ArticlePDF Available

Mesh‐independent matrix cracking and delamination modeling in laminated composites

Authors:

Abstract

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 withouttextita priori knowledge 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.
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, 1426] 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=1un
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=(1d)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+(S2Y2)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=(TT0)a.
Here, TT0is 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)dVv
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
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.
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)+
N1
n=1
nA=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|gEV
|∇ ˜
H|gE0
sym V
|∇ ˜
H|gE0
000
U(1)
U(2)
U(3)
,(31)
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 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)
nu(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,nun,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˜
Hn0z=zn
gEnn ˜
Hn
0z=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+1z=zn
gEnn+1(1˜
Hn)˜
Hn+1z=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˜
Hnz=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.
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.
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)=ui1
x(0,y,z)iand ui
x(L,y,z)=ui1
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.
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 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=Pi1+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
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.
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 0.4×106
22,33(1/C) 3×1052.54 ×105
TT0(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
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 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 45plies, 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
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 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 90plies, the 90 and 45plies and the 45 and 0plies.
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
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
matrix crack. The inset figures in the lower left corners of Figure 8(a, b) display the fiber direction
stress distribution in the 0ply. This stress distribution indicates the shape of the delamination
forming at the interface between the 45 and 0plies. The region of elevated stress in the 0ply
corresponds to the delaminated area since the 45ply is cracked and does not carry load, and
the load in this area is fully transferred through the 0ply. 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 45crack 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)=1expv
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
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 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 90ply, followed by cracking in the 45 and 45plies. 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 90ply (dark filled triangles), the 45ply
(lighter filled triangles) and the 45ply (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 0plies 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 90ply, 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 d10/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
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
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 0plies 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
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 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
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 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 90plies are accompanied by slight
delaminations. In the case with the thin set of 90plies, 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
... Because of these favorite characteristics, laminated composite structures have been used in various industries like aerospace engineering, naval engineering, and civil engineering [4][5][6]. Delaminations as the most common damages in laminated composite structures are an interesting research area in literature [7]. Valdes et al. [8] conducted an investigation on delamination detection in composite laminates from variations of their modal characteristics. ...
Chapter
Delamination is one of the most common damages in laminated composite structures. This damage is usually created during manufacturing. Therefore, delamination detection is essential to prevent structural failure in operational conditions. This study proposes a new delamination detection technique by combining the one-dimensional and two-dimensional discrete wavelet transforms. Since delamination is boundary damage, differentiation of its boundaries is significant, but challenging, and the conventional two-dimensional wavelet transformations have weaknesses in overcoming this challenge in some cases. The main idea of the proposed technique is to combine the ability of one-dimensional discrete wavelet transform with two-dimensional discrete wavelet transform to increase the accuracy of delamination detection. Findings show that the proposed technique can significantly improve delamination detection accuracy.KeywordsDelamination detectionBoundary separation techniqueWavelet transformsRectangular laminated composite plates
... Discrete damage models can be classified as nodal-enrichment methods (X-FEM, regularized extended finite element method (RX-FEM) [32][33][34][35], variational multiscale cohesive method (VMCM) [36,37], phantom node method (PNM) [38], floating node method (FNM) [39][40][41]) and cohesive network models [42][43][44][45][46][47][48][49]. They usually offer high-fidelity prediction of matrix cracks. ...
Article
In this paper, a finite element-based framework is presented to model the probabilistic progressive failure of fiber-reinforced composite laminates with high fidelity and efficiency. The framework is based on the semidiscrete modeling approach that can be seen as a good compromise between continuum and discrete methods. The enhanced semidiscrete damage model (ESD2M) tool set comprises a smart meshing strategy with failure mode separation, a new version of the enhanced Schapery theory with a novel generalized mixed-mode law, and a novel probabilistic modeling strategy. These three joined components make the model efficient in capturing failure modes such as matrix cracks, fiber tensile failure, and delamination, as well as their interactions with high fidelity, while taking material nonuniformities into account. The model capabilities are demonstrated using single-edge notched tensile cross-ply laminates as an example. The ESD2M was not only capable of capturing the complex damage progression but also provided insights and explanations for some of the failure events observed in the laboratory. The presented framework efficiently integrates failure mode predictions with probabilistic modeling and enables Monte Carlo simulations to predict the ultimate failure strength with good accuracy, as well as its scatter.
