Date post: | 05-Apr-2018 |
Category: |
Documents |
Upload: | vu-dinh-quy |
View: | 223 times |
Download: | 0 times |
of 12
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
1/12
A criterion for modelling initiation and propagationof matrix cracking and delamination in cross-ply laminates
J.-L. Rebiere a,*, D. Gamby b
a Institut dAcoustique et de Mecanique, Universite du Maine, Avenue Olivier Messiaen 72085 Le Mans Cedex 9, Franceb Laboratoire de Mecanique et Physique des Materiaux, ENSMA, Teleport 2, 1 Avenue Clement Ader, BP 40109,
86961 Futuroscope Chasseneuil Cedex, France
Received 27 November 2003; received in revised form 25 March 2004; accepted 30 March 2004
Available online 14 May 2004
Abstract
A variational approach is used to model the behaviour of composite cross-ply laminates damaged by transverse, longitudinal
cracking and delamination. An energetic criterion is proposed. It is based on the strain energy release rate associated with each of the
three damage modes. The first part of this paper is concerned with the modelling of the transverse and longitudinal cracking. In the
secondpart, a model forstudying delamination damageis presented. The numerical results show that these models provide a consistent
level of accuracy for a variety of thin laminate material systems and configurations, with various combinations of delaminations and
matrix cracks. In this paper several numerical simulations meant to describe initiation for each damage mode are proposed. The es-
timation of damagemodescontributions is achieved fortwo thin laminatesin order to predict theevolution of damagemode transition.
2004 Elsevier Ltd. All rights reserved.
Keywords: A. Polymer-matrix composites; B. Matrix cracking; C. Delamination; Damage mechanics; D. Life prediction
1. Introduction
The ultimate failure of a laminate follows the occur-
rence of two or three damage mechanisms and fibre
breaking. Usually, these three main damage modes are,
first, transverse cracking, later longitudinal cracking
and/or delamination. Experimentally, it was observed
that the order and initiation time of each damage mode
are governed by the following parameters: the laminate
geometry, for example the thicknesses of the different
layers [1,2], the nature of the fibre/matrix constituents,
the loading history and the cycle of fabrication [3,4]. The
first part of this study investigates the influence of matrix
cracking (transverse and longitudinal) on the mechanical
properties of a cross-ply laminate. In the second part,
delamination is studied and in the third part examples of
damage mode succession are proposed. This study was
prompted by experimental results [513]. In experimen-
tal loading conditions (monotonic and fatigue tests), the
results [513] show that the first damage mode is usually
transverse cracking. Two particular states can cha-
racterise this damage mode: its initiation or occurrence
of the first transverse crack called first ply failure(FPF) on one hand and the limiting state when no more
transverse crack can be created, named characteristic
damage state (CDS) on the other. Afterwards, it was
observed that the nature of the second damage mode
depends on the three above parameters. For example, in
a thick laminate, the authors of [59] observed the ini-
tiation and evolution of delamination. Ply separation is
caused by the increase of the normal stress rzz and of the
interlaminar stress rxz. For thin laminates, the damage
mode succession is different. Some authors [5,1014]
observed that the second damage mode, which follows
transverse cracking, is longitudinal cracking. In this
case, local delamination appears between 0 and 90
layers, near the crossing of longitudinal and transverse
cracks, only when longitudinal cracks are widespread. In
each case, the accumulation of the different damage
* Corresponding author. Tel.: +33-2-43-83-34-75; fax: +33-2-43-83-
31-49.
E-mail addresses: [email protected] (J.-L. Rebiere),
[email protected] (D. Gamby).
0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compscitech.2004.03.008
Composites Science and Technology 64 (2004) 22392250
www.elsevier.com/locate/compscitech
COMPOSITES
SCIENCE AND
TECHNOLOGY
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
2/12
modes (two or three damage modes present within the
laminate volume) causes fibre breaking in the 0 layers.
All fibre breaks entail splitting which appears just
before the ultimate failure of the laminate.
For modelling the strain/stress relationship during
damage growth, analytical and numerical approaches
have been proposed. Several models describe the initia-
tion of the first damage mode. They mainly rely on some
stress field distribution and a relationship between
loading and crack density is usually proposed. The sim-
plest models, called shear lag analysis [9,1518], aregenerally displacement-based approaches. Other models
such as variational approaches, whose principles are
explained in [19,20], use the principle of minimum com-
plementary energy [2126]. Other studies rely on the fi-
nite element method [2729]. Alternative models are
based on phenomenological approaches [3035], self-
consistent analysis [36,37] or approaches that use specific
aspects of the cracks patterns [38]. Local delamination is
often described by two-dimensional models. This is the
case of the finite element study of Wang et al. [12]. We
can also cite the works of Nairn and Hu [39], based on a
variational approach in which the interaction between
transverse cracks and local delaminations, which appear
near crack tips, is described. Hashin [40] analyses lon-
gitudinal and transverse cracking through a variational
model, with a restrictive hypothesis of constant normal
stress distribution through the thickness of each dam-
aged layer. Binienda et al. [41], who propose a finite el-
ement approach, use the same energy criterion. For
modelling the initiation of the second and/or third
damage mode, several criteria were used. For instance, to
our knowledge, no criterion has been proposed for the
initiation and growth of longitudinal cracking, except in
the works of Binienda et al. [41] who computed the strain
energy release rate in a cross-ply laminate with a pre-
existing longitudinal crack. The main reason for this lack
of attention is that longitudinal damage appears only in
laminates having some specific thickness ratio and
stacking sequence. Moreover, when longitudinal matrix
cracking appears, it is generally shortly before the end of
the laminate life. On the opposite, for describing de-
lamination damage evolution, several criteria have been
proposed. Most of them either involve local stress values
or strain energy release rate associated with the damaged
area (or a parameter related to the damage surface). Theworks of Schon et al. [42] are based on an energy ap-
proach. For these authors, the strain energy release rate
is a good measure of the material resistance to delami-
nation growth. Two different delamination modes can be
observed according to the loading history, monotonic
loading (sudden loading) or fatigue loading. The above
authors conducted tests on DCB specimens. They
showed that the strain energy release rate associated with
delamination is not the same for static and fatigue tests.
