Materials Science and Engineering A250 (1998) 249 – 255

Damage modelling for a laminated ceramic composite

A. Gasser a,1, P. Ladeveze a,*, P. Peres b
a
Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure de Cachan/CNRS/Uni6ersité Paris 6, 61, A6enue du Président Wilson,
94235 Cachan Cedex, France
b
Aérospatiale, Le Haillan, 33165 Saint-Medard-en-Jalles, France

Abstract

A general theory of damage mechanics is applied to a fibre-reinforced ceramic matrix composite, SiC/MAS-L, in order to
describe its non-linear mechanical behaviour up to failure. This study is limited to the case of quasi-monotonic loading at room
temperature. The model, which uses an anisotropic damage theory previously applied to other composites, is a mesoscopic-scale
model which has been developed using tension–compression tests on different stacking sequences and applies in the case of
multiaxial loading. It includes the marked differences observed between mechanical behaviour in tension and in compression, and
is also able to predict the failure values. © 1998 Elsevier Science S.A. All rights reserved.

Keywords: Anisotropy; Damage; Laminate

1. Introduction between macro and micro), where damage phenomena

can be described in a fairly simple way. The proposed
The purpose of the present study is to model the ply model is an extension of the one built for carbon/
non-linear mechanical behaviour up to failure, in the epoxy composites [3].
case of monotonic loading at room temperature, of a Several combined tension–compression tests were
fibre-reinforced ceramic matrix composite made by carried out on different stacking sequences. These tests
Aérospatiale, France. For this type of material, dam- showed that the composite has an elastic-damageable
age, that is to say the more or less gradual development behaviour in tension, just as in compression on [945°],
of micro-voids and micro-cracks, is the major mechani- and an elastic–brittle behaviour in compression in the
cal phenomenon. The classical theory of isotropic dam- fibre direction. Its behaviour is thus not the same in
tension and in compression. These tests allow an iden-
age, developed for metallic materials, does not offer a
tification of the model.
satisfactory framework for the study of composites,
The completely identified model is able to predict the
where there is no single damage mechanism, but sev-
behaviour of any laminated stacking sequence of the
eral. These mechanisms are highly anisotropic and
composite under uniaxial tension–compression loading,
present a strong unilateral feature (different behaviour
as well as under complex loading, biaxial for example.
in tension and in compression), depending on whether
the micro-defects are closed or open.
These different aspects are included in the general
approach proposed by Ladevèze [1] which has already 2. Material description
been applied to other composites, like ‘carbon/carbon’
[2], ‘carbon/epoxy’ [3], ‘woven SiC/SiC’ [4], and is used The material is a laminated SiC/MAS-L composite
herein. The scale selected for modelling is the scale of (made by Aérospatiale [5]), with silicon carbide fibres
the elementary ply, named the meso-scale (intermediate (SiC-Nicalon) and glass matrix of MAS-L (0.5 MgO,
Al2O3, 4 SiO2, 0.5 Li2O). The fibre stiffness (200 GPa)
is higher than that of the matrix (75 GPa), and cracks
* Corresponding author. Tel.: +33 1 47402238; fax: 33 1
47402240. appear first in the matrix. The failure surfaces of speci-
1
Present address: ESEM, 8, rue Léonard de Vinci, 45072 Orleans mens submitted to a tension loading show a significant
Cedax 2, France. pull-out (until a length of 10 mm, see Fig. 1), which is

0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0921-5093(98)00598-X
250 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255

Fig. 1. Failure surfaces of tension specimens [0 – 90°]2s and [9 45°]2s.

