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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng (2011)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3195

Mesh-independent matrix cracking and delamination modeling

in laminated composites

Endel V. Iarve1,2,∗,†, Mark R. Gurvich3, David H. Mollenhauer1,CherylA.Rose

4

and Carlos G. Dávila4

1Air Force Research Laboratory,2941 Hobson Way,WPAFB,OH 45433,U.S.A.

2University of Dayton Research Institute,300 College Park,Dayton OH 45469,U.S.A.

3United Technologies Research Center,411 Silver Lane,MS 129-73,East Hartford,CT,U.S.A.

4NASA Langley Research Center,Hampton,VA 23681,U.S. A.

SUMMARY

The initiation and evolution of transverse matrix cracks and delaminations are predicted within a mesh-

independent cracking (MIC) framework. MIC is a regularized extended ﬁnite element method (x-FEM)

that allows the insertion of cracks in directions that are independent of the mesh orientation. The Heaviside

step function that is typically used to introduce a displacement discontinuity across a crack surface is

replaced by a continuous function approximated by using the original displacement shape functions. Such

regularization allows the preservation of the Gaussian integration schema regardless of the enrichment

required to model cracking in an arbitrary direction. The interaction between plies is anchored on

the integration point distribution, which remains constant through the entire simulation. Initiation and

propagation of delaminations between plies as well as intra-ply MIC opening is implemented by using

a mixed-mode cohesive formulation in a fully three-dimensional model that includes residual thermal

stresses. The validity of the proposed methodology was tested against a variety of problems ranging

from simple evolution of delamination from existing transverse cracks to strength predictions of complex

laminates without aprioriknowledge of damage location or initiation. Good agreement with conventional

numerical solutions and/or experimental data was observed in all the problems considered. Published

2011. This article is a US Government work and is in the public domain in the USA.

Received 9 June 2010; Revised 4 February 2011; Accepted 19 February 2011

KEY WORDS: composite; mesh independent cracking; delamination; failure

INTRODUCTION

The utilization of composite materials has spread rapidly from mostly military applications to a

broad range of applications in the civil sector and most notably in commercial aviation. The service

regimes for civil aircraft differ from those in military applications in that commercial aircraft are

subjected to loads that are characterized by long-term moderate loadingand environmental exposure

as opposed to extreme loading and relatively short service duty for the military applications.

Increasing the service hours of composite structures increases the importance of understanding

the subcritical damage initiation, growth and interaction phenomena. This brings to the forefront

the need for the development of methodologies dealing with detailed modeling of the processes

of damage accumulation and evolution in composite materials. Fracture in a composite structure

is the result of the evolution of discrete damage events such as ﬁber/matrix debonding, matrix

∗Correspondence to: Endel V. Iarve, Air Force Research Laboratory,2941 Hobson Way,WPAFB,OH 45433,U.S.A.

†E-mail: endel.iarve@wpafb.af.mil

Published 2011. This article is a US Government work and is in the public domain in the USA.

E. V. IARVE ET AL.

cracking, delamination between plies and ﬁber failure. These damage modes evolve in various

combinations that depend on the stacking sequence and ply thickness and cause redistributions of

stresses in the failing composite. Some combinations may reduce the stress concentrations, and

others may precipitate the fracture.

Signiﬁcant progress has been achieved to date in developing numerical approaches and under-

lying constitutive models for initiation and propagation of speciﬁc damage modes. Intra-ply damage

modes have been investigated primarily within the framework of continuum damage mechanics

(CDM) [1, 2], while delamination has been studied extensively using interface fracture modeling

techniques such as cohesive zone models [3–5] and virtual crack closure techniques (VCCT) [6, 7].

A critical distinction between the CDM and interface fracture models exists in the approach in

which a displacement discontinuity is represented; i.e. the CDM methodology replaces the displace-

ment discontinuity with local volumetric stiffness degradation, whereas the interface fracture-based

techniques directly include the kinematics of the displacement jump. The value of the CDM

methodology is the ability to describe the stiffness response of a laminate containing signiﬁcant

amounts of matrix damage. Among the limitations of these models is their inability to describe

accurately local effects of the interaction between various damage modes and local effects of stress

redistribution in a damaged area. These deﬁciencies are particularly evident when the observed

fracture mode exhibits matrix splitting and pullouts [8, 9] or when the fracture is characterized by

strong coupling between transverse matrix cracking and delamination [10, 11].

To address these difﬁculties within the CDM framework, several solutions have been proposed.

A phenomenological mesoscopic model composed of stacks of alternating homogenized plies

and damageable interfaces was proposed by Allix et al. [11] for modeling impact damage in

laminated composites. The kinematic interaction between the intralaminar CDM and the interfaces

was represented using empirical coupling parameters. The coupled analysis method was shown

to reproduce correctly the ‘double-helix’ damage cone observed in the experiments. Transverse

cracking and delamination interaction has been studied by Ladevèze and coworkers using a non-

local meso-model where the coupling between intralaminar and interlaminar damage variables is

deﬁned by an energy equivalence between a meso- and a micro-model in which the cracks are

introduced explicitly [12, 13]. With such a coupling, the prediction of delamination induced by

transverse cracking was quite accurate.

Although it is conceivable that non-local or other type of non-classical CDM models could

address limitations of classical CDM approaches, difﬁculties with implementation of the non-local

strategies has resulted in recent emphasis on the development of computational methodologies

to explicitly model evolving displacement discontinuities [9, 14–26] in solids. Since delamina-

tion surfaces in composite materials coincide with morphological features such as ply interfaces,

delaminations can be represented by the opening of doubled nodes using existing ﬁnite element

(FE) tools. Matrix cracks, on the other hand, are not amenable to straightforward general treatment

within the framework of traditional FE tools. Indeed, the creation of a FE mesh for a composite

laminate that accommodates the boundaries and features such as holes and cracks with different

orientation in different plies, is a formidable challenge.

Several methodologies have been proposed for modeling the kinematics of arbitrary cracking.

The evolution of a crack front can be captured by traditional FE modeling combined with adaptive

remeshing techniques [14]. Such techniques have been successful in predicting complex crack

evolution in metallic structures. However, application of these techniques to laminated composites,

where cracks form in different plies in adjacent locations, require remeshing of various volumes

under multiple mesh compatibility constraints.

An alternative approach to modeling crack-induced displacement discontinuities involves mesh-

independent crack modeling techniques. Early works devoted to mesh-independent modeling of

matrix cracking in composite laminates include [15, 16]. Over the last decade, a signiﬁcant effort

has been devoted to further develop the ideas in the pioneering work of Moës et al. [17], in which

the concept of the eXtended Finite Element Method (x-FEM) was proposed. Although most of the

research has been devoted to arbitrary crack propagation in isotropic materials, recent applications

of the x-FEM method to composite materials include delamination modeling and textile composite

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

architecture representation [18]. Huynh and Belytschko [19] provide a review of contemporary

development of the ﬁeld as well as novel applications of x-FEM to interfacial cracking analysis

in two- and three-dimensional settings.

Modeling a matrix crack that propagates parallel to the ﬁber direction in a ply is conceptually

straightforward using x-FEM. However, it is more difﬁcult to model networks of matrix cracks

in a laminate where the fracture planes of matrix cracks in individual plies intersect at common

interfaces and can cause delaminations that link the matrix cracks through the thickness. Within the

traditional x-FEM approach, the difﬁculty in modeling linked networks of multiple cracks could

be addressed by developing a special enrichment for multiple crack situations or by connecting

two enriched/cracked elements. Such connections were recently accomplished in a quasi-two

dimensional formulation [20, 21], following the approach of Hansbo and Hansbo [22].

Another direction in which the x-FEM is being developed is the regularized extended Finite

Element Method (Rx-FEM) [23–27], where the step function used in x-FEM approaches to describe

the crack surface is replaced by a continuous function. Iarve [23] proposed the use of displacement

approximation shape functions to approximate the step function and thus maintain the Gauss

integration schema for element stiffness matrix computation, without regard to cracking orientation.

In this case, the Gauss integration points of the initial approximation may be used for integration

of the enriched functions, providing a framework for connecting plies where the matrix cracks can

propagate in arbitrary directions.

