m=1,2)
adjacent to Γ. The shape function matrix Nmcan be represented as
where Nm
J(J=1,2,3,4) are the shape functions of four-noded quadrilateral bulk elements. Af-
ter we introduce the discretized form for the approximation spaces into the variational form in
Equation (13), it leads us to the following discrete equation of equilibrium in the residual form
R(u) =fext (fb
int +fc
Note that the residual vector contributes to the RHS term in the Abaqus UEL subroutine. Neglect-
ing body forces, fext is obtained by assembling the element contributions from traction boundary
conditions on the Neumann boundary
fext =
NmThmdΓefor m=1,2.(21)
indicates the matrix (or vector) assembly of the global system from the element matrices
(or vectors) in entire computational domain. The internal bulk force vector fb
int is assembled as
int =
edefor m=1,2,(22)
where Bmis the strain-displacement relationship matrix, and Cmis the elasticity matrix in Voigt
notation. The strain-displacement relationship matrix is given by
where Nm
J,1denotes the derivative of the shape function Nm
Jwith respect to x1in m. The cohesive
element contribution to the internal force vector fc
int is assembled as
int =fstabilized +fconsistency =ZΓe
NTSt dΓe+ZΓe
where Sis the 2 ×2 stabilization matrix, Iis the 2 ×2 identity matrix, t= [tτ,tn]Tis the cohesive
traction vector that is a function of the displacement jump vector [[u]],Tis the stress transformation
matrix and σγis the weighted stress vector in Voigt notation for in-plane stress components.
Let us now define the element tangent matrix Kthat contributes to the AMATRX term in the
Abaqus UEL subroutine. The element tangent matrix is obtained by assembling the contributions
of the bulk and cohesive tangent matrices, that is,
BTCmBdefor m=1,2 (26)
Kc=Kstabilized +Kconsistency =
[[N]]TSM[[N]] dΓe+
In the above equation: the jump in shape function matrix [[N]] is represented as follows
[[N]] = N10N20N20N10
where Nm(m=1,2) represents the linear shape functions evaluated at the interface Gauss points
(GPs); the weighted shape function gradient matrix Bγis given by
where B+and Bare matrices containing gradient of the shape functions calculated at the interface
GPs from the neighboring bulk domains of the cohesive element; the stress transformation matrix
Tis defined as
S2C22CS ,(30)
where Cand Srepresent cosθand sin θ, respectively, and θis the inclination of the cohesive
element with the x1coordinate direction; the cohesive stiffness matrix Mand the stabilization
matrix Sare defined as
Figure 3 gives an overview of the implementation of the proposed approach in commercial FE
software Abaqus. The presence of weighted average of stress and shape function derivatives across
the interface implies that the computation of cohesive element tangent matrices and the residual
vectors depends on the displacement shape functions associated with their nodes as well as those
in the two neighboring bulk elements. In our scheme, we calculate these quantities at the interface
GPs in the UELMAT subroutine for the bulk elements, and then pass them to the UEL subroutine
for the cohesive elements using global modules. These set of calculations are done separate from
the standard loop over bulk GPs for assembling the bulk stiffness matrix and right hand side (RHS)
vector. The element tangent matrix is unsymmetric and Kconsistency has the dimension of 8×16
(the number of rows correspond to the interfacial degrees of freedoms (DoFs) and the number of
columns correspond to the interfacial and adjacent bulk element DoFs) for the setup (Fig. 4). As
a cohesive interface element has only four nodes associated with it and two DoFs defined at each
of them, it can only assemble a stiffness matrix of size 8×8. To resolve this implementation issue,
we use “dummy” elements (elements IV-VII in Fig. 4) in the UEL subroutine that facilitate the
partition and assembly of the stiffness matrix in Abaqus.
In this section, we first verify the proposed stabilized finite element formulation, by performing
uniaxial tension test and comparing numerically obtained solution with the analytical solution.
Next, we simulate mode I and mixed mode bending tests using the proposed methodology to
demonstrate the its applicability to large scale complex problems. All simulations are performed in
two dimensions assuming plane strain conditions. We assume linear, elastic, and isotropic material
Figure 3: Abaqus flowchart showing interaction between UELMAT and UEL
1 2
3 4
5 6
7 8
3 4
7 8
5 6
7 8
1 2
3 4
1 2
5 6
1 2
3 4
5 6
7 8
3 4
5 6
Figure 4: Bulk (I and II), cohesive (III), and dummy (IV-VII) elements
behavior for the simulations. Table I shows a summary of the material properties considered for
each of the test cases.