Article
A methodology was developed and validated to quantify the uncertainty for advanced progressive damage models for composites. It relies on a pragmatic approach entailing the definition of efficient emulators, the use of state-of-the-art computational models, and the employment of bootstrapping statistic techniques. The proposed methodology was calibrated on numerical results obtained running a limited amount of virtual experiments (five for each configuration) of unnotched and open-hole specimens in tension and compression. The structural strength was taken as the quantity of interest, and a methodology was proposed and validated to determine its distribution and associated statistics.
Article
The durability and residual load carrying capacity of composite materials is of critical importance for increasing their applicability. Regularized eXtended Finite Element Method framework for discrete modeling of damage evolution and interaction in laminated composites has been extended to increase the limit of crack density for a given mesh size. The formulation allows multiple twining of original nodes while the displacement discontinuity is represented by a pair of element twins maintaining the Gauss integration schema of the original element. Residual Strength Tracking methodology was applied for mesh independent crack insertion as well as in the initiation phase of the fatigue Cohesive Zone Model (CZM). An automatic cycle jump step selection algorithm within implicit framework was implemented to provide solution stability. The predictions were compared with the experimental data for three different open hole composite laminates under tension-tension fatigue and showed excellent agreement.
Article
In the paper presented a study of the fracture process of explosive welded layered material AA2519-AA1050-Ti6Al4V (Al-Al-Ti laminate) at ambient (+20 ℃) and reduced (-50 ℃) temperatures. The tensile tests and fracture toughness tests were conducted for both types specimens made of base materials plates and of laminated plate. During tensile tests performed on flat specimens the signals of loading force (P) and specimen extension (uext) were recorded. The signals of loading force (P), specimen deflection (udef) and the crack mouth opening displacement (COD) were recorded while performing the tests on fracture toughness of materials on SENB type specimens. The breakthroughs of the tested specimens were observed by scanning electron microscope (SEM). The results obtained during the experimental tests indicate the complicated nature of the cracking process of specimens from the Al-Al-Ti laminate. The deviation from linearity was observed during loading on the sections where it should be linear. There were also slightly decreases in the force value during the load. During macro observation of the fractured surfaces of SENB specimens, delamination cracks between the Al and Ti layers are clearly visible. The size of these cracks increases as the test temperature decreases. The SEM observations clearly indicate that the delamination crack is formed in the AA1050 layer joining the Ti6Al4V and AA2519 base materials. Precise observation by SEM allowed to establish that the development of a delamination crack begins with different type of particles fracture and material discontinuous in narrow strip (20-40 mm) near at Ti6Al4V material. Then the delamination crack develops in the layer of AA1050 material according to a shear mechanism. In order determine the stress and strain distributions in tested specimens the they numerical models were developed and simulation of loading were performed by ABAQUS program. The results of stress and strain distributions obtained clearly shows on large differences they values in the different layers of the laminate. Maximal differences of stress levels occurs between layers Ti6Al4V and AA1050 materials. Due to the fact that the material of the connecting layer (AA1050) has a low level of yield strength, a very high level of plastic strain occurs. Based on the results obtained during the experimental tests, SEM observation of fracture surfaces and numerical calculations, it can be concluded that the weakest area in the tested laminate is the connecting layer AA1050. Namely is it, a thin strip between AA1050 and Ti6Al4V, which consists of metal base particles, intermetallic compounds, their oxides and discontinuities of the material in form of voids. When using this type of laminate, special attention should be paid to the strength of the AA1050 bonding layer.
Article
This paper presents and validates a new local to global (L2G) FEM approach that can analyze multiple, interactive fracture processes in 2D solids with improved numerical efficiency and robustness. The method features: 1) forming local problems for individual and interactive cracks; and 2) parallel solving local problems and returning local solutions as part of the trial solution for global iteration. It has been demonstrated analytically (through a simple 1D problem) and numerically (through several benchmarking examples) that, the proposed method can substantially improve the robustness of the global solution process and significantly reduce the costly global iteration for convergence. The demonstrated improvement in numerical efficiency is up to 20∼40% for mildly unstable problems. For problems with severely unstable crack initiation and propagation, the improvement can be more significant. This new method is readily applicable to other popular methods such as the extended FEM (X-FEM), Augmented FEM (A-FEM) and Phantom-node method (PNM).
Article
In this study, the non-local continuum damage model is further developed for progressive failure analysis of laminated composites. Characterized by two internal length scales, an orthotropic non-local integral strategy is implemented to model the constrained matrix damage along fiber direction. The introduction of spatial averaging may avoid spurious strain localization and thus ensure the physically-meaningful energy dissipation during damage evolution process. Compared to standard damage descriptions, the pathological mesh sensitivity and mesh orientation bias are effectively alleviated, leading to objective and accurate solutions. Moreover, an interactive damage transfer scheme is proposed to capture the strong interaction between matrix cracking and interface delamination, so that faithful predictions on the complex failure mechanisms of composite laminates are guaranteed. Performance of the present model and its superiority over conventional methods are demonstrated through several numerical examples, including the typical tensile failure of notched and unnotched laminates.