The purpose of the present manuscript is to propose a numerical approach capable of modeling

the origination and evolution of complex matrix cracking and delamination networks in laminated

composites without any prior knowledge or assumptions regarding the locations of damage initi-

ation. The method is based on a combination of a cohesive interface damage model proposed by

Turon [5] for modeling the delaminations, and a regularized x-FEM (Rx-FEM) method proposed

by Iarve [23]. In the present paper we extend this method to allow for multiple matrix cracks in

a ply as well as connectivity and subsequent delamination between plies with arbitrary ﬁber and

matrix cracking direction. A description of the modeling methodology is presented ﬁrst, including

discussions of the implementation of the cohesive zone for damage propagation, the Rx-FEM

methodology for matrix cracking and a speciﬁc discussion on the propagation of an Rx-FEM crack.

It also includes a discussion of the critical subject of the interaction between matrix cracking and

delaminations. This is a signiﬁcant feature of the proposed method and is enabled by the fact that

the shape functions in the adjacent plies share integration points before and after enrichment and

thereby their products can be accurately computed regardless of the orientations of the transverse

matrix cracks in the adjacent plies. This serves as a basis for a robust numerical approach for

modeling complex networks of matrix cracks and delaminations.

The theoretical section is followed by a series of validation studies where the methodology

is tested against traditional numerical implementations and/or experimental observation. These

include a numerical study to verify the performance of the methodology for delaminations

emanating from a transverse crack, the numerical/experimental examination of unnotched quasi-

isotropic tensile coupon strength with and without predetermined cracking patterns, a comparison

of numerical and experimental results of the effects of ply thickness on crack distribution and

delamination patterns and ﬁnally a numerical examination of the variation of delamination initia-

tion location due to differing off-axis plies. In all cases, the proposed methodology was validated

against the respective numerical and/or experimental observations.

TRANSVERSE CRACKING AND DELAMINATION MODELING

A simpliﬁed damage progression sequence of coupled transverse matrix cracking and interlaminar

delamination is shown in Figure 1(b)–(d) for the case of a laminated plate subjected to a tensile load.

Initially, the laminate is undamaged (Figure 1(a)). As a result of the load application, transverse

matrix cracks form in different plies of the laminate, as shown in Figure 1(b). In the absence

of a stress concentration, the locations of the initial matrix cracks are random, and cannot be

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Figure 1. Idealized damage progression sequence in a laminated composite plate subjected to tensile

loading: (a) initial stage without damage; (b) matrix cracking stage; (c) delamination stage, linking up

matrix cracks in various plies and (d) specimen fracture.

known apriori. As the load increases, new cracks appear and the spacing between them becomes

increasingly deterministic. At some value of the applied load, delaminations initiate from the

matrix cracks (Figure 1(c)). These delaminations can connect matrix cracks in adjacent plies,

which can cause the disintegration of the laminate. The failure scenario outlined above is intended

for illustration purposes only, and simpliﬁes the actual damage progression and failure process in

which there is no separation of the transverse matrix cracking and delamination phases. The ﬁber

failure mode will not be treated in the present manuscript and is deferred to future work.

The approach proposed herein for modeling networks of multiple parallel transverse matrix

cracks within individual plies of a laminate and delaminations between plies couples a regularized

mesh-independent crack (MIC) modeling technique [23] for arbitrary transverse matrix cracks and

a cohesive model for the delamination between plies. In the regularized MIC formulation proposed

in [23], the step function used in traditional x-FEM approaches to construct local enrichment for a

crack discontinuity is replaced with a continuous function that is approximated by the same shape

functions as those used for the initial displacement approximation. The surface of each crack is

replaced with a gradient zone (a volume where the gradient of the approximate step function is

nonzero) and the surface fracture energy is replaced with the cohesive energy in the gradient zone.

A ﬂow chart of a typical simulation is shown in Figure 2. A simulation begins without any initial

matrix cracks. As the loading is increased, matrix cracks oriented parallel to the ﬁber direction

are inserted according to a failure criterion. In the present paper, the LaRC03 failure criterion [28]

is used. The criterion is evaluated at each integration point and, if the criterion is exceeded, a

matrix crack oriented in the ﬁber direction is added. The crack is inserted using the displacement

enrichment necessary to model the displacement jump. The magnitude of the jump is initially

zero and is controlled by an interface cohesive law [5]. The same cohesive law is used at the ply

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

Figure 2. A ﬂow chart of a typical failure simulation.

interfaces to represent potential delamination surfaces. A Newton–Raphson procedure is applied

to ﬁnd the equilibrium solution at each load step of the implicit incremental solution.

The following sections describe the formulation of the MIC modeling technique. First, the

cohesive model used for delamination and matrix crack propagation is described. Next, development

of the regularized MIC modeling technique for transverse cracks within a single ply is presented.

This is followed by discussion of the use of Turon’s [5] surface-based cohesive model to describe

the MIC crack propagation characteristics. Finally, the system of equations for modeling the

interaction of transverse matrix cracks and delaminations in a multi-layered composite laminate

are derived.

Cohesive model for delamination and matrix crack propagation

We begin by discussing the mixed-mode cohesive zone interface fracture model proposed by Turon

et al. [5]. The brief description below is given for completeness of the present formulation and

the reader is referred to reference [5] for full details. We consider an arbitrary point at the crack

interface with a normal vector nand a displacement jump vector u. Our goal is to describe the

fracture energy of separation of an arbitrary interface point, which then will be used in deriving the

governing equations from the minimum potential energy principle. This discussion will directly

pertain to the delamination propagation between plies and will be later utilized for matrix crack

propagation as well.

The cohesive energy can be written in the invariant form as a function of the norm of the

displacement jump =uand a mode mixity parameter, B

B=1−un

2

2

,(1)

where un=(u·n) is the normal, mode I component of the displacement jump, and the brackets

x= 1

2(x+|x|) represent the McAuley operator. The parameter Bis equal to 0 for mode I propa-

gation, and is equal to 1 for mode II propagation. The functional shape of the fracture energy as

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

a function of the displacement gap is deﬁned by the relationship between the cohesive tractions

and the displacement jump, which is assumed to have the form:

s=(1−d)Ku+dKunn,(2)

where Kis a high initial penalty stiffness and dis the damage parameter. The ﬁrst term in

Equation (2) represents the crack cohesive force, and the second term prevents interpenetration of

the crack surfaces. A bilinear relationship is assumed for the magnitude ()=|s|of the cohesive

traction vector, deﬁned in (2), such that d=0if<0and d=1if>f. The initial value of the

displacement jump, beyond which the interface failure begins, is deﬁned as

0=0/K,(3)

where 0is the cohesive strength. The cohesive strength depends on the mode mixity parameter Bas

(0)2=Y2+(S2−Y2)B,(4)

where Yand Sare the interfacial normal and shear strengths, respectively, and is an experimentally

determined inﬂuence parameter [5]. The fracture energy density g(,B) is the area under the ()

curve, so that

g(,B)=

q=0

(q,B)dq.(5)

To ensure the correct crack propagation characteristics, the ﬁnal value of the displacement jump,

f, is deﬁned so that the following condition is satisﬁed

g(f,B)=Gc(B),(6)

where the critical energy release rate (ERR), Gc, or fracture toughness, is assumed to be a function

of the mode mixity as follows [5]:

Gc=GIc +(GIIc −GIc)B(7)

and GIc and GIIc are experimentally measured fracture toughness values. In the case of a bi-linear

() relationship, the ﬁnal value of the displacement jump is determined by the initial value of the

displacement jump and the fracture toughness as

f=2Gc/(K0).(8)

All parameters entering the analysis, such as the fracture toughness and strength values, are material

properties that can be measured by using standard test methods.

The fracture energy associated with a delamination between plies is computed by integrating the

fracture energy over the interface between plies nand n+1, designated by the horizontal surface

z=zn, and is given by

n=z=zn

g(,B)dS.(9)

In the case of a delamination crack, the normal vector to the crack surface is (0,0,1), where the

x-andy-axes of the Cartesian coordinate system are oriented in the in-plane directions, and the

z-axis is oriented in the thickness direction. The displacement jump at the interface surface z=zn

is computed by using the enriched displacement approximations in the adjacent plies, nand n+1,

provided subsequently in Equation (14). The displacement jump vector between the plies n+1

and nis denoted as un.