proof of a weak fibre/matrix interface. Several laminate The next step is to define two damage functions: d(n6 ),
stacking sequences were available: [0°]8, [0 – 90°]2s, [ 9 the stiffness loss of E(n6 ), and d(n6 ), the stiffness loss of
45°]2s and [9 67.5°]2s. g(n6 ):
E0(n)− E(n) g0(n)− g(n)
d(n)= g(n)= (3)
E0(n) g0(n)
3. Mechanical behaviour modelling
where ‘0’ indicates the initial values (no damage). The
3.1. Anisotropic damage functions d and d can be represented in space by two
surfaces Sd and Sd, called damage surfaces, and defined
The idea, credited to Kachanov [6] and Rabotnov [7] by:
for isotropic materials, is to use the loss of stiffness (i.e.
Young’s modulus decrease) as a damage indicator. In %d = n · d(n) %d = n · d(n). (4)
the case of anisotropic materials (for example, com-
posites), the problem is more complex. Consider a They allow for the characterisation of the state of
damaged material, submitted to a stress perturbation damage of the material. Different damage kinematics
s* in the direction n6 of the space, with no further can be obtained when modelling these surfaces with
anelastic strains due to this stress perturbation. The different shapes (spheres, ellipsoids, or more complex
strain perturbations associated with s* and n6 are: surfaces).
o *(n
L 6 ): the longitudinal strain in the n6 direction, linked
to the perturbation in the n6 direction,
o *(n
T 6 ): the transversal strain in the T 6 direction, linked
to the perturbation in the n6 direction,
o *(n
T 6 ); the transversal strain in the T6 ’ direction, linked
to the perturbation in the n6 direction
with:
o*L (n)=n to*n
o*T (n)=T to*T
o*T (n)=T %to*T % (1)
where T6 , T6 % are two transversal directions, and (n6 , T6 ,
T6 %) is a base. Two moduli can be defined in the n6
direction: the Young’s modulus E(n6 ) and the volumic
modulus g(n6 ):
s* s*
E(n)= g(n) = . (2)
o*L (n) o*L (n) + o*T (n) + o*T’ (n)
E(n6 ) and g(n6 ) are independent, and they completely Fig. 2. [0 – 90°]2s laminate slice (the width of the photo corresponds to
define the Hooke tensor of the material [1]. 2.1 mm).
 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255 251

Fig. 5. Elementary ply.

trix debonding, can be described in a fairly simple

way.

3.3. Ply modelling

The proposed meso-model described in [9] and [10]

has been inspired by work conducted on carbon/epoxy
Fig. 3. [ 945°]2s laminate slice (the width of the photo corresponds to composites [3]. The damage variables can be linked to
2.1 mm). physical parameters, as well as to the rate of cracking
[11]. In this study however, the loss of stiffness was
3.2. Meso-scale used. Three scalar damage variables (assumed to be
constant throughout the thickness of the ply) are used:
Observations by optical microscopy (Figs. 2 and 3) d1 in the longitudinal direction (associated with cracks
show that a laminate is a stacking sequence of identical, perpendicular to the fibre direction), d2 in the transverse
homogeneous elementary plies (fibres and matrix) and direction, and d in shear (associated with cracks parallel
interlaminar interfaces (matrix) (Fig. 4). If we call meso to the fibre direction, and fibre/matrix debonding). The
an intermediate scale between micro (fibre, matrix) and model should account for the following experimental
macro (the structure), the elementary plies and the observations: a different behaviour between tension and
interfaces are named meso-constituents. In the follow- compression, and a constant ratio between n12 and E1.
ing, only the bulk mechanical behaviour is studied and It was then proposed for the strain energy of an ele-
interfacial phenomena, e.g. delamination which occurs mentary ply, in a plane stress state, in the following
mostly near free edges and macro-flaws, are not consid-

form:
ered. Delamination has already been studied on other
1 s112+ s 2 n0 s 2
composites [8]. Therefore, when the mechanical be- Ed = + 110 + − 2 120 s11s22 + 220 −


0
haviour of the elementary ply is determined, the me- 2 E 1(1−d1) E1 E1 E2
chanical behaviour of any laminate stacking sequence is s222+ s 212
found using the classical laminate theory. Some re- + + (5)
E 02(1−d2) (1−d)G 012
marks are relevant at this point:
“ The ply behaviour is independent of its environment,
where s + = s if s] 0 and s + = 0 if s B0, s −
i.e. indifferent to the presence of another ply (and of = s if sB 0 and s − = 0 if s] 0, ‘0’ indicates the
its fibre direction). However, the stresses in the ply initial value, and 1 and 2 are the fibre direction and the
do depend on the other ply orientations (via the transverse direction, respectively (Fig. 5). The notation
classical laminate theory). s + permits splitting the strain energy into a ‘tension’
“ After testing several laminate stacking sequences in
energy and a ‘compression’ energy, according to
tension, it was observed that the specimen failure whether the cracks are closed or open, and serves to
always occurs parallel to one of the fibre directions separate the tension behaviour from the compression
(Fig. 1). This would seem to show the predominance behaviour. The strength variables (Y1, Y2, Y) are intro-
of the ply co-ordinate system that will be used in the duced as being thermodynamically associated with d1,
next paragraph to model the ply’s behaviour. d2 and d, respectively:

)
“ At the meso-scale the main damage phenomena,
which are matrix cracking, fibre failure and fibre/ma- ([Ed ] [s 2 ]
Y1= = 0 11 + 2

)
(d1 s constant 2E 1(1−d1)
([Ed ] [s 2 ]
Y2= = 0 22 + 2 (6)

)
(d2 s constant 2E 2(1−d2)
([E d ] [s 2 ]
Y= = 0 12
(d s constant 2G 12(1−d)2
Fig. 4. Laminate modelling.
252 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255

where [ ] shows the mean value in the thickness. This

point plays an important role in failure prediction
[12]. The driving forces of the damage variables are
the maximum values of Y1, Y2 and Y, i.e. d1, d2 and
d are functions of Y1, Y2 and Y. For the carbon/
epoxy composite [3], only two damage variables (d2
and d) were used. They were functions of Y2 and Y,
with a coupling parameter b. For the material stud-
ied here, to develop a model as simple as possible
the first stage is to test simplified laws of damage
evolution (with only one coupling), as inspired from
the previous carbon/epoxy study. This separates the
behaviour in the fibre direction from the transverse
and shear behaviours:
Ád1 =f(Y %1)
Ã
Íd2 =g(Y %2)
à Fig. 6. Tension – compression test at 0° on unidirectional SiC/MAS-L
Äd= h(Y %2) (longitudinal and transversal strains).

with
Á dinal Young’s modulus E 01 = 121 GPa, the initial
à Y %1 = max
t5t
Y1 t Poisson’s ratio n 012 = 0.28, and the function f (as a
Í power law function, see Fig. 7).
ÃY %2 =max Y t +bY2
t5t t
Ä 4.2. Test at 0 ° on a [0 – 90 °]2s laminate

where b is a coupling coefficient of the material and The tension–compression test at 0° on a [0–90°]2s
t is time. laminate (Fig. 8) shows the same type of behaviour
as for the unidirectional material. In this test the 0°
ply is essentially submitted to longitudinal stresses
4. Identification of model parameters and the 90° ply to transversal stresses. Since the lon-
gitudinal ply behaviour is known (see Section 4.1),
The determination of the ply behaviour is not this test provides the transversal ply behaviour using
straightforward because it is not possible to test only the classical laminate theory (same strains in each
one layer. Furthermore, the behaviour of a ply is layer). Hence, the initial transversal Young’s modulus
not the same alone as it is in a laminate stacking E 02 = 57 GPa was obtained. Y has also been set
sequence (for example, an elementary ply is more equal to zero (no shear stresses in the plies for this
sensitive to flaws). The solution considered is to per- test). Therefore, d2 versus Y6 can be plotted and the
form macro-tests (in tension – compression) on differ- function g determined as a power law function (Fig.
ent laminate stacking sequences in which each ply is 9).
submitted to complex loading conditions. Three of
these laminates are necessary in order to identify the
parameters in the present model.

4.1. Test at 0 ° on a unidirectional composite

In the fibre direction, the unidirectional material

exhibits a brittle, linear-elastic behaviour in compres-
sion, and an elastic and damageable behaviour in
tension (Fig. 6). Furthermore, the evolution of the
transverse strain is quasi-linear, i.e. the fraction n12/
E1 is approximately constant. In this test, the ele-
mentary ply is only submitted to longitudinal
stresses. Thus, it allows for the determination of the Fig. 7. Evolution law of d1 versus Y6 (points: tests, solid line: identifi-
longitudinal ply behaviour, namely the initial longitu- cation).
 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255 253

Fig. 8. Tension –compression test on a [0–90°]2s laminate. Fig. 10. Tension – compression test on a [9 45°]2s laminate.