Uniaxial Tension Test
To perform the uniaxial tension test, a vertical displacement (δ) is applied at the upper two nodes
(i.e., nodes 7 and 8) of the model (leftmost diagram in Fig. 4) , and the bottom nodes (i.e., nodes
1 and 2) are constrained using pinned/roller boundary conditions. The computational domain con-
sists of two bilinear quadrilateral plane strain elements (CPE4), and a user defined zero thickness
cohesive spring element at the interface. To establish the accuracy of the formulation, the differ-
ence between the computed values of the vertical displacements at the middle nodes (i.e., nodes
3, 4, 5, 6) and the corresponding theoretical values for the infinitely stiff case is evaluated. Our
numerical results reported in Table II indicate that the proposed formulation guarantees stability
and accuracy (close to machine precision) for large values of cohesive stiffness (i.e., for cohesive
Material parameter E νGIC GIIC σmax τmax
(units) (N/mm2) (N/mm) (N/mm) (N/mm2) (N/mm2)
Patch test 1.00E+05 0.35 0.28 - 5.7 -
Mode I 1.00E+05 0.35 0.28 - 5.7 -
Mode II 1.00E+05 0.35 - 4 - 57
Mixed mode 1.00E+05 0.35 4 4 57 57
Cohesive Stiffness (αn) (N/mm) % Error
1.00E+03 96.61
1.00E+05 22.17
1.00E+08 2.84E-02
1.00E+12 2.85E-06
1.00E+15 2.85E-09
1.00E+20 1.39E-14
1.00E+100 9.71E-14
stiffness up to 10100 N/mm). As expected the error between the computed and theoretical values
decreases with an increase in the cohesive stiffness.
Mode I: Double Cantilever Beam (DCB) Test
Figure 5: Geometry and boundary conditions for the mode I double cantilever beam (DCB) test. The dimensions are:
L=100mm,H=4mmanda0= 25 mm
Fig. 5 shows the set up of the double cantilever beam test. Fixed boundary condition is ap-
plied at the right end of the beam. A displacement is applied at the upper and lower nodes at the
left end to initiate mode I delamination and the corresponding load is recorded. The simulation
is displacement controlled so as to capture the softening portion of the load-displacement curve.
The computational mesh consists of bilinear quadrilateral elements (CPE4) and user-defined zero
thickness cohesive elements. The load vs upper left node displacement curves for the isotropic ma-
terial from the traditional CZM approach, proposed stabilized approach, and the analytical LEFM
solution ([24]) are shown in Fig. 6. It can be seen in Fig. 6 that:
1. In traditional cohesive zone formulations, if the initial cohesive stiffness is very high, traction
oscillations are commonly observed as a result of numerical instability in the formulation.
2. The proposed formulation alleviates numerical instability, consequently, the post-peak load-
displacement curve is free of oscillations for any choice of initial cohesive stiffness.
3. The initial portion of the load-displacement curve obtained from proposed method shows an
excellent match with the analytical curve.
4. The peak load obtained from the numerical simulation approaches the analytical (i.e., LEFM)
peak load value as the cohesive strength is increased.
(a) Traditional CZM formulation (b) Stabilized finite element formulation
Figure 6: Load vs. displacement curves for an isotropic material from the mode-I delamination:
(a) traditional CZM formulation shows instability for large stiffness values; (b) the stabilized for-
mulation alleviates instabilities for large stiffness values.
Mixed Mode: Mixed Mode Bending (MMB) Test
The MMB test setup is shown in Fig. 7 (proposed by [25] as an alternative to the original configu-
ration proposed in [26]). The beam of length 2L is simply supported at the lower left and right end
nodes. The forces applied on the upper and lower arm are obtained from superposition of mode I
and mode II are given as:
where c is the length parameter that decides the ratio between the two forces, and thus defines the
mixed mode ratio. The computational mesh for the beam is similar to that used for mode I. The
load (Pu) vs displacement (δ) curves for the isotropic material from the traditional CZM, proposed
stabilized method and the analytical LEFM solution [24] are shown in Fig. 8. Displacement control
is used for the simulation in this case as well. The results reaffirm the applicability of the proposed
stabilized approach.
Figure 7: Geometry and boundary conditions for the mixed mode bending (MMB) test. The
dimensions are: L = 100 mm, H = 4 mm and a0= 25 mm
In this work, we propose a unified formulation based on the Nitsche method that is equally ef-
fective for any specified stiffness of a cohesive law. We showed that when the cohesive stiffness
approaches zero, the formulation collapses to a traditional finite element formulation with the cohe-
sive law enforced as a Neumann boundary condition; on the other hand, as the stiffness approaches
a large value, the proposed approach becomes identical to that of a standard Nitsche method and the
cohesive law is enforced as a kinematic constraint. Thus, the key advantage is that it generalizes the
Nitsche approach to a traction-separation law of any arbitrary initial stiffness and provides a unified
way to treat such problems in a variationally consistent and stable manner. The stabilized finite
element formulation would naturally extend to general nonlinear forms of traction-separation rela-
tionships, although we only use the bilinear cohesive law. We performed several numerical studies
to demonstrate the advantages of the proposed approach over the standard CZM approach through
(a) Traditional CZM formulation (b) Stabilized finite element formulation
Figure 8: Load vs. displacement curves for the mixed mode bending test. Similar to the results
of mode-I test, traditional CZM approach shows instability for large stiffness but the proposed
stabilized approach is free of that
two benchmark problems: mode-I and mixed-mode delamination tests. The load-displacement
curves obtained from these tests for traditional CZM formulation show instability and oscillation
for large stiffness values, whereas, the stabilized formulation results are free of any numerical in-
stabilities or oscillations. Moreover, results obtained from the stabilized formulation show much
better agreement with the analytical solution in comparison to the traditional CZM formulation.