MIC of matrix cracks within a Ply

The goal of the present study is to model networks of transverse matrix cracks and delaminations

in a composite laminate. A single ply containing s=1,...,Scracks parallel to the ﬁber direction

deﬁned by the angle is discussed ﬁrst. Consider a partition of unity set of continuous 3D basis

functions Xi(x) and a displacement approximation on the domain of interest Vcorresponding to

one ply in the laminate,

u(x)=

i∈

Xi(x)Ui(10)

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

where Uiare displacement approximation coefﬁcients, not necessarily associated with nodal

displacements, and is the set of all index values. Next consider a crack appearing in this volume,

with a surface deﬁned by means of the signed distance function deﬁned as:

f(x)=sign(n(¯

x)·(x−¯

x)) min

¯x∈

x−¯

x(11)

where n(¯

x) is the normal to the crack surface at the point ¯

x. The traditional x-FEM strong

discontinuity formulation is based on element enrichment with displacement modes discontinuous

over the crack surface. The discontinuity is obtained by multiplying the shape functions by the

Heaviside step function H(f(x)). In the regularized formulation [23] the Heaviside step function

is replaced with a continuous function ˜

H(x)

˜

H(x)=

i∈

Xi(x)hi(12)

where Xi(x) are the same shape functions as in Equation (10). This function is equal to 0 or 1

everywhere, except in the vicinity of the crack surface. The coefﬁcients hiare calculated as follows

hi=1

21+VXi(x)f(x)dV

VXi(x)|f(x)|dV(13)

This deﬁnition involves only continuous functions so the integrals can be calculated by using

standard Gauss quadratures. The coefﬁcients hiare equal to 0 or 1 if the signed distance function

does not change sign in the support domain of the shape function Xi(x). On the other hand, when

the crack crosses the support domain of the shape function, then 0<hi<1. Denoting the set of

such index values (for which hiis not equal to 0 or 1) by , the enriched approximation for the

domain Vwith an arbitrary crack is deﬁned in the following form

u=˜

Hu(1) +(1−˜

H)u(2) +u(3) (14)

u(1) =

i∈

XiU(1)

i,u(2) =

i∈

XiU(2)

i,(15)

and

u(3) =

i∈/

XiU(3)

i(16)

where the spatial argument has been omitted for conciseness. The displacement approximation

given in Equation (14) contains the enrichment in the crack region via displacement ﬁelds u(1)

and u(2) as well as the unchanged displacement ﬁeld, u(3), away from the crack. Equations (14)–

(16) deﬁne the enriched displacement approximation by replacing each original shape function Xi

inﬂuenced by the crack, i∈, with two shape functions, ˜

HX

iand (1−˜

H)Xi. This approximation

was applied in Reference [23] in conjunction with a higher order C0displacement approximation

(p-elements), as well as with a B-spline approximation of displacements, where the coefﬁcients

Uido not correspond to nodal displacements. The bookkeeping of the connectivity is transparent,

and the two copies, ˜

Hand (1−˜

H), of the shape function do not interact. They are only connected

to alike ˜

Hor (1−˜

H) multiple copies of other enriched shape functions if their supports overlap.

For multiple cracks in a single ply, the function Fis used in Equation (13) instead of fto

deﬁne the set of enriched shape functions

F=

S

s=1

fs.(17)

The product function Fmaintains some key features of the signed distance function for each of the

scracks in the ply under consideration. This function changes sign over the surface of each crack,

and its gradient ∇Fis perpendicular to the face of each crack. As follows from Equation (13),

the enrichment is triggered by the change of sign of f, and thus replacing Equation (11) with

Equation (17) extends the previous discussion to multiple parallel MICs in each ply. However,

when a single signed distance function is used to represent multiple physical cracks, the crack

spacing is limited by the size of the mesh and exceeds the span of a shape function.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

The limitation on minimum cracking distance for a given mesh size can be reduced by using

individual step functions for individual cracks. Iarve used this approach to investigate the effect of

matrix cracks and delaminations around open holes in composite laminates on stress redistribution

in the ﬁber direction [8, 29]. Damage progression was not modeled and the mesh-independent

technique was used to place predeﬁned matrix cracks and delaminations. To represent such damage

networks with displacement discontinuity surfaces crossing each other, the displacement shape

functions have to be quadrupled for two crossing cracks, and reproduced 2ntimes for ncrossing

cracks. Such a representation is useful in the case of parallel cracking networks as well, because

it allows the minimum crack spacing distance for a given mesh size to be signiﬁcantly reduced.

This development will be pursued in the future.

The strain energy in the volume Vwith displacement approximation given by Equation (14) is

written as

W=1

2V

{˜

H(e(1) −e)Tr(1) +(1−˜

H)(e(2) −e)Tr(2)}dV

+V

{˜

H(e(1) −e)Tr(3) +(1−˜

H)(e(2) −e)Tr(3)}dV+1

2V

{(e(3) −e)Tr(3)}dV,(18)

where superscript T denotes the transpose operation, r(k)are elastic stress tensors, and e(k)and

eare the total and the nonmechanical strain tensors, respectively. In the global xyz coordinate

system Hooke’s law reads

r(k)=C(e(k)−e),(19)

where Cis the elastic orthotropic stiffness tensor (see Reference [30] for engineering constant and

ply coordinate transformation expressions).The non-mechanical strain is required to account for the

residual stress state arising in a composite laminate from the curing process, and is e=(T−T0)a.

Here, T−T0is the difference between the cure temperature and room temperature, and ais the

tensor of thermal expansion coefﬁcients in the global coordinate system. The strain tensors e(k)

are computed from displacement ﬁelds u(k)in Equation (14) for k=1,2,3, so that

ε(k)

ij =u(k)

i,j+u(k)

j,i.(20)

The strain energy given in Equation (18) does not include the energy associated with the propagation

of transverse cracks.

Propagation of transverse matrix cracks

In a regularized formulation, the propagation of transverse matrix cracks using MIC is governed

by the constitutive properties in the gradient zone, which is deﬁned as a volume where |∇ ˜

H|=

∇˜

H·∇ ˜

H>0. In Reference [26], this zone is treated as a transition region with constitutive

properties derived from the bulk properties. In the present formulation, however, the cohesive

constitutive relationship [5] developed for interface fracture modeling is inserted directly into the

gradient zone of the regularized formulation. In the traditional surface fracture cohesive constitutive

formulation, the fracture energy is a function of point-wise crack-opening displacements at the crack

surface and is deﬁned by the area under the cohesive traction versus crack-opening displacement

relation. In the case of the regularized formulation, the two-dimensional crack surface is replaced

with a gradient zone and the fracture energy balance has to be expressed through volume integration.

Such an expression can be obtained by using Dirac’s delta function of the signed distance function

of the crack surface. Consider a sub-volume vof a ply which contains a crack deﬁned by its

signed distance function f. The surface area of this crack enclosed in vcan be obtained as

Sv=v

D(f)dV,(21)

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

where D(f) is the Dirac’s delta function of the signed distance function. One can also establish

that for an arbitrary continuous function g(x) deﬁned in volume v, a relationship between the

surface integral over the crack surface (⊂v) and a volume integral exists so that

∈v

g(x)dS=v

g(x)D(f)dV.(22)

This relationship can be readily established by applying Equation (21) in small adjoining volumes

encompassing surface to develop the integral sums representing the left- and right-hand sides of

Equation (22). In the case of the regularized formulation, the approximate value of the right-hand

side is computed by replacing the Dirac’s delta function of the signed distance function of the

crack surface, D(f), by the gradient of the approximate step function |∇ ˜

H|. The continuous

function g(x) deﬁned over the volume is replaced with the point-wise fracture energy of crack

opening. Therefore, the fracture energy required for crack surface opening within the arbitrary

sub-volume vis equivalent in the regularized and conventional crack surface formulations and can

be calculated as

v

g(x)D(f)dV≈v

g(x)|∇ ˜

H|dV.(23)

Note that the volume vmust be sufﬁciently large with respect to the gradient zone of the regularized

step function for the crack surface area computed by using the step function gradient to be equal

to the exact crack surface area. It is also important to mention that in the case of multiple cracks

within a ply, the right-hand side of Equation (23) requires no modiﬁcations, provided that the

approximate step function is computed by using Equations (13) and (17).

In the traditional interface cohesive formulation, the fracture energy is a function of the crack

opening displacement and is also dependent on the ratio of the opening mode I displacement

(perpendicular to the crack surface) and the shearing mode II displacement (tangential to the crack

surface). To separate these modes, the direction of the normal to the crack surface needs to be

known at all points. In the regularized formulation, the displacement jump and the normal vector

are deﬁned at all points of the gradient zone as

u=u(1) −u(2) and n=∇ fa,(24)

where u(1) and u(2) are deﬁned by using the displacement ﬁelds in the enriched approximation

given in Equation (14). The fracture energy of the crack opening g(x) is considered homogeneous

and therefore dependent upon the spatial coordinate xonly as a function of the displacement jump

and the normal vector to the crack surface, so that g(x)=g(u,n). Considering Equation (23) and

extending the arbitrary volume to the volume Vof the entire ply, the fracture energy of multiple

cracks propagating within a ply is expressed as

M=V

g(,B)|∇ ˜

H|dV.(25)

The number of cracks, if any, does not enter Equation (25) explicitly. However, since a single

signed distance function, Equation (17), has been used to represent multiple parallel cracks, the

number of gradient zones with the associated fracture energy, contributing to the step function, is

equal to the number of cracks.