4.3. Test on a [945 °]2s laminate 5. Prediction and validation

The test on a [9 45°]2s laminate shows an elastic-dam- 5.1. Laminate failure prediction
ageable behaviour both in tension and in compression
(Fig. 10). In compression, the plies are submitted to shear The model is able to predict not only the damage
stresses and compressive longitudinal and transversal evolution but also the failure values of the stress using
stresses. Since these do not imply any damage (see Figs. a multi-criteria approach. In tension, laminate failure
6 and 8), this test allows for the identification of the ply occurs at the instability point (s =0 with o"0, which
shear behaviour. Indeed, we have (with L the tension represents the maximum of the curve sL versus oL). In
direction, and T the direction perpendicular to L): compression, both the instability criterion and a brittle
failure criterion are used. Fig. 12 shows that a very simple
sL oL −oT
s12 = o12 = . (8) criterion is able to predict the failure strength for all
2 2
laminate stacking sequences [9u°]s in tension. In com-
Thus, from a plot of s12 versus o12 the initial value of pression, the value given by the brittle failure criterion
the shear modulus G 012 =31 GPa is obtained. Further- was not determined, because the failures obtained during
more, the compression part of the test allows for the the tests were caused by buckling.
determination of the function h (as a power law function,
see Fig. 11), because: 5.2. Test predictions
Y2 = 0 (s22 + =0). (9)
The validation of the model for uniaxial tests should
In tension, the plies are submitted to shear stresses and be performed with a laminate stacking sequence not used
positive longitudinal and transversal stresses. Conse- in the parameter identification procedure, e.g. a [9
quently, the value of the coupling parameter can be 67.5°]2s laminate, in which the plies are essentially sub-
determined as b=66. mitted to shear and to transversal stresses. The

Fig. 9. Evolution law of d2 versus Y6 % (points: tests, solid line: Fig. 11. Evolution law of d versus Y6 2% (points: tests, solid line:
identification). identification).
254 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255

Fig. 12. Comparison between experimental data and the model’s

prediction for the failure stress (solid line) for [ 9 u°]s laminates in
tension.

results of a tension – compression test on such a lami-

nate are given in Fig. 13. The important point to
observe is that the stiffness increases in compression, Fig. 14. Comparison between experimental curves (solid line) and
due to closing of initial cracks. These cracks could predictions (dotted line). The stars indicate the failures calculated.
also be seen before the test, especially near the speci-
men edges. Therefore, since the composite was al-
Since the ply is more sensitive to shear and transver-
ready damaged before testing, it could not be used
sal stresses, it is assumed that only d and d2 are
to directly validate the model. Fig. 14 indeed shows
non-zero. With d= d2 = 0.55 (which is a sizeable
a very good agreement between tests and predictions
value), the non-linear evolution and the failure value
for longitudinal and transversal strains, both in ten-
of the [9 67.5°]s test were found (Fig. 15).
sion and in compression, except for the [9 67.5°]2s
laminate which was initially cracked (the predicted
initial Young’s modulus was twice the experimentally 5.4. Damage surfaces
observed value).
In Section 3.1, the damage surfaces linked to d(n6 )
and d(n6 ) were introduced. It is now possible using
5.3. Test prediction for the [967.5 °]s laminate the model to plot the intersections between these sur-
faces and the plane (1,2) for different laminates: [0°]
Since the [967.5°]s laminate was initially cracked, (Fig. 16) and [ 945°] (Fig. 17). These plots show the
one could try to incorporate the initial damage into anisotropy of the damage in the plane of each ply.
the model. The problem then is to know the break-
down of damage among the three damage variables.

Fig. 15. Comparison between test (solid line) and predictions both
with (curve 1) and without (curve 2) initial damage for a [ 9 67.5°]s
Fig. 13. Tension –compression test on a [9 67.5°]2s laminate. tension test.
 A. Gasser et al. / Materials Science and Engineering A250 (1998) 249–255 255

Complementary tests are necessary to verify the

model under complex loading conditions (biaxial, etc.).
The next step could be the extension of the model to
predict the material behaviour under temperature load-
ing (studied in [13]) or under cyclic loading (studied in
[14]).

Fig. 16. Damage surfaces Sd and Sd for a tension test on a [0°] Acknowledgements
laminate (the horizontal direction is the fibre direction) at failure, in
the plane of the laminate. This study was carried out within the French scien-
tific group GS4C ‘Thermomechanical behaviour of ce-
ramic matrix composites with fibres’, supported by the
CNES, CNRS, DRET, MRT, Aérospatiale, SEP and
SNECMA.

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