Other models involve critical stress values. Marion
et al. [43] propose a quadratic stress criterion using the
value of the interlaminar stress at a characteristic dis-
tance of the interface. Leguillon et al. [44] compare the
stress criterion of [44] with a stress criterion based on the
mean shearing stress value only. There is a major ob-
stacle to the use of a delamination criterion based on a
maximum stress value along a debonding edge [44]. In a
model with homogenised layers and perfect interfaces,
the stress field is singular as already mentioned: stress
components take on infinite values at the intersection of
the interface and free edges. Even if these values remain
finite when computed, they are irrelevant. To overcome
this problem, Whitney and Nuismer [45] introduce a
characteristic length like Marion et al. [43]. Kim and
Nomenclature
a longitudinal half crack spacing
Ad interlaminar delaminated area
Af intralaminar cracked area
b transverse half crack spacingdx delamination length in the longitudinal di-
rection
dy delamination length in the transverse direction
k constraint parameter
h ply thickness
G strain energy release rate
GFT strain energy release rate associated with
transverse cracking
GFL strain energy release rate associated with
longitudinal cracking
Gdx , Gdy strain energy release rate associated with
delaminated length dx or dy
Gc critical strain energy release rate
Gcrf cracking critical strain energy release rate
Gcrd delamination critical strain energy release
ratek ply index
L1 laminate length in x direction
L2 laminate length in y direction
m number of longitudinal cracks
n number of transverse cracks
Sijkl local compliances
t0 0 layer thickness
2t90 90 layer thickness
Ud deformation energy of the whole laminate
Ucel deformation energy of the unit damaged
cell
V volume of the half unit cell
2240 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
3/12
Sony [46] connect this distance with the layer thickness
modified by a factor varying from one to two. Brewer
and Lagace [1] and Lecuyer [47] assume that this distance
should be independent of the layer thickness, and that
the factor must be determined from experimentations.
We can also cite the works of Diaz-Diaz and Caron [48]
who propose a model for laminates with free edges inuniaxial loading. In their approach, the edge singularity
is smoothed out. A simple shear stress criterion is pro-
posed and validated for two hns laminates. Theseauthors obtain critical values of the interlaminar shear
stress that depend on the thickness ratio of the layers but
do not depend on the layer orientations. To confirm this
result, they also performed a calculation with a criterion
based on the strain energy release rate.
In this section, we report experimental observations
concerning the different damage modes. Various ana-
lytical and numerical models are proposed for the
analysis of the stress field during the evolution of the
damage. Some damage criteria, a stress based approach
and an energetical approach are described. In this arti-
cle, a damage initiation and growth is proposed.
A variational approach gives the stress field necessary to
derive the strain energy release rate associated to each
damage mode. The influence of the three damages
modes (transverse cracking, longitudinal cracking and
delamination) is studied. Using the proposed model, our
objective is to predict the successive occurrence of sev-
eral damage mechanisms during the life of a laminate.
We also studied the influence of laminate architecture on
damage mechanisms and life prediction.
2. Matrix cracking modelling
The proposed analytical model is based on a varia-
tional approach for 0m; 90ns cross-ply laminates
(Fig. 1). The parameter related to the lay up architecture
is kk t0=t90 m=n where t0 is the thickness of the 0layer and 2t90 is the thickness of the 90 layer. Experi-
mentally, as explained in the previous section, the fol-
lowing succession of damage modes can be observed.
The transverse cracking is the first damage mode oc-
curring in the 90 layers. The cracks are supposed tohave a rectangular plane geometry and all cracks span
the whole width of the laminate plate and the whole
thickness of the 90 layers. Different damage mecha-
nisms are observed in the second damaging step which
occurs in this type of laminate. It can be delamination or
longitudinal cracking. Under some circumstances, lon-
gitudinal cracks appear in the 0 layers and they are
supposed to obey the same hypotheses as transverse
cracks. The distribution of longitudinal and transverse
cracks as well is supposed to be uniform in the two
x and y directions.With the previous hypotheses related to the trans-
verse and longitudinal distributions and geometry of the
cracks the laminate damage can be described by the
unit damaged cell displayed in Fig. 2. This unit
damaged cell is situated between two consecutive
transverse cracks and two consecutive longitudinal
cracks. The geometrical hypotheses will be described
later.
The variational approach is based on the proper
choice of a statically admissible stress field. The starting
point is the distribution used first by Vasilev and
Duchenco [21], later by Hashin [22] and then by Varna
and Berglund [25]. However, we also take into account
the variation of the stress field through the thickness ofthe laminate damaged by transverse and longitudinal
cracks.