The discussion in the present section pertains to cracking in a single ply. All the quantities

including displacements, normal vectors and step functions are deﬁned within a ply and thus need

to be indexed by the ply number n=1,...,N. This index has been omitted above. However, for

the purpose of deriving the set of ﬁnal equations, the fracture energy of matrix cracking in each

ply Mn,n=1,...,Nwill be required.

Modeling interacting transverse matrix and delamination cracks

The equilibrium conditions for a body that contains multiple matrix and delaminations can be

obtained in terms of the displacement approximation coefﬁcients (Equations (15) and (16)) by

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

setting the ﬁrst variation of the potential energy to zero. Combining Equations (9), (18) and (25)

and summing over all plies and interfaces gives,

N

n=1

(Wn+Mn)+

N−1

n=1

n−A=0,(26)

where Ais the work of external tractions applied at the laminate boundary. The lower indices in

the strain energy, Wn, and the MIC fracture energy, Mn, designate that they are computed for the

nth ply. The subscript on the delamination fracture energy, n, designates that it is computed for

the interface between the nth and (n+1)th ply. Performing the variation results in the following

system of equations

(W+M+U)U=P+N.(27)

The vector of unknowns is arranged by ply in the order UT=(U(1)

1,U(2)

1,U(3)

1.....U(1)

N,U(2)

N,U(3)

N)T.

The matrix Wand the right-hand side vector Nare obtained from variation of Wnand are the elastic

stiffness matrix and the mechanical load vector, respectively. Computation of the components of

the matrix Wand the vector Nis similar to that for the un-enriched approximation (10) with the

exception that the shape functions Xifor the enriched displacement approximation coefﬁcients

U(1) and U(2) are multiplied by ˜

Hand (1−˜

H), respectively. These shape functions are complete

polynomials on the mesh cells and their integration is performed on a ﬁxed Gauss grid even though

the step functions ˜

Hchange. The general structure of the Wmatrix is as follows

{W}=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

V

˜

HBCBT0V

˜

HBCBT

0V

(1−˜

H)BCBTV

(1−˜

H)BCBT

sym sym V

BCBT

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎝

U(1)

U(2)

U(3)

⎞

⎟

⎟

⎠

(28)

where Bis the derivative matrix, so that e(k)=BU(k)and Cis the stiffness matrix. Note that

Equation (28) is written for each ply and all matrices and step functions must have the ply index,

which is omitted. We also indicate at the right that the matrix Wwill be multiplied (thus the

vertical line) by the respective displacement approximation coefﬁcient vectors in the ﬁnal system

of equations, Equation (27). The volume integration is performed at Gauss points of the original

mesh, since all of the functions in Equation (28) are continuous functions. Variation of the strain

energy also yields the non-mechanical load vector in the form

{P}=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

V

˜

HBCe

V

(1−˜

H)BCe

V

BCe

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.(29)

Consider the Mmatrix next. Variation of the MIC fracture energy can be written as

M=V

|∇ ˜

H|*g

*(,B)=V

|∇ ˜

H|*g

*(,B)u(u)

,(30)

where the displacement jump is given by Equation (24). The following form for the Mmatrix is

obtained

{M}=⎛

⎜

⎜

⎜

⎜

⎜

⎝

V

|∇ ˜

H|gE−V

|∇ ˜

H|gE0

sym V

|∇ ˜

H|gE0

000

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎝

U(1)

U(2)

U(3)

⎞

⎟

⎟

⎠

,(31)

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Int. J. Numer. Meth. Engng (2011)

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MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

where Eis a banded matrix of shape function cross-products, {E}ij =XiXj,andg=

(1/)(*g/*)(,B) is introduced for conciseness. Analogous to Equation (28), Equations (29) and

(31) are written for a single ply and must be added to the global system of equation as diagonal

blocks.

The delamination fracture energy matrix involves the interaction of displacement vectors in

two neighboring plies. Consider the delamination fracture energy between plies nand n+1. In

this case

n=z=zn

*g

*(,B)=z=zn

gun(un),(32)

and the displacement jump is

(un)=˜

Hn+1u(1)

n+1+(1−˜

Hn+1)u(2)

n+1+u(3)

n+1−˜

Hnu(1)

n−(1−˜

Hn)u(2)

n−u(3)

n.(33)

Note that the integral in Equation (32) is a surface integral. The delamination fracture energy

variation ties together the displacement approximation in two adjacent plies and results in the

following matrix

{n}=un,n−un,n+1

sym un+1,n+1

⎛

⎜

⎜

⎝

U(1)

n

U(2)

n

U(3)

n

⎞

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

U(1)

n+1

U(2)

n+1

U(3)

n+1

⎞

⎟

⎟

⎟

⎠

,

(34)

where

{un,n}=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

z=zn

gEnn ˜

Hn˜

Hn0z=zn

gEnn ˜

Hn

0z=zn

gEnn(1 −˜

Hn)(1−˜

Hn)z=zn

gEnn(1 −˜

Hn)

sym sym z=zn

gEnn

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(35)

{un,n+1}=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

z=zn

gEnn+1˜

Hn˜

Hn+1z=zn

gEnn+1(1−˜

Hn)˜

Hn+1z=zn

gEnn+1˜

Hn+1

z=zn

gEnn+1˜

Hn(1−˜

Hn+1)z=zn

gEnn+1(1−˜

Hn)(1−˜

Hn+1)z=zn

gEnn+1(1−˜

Hn+1)

z=zn

gEnn+1˜

Hnz=zn

gEnn+1(1−˜

Hn)z=zn

gEnn+1

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,

(36)

The matrix un+1,n+1is similar to the matrix un,n, with all the indices replaced by n+1, since

it contains the surface displacement of the (n+1)th ply. The shape function product matrix Ein

Equations (35) and (36) has also been indexed so that in Equation (35) Eis the product of the

displacement approximation functions in the nth ply and in Equation (36) Eis the product of

displacement approximation functions in the nth and (n+1)th plies; i.e. {Enn}ij =Xply=n

iXply=n

j

and {Enn+1}ij =Xply=n

iXply=n+1

j. The surface integrals are computed using Gauss integration by

sampling the shape functions at the surface between the plies. All shape functions are complete

polynomials on all the mesh cells and integration of their cross-products is straight forward.

The system of Equations (27) contains highly nonlinear components Mand U,wherethe

nonlinearity is brought in by the cohesive energy function g. The system of equations is solved

by using the Newton–Raphson (NR) method at each step.

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

VERIFICATION MODELS

Numerical results devoted to veriﬁcation and application of the proposed methodology for failure

prediction in laminated composites are presented below. Five different cases are examined, illus-

trating different aspects of failure in composite materials. First, a delamination test called the

Transverse Crack Tension test is considered for verifying the performance of the methodology for

delaminations emanating from a transverse crack. Second, the strength of a quasi-isotropic tensile

coupon is determined using a limited number of pre-deﬁned matrix cracks. Third, the strength of

a quasi-isotropic tensile coupon is again determined, but this time without any prior knowledge

of the cracking pattern or restriction on the number of cracks. Fourth, the effects of ply thickness

on crack distribution and delamination patterns was examined and compared with experimental

results. Finally, the differences in the evolution of the delamination fronts in tensile coupons with

differing off-axis plies are examined.

Boundary conditions

The following loading conditions will be used through the ﬁve example cases described below.

Consider a multilayered composite plate consisting of Northotropic layers with in-plane dimen-

sions Land Win the x-andy-directions, respectively. Let the thickness of the plate be H

(z-direction) as shown in Figure 3. Each ply represents an orthotropic material, which is char-

acterized by engineering stiffness constants Eij,Gij,vij and thermal expansion coefﬁcients ij

(i,j=1,2,3). The direction x1coincides with the ply ﬁber direction and the angle between the

direction of the global coordinate xand the ﬁber direction x1in a given ply is called the ply

orientation. Tensile loading in the x-direction is applied by incrementing the displacement uxat

the edges x=0,L,sothat

ui

x(0,y,z)=ui−1

x(0,y,z)−iand ui

x(L,y,z)=ui−1

x(L,y,z)+i,(37)

where iis a constant and iis the loading step number. An incremental formulation is required to

properly account for the thermal curing stresses prior to mechanical loading. The displacement ﬁeld

u0

xappearing in Equation (37) is computed by solving a thermal–mechanical expansion problem

with boundary conditions to simulate free expansion and restrict rigid body motion, i.e.

u0

x(0,0,0)=0,u0

y(0,W,0)=0andu0

z(x,y,0)=0.(38)

The incremental loading boundary conditions (37) are supplemented with constraint conditions on

the other displacement components at the lateral edges x=0andL,sothat

ui

y(0,y,z)=u0

y(0,y,z)andui

y(L,y,z)=u0

y(L,y,z),

ui

z(0,y,z)=u0

z(0,y,z)andui

z(L,y,z)=u0

z(L,y,z),

(39)

Figure 3. Generic multilayered composite plate showing the global and material

coordinate systems and dimensions.

