A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross...

7/31/2019 A Criterion for Modelling Initiation and Propagation of Matrix Cracking and Delamination in Cross Ply Laminates Re

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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

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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


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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


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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.



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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.



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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.



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


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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.



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



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.



11/12

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