The stress field in the two layers of the laminate has
the following form:
2 a
2 b
t0
2 t90 2 h
LongitudinalCracks
TransverseCracks
TriangularDelaminatedArea
Uniaxial loading
z
y
x
Fig. 1. Laminate damaged by transverse and longitudinal cracks and delamination.
J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250 2241
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
4/12
rTkij r
0kij r
Pkij : 1
For an undamaged laminate loaded in the x direction,
the layers are in an uniform plane stress state r0kij ob-
tained by the laminate plate theory, where k is the ply
index k 0; 90. Orthogonal cracks induce stressperturbations in the 0 and 90 layers which are denoted
r
Pk
ij .In order to verify all the following boundary condi-
tions, we must use the hypothesis of uniform stress
distribution in the thickness of the 90 damaged layer. In
the 0 layers, the stress distribution through the thick-
ness is not uniform. The normal stresses have the fol-
lowing form:
r90xx r090xx 1 /1x; r
90yy r
090yy 1 w1y;
r0xx r00xx 1 /2xuz; r
0yy r
00yy 1 w2y:
2
The unknown functions are /1x, /
2x, w
1y, w
2y
and uz. The overall equilibrium conditions in thedamaged laminate give:
r090xx t90 r00xx t0 r
90xx t90 r
0xx t0 r0h;
r090yy t90 r00yy t0 r
90yy t90 r
0t0 0:
3
Using dimensionless quantities, x x=t90, y y=t90,z z=t90, h h=t90, a a=t90, b b=t90 and k t0=t90 inthe previous Eqs. (2) and (3), we obtain:
r90xx r090xx 1 /x; r
90yy r
090yy 1 wy;
r0xx r00xx
r090xx
k
/xuz; r0yy r00yy
r090yy
k
wy:
4
Eq. (3) will be verified if the following condition is
imposedZh1
uz dz k: 5
The three sets of boundary conditions presented in
Fig. 2 are:
Antisymmetric shear stress distribution:
r90xz x; y; 0 r90yz x; y; 0 0: 6
Traction continuity across the 0/90 interface:
r90xz x; y; 1 r0xz x; y; 1;
r90yz x; y; 1 r0yz x; y; 1;
r90zz x; y; 1 r0xz x; y; 1:
7
The upper face of the laminate at z h is stress free:
r0xz
x;
y;
h 0;
r0yz
x;
y;
h 0;
r0zz x; y;h 0:
8
In this model, rxy is neglected in the whole laminate.
This hypothesis was brought out after several numerical
simulations with a finite element model and other
models [4]. So, with this hypothesis, the stress field in the
two layers of the damaged laminate is as follows:
The stress field in the 90 layer is such that
r90xx r090xx 1 /x;
r90yy r090yy 1 wy;
r
90
zz r090
xx
d2/x
dx2 R
z2
2
r090
yy
d2wy
dy2
h z
2 ;
r90xy 0;
r90yz r090yy
dwy
dyz;
r90xz r090xx
d/x
dxz:
9
The stress field in the 0 layers has the following form:
r0xx r00xx
r090xxk
/xuz;
r0yy r00yy
r090yy
k
wy;
r0zz r090xxk
d2/x
dx2uIIz
r090yy
2k
d2wy
dy2h z
2;
r0xy 0;
r0yz r090yy
k
dwy
dyh z;
r0xz r090xxk
d/x
dxuIz:
10
The constant R is obtained with the continuity Eq. (4)
and is such that R uII1k
12
with uI
Ruzdz and
uII RuIzdz.
2 a
2b
t0
2 t90
Transverse CracksLongitudinal Cracks
Fig. 2. Unit damaged cell with transverse and longitudinal cracks.
2242 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
5/12
The boundary conditions for the stress field in the
damaged unit cell are:
/a wb 1; /0a w0b 0;
where
/
0
x
d/x
dx ; /
00
x
d2/x
dx2 and
w0y dwy
dy; w00y
d2wy
dy2; 11
uIh uIIh 0; uI1 k; uII1 k R 1
2
:
The complementary energy functional has the fol-
lowing form for a half unit cell, in a laminate subjected
to a tensile loading in the 0 direction
Uc
Zv
Sijklrijrkl dv; 12
where Sijkl are the local compliances, rij is the admissiblestress field and V is the volume of the half unit cell such
that jxj6 a, jyj6 b and jzj6 h. Hashin [22] showed thatfor any elastic body containing cracks, the comple-
mentary energy can be expressed in the form
Ud Uc U0. All the details concerning the expressionsof the complementary energy and the /, w and u
functions are given in Appendix A.