Currently, the formulation is only implemented assuming linear isotropic elastic behavior in the
bulk material domain and our future work will focus on extending the proposed method to model
static delamination in transversely isotropic laminated composites. Another direction of future
work is to model high cycle fatigue delamination of composites.
GG and RD gratefully acknowledge the funding support from the Office of Naval Research award
#N0014-17-12040 (Program Officer: Mr. William Nickerson)
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... 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 implementation of the proposed formulation in the commercial software ABAQUS using user-defined subroutines; third, we will present numerical examples demonstrating the ability of the formulation to remedy spurious traction oscillations; and, finally, we conclude with a few closing remarks. ...
... 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 S we obtain ...
... 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. ...
... Based on previous equations, the stiffness of the normal and transvers displacement components of loads applied becomes as showing in equations (10) and (11), respectively [17]. ...
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One of the main failure modes of laminate structures is interfacial failure by interlaminar failure or debonding. Interlaminar failure can be simulated with a model called cohesive zone approach A key ingredient of a cohesive zone approach is a traction-separation law that describes the softening in the crack zone near the tip of interlaminar failure. This simulation utilized the implementation of a cohesive zone approach with traction-separation laws, which implemented within the thin elastic layer feature of the Solid Mechanics interface in a COMSOL Multiphysics analysis software’s on a sample of composite material APC-2/AS-4. The capabilities of the cohesive zone approach to predict mixed mode effect in the beginning and propagation of the crack are demonstrated in a model of mixed-mode bending test. Both load and displacement are measured at the crack interface. The maximum load that carried by the sample used was 257.8 N. Whereas the maximum displacement was 6 mm as a load-displacement curve was shown.
... Currently, the user element subroutines 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]. ...
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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 stabilized 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 displacement increment in mixed-mode fracture simulation, owing to its stability and accuracy.
Conference Paper
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We perform delamination analysis in laminated composites in 2D and 3D using the discrete damage zone model within the framework of the finite element method. In this approach, springlike elements are placed at the laminate interface and damage laws are used to prescribe both interfacial spring softening and bulk material stiffness degradation to study crack propagation. The irreversibility of damage naturally accounts for the subsequent reduction of material stiffness once the material is loaded beyond the elastic limit. The model is implemented in the commercially available ABAQUS software via the user element subroutine (UEL). Numerical results for 2D mixed-mode and 3D mode-I delamination are presented. The results for the benchmark examples show good agreement with those obtained from virtual crack closure technique (VCCT), which validates the method. This discrete method is particularly suitable when the material nonlinearities in the continuum surrounding the crack tip are significant.
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Computer implementation of the hybrid discontinuous Galerkin/cohesive zone model method (dG/CZM) for crack modeling is presented. The dG/CZM constitutes a (volumetric) locking free single field yet highly scalable method for nonlinear solid/fracture mechanics problems, particularly dynamic fracture analyses with crack branching and fragmentation. In the dG/CZM cohesive interface elements are placed at interelement boundaries prior to the simulation of which artificial compliance is removed by a dG formulation. The formulation is switched to a standard extrinsic cohesive crack model upon satisfaction of a failure criterion. We provide details on the preprocessing step where interface elements are inserted into the finite element mesh and on the computation of the internal force vector and the tangent stiffness matrix. Various examples that consist of (in)compressible elasticity, microcracking of fiber reinforced composite materials and dynamic fracture are investigated to verify the model and its implementation. This paper is addressed to researchers who would like to have a quick working implementation of dG/CZM methods.
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In this study, criteria on the artificial compliance due to intrinsic cohesive zone models are presented. The approach is based on a micromechanical model for a collection of cohesive zone models embedded between each mesh of a finite element-type discretization. The overall elastic behaviour of this cohesive volumetric medium is obtained using homogenization techniques and is given in a closed-form as function of bulk properties of the relevant material and mesh parameters (the mesh type and size). Practical criteria are obtained for the calibration of the cohesive stiffnesses bounding the additional compliance inherent to intrinsic cohesive zone models by lower value. For isotropic planar discretizations (e.g. Delaunay mesh), a rigorous bound is derived whereas convenient estimates are given for non-isotropic discretizations (e.g. regular mesh).