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

Figure 4. Conventional ﬁnite element model (FE) and mesh-independent crack model (MIC) of the

transverse crack tension specimen (TCT): (a) Conventional FE model and (b) MIC model.

which means that all the displacement components, except the axial displacement, are ﬁxed at the

loaded edges. In this setting the external loading vector Piat the ith load step is

Pi=Pi−1+iE(40)

where Eis the unit displacement loading vector. The vector P0is the edge displacement expansion

vector resulting from solving the thermal–mechanical expansion problem (Equation (38)).

Veriﬁcation 1: transverse crack tension test (TCT)

The TCT specimen, described in reference [31], was examined to evaluate the accuracy of the

proposed methodology for predicting the growth of a delamination that emanates from a transverse

crack. The TCT specimen consists of three unidirectional (=0) plies with thickness t,2tand t,

respectively, that is subjected to axial tension loading. The ﬁbers of the middle ply are cut through

the width at the mid-length, creating a crack across the specimen. In this case no thermal pre-stress

is considered since all plies have the same orientation and no mismatch of thermal expansion

properties between plies is present. When the loading is applied through displacement increments,

the applied stress initially increases linearly until delaminations between the middle ply and the

top and bottom plies initiate from the crack in the middle ply. These delaminations then propagate

in a stable manner while the applied stress remains constant. Finally, after the delaminations reach

the grips, the load starts increasing again with increasing end displacement, but with a different

slope.

The problem was solved using a conventional ﬁnite element (FE) model, shown in Figure 4(a),

and a MIC model, shown in Figure 4(b). Figure 4 shows the entire laminate thickness, although

only half of the laminate thickness was modeled, i.e. two plies with equal thickness twere modeled

and a zero z-displacement condition was applied on the surface z=0. The shaded regions in

Figure 4(a), (b) shows the initial middle ply crack. In the case of the conventional FE model, the

crack in the middle ply is aligned with a mesh line and is simply modeled by using unconnected

double nodes. For the MIC model, a curved non-uniform mesh was used to demonstrate the mesh

independence of the approach.

In the MIC model, the middle ply crack was inserted at the start of the analysis such that it is not

aligned with the mesh cell boundaries. In this and other cases, linear displacement approximation

was used. The total number of axial intervals in the two models is equal to 120, with 32 intervals

through the width and 1 through each ply thickness. However, due to the mesh non-uniformity of

the MIC model, the local density of the MIC mesh near the delamination tip varies signiﬁcantly,

i.e. hmin/hmax =1/24. In both models, the delaminations between the plies were modeled by using

the cohesive technique previously described. The material properties used in this analysis and in

other analyses presented in this paper are summarized in Table I. The composite system used

in TCT testing in Reference [31] was T300/914C. The load versus applied displacement curves

predicted by the two models are shown in Figure 5 and are nearly identical.

To verify the ability of a MIC model to represent the stress singularity represented by a MIC

crack, the stress components zx and zz at the tip of the cut-ply crack were examined and compared

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Table I. Material properties used in the analyses.

T300/914C IM7/8552 T300/976

Reference [31] Reference [9] Reference [32]

E11 (GPa) 139.9 161 138

E22,E33 (GPa) 10.1 11.38 10.3

G23(GPa) 3.7 3.98 3.1

G12,13 (GPa) 4.6 5.17 5.5

23 0.436 0.436 0.66

12,13 0.30.320.3

11(1/◦C) — 0 0.4×10−6

22,33(1/◦C) — 3×10−52.54 ×10−5

T−T0(◦C) — −150 −125

Yt(MPa) 80 60 37.9

Yc(MPa) 300 260 200

S (MPa) 100 90 100

GIC (J/m2) 120 200 157

GIIC (J/m2) 500 1000 315

Figure 5. Load versus displacement curves of the TCT specimen predicted using the

conventional FE model and the MIC model.

with a standard FE model, as shown in Figure 4(a). For the present comparison, the MIC model

had the exact same mesh distribution and arrangement as in Figure 4(a), but the cut-ply crack

was inserted using the MIC methodology. Figure 6 shows the distribution of these two stress

components normalized by the applied stress. The similarity in the results indicates that the stress

ﬁeld obtained from the MIC model is predicted accurately. Also, since the delaminations are

propagated using a cohesive zone, the nature of the singular stresses at the tip of the cut-ply crack

is not particularly important. It is, however, illustrative that the stress concentration predicted by

Rx-FEM and regular FEM is very similar for a given mesh reﬁnement.

Veriﬁcation 2: quasi-isotropic strength with pre-determined cracking

An important phenomenon facilitated by matrix cracking during the fracture process in composite

laminates is the so-called delamination jump. A simulation of such scenario was recently performed

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

Figure 6. (a) xz and (b) zz normalized stress distribution at the delamination surface

in the vicinity of the transverse crack tip.

Figure 7. A three-crack representation of matrix crack damage in a [454/904/−454/04]sIM7/8552

laminate used by Hallett et al. [9] to predict the delamination failure load.

in Hallett et al. [9] as part of a detailed investigation of failure mechanisms in quasi-isotropic

laminates as a function of ply thickness and other dimensional parameters. Experimentation as

well as high-ﬁdelity FE analysis were performed on IM7/8552 specimens with stacking sequence

[45m/90m/−45m/0m]sfor m=1,...,8, that were subjected to uniaxial tension. Observation of the

test specimens during testing and examination of the failed specimens showed signiﬁcant inﬂuence

of matrix cracks and delaminations on the ﬁnal failure. It was observed that the delaminations

were, in all cases, caused by the free edge and matrix cracking interactions. The classical free

edge delamination initiation loads (when no matrix cracking is taken into account) were estimated

and found to be signiﬁcantly higher than the delamination thresholds measured experimentally.

Furthermore, it was observed that increasing the ply thickness or decreasing the specimen width

lowers the delamination failure load.

In the present paper, we consider the m=4 case, in which the plies are approximately 0.5 mm

thick. This laminate exhibits a complex delamination failure progression sequence, with a delam-

ination initiating at the outer 45/90 interface and stepping to the inside 90/−45 and −45/0

interfaces via matrix cracks in the 90 and 45◦plies, respectively. Hallett et al. experimentally

observed small load drops accompanying the delamination at the 90/−45 interface, and a relatively

signiﬁcant load drop accompanying the delamination at the −45/0 interface. The delamination at

the −45/0 interface extended nearly to the grips, essentially leaving a unidirectional specimen.

The load corresponding to the delamination at the −45/0 interface was taken as the failure load,

and is referred to subsequently as the delamination failure load.

Hallett et al. [9] identiﬁed from the experimental data a relatively simple matrix cracking pattern

that was incorporated in a ﬁnite element model that allowed predicting the delamination failure

load. This matrix cracking pattern is shown in Figure 7, and consists of one matrix crack in each

ply. This pattern was modeled in [9] by a 3-D FEA model, where all three cracks were incorporated

in the mesh and were considered open from the beginning of the analysis. The delaminations

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Figure 8. Delamination jump predictions in a [454/904/−454/04]squasi-isotropic laminate from a MIC

analysis of the three-crack model: (a) delamination jump from the 45/90 interface to the 90/−45 interface

and (b) delamination jump from the 90/−45 interface to the −45/0 interface.

between plies were modeled by inserting cohesive zone elements on the interfaces between the 45

and 90◦plies, the 90 and 45◦plies and the 45 and 0◦plies.

To verify the proposed MIC methodology, the simulation conducted by Hallett et al. [9] was

repeated by predeﬁning one open MIC in each ply of the above quasi-isotropic laminate, so that

the projections of the ˜

H=0.5 planes of the respective signed distance functions form the matrix

cracking pattern used by Hallett as shown in Figure 7. A regular mesh with 80 by 20 intervals

in the xy-plane and 2 intervals through the thickness of each ply was used. Cohesive surfaces are

included at all ply interfaces in the model to allow for the prediction of delamination initiation

and propagation.