The strain energy release rate G associated with the
initiation and development of damage for a given stress
state is defined by:
G
d ~Ud
dA !
r; 13
where ~Ud is the strain energy of the whole laminate and
A is the damaged area. Let L1 denote the length of the
laminate in the x direction, L2 being its width in the y
direction. The strain energy of the whole laminate and
the numbers n and m of transverse and longitudinal
cracks, respectively, are such that
~Ud nmUcel; 14
where
n L1
2at90; m
L2
2bt90: 15
The intralaminar (transverse and longitudinal) cracked
area Af is such that
Af L1L21
a
1b
: 16
We will distinguish between the strain energy release
rates associated with different damage mechanisms. The
strain energy release rate associated with transverse
cracking is denoted GFT. The strain energy release rate
related with longitudinal cracking is denoted GFL. The
GFT and GFL expressions are:
GFT d~Ud
dAf
d ~Ud
da
da
dAf; GFL
d~Ud
dAf
d ~Ud
db
db
dAf: 17
Using Eqs. (14)(17), the strain energy release rates
associated with transverse or longitudinal damage are
such that
GFT 12bt290
Ucel
a dUcelda
;
GFL 1
2at290kUcel
b
dUcel
db
:
18
3. Delamination modelling
Experimentally, local delamination can also appear in
some laminates [43,46]. According to the proposed
model, the delaminated area is supposed to have a tri-
angular shape (Figs. 3 and 4). This damage occurs at the
intersection of the longitudinal and transverse cracks.
Experimental results confirm that the initiation of local
delamination takes place at the 0/90 interface, near
transverse and longitudinal crack tips and the intensity
of the interlaminar stresses is enhanced close to the
crack planes. The damaged laminate can be represented
by the unit delaminated damaged cell displayed in
Fig. 3. In the unit delaminated damaged cell the del-
aminated area consists of two distinct areas at the 0/90
interface: In Fig. 4, area I is the undamaged area and
delaminated area II has a triangular shape. The dela-
minated area is defined by the dx and dy parameters. Thehypotheses and derivation are explained in [49]. In [47],
the authors used the same hypotheses in a multipartic-
ular model.
With the previous hypotheses, the stress field in the
delamination area is such that
r90xx r90yy r
90zz r
90xz r
90yz 0: 19
Therefore, the stress field in the delamination area
(triangular area II) is such that
/x wy 1; /0x w0y 0; /00x w00y 0:
20
2 a
2 t90
t0
2 b
TransverseCracks
LongitudinalCracks
4 Triangular Delaminatedareas
Fig. 3. Unit damaged cell with transverse and longitudinal cracks and
triangular areas of the delaminated 0/90 interface.
J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250 2243
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
6/12
Taking into account the boundary conditions in thetriangular area, the stress field is reduced to the single
component:
r0xx x; y;z r00xx
r090xxk
uz: 21
To summarise, when the three damage modes are
present, the stress field has one non-zero component
only (21) in the delaminated zone. In the rest of the
laminate the stress field is given by Eqs. (10) and (11).
Due to symmetry with respect to the laminate mid-
plane z 0, the complementary energy of the half unit
cell 0 < z< h, for a laminate subjected to tractionboundary conditions, is still defined by:
Uc 1
2
ZV
Sijklrijrkl dv; 22
where Sijkl are the local compliances, rij is the admissible
stress field and V is the volume of the half unit cell such
that jxj6 a, jyj6 b and jzj6 h.The unit damaged cell, where the three damage modes
are present, is schematised in Fig. 3. The strain energy in
the unit cell is the sum of the strain energies in the non-
delaminated portion (area I) and in the delaminated
portion (triangular area II). In the non-delaminated por-tions (area I), sub-regions are used for calculating the
energy. The strain energy expressions are detailed in [49].
In the non-delaminated region, the strain energy expres-
sion appears in Eq. (23). In the delaminated portion (area
II), the complementary energy is given by Eq. (24), using
the normal stress expression (21):
UId 12Uda; b: Uda; b dy Uda dx; b
Uda dx; b dy; 23
UIId dxdyt390EL
kr002
xx
" 2r00xx r
090xx
r0902xx
k2
Zh
1
u2zdz
#:
24
For a laminate degraded by the three damage modes,
the strain energy of the half unit cell is
Ucel UId U
IId : 25
As in Section 2, pertaining to transverse cracking
damage, the strain energy release rate Gassociated with
the initiation and development of the delamination
damage for a given stress state is defined by:
Gd ~Ud
dA
!r
; 26
where ~Ud is the deformation energy of the whole lami-
nate, and A is the delaminated area. Let L1 denote the
length of the laminate in the x direction, L2 being its
width in the y direction.
The strain energy of the whole laminate is such that:
~Ud nmUcel: 27
The numbers n and m of transverse and longitudinal
cracks are defined in (15). The delaminated area Ad is
such that:
Addx; dy Ad L1L2dxdy
2ab
!: 28
The strain energy release rates associated with de-
lamination in the x and ydirections are denoted Gdx and
Gdy respectively. They mainly depend on the delami-
nated lengths dx and dy. When dx (respectively dy) alone
is varied, we get
Gdx d ~Ud
ddx
ddx
dAd; Gdy
d ~Ud
ddy
ddy
dAd: 29
dy
dy
b-dy
b-dy
dx dxa-dx
x
y
Area I
triangles
Area II
a-dx
4 delaminated
Fig. 4. Schematic triangular areas of the delaminated 0/90 interface.
2244 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
7/12
From Eqs. (16), (25), (28), (29), we obtain the strain
energy release rates associated with delamination in the
x and y directions denoted Gdx and Gdy, respectively:
Gdx 1
2dyt290
dUcel
ddx; Gdy
1
2dxt290
dUcel
ddy: 30
4. Results
4.1. Initiation of the first damage mode
The results of Table 1 show the influence of the 90
ply thickness on the first ply failure in a cross-ply lam-
inate as computed with the proposed model. Several
numerical simulations were achieved with other models
and the analyses of the results converge to the same
conclusion. It is easier to create transverse cracking in a
laminate containing thicker 90 layer. A lot of experi-
mental results, on cross-ply laminates submitted to axialloading, show that during the loading progression, a
second damage mode usually appears after the trans-
verse cracking.