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A Discontinuous Galerkin (DG) interface treatment embedded in a Continuous Galerkin formulation is presented for simulating the progressive debonding of bi-material interfaces. The method seamlessly tracks the progression from perfect adhesion to interface softening and finally to complete separation without resorting to node-to-node springs or deletion of connectivities. While the formulation is inspired by an augmented Lagrangian approach, it does not introduce multiplier fields to enforce the continuity conditions, resulting in a pure displacement method amenable to traditional symmetric positive-definite solvers. The study of fibrous composites serves as the application in the present work. The constitutive behavior of the interface is modeled within an energetic framework from which the initiation criterion and softening response emanate. Numerical simulations are conducted for various loading cases on a fiber-matrix unit cell that highlight the performance of the method on crude meshes.
A discrete damage zone model is developed to describe the mode-mix ratio and temperature dependent delamination of laminated composite materials under high cycle fatigue loading within the framework of the finite element method. In this approach, discrete nonlinear spring elements are placed at the finite element nodes of the laminate interface, and a combination of static and fatigue damage growth laws is used to define its constitutive behavior. The model is implemented in the commercial software Abaqus using the user element subroutine. The static damage model parameters are estimated from fracture mechanics principles, whereas the fatigue damage model parameters are calibrated by fitting the numerical results to published experimental data. A quadratic relation is proposed to describe the non-monotonic variation of fatigue damage model parameters with mode-mix ratio. Next, an Arrhenius relation is proposed for the temperature dependence of fatigue damage, in addition to the incorporation of the temperature dependence of critical fracture energy. The model is convergent upon mesh refinement; however, for accurate prediction the mesh size used for model calibration should be sufficiently small. The model predicted fatigue crack growth rates are in agreement with those obtained from a quadratic relation for the Paris law parameters for variable mode mix conditions, thus verifying the approach. While the model captures the temperature effects on delamination for mode I and 50 % mode II, our prediction deviates from experiments for pure mode II, since the corresponding damage mechanism entirely changes with temperature.
This article investigates the sensitivity of cohesive zone models (CZMs) for high-cycle fatigue delamination in relation to constituent static parameters, namely, the cohesive strength and stiffness, whose values are frequently calibrated by curve fitting or selected for convenience without any physical basis. After reviewing the damage mechanics formulation of mixed-mode CZMs for static (monotonic) loading in bilinear, exponential, and polynomial cohesive laws, the source of uncertainty arising from the calibration or selection of static parameters is remarked. The formulation of the CZMs for high-cycle fatigue loading using interface separation, strain, and strain energy release rate (SERR) based fatigue damage rate functions is discussed. Several numerical studies are conducted to explore the sensitivity of CZMs for fatigue delamination in relation to static cohesive parameters and to the shape of the cohesive law under mode I and mixed-mode loading. The performance of the CZMs is also investigated for additive and non-additive decomposition of total damage into its static and fatigue components, and for constrained and unconstrained damage update strategies in the vicinity of the crack tip. Numerical studies illustrate that a CZM employing the separation or strain based fatigue damage rate function is highly sensitive to phenomenological cohesive strength and stiffness parameters, whereas a CZM employing the SERR based damage rate function is minimally sensitive to the same static parameters. While the shape of the static cohesive law does not affect fatigue crack growth rate predictions, studies show that cohesive laws with higher-order smoothness can better describe linear Paris regime behavior. The main conclusion of this article is that incorporating a SERR based fatigue damage rate function into a CZM with higher-order smoothness leads to a more robust approach for simulating high-cycle fatigue delamination of laminated composite materials.
We give a review of J. Nitsche’s method [Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971; Zbl 0229.65079)] applied to interface problems, involving real or artificial interfaces. Applications to unfitted meshes, Chimera meshes, cut meshes, fictitious domain methods, and model coupling are discussed.
In order to achieve realistic cohesive fracture simulation, a parallel computational framework is developed in conjunction with the parallel topology based data structure (ParTopS). Communications with remote partitions are performed by employing proxy nodes, proxy elements and ghost nodes, while synchronizations are identified on the basis of computational patterns (at-node, at-element, nodes-to-element, and elements-to-node). Several approaches to parallelize a serial code are discussed. An approach combining local computations and replicated computations with stable iterators is proposed, which is shown to be the most efficient one among the approaches discussed in this study. Furthermore, computational experiments demonstrate the scalability of the parallel dynamic fracture simulation framework for both 2D and 3D problems. The total execution time of a test problem remains nearly constant when the number of processors increases at the same rate as the number of elements.
Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57:891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.