Figure 8 displays two key events predicted by the fracture simulation using MIC: the delamination

jump from the 45/90 interface to the 90/−45 interface, and then the event causing complete

delamination of the specimen, i.e. the delamination jump from the 90/−45 interface to the −45/0

interface. Note that the results in Figure 8 were obtained from a FE visualization of the continuous

displacement ﬁeld (14) and, therefore, do not reﬂect the actual displacement discontinuity of the

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Int. J. Numer. Meth. Engng (2011)

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MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

matrix crack. The inset ﬁgures in the lower left corners of Figure 8(a, b) display the ﬁber direction

stress distribution in the 0◦ply. This stress distribution indicates the shape of the delamination

forming at the interface between the 45 and 0◦plies. The region of elevated stress in the 0◦ply

corresponds to the delaminated area since the 45◦ply is cracked and does not carry load, and

the load in this area is fully transferred through the 0◦ply. The shape of the delamination shown

in Figure 8(b) is identical to the shape predicted by Hallett et al. [9] using a conventional FE

simulation. Additionally, the load levels at which the delamination jumps were predicted to occur

are very close to those predicted by the FE analysis in [9]. The origination of the delamination

at the 45/90 interface at the intersection of the free edge and the 45◦crack was predicted by

the present analysis at 380 MPa as compared to the 340–414 MPa range predicted in [9]. The

maximum load carried by the laminate before complete delamination at the −45/0 interface was

predicted by the present analysis at 502 MPa compared to approximately 542MPa predicted in [9].

In both analyses, ﬁber failure was not taken into account so that the laminate continued to carry

load after complete delamination at the −45/0 interface.

Veriﬁcation 3: quasi-isotropic strength without a priori crack knowledge

The preceding examples provide quantitativeveriﬁcation of the accuracy of modeling the interaction

between predeﬁned MIC and delamination cracks. The goal of the present developmentis to provide

a tool for predicting strength in arbitrary laminated composites without any prior knowledge of

the crack locations and damage patterns. Example strength predictions are presented below for

several ﬂat laminates under axial tension loading. In the examples presented, strength predictions

are made assuming a pristine laminate at the start of loading. As the load is increased, the LaRC03

failure criterion [28] is evaluated at the end of a load increment and, if the criterion is exceeded,

a MIC is automatically inserted. Cohesive surfaces are included at all ply interfaces in the models

to allow for delamination initiation and propagation.

The initial stress ﬁelds (prior to crack initiation) in the ﬂat laminates under axial tension

conditions are highly uniform in the x-direction, and therefore, the initial crack insertion locations

have a tendency to cluster, depending on the last digit of computer number representation. To obtain

more realistic initial crack insertion patterns and mimic inevitable statistical material variability,

quasi-random strength properties were generated across the coupon volume, according to Gurvich

[33]. Distributions of random transverse strength properties Yt(tensile), Yc(compressive) and shear

strength, S, were assumed to follow a classical two-parameter Weibull law deﬁned as

P(X)=1−exp−v

V0X

Axx,X=Yt,Yc,S(41)

where Axand xare scale and shape parameters, respectively, of the corresponding strength

properties. The value of x=12 was assumed for all the strength distributions, based on transverse

tensile strength scaling in carbon epoxy composites [34]. Using average strength values, shown in

Table I, scale parameters were calculated as

Ax=X/(1+1/x),X=Yt,Yc,S(42)

To ensure mesh independence of generated quasi-random strength values [33], the calculated

strengths were normalized by the reference volume V0=6250mm3(typical of an 8 ply unidirec-

tional composite, which is used for transverse strength measurement) and by the corresponding

local volumes, vi. Since the failure criterion was applied at individual integration points, a ‘local

volume’ viwas associated with each integration point as a product of the Gauss weight and Jacobian

so that INP

i=1i=Vn,whereINP is the total number of integration points and Vnis the volume of

the nth ply. Note that the random distribution of strength (41) was utilized only for crack initiation

purposes. There was no variability introduced in the cohesive law. This simpliﬁed deﬁnition is

used below for the method demonstration purposes only and requires modiﬁcation for systematic

studies of stochastic strength distribution effects on the apparent strength of composites.

The methodology proposed in the present document was applied to perform the fracture analysis

of the quasi-isotropic laminate considered in the previous section, but without any predeﬁned

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Figure 9. Axial stress versus displacement curve of the [454/904/−454/04]squasi-isotropic laminate

predicted with the three-crack model and a model with automatic crack insertions.

matrix cracking pattern. A typical load versus displacement curve obtained for one realization of

the stochastic strength ﬁeld is shown by the dotted line in Figure 9. The predicted load versus

displacement curve obtained with the idealized three-crack model is also shown in Figure 9 by

the solid line. The ﬁrst failure events predicted using the MIC model correspond to cracking in

the 90◦ply, followed by cracking in the 45 and 45◦plies. The cracking events at higher loads

occur simultaneously in all plies. The triangular symbols along the load versus deﬂection curve

designate the loads at which cracks are inserted in the 90◦ply (dark ﬁlled triangles), the 45◦ply

(lighter ﬁlled triangles) and the 45◦ply (open triangles). The maximum number of cracks inserted

in each ply was limited to 15. The analysis methodology correctly predicted both the sequence of

the delamination progression, the delamination failure load and the multistep load drop behavior

before the ﬁnal specimen failure that was experimentally observed in [9]. The average delamination

failure load determined from the experiments is 452 MPa, which is approximately 10% lower than

the predicted value. The load carried after the delamination failure load corresponds to the load

carried by the 0◦plies up to the point of ﬁber failure.

The delamination evolution process predicted by the automatic crack insertion simulation has

the same general features as predicted by the idealized three crack model, namely the delamination

evolution process initiates at the 45/90 interface at the intersection of matrix cracks and the free

edges and the ﬁnal delamination at the −45/0 interface extends over the majority of the specimen,

and results in a visible load drop. There are, however, apparent differences in the delamination

propagation process predictions of the two models. In the idealized three crack model delaminations

on the 45/90, 90/45 and −45/90 interfaces developed sequentially. In the case of automatic

crack insertion, the sequential development of delaminations was less pronounced. Although the

45/90 surface was ﬁrst to exhibit delaminations, delaminations on the surface continued to appear

and propagate after the 90/45 surface delaminations initiated and propagated. The z-direction

displacement contours obtained from the MIC simulation at two load levels are shown in Figure 10.

At a load level of 406MPa, accumulation of matrix cracking is clearly visible in the outside 45◦

ply, as shown in Figure 10(a). The displacement contours shown in Figure 10(b) were obtained at

a load level of 495 MPa, immediately prior to the full delamination of the −45/0 interface and the

associated load drop. Figure 10(b) displays a massive delamination at the −45/0 interface as well

as a number of evolving smaller delaminations that initiate from the free edge and matrix crack

intersections. This complex process results in a multistep softening of the load versus displacement

curve shown in Figure 9. In contrast, the load versus displacement curve for the three-crack model

shown by the solid line in Figure 9 exhibits a sharp load drop and no softening behavior due to

subcritical damage.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

Figure 10. Vertical (z-direction) displacement map for the [454/904/−454/04]slami-

nate with multiple cracks and delaminations: (a) applied axial stress equal to 406MPA

and (b) applied axial stress equal to 495 MPA.

Veriﬁcation 4: ply thickness and crack density effects

One of the key factors affecting the matrix cracking and delamination failure mode in laminated

composites is the ply thickness. A systematic experimental study of the delamination failure as a

function of ply thickness was conducted by Crossman and Wang [35]. A T300/934 [±25/90n]s

laminate family with n=1,...,8 was subjected to uniaxial tensile loading, perpendicular to the 90◦

ply, and failure loads and patterns were carefully documented. The delamination patterns (hatched

lines) and crack densities (spacing between horizontal lines) can be observed in Figure 11 for three

load levels and for two thicknesses of the 90◦ply, namely n=3 and 8.

The differences observed in the shape of the delamination in the two laminates are evident: in

the n=8 case, the delamination is funnelling off the individual matrix cracks, and, in the case of

thinner plies, the delamination spreads over multiple transverse cracks, and is referred to as ‘oyster

shaped’ in Reference [35]. A signiﬁcant difference between the two cases is also seen in the 90◦

ply transverse crack density.

The results of simulating the tensile loading in these two laminates are presented in Figure 12.