4.2. Initiation of the second damage mode
First of all, the second damage mode can succeed
transverse crack damage or coexist with it. This second
damage mode is generally delamination at the 0/90
interface or longitudinal matrix cracking in the 0 layers.
All the studies conducted on this subject prove that the
nature of the second damage mode is strongly influencedby the architecture of the laminate (ply thickness and
constraint parameter) and the nature of the laminate
system constituents (fibre and matrix). When studying
the influence of the damage process on the degradation
of the laminate mechanical properties, it is necessary to
propose a model which is able to predict the initiation
and the propagation of the different damage mode
during loading development. The numerical results dis-
played in Tables 2 and 3 give the mean strain value
necessary for the initiation of longitudinal matrix
cracking or delamination computed for a 02; 90nslaminate made of the carbon/epoxy T300/934 material
system. The numerical simulations pertaining to the
model are compared with experimental results from
Wang et al. [12] and the finite elements results of [12].
4.3. Thickness ply influence on initiation of longitudinal
matrix cracking
The experimental results related to the ultimate fail-
ure of the 02; 90s laminate (Table 2) show that thesecond damage mode does not appear in this laminate.
For other types of laminates containing a more impor-
tant number of 90 plies, the initiation of a second
damage mode is observed. It can be longitudinal matrix
cracking or delamination. As is well known, the initia-
tion of delamination or longitudinal matrix cracking is
easier in a laminate containing a thick 90 layer. The
numerical results from the model are in good agreement
with the experimental results from Bailey et al. [50] who
studied the ply thickness influence on the initiation of
longitudinal cracks in a glass/epoxy laminate. On Fig. 5,numerical results from our model are compared with
experimental data from [50]. For a given thickness of the
0 layer, it is easier to create a longitudinal cracking
damage in a thick 90 layer (Fig. 5(a)). A similar remark
can be made for 0 plies. For a given thickness of the 90
layer, the risk to initiate a longitudinal cracking in-
creases with the 0 layer thickness (Fig. 5(b)).
4.4. Life prediction in [02; 902]s and [02; 904]s laminates
The deformation thresholds pertaining to the initia-
tion of the three damage modes are displayed in Figs. 6
and 7 for an equilibrated laminate 02; 902s and in Figs.8 and 9 for a 02; 904s laminate. The numerical simula-tions are achieved for a carbon/epoxy T300/934 system;
see Table 4 for the constant of the material. Once more,
it can be conclude that it is easier to damage a laminate
containing a thick 90 layer. The initiation of the three
damage modes appears later in the 02; 902s laminatethan in the 02; 902s laminate. In Fig. 7, we can observethe initiation of delamination during the propagation of
the transverse cracking damage (Fig. 6). Before the ul-
timate failure of the specimen, longitudinal matrix
cracking appears in the 0 plies. At the end of the test,
Table 1
Mean stress value (MPa) at initiation of transverse matrix cracking
n in 0; 90n; 0 Model Experimental [12]
n 1 908 915n 2 537 540n 3 418 430n 4 297 305
Table 2
Applied mean strain value e0 % at initiation of longitudinal matrixcracking in a 02=90ns carbon/epoxy T300/934 laminate system
n in 02; 90ns Mode l FEM [12] Expe rimental [12]
n 1 1.35 >1.2 n 2 0.98 1.05 0.92n 4 0.74 0.78 1.2 n 2 0.95 1.09 >0.92n 4 0.65 0.75
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
8/12
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 (%)
CrackDensity(cm
-1)
Longitudinal Cracks
Transverse Cracks
Fig. 6. Damage mechanism in a 02=902s carbon/epoxy T300/934laminate: matrix cracking.
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
(%)
CrackDensity(cm
-1)
Longitudinal Cracks
Transverse Cracks
Fig. 8. Damage mechanism in a 02=904s carbon/epoxy T300/934laminate: matrix cracking.
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 (%)
Delamination
(mm)
Delamination width d
Delamination length dx
y
Fig. 9. Damage mechanism in a 02=904s carbon/epoxy T300/934laminate: delamination in x and y directions.
0.8
1
1.2
1.4
1.6
1.8
0.5 1 1.5
t 0 (mm)
FL(%)
Experiment Bailey [50]
3D Model
Bailey [50]
0.8
1
1.2
1.4
1.6
1.8
0.50 0.75 1.20 1.25
t90 (mm)
FL
(%)
3D Model
Bailey [50]
Experiment
Bailey [50]
(a) (b)
Fig. 5. Mean strain value e0% at initiation of longitudinal cracking in a glass/epoxy laminate. (a) 02;5=90ns laminate with t0 0:5 mm (b)0m=902:5s laminate with t90 0:5 mm.
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 (%)
Delamination
(mm)
Delamination width d
Delamination length d
y
x
Fig. 7. Damage mechanism in a 02=902s carbon/epoxy T300/934laminate: delamination in x and y directions.
2246 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
9/12
the longitudinal crack density is about 5 cm1 for the
02; 902s laminate and about 10 cm1 for the 02; 904s
laminate. The delamination evolution is not the same for
the two laminates. In the 02; 902s laminate, the evolu-tion is different in the x and y directions. At the end of
the test, for this 02; 902s laminate, the delaminated
length is 16 mm in the x direction and 10 mm in the ydirection (Fig. 7), contrary to the 02; 904s laminatewhere delaminated lengths are equivalent (Fig. 9).
5. Conclusion
Using the proposed energetic model, our objective
was to predict the occurrence of several damage mech-
anisms in cross-ply laminates. The beginning of the
damaging process is well described when compared to
Wangs experiments [12]. We propose some results about
the initiation and propagation of the different damagemodes, longitudinal matrix cracking in the 0 plies and
delamination. As an example, the lifes of an equilibrated
02; 902s laminate and a 02; 904s laminate have beendescribed and the successive damage mechanisms for
these laminates have been predicted. We have also been
able to bring out the propagation of transverse cracks
during the initiation of delamination, the development of
the triangular shape of delamination and the fact that the
delaminated length is not the same in the x and y direc-
tions of the laminate. Other examples, not presented
here, showed that the damage mechanism succession can
be different. The nature of the material system and the
laminate architecture, represented here by 90 and 0
layers thicknesses, are very important parameters.
Appendix A
Uc
Zv
Sijklrijrkl dv:
Sijkl are the local compliances, rij is the admissible stress
field and V is the volume of the half unit cell such that
jxj6 a, jyj6 b and jzj6 h.
Ud Uc Uo;
where Uo denotes the constant complementary energy of
the uncracked laminate, and Uc is the variation of en-
ergy due to damage in the laminate. Using the density of
complementary energy Wk in each layer, the energy
perturbations due to cracks can be written as:
Uc
ZV90
W90 dv
ZV0
W0 dv:
Substituting Eqs. (9) and (10) into these expressions, we
obtain:
Uc 1
2r20t
390
Za
a
Zbb
A0k2x/
2h
2B0kxky/w C0k2yw
2
A1k2x/
02 B1k2yw
02 A21k2x//
00 A21kxky/w00
B12kxkyw/0 B22k
2x/
02 B1k2yww
0 C01k2x/
002
C02k2yw
002
C03kxkyw00
/00i
dxdy
with
A0 1
ET
1
EL; A0
mTT
ETk
2
3
mLT
EL
2I5
k2;
B0 mLT
EL1
I3
k2
; B21 2
mLT
ELk
1
6
2
mTT0
ET
I6
k2;
C0 1
EL
1
kEL; B22
mLT
ELk
2
3
mTT0
ET
k
3
A1 13GTT0
I2k2GLT
; C01 1ET
k2
k3
120
I7
20k2ET;
B1 1
3
1
GLT
1
GTT0
; C02
3k3 15k2 20k 8
60ET;
A21 mTT0
ETR
1
6
2
mLT
EL
I4
k2;
C03 1
ETRh
1
3R
h
2
1
10
I8
k2ET;
and
kx r090xxr0
; ky r090yy
r0;
/0 o/
ox; /00
o2/
ox2; /0
o/
ox; /00
o2/
ox2:
The parameters Ii i 1; 8 are defined by:
I1
Zh1
u2zdz; I2
Zh1
uI2
zdz; I3
Zh1
uzdz;
I4
Zh1
uzuIIzdz;
Table 4
Material constants for a T300/934 unidirectional ply [12]
Property SI unit
ELL 144.8 GPa
ETT, Ezz 11.7 GPa
mLT , mLz 0.3
mTz 0.54
GLT, GLz 6.5 GPaGTz 3.5 GPa
t, nominal ply thickness 0.132 mm
Gcrf 228 J m2
Gcrd 158 J m2
J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250 2247
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
10/12
I5
Zh1
uzh z2dz; I6
Zh1
uIIzdz;
I7
Zh1
uII2
zdz; I8
Zh1
uIIzh z2
dz:
The EulerLagrange differential equations are:
d4/
dx4 p1
d2/
dx2 q1/
kx
ky
B0
C01
1
2b
Zbb
wydy 0;
d4w
dy4 p2
d2w
dy2 q2w
kx
ky
B0
C02
1
2a
Za
a
/xdx 0;
where
p1 A21 A1
C01; q1
A0
C01; p2
B22 B1C02
; q2 C0
C02;
/x D1f1x F1g1x m1w;
wy D2f2y F2g2y m2/:
The constants Di, Fi, / and w are such that:
D1 1 m1wg
01a
f1ag01a f0
1ag1a;
F1 1 m1wf
01a
f1ag01a f0
1ag1a;
D2 1 m2/g
02b
f2bg02b f02
bg2b;
F2
1 m2/f
02b
f2bg02b f02bg2b;
/ x1 m11 x1x2
1 m1m21 x11 x2;
w x2 m21 x2x1
1 m1m21 x11 x2;
and
m1 kyB0
kxA0; m2
kxB0
kyC0:
The functions fi, gi and xi depend of the signe:
4q1
p2i
i
1;
2
If 4q1Pp2i
fiui coshaiui cosbiui;
giui sinhaiui sinbiui;
xi 2aibicosh2airi cos2biri
ria2i b2i ai sin2biri bi sinh2airi
:
If 4q16p2i with pi < 0:
fiui coshaiui;
giui coshbiui;
xi b2i a
2i sinhairisinhbiri
aibiribi coshairisinhbiri ai sinhairicoshbiri;
where:
ai q1=4i cos hi
2
; bi q
1=4i sin h
i
2
;
hi Arctg
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4qi
p2i 1
s; ai;bi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipi
2 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
4qi
p2i
s !vuutwith u1 x, u2 y, r1 a, r2 b and i 1, 2.