Some required material properties are not provided in Reference [35]. Therefore, the simulations

were performed using the material properties for T300/936 provided in Table I. To illustrate the

damage development process in detail, damage variable contours for both transverse matrix cracks

and delaminations are plotted on the undeformed geometry. The areas of delamination correspond

to interfaces where the value of the damage variable dexceeds 0.995. The process of delamination

within the cohesive zone framework is a continuous process in which the damage variable changes

from 0 to 1. However, the relationship between the magnitude of the displacement jump and

the value of the damage parameter is a function of the constitutive model used in the analysis.

For the bilinear constitutive model [5] used in the present paper, this relationship is nonlinear

and a displacement jump of approximately 0.5fcorresponds to d∼1−0/f.Since0/f1

only those values of the damage parameter very close to 1 correspond to a displacement gap

on the order of magnitude of f. Therefore, a threshold value of 0.995 was chosen for display.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Figure 11. Schematics of the fracture sequence in (25/−25/90n)slaminates: (a) just prior to

delamination; (b) subsequent to delamination; and (c) just prior to ﬁnal failure. ‘Reprinted with

permission from ASTM STP 775 Damage in Composite Materials [35], copyright ASTM Inter-

national, 100 Barr Harbor Drive, West Conshohocken, PA 19428’.

(a) (b) (c)

Figure 12. Predicted delamination patterns in: (a) [±25/908]sand (b), (c) [±25/903]s

laminates. Light shaded areas are delaminations between 90 and −25 plies and the

dark shaded corresponds to 25/−25 interface delamination.

In the case of transverse matrix cracks, the surface where ˜

H=0.5andd>0.995 is displayed in

each ply.

Predicted matrix cracking and delamination patterns for the laminates with n=8andn=3are

shown in Figure 12. Light shaded areas correspond to predicted delaminations at the 90/−25

interface, and dark shaded areas correspond to predicted delaminations at the 25/−25 interface.

The state of cracking and delamination just before complete failure of the [±25/908]slaminate

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

is shown in Figure 12(a). It can be observed that the shape of the delamination in Figure 12(a)

is very similar to the experimentally observed funnel-type delamination shown in Figure 11. In

addition, in both the experimental observations [35] and the predictions, thin delamination areas

accompany all matrix cracks.

The delaminations in the [±25/903]slaminate evolve extremely rapidly before failure.

A sequence of two states of delamination at nearly identical load levels is shown in Figure 12(b), (c).

It can be observed that the predicted density of matrix cracking for n=8 is signiﬁcantly lower

than predicted for the thinner n=3 case. In addition, the delaminations predicted for n=3 cover

multiple transverse cracks and have shapes consistent with the experimental results shown in

Figure 11. The extent of the delamination in Figure 12(b) is very similar to that in Figure 11.

It is likely that the larger predicted delamination on Figure 12(c) corresponds to an unstable

equilibrium state, which is unlikely to be caught in the experiment due to the very sudden failure

process after delamination initiation.

Veriﬁcation 5: delamination initiation variations

In the case of the quasi-isotropic laminates considered above, the delamination evolution process

initiated from the intersection of matrix cracks and the free edges, leading to eventual disintegration

of the laminate. It is of interest to evaluate the present methodology for characterizing the process

of matrix crack-induced damage accumulation in laminates with different ply orientations, where

the delamination and matrix cracking evolution and interaction patterns may vary. A number of

angle-ply laminate conﬁgurations were experimentally and analytically investigated by Johnson

and Chang [36]. The T300/976 graphite ﬁber material system (see Table I for ply level properties)

was used. Tensile failure of a [±45/90]slaminate and a [±602]slaminate, both considered in

Reference [36], with a ply thickness of 0.127 mm are considered below. These laminates do not

contain any 0◦plies and completely lose their load-carrying capacity as a result of matrix cracking

and delamination.

Predicted matrix crack and delamination damage evolution patterns are provided for the

[±45/90]slaminate and for the [±602]slaminate in Figures 13 and 14, respectively. Damage

patterns are shown at three load levels, including the load level immediately preceding the

simulated ﬁnal failure. To illustrate the damage development process in detail, damage variable

contours for both MICs and delaminations are plotted on the undeformed geometry. In all cases

damage variable values of d>0.995 are displayed to indicate the locations of displacement

discontinuity. For both laminates considered, all the matrix cracks quickly grow through the width

of the specimen, except for the cracks that develop in the very early stages of loading (not shown).

The general damage evolution process of the [±45/90]sspecimen is similar to the process for

the quasi-isotropic laminate considered above. Triangular shape delaminations initiate in multiple

locations on the +45/−45 and −45/90 interfaces at the matrix crack and free edge intersections,

as shown in Figure 13(a), (b). As the loading continues, the delaminations grow inward and expand

in size until they connect the two edges, and the interfaces via matrix cracks, at which point the

specimen fails.

The failure process in the [±602]sspecimen is starkly different compared to the previous

laminate. Delamination initiation and propagation is not anchored around the outer edges of

the specimen as in the [±45/90]sspecimen. Delaminations in this case initiate in the interior of the

specimen at the matrix crack intersections, as shown in Figure 14(a). As the load is increased the

delamination grows in the interior of the specimen. Figure 14(b) shows a delamination band of

almost uniform length through the entire width of the specimen, which then extends and allows the

matrix cracks to separate the plies (Figure 14(c)). This difference in failure mechanisms between

the two laminates has been observed experimentally in [36].

The ability to address various failure mechanisms arising in nontraditional composite laminates

without modifying the analysis framework and/or mesh is a critical advantage of x-FEM tech-

nology. Such capability is becoming increasingly important with aerospace companies focused on

increasing structural efﬁciency of composites and breaking away from traditional laminate design.

Published 2011. This article is a US Government

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

Figure 13. Predicted damage at increasing load levels in a [45/−45/90]slaminate: (a) delamination

initiation; (b) intermediate stage; and (c) immediately preceding ﬁnal failure.

CONCLUSIONS

A fully three-dimensional analysis methodology is proposed for modeling complex matrix cracking

and delamination networks in laminated composites. The proposed methodology is based on a

regularized x-FEM formulation [23] for MIC modeling of arbitrary transverse matrix cracks, and a

cohesive formulation to model delaminations between plies. Veriﬁcation studies include simulation

of delamination initiation from matrix cracks, modeling of delamination jumps from one ply

interface to another, prediction of the effects of ply thickness on delamination shape and transverse

crack density and examination of delamination initiation variations due to varying ply orientations.

In each of the veriﬁcation studies, good agreement between experimental observations and/or other

computational techniques and the MIC modeling methods described above was shown.

Delamination evolution emanating from transverse cracking was examined in detail by modeling

a mode II fracture specimen called the transverse crack tension specimen. Comparison between

Published 2011. This article is a US Government

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Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

Figure 14. Predicted damage at increasing load levels in a [602/−602]slaminate: (a) delamination

initiation; (b) intermediate stage; and (c) immediately preceding ﬁnal failure.

a standard FE approach and an approach where the transverse crack was modeled using a MIC

yielded nearly identical results, even in the case when the MIC model had an extremely skewed

mesh. This modeling effort was used to also verify the stress predictions near the singularity

of the transverse crack, again yielding very good agreement between standard FE and the MIC

methodologies.

Damage evolution and failure prediction of quasi-isotropic composite laminates subject to

uniaxial tension was performed. Thermal residual stresses were also included in the analysis.

A case with a limited number of pre-deﬁned cracks was modeled and compared with numerical

results using a standard FE approach, yielding an almost exact match in behavior. A case with no

preliminary information of any kind regarding the locations of damage onset and the sequence of

damage progression was compared with the ﬁrst case and experimental data. This model required

a random variation of strength properties in the analysis due to the initially uniform nature of the

stress ﬁeld arising in composite laminates under axial tension. Good agreement between the anal-

ysis predictions and experimental data was observed both in terms of the predicted delamination

evolution sequence and the predicted failure.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

E. V. IARVE ET AL.

The variation of delamination shape and crack density due to ply thickness was studied by

modeling a [±25/908]slaminate and a [±25/903]slaminate and comparing to experimental data.

In the case with the thick set of central plies, the delaminations are shaped like funnels across the

specimen, and the sparse number of transverse cracks in the 90◦plies are accompanied by slight

delaminations. In the case with the thin set of 90◦plies, the delaminations are broader ‘oyster-like’

shapes and the crack density is considerably higher. These experimentally observed delamination

shape and crack density differences were matched quite well by the MIC modeling methodology

predictions.