A.1. Model I
The function uz can be taken in the form:
uz kDcoshDh z
sinhDk;
uIz ksinhDh z
sinhDk;
uIIz kcoshDh z 1
D sinhDk;
the parameter R is such that:
R 1
2
coshDz h 1
D sinhDk:
This function is independent of the damage state of
the laminate. The problem is thus reduced to mini-mizing a function with only one unknown parameter
UcD.
A.2. Model II
In order to analyse the influence of function uz onthe stress field distribution in the unit cell, this function
is taken in the form of a second order polynomial in z
with only one unknown parameter D. Taking into ac-
count the boundary and continuity conditions, the
function uz is such that:
uz 3D
k2z h
2 1 D;
uIz D
k2z h
3 1 Dz h;
uIIz D
4k2z h
4 1 Dz h
2:
The related parameter R is now: R 1k2
Dk4
.
As in Model 1, the determination of uz is reducedto minimizing a function of only one parameter
UcD.
2248 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
11/12
References
[1] Brewer J-C, Lagace PA. Quadratic stress criterion for initiation of
delamination. J Compos Mater 1988;(22):114155.
[2] Leguillon D. A method based on singularity theory to
predict edge delamination of laminates. Int J Fract 1999;
(10):10520.
[3] Sicot O, Gong X, Cherouat A, Lu J. Importance des contraintesresiduelles sur le comportement mecanique dun composite
stratifie. Ann Compos (AMAC) 1991;(1):659.
[4] Rebiere J-L, Maatallah M-N, Gamby D. Initiation and growth of
transverse and longitudinal cracks in composite cross-ply lami-
nates. Compos Struct 2001;53(2):17387.
[5] Urwald E. Influence de la geometrie et de la stratification sur
lendommagement par fatigue de plaques composites carbone/
epoxyde. These de doctorat, Universitede Poitiers; 1992 [in
French].
[6] Wang ASD, Crossman FW. Initiation and growth of transverse
cracks and edge delamination in composite laminates Part I: An
energy method. J Compos Mater 1980;Suppl 14:7187.
[7] Xu LY. Interaction between matrix cracking and edge delamina-
tion in composite laminates. Compos Sci Technol 1994;(50):469
78.[8] Highsmith A-L, Reifsnider K-L. Internal load distribution effects
during fatigue loading of composite laminates. Composite mate-
rials: fatigue and fracture. In: Hahn HT, editor. ASTM STP 907,
Philadelphia; 1986. p. 23351.
[9] Han YM, Hahn HT. Ply cracking property degradation of
symmetric balanced laminates under general in-plane loading.
Compos Sci Technol 1989;35:37797.
[10] Jamison R-D, Schulte K, Reifsnider K-L, Stinchcomb W-W.
Characterization and analysis of damage mechanisms in tension
tension fatigue of graphite/epoxy laminates, ASTM STP 836;
1984. p. 2155.
[11] Stinchcomb W-W. Non destructive Evaluation of Damage Accu-
mulation Process in Composite Laminates. Compos Sci Technol
1986;25:10318.
[12] Wang ASD, Kishore NM, Li CA. Crack development in graphite-epoxy cross-ply laminates under uniaxial tension. Compos Sci
Technol 1985;24:131.
[13] Takeda N, Ogihara S. Initiation and growth of delamination from
the tips of transverse cracks in CFRP cross-ply laminate. Compos
Sci Technol 1994;52:30918.
[14] Ogihara S, Takeda N. Interaction between transverse cracks and
delamination during damage process in CFRP cross-ply laminate.
Compos Sci Technol 1995;54:395404.
[15] Steif PS. Parabolic shear-lag analyses of a 0=90s laminate. InTransverse crack growth and associated stiffness reduction during
the fatigue of a simple cross-ply laminate. Ogin SL, Smith, PA,
Beaumont, editors. Report CUED/C/MATS/TR 105, Cambridge
University; 1984.
[16] Law N, Dvorak GJ. Progressive transverse cracking in composite
laminates. J Compos Mater 1988;22(Oct.):90016.[17] Lim SG, Hong CS. Effect of transverse cracks on the thermome-
chanical properties of cross-ply laminated composites. Compos
Sci Technol 1989;34:14562.
[18] Lee JW, Allen DH, Harris CE. Internal state variable approach
for predicting stiffness reduction in fibrous laminated composites
with matrix cracks. J Compos Mater 1989; 23:127391.
[19] Reissner E. On a variational theorem in elasticity. J Math Phys
1950;29:905.
[20] Lematre J, Chaboche J-L. Mecanique des materiaux solides. 2nd
ed. Paris: DUNOD; 1988.
[21] Vasilev VV, Duchenco AA. Analysis of the tensile deformation of
glass-reinforced plastics. Translated from Mekhanica Polimerov,
Jan/Feb. 1970;(1):1447.
[22] Hashin Z. Analysis of cracked laminates: a variational approach.
Mech Mater 1985;(4):12136.
[23] Nairn JA. The strain energy release rate of composite micro
cracking: a variational approach. J Compos Mater 1989;23:1106
29.