Two angle-ply laminates were considered: a [45/−45/90]slaminate and a [602/−602]slami-

nate. An apparent difference in the failure mechanism between the [45/−45/90]slaminate and

the [602/−602]slaminate was observed where the [45/−45/90]sexhibited delaminations initi-

ating at the intersections of matrix cracks and the free edges while the [602/−602]sexhibited

delaminations initiating throughout the width of the specimen at crack intersections. These same

trends have been observed experimentally and are reported in the literature.

The application of the Rx-FEM methodology in conjunction with cohesive zone modeling of

ply delamination has allowed the modeling of the extremely complex phenomenon shown in the

veriﬁcation test cases. This methodology has been shown to be numerically robust and, due to the

constancy of the integration scheme during the addition of cracks, has allowed the easy connection

of neighboring plies through cohesive zones. It is clear that the value of a modeling scheme such

as the one proposed and demonstrated is in the ability to link the complex interactions of matrix

cracking with the initiation and evolution of delaminations throughout the composite material.

ACKNOWLEDGEMENTS

The work was funded under NASA AAD-2 contract number NNX08AB05A-G and partially by AFRL

contract FA8650-05-D-5052 with the University of Dayton Research Institute.

REFERENCES

1. Maimí P, Camanho PP, Mayugo JA, Dávila CG. A continuum damage model for composite laminates: part

I—constitutive model. Mechanics of Materials 2007; 39:897–908.

2. Maimí P, Camanho PP, Mayugo JA, Dávila CG. A continuum damage model for composite laminates: part

II—computational implementation and validation. Mechanics of Materials 2007; 39:909–919.

3. Alfano G, Crisﬁeld MA. Finite element interface models for the delamination analysis of laminated composites:

mechanical and computational issues. International Journal for Numerical Methods in Engineering 2001;

50:1701–1736.

4. Jiang WG, Hallett SR, Green BG, Wisnom MR. A concise interface constitutive law and its application to scaled

notched tensile specimens. International Journal for Numerical Methods in Engineering 2007; 69:1982–1995.

5. Turon A, Camanho PP, Costa J, Dávila CG. A damage model for the simulation of delamination in advanced

composites under variable-mode loading. Mechanics of Materials 2006; 38:1072–1089.

6. Krueger R. The virtual crack closure technique: history, approach and applications. NASA/CR-2002-211628,

2002.

7. Deobald LR, Mabson GE, Dopker B, Hoyt DM, Baylor J, Greasser D. Interlaminar fatigue elements for crack

growth based on virtual crack closure technique. 48th AIAA/ASME/ASCE/AHS/ASC Structures,Structural

Dynamics,and Materials Conference, Honolulu, Hawaii, 23–26 April 2007.

8. Iarve EV, Mollenhauer D, Kim R. Theoretical and experimental investigation of stress redistribution in open hole

composite laminates due to damage accumulation. Composites Part A 2005; 36:163–171.

9. Hallett SR, Jiang WG, Khan B, Wisnom MR. Modeling the interaction between matrix crack and delamination

damage in scaled quasi-isotropic specimens. Composites Science and Technology 2008; 68:80–89.

10. Van der Meer FP, Sluys LJ. Continuum models for the analysis of progressive failure in composite laminates.

Journal of Composite Materials 2009; 40:2131–2156.

11. Allix O, Guedra-Degeorges D, Guinard S, Vinet A. Analyse de la Tenue aux Impacts à Faible Vitesse et Faible

Énergie des Stratiﬁés Composites par la Mécanique de l’Endommagement. Mécanique and Industries 2000;

1(1):27–35.

12. Ladevèze P, Lubineau G, Marsal D. Towards a bridge between the micro- and mesomechanics of delamination

for laminated composites. Composites Science and Technology 2006; 66:698–712.

13. Lubineau G, Ladevèze P. Construction of a micromechanics-based intralaminar mesomodel, and illustrations in

ABAQUS/Standard Computational. Materials Science 2008; 43:137–145.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme

MESH-INDEPENDENT MATRIX CRACKING AND DELAMINATION MODELING

14. Wawrzynek PA, Ingraffea AR. An interactive approach to local remeshing around a crack tip. Finite Elements

in Analysis and Design 1989; 5(1):87–96.

15. Fish J, Markolefas S. The s-method of the ﬁnite element method for multilayered laminates. International Journal

for Numerical Methods in Engineering 1992; 33:1081–1105.

16. Iarve EV. Three-dimensional stress analysis in open hole composite laminates containing matrix cracks. AIAA-

98-1942, 1998.

17. Moës N, Dolbow J, Belytschko T. A ﬁnite element method for crack growth without remeshing. International

Journal for Numerical Methods in Engineering 1999; 46:601–620.

18. Belytschko T, Parimi C, Moës N, Sukumar N, Usui S. Structured extended ﬁnite element methods for solids

deﬁned by implicit boundaries. International Journal for Numerical Methods in Engineering 2003; 56:609–635.

19. Huynh DBP, Belytschko T. The extended ﬁnite element method for fracture in composite materials. International

Journal for Numerical Methods in Engineering 2009; 77:214–239.

20. Van der Meer FP, Sluys LJ. A phantom node formulation with mixed mode cohesive law for splitting in laminates.

International Journal of Fracture 2009; 158(2):107–124.

21. Ling DS, Yang QD, Cox BN. An augmented ﬁnite element method for modeling arbitrary discontinuities in

composite materials. International Journal of Fracture 2009; 156:53–73.

22. Hansbo A. An unﬁtted ﬁnite element method for the simulation of strong and weak discontinuities in solid

mechanics. Computer Methods in Applied Mechanics and Engineering 2004; 193:3523–3540.

23. Iarve EV. Mesh independent modeling of cracks by using higher order shape functions. International Journal

for Numerical Methods in Engineering 2003; 56:869–882.

24. Patzak B, Jirásek M. Process zone resolution by extended ﬁnite elements. Engineering Fracture Mechanics 2003;

70:957–977.

25. Benvenuti E, Tralli A, Ventura G. A regularized x-FEM model for the transition from continuous to discontinuous

displacements. International Journal for Numerical Methods in Engineering 2008; 74:911–944.

26. Benvenuti E. A regularized x-FEM framework for embedded cohesive interfaces. Computer Methods in Applied

Mechanics and Engineering 2008; 197:4367–4378.

27. Oliver J, Huespe AE, Sánchez PJ. A comparative study on ﬁnite elements for capturing strong discontinuities:

E-FEM vs X-FEM. Computer Methods in Applied Mechanics and Engineering 2006; 195:4732–4752.

28. Dávila CG, Camanho PP, Rose CA. Failure criteria for FRP laminates. Journal of Composite Materials 2005;

39(4):323–345.

29. Mollenhauer D, Iarve EV, Kim R, Langley B. Examination of ply cracking in composite laminates with open-holes:

a Moiré Interferometric and Numerical Study. Composites Part A 2006; 37:282–294.

30. Whitney JM. Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing: Lancester, PA, U.S.A.,

1987; 339.

31. König M, Krüger R, Kussmaul K, von Alberti M, Gädke M. Characterizing static and fatigue interlaminar

fracture behaviour of a ﬁrst generation graphite/epoxy composite. In Composite Materials:Testing and Design,

ASTM STP 1242, Hooper SJ (ed.), vol. 13. American Society for Testing and Materials: West Conshohocken,

PA, U.S.A., 1997; 60–81.

32. Liu S, Kutlu Z, Chang F-K. Matrix-induced delamination propagation in graphite/epoxy laminated composites

due to a transverse concentrated load. In Composite Materials:Fatigue and Fracture,4,ASTM STP 1156,

Stinchkomb W, Ashbaugh NE (eds). American Society for Testing Materials: Philadelphia, 1993; 86–11.

33. Gurvich MR. Strength size effect for anisotropic brittle materials under random stress state. Computer Science

and Technology 1999; 59(11):1701–1711.

34. Iarve EV, Mollenhauer D, Kim R. Delamination onset prediction in joints by using critical Weibull failure volume

method. Proceedings of ICCM-15, Durban, SA, 2005.

35. Crossman SW, Wang ASD. The dependence of transverse cracking and delamination on ply thickness in

graphite/epoxy laminates. Damage in Composite Materials. ASTM International: West Conshohocken, PA, U.S.A.,

1982; 118–139.

36. Johnson P, Chang FK. Characterization of matrix crack-induced laminate failure—part I: experiments. Journal

of Composite Materials 2001; 35(22):2009–2034.

Published 2011. This article is a US Government

work and is in the public domain in the USA.

Int. J. Numer. Meth. Engng (2011)

DOI: 10.1002/nme