[24] Pijaudier-Cabot G, Dvorak GJ. A variational approximation of
the stress intensity factors in cracked laminates. Eur J Mech A:
Solids 1990;9(6):51735.
[25] Varna J, Berglund LA. Thermo-elastic properties of composite
laminates with transverse cracks. J Compos Technol Res
1994;16(1):7787.
[26] Rebiere J-L. Modelisation du champ des contraintes cree par des
fissures de fatigue dans un composite stratifie carbone/polymere.
These de doctorat, Universite de Poitiers; 1992 [in French].
[27] Herakovich CT, Aboudi J, Lee SW, Strauss EA. Damage in
composite laminates: effects of transverse cracks. Mech Mater
1988;(7):91107.
[28] Talreja R, Yalvac S, Yats LD, Wetters DG. Transverse cracking
and stiffness reduction in cross ply laminates of different matrix
toughness. J Compos Mater 1992;26(11):164463.
[29] Gamby D, Henaff-Gardin C, Rebiere J-L. Modelling of the
damage distribution along the width of a composite laminate
subjected to a tensile fatigue test. In: Proceedings of localized
damage II. Southampton, UK: Elsevier Applied Science; 1992. p.
31525. 13 July.
[30] Talreja R. Internal variable damage mechanics of composite
materials. Symposium on Yielding, Damage and Failure of
Anisotropic Solids Grenoble, France; 1987. p. 248.
[31] Pyrz R. A micromechanically-based model for composite mate-
rials with matrix cracks. Compos Eng 1992;2(8):61929.
[32] Allen DH, Groves SE, Harris CE. A thermomechanical consti-
tutive theory for elastic composites with distributed damage. Part
I theoretical development. Report MM-5023-85-17, Mechanics
and Materials Center, Texas A&M University; 1985.
[33] Allix O, Ladeveze P, Le Dantec E. Modelisation de lendomm-
agement dun pli elementaire des composites stratifie s. In:
Fantozzi G, Fleishman P, editors. Proceedings of the 7th Jurnees
Nationales des Composites, Lyon, Nov. Paris: AMAC; 1990. p.111524.
[34] Thionnet A. Prevision de lendommagement sous chargements
quasi-statiques et cycliques des structures composites stratifie s.
These de doctorat, Universite de Paris 6; 1991 [in French].
[35] Chow CL, Yang F. Elastic damage analysis of interlaminar stress
distribution in symmetrical composite laminates with edge delam-
ination cracks. Proceedings of the Institution of Mechanical
Engineering, Part C. J Mech Eng Sci 1994;208(1):111.
[36] Adali S, Markins RK. Effect of transverse matrix cracks on the
frequencies of unsymmetrical, cross-ply laminates. J Franklin Inst
1992;329(4):65565.
[37] Yeh JR. The mechanics of multiple transverse cracking in
composite laminates. Int J Solids Struct 1992;25(12):144555.
[38] Kaw A, Besterfield GH. Mechanics of multiple periodic brittle
matrix cracks in unidirectional fiber-reinforced composites. Int JSolids Struct 1992;29(10):1193207.
[39] Nairn JA, Hu S. The initiation and growth of delamination
induced by matrix microcracks in laminated composites. Int J
Fract 1992;57:124.
[40] Hashin Z. Analysis of orthogonally cracked laminates under
tension. J Appl Mech 1987;25:8729.
[41] Binienda WK, Hong A, Roberts GD. Influence of material
parameters on strain energy release rates for cross-ply laminate
with pre-existing transverse crack. Compos Eng 1994;4(12):1167
210.
[42] Schon J, Nyman T, Bom A, Ansell H. A numerical and
experimental investigation of delamination behaviour in the
DCB specimen. Compos Sci Technol 2000;60:17384.
J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250 2249
7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re
12/12
[43] Marion G, Rospars C, Harry R, Lecuyer F. Prediction de
lamorcage du delaminage des stratifies croise s. Ann Compos J
Sci Tech: Delaminage AMAC 2000;(1):2935.
[44] Leguillon D, Marion G, Harry R, Lecuyer F. The onset of
delamination at stress-free edges in angle-ply laminates
analysis of two criteria. Compos Sci Technol 2001;61(3):377
82.
[45] Whitney JM, Nuismer RJ. Stress fracture criteria for laminated
composites containing stress concentration. J Compos Mater
1974;8:25365.
[46] Kim RY, Sony SR. Experimental and analytical studies on the
onset of delamination in laminated composites. J Compos Mater
1984;18:706.
[47] Lecuyer F. Etude des effets de bord dans des structures minces
multicouches. These de doctorat, Universite de Paris 6; 1991 [in
French].
[48] Diaz-Diaz A, Caron J-F. Criterion of delamination initiation in
multilayered composites. Ann Compos J Sci Tech: Delaminage
AMAC 2000;(1):3744.
[49] Rebiere J-L, Maatallah M-N, Gamby D. Analysis of damage
mode transition in a cross-ply laminate under uniaxial loading.
Compos Struct 2002;55(2):11526.
[50] Bailey J-E, Curtis PT, Parvizi A. On the transverse cracking and
longitudinal splitting behaviour of glass and carbon fibre rein-
forced epoxy cross ply laminates and the effect of Poisson and
thermally generated strain. Proc R Lond A 1979;(366):599623.
2250 J.-L. Rebiere, D. Gamby / Composites Science and Technology 64 (2004) 22392250