,:-~...

~-~>--,-~-,~
Composites Science and Technology 51 (1994) 35-42 ~,: ';.. ,'.
~3~ .:+~;~

M O D E L L I N G A N D IDENTIFICATION OF THE
M E C H A N I C A L B E H A V I O U R OF COMPOSITE L A M I N A T E S
IN COMPRESSION

O. Allix
Universit~ D'Evry Val d'Essone IUT/GMP, 2 Cours Monseigneur Romdro 91000 Evry, Laboratoire de M~canique et
Technologie ENS de Cachan, CNRS, Universitd Paris 6, 61 Avenue du Pr~sident Wilson, 94235 Cachan Cedex, France

P. Ladev6ze*
Laboratoire de Mgcanique et Technologie ENS de Cachan, CNRS, Universit~ Paris 6, 61 Avenue du President Wilson,
94235 Cachan Cedex, France

&

E. Vittecoq
Laboratoire de M~canique de Lille USTL, ENSAM, EC Lille URA, CNRS, D1441 Cit~ Scientifique, BP 48,
59651 Villeneuve d'Ascq Cedex, France

(Received 25 February 1993; accepted 28 June 1993)

Abstract Refs 7 and 8 in order to describe the behaviour of

It is shown that mechanical compression behaviour in fibrous laminates under shear loading.
the fibre direction for laminated carbon/epoxy structure Laminates under tension in the fibre direction have
can be predicted with an elastic non4inear model. For an elastic and brittle behaviour. However, a different
that purpose, a specific bending test has been and strongly non-linear behaviour appears after
developed. A model for any stacking sequence is compression and bending tests. 9 This behaviour is
proposed and verified in several tension, compression usually determined by standard compression tests
and bending tests. Finally, a possible explanation using whose results are quite dispersed m'~ and thus difficult
the model proposed by Lee and Harris is developed. to understand and model.
In this paper the authors show that the differences
between tension and compression can be easily
Keywords: carbon/epoxy laminates, compression be-
modelled, at the meso-scale of the single layer, by
haviour, four-point bending test, mechanical modell-
considering that the mechanical behaviour in the fibre
ing, Lee-Harris micro-mechanical model
direction is different in tension than in compression.
In tension it is elastic, brittle and linear; in
compression it is always elastic and brittle but
1 INTRODUCTION non-linear.
The behaviour of carbon/epoxy laminates under This approach allows the behaviour of any
tension has been widely studied and is thus well laminates to be predicted quite satisfactorily for a
known. A mechanical model of the anisotropic wide range of loadings. In particular, there is a good
damageable and inelastic behaviour of the elementary agreement between the modelled results and those
ply of carbon/epoxy laminates has been proposed and obtained by tension, compression and bending tests.
identified, t-~ The damageable behaviour due to This study relies mainly on a specially developed
distributed cracks has been studied in Refs 4-6. four-point bending test which allows us precisely to
Visco-inelastic-like models have been proposed in identify mechanical behaviour in compression. ~2 An
original bending set-up with flexible elements
* To whom correspondence should be addressed. minimises parasitic loadings. Two carbon/epoxy
materials have been studied, viz T300/914 and
Composites Science and Technology 0266-3538/94/$07.00 IM6/914 laminates.
© 1994 Elsevier Science Limited. In the first part of this paper an improvement of the
35
36 O. Allix, P. Ladev~ze, E. Vittecoq

meso-modelling of carbon/epoxy laminates described with

in Ref. 3 is proposed. An additional parameter which
H+(x)=l if x > 0 ; H+(x)=O otherwise
characterises the loss of rigidity in compression in the
fibre direction is introduced. Its identification is then H-(x)=l if x < 0 ; H-(x)=O otherwise
discussed and the model is compared to different
Here, d and d' are scalar damage variables related to
experimental results. In Section 5 a micro-modelling the loss of rigidity in shear and in transverse loading.
procedure, proposed by Lee and Harris, 1~ is applied They are assumed to be constant throughout the ply
and identified. It is shown that a small-angle fibre thickness. This dispenses with the main computational
misalignment, which is the basic assumption of this difficulties, such as mesh dependency. 2~-22 The
micro-modelling, explains both the loss of rigidity and 'transverse' behaviour is unilateral. It remains
the elastic behaviour which characterise the mechani- unchanged in the case of a transverse compression
cal compression behaviour in the fibre direction. loading because, in that case, the micro-defects are
Other explanations are based on an assumption of mainly closed. Thus, the elastic strain has the
micro-buckling. This idea, initially developed by
following expression:
Rosen, I~ has been widely studied and the correspond-
ing micro-model has received many refinements. ~5-~7 (o o,,
e l l = -~'~ -- --g~F-
Micro-experiments are also done in order to ~ Et El
characterise the behaviour in compression of the fibre
itself. ~s e~= OEa ~ te~2=-~+~o 1 022
80 la.a' 1 z( - d'H+(°~=))
2 MESO-SCALE MODELLING OF LAMINATES Ee ~ O12 ~

The authors consider that any laminate can be defined k ~z- 2G72(1 - d)
by two elementary constituents: In the case of carbon/epoxy, E~ >> v]~zZE~. With this
(i) the elementary ply (supposed to be unidirec- assumption, the inverse relationship may be written as
tional); and follows:
(ii) the interlaminar interface which is a mechanical
8Ea~
surface depending on the two fibre directions of
o = ~Oe [a d' ~
the upper and lower plies. 19
0 c
The interlaminar interface influence is seen near ~0~ = E~z~ + ~[}E 2 (0 1 - d ~H + (~22))z~2
c c

0 0
edges, mainly in the case of delamination, or near {o22 = v,zE:(1 - d t H + (e=))e,,
C c 0
+ E=(1 - d ~H + (ez=))e2:
e c

macro-defects. This influence can be neglected since [ o,2 = 2G]}z(1 - d)e~2

the present paper deals with the modelling of the
elementary ply only. where the strain energy is considered to be a function
of the elastic strain:
2.1 Model of the elementary ply
A plane stress state is assumed and thus only the ~ +- 2v]~zE~(1 - d ,H +(ez2))e,,ez2
2Ea = e,e,,-° ~ ~
plane components of strain are introduced. In what + - a'H +(e=))e=e + 46]' (1 -
follows, subscripts 1 and 2 designate, respectively, the
fibre direction and the transverse direction (Fig. 1).
2.1.2 Damage evolution law
2. I. 1 Free energy and state equation in the case of a d and d' are two scalar damage variables which are
plane stress state constant throughout the ply thickness. The associated
The strain energy of the single layer has the following forces to d and d', namely Ya and Y~, are defined by:
expression:
al&l
o ,1 V']2 -- .+ 0"222
Ya - oa o.d' -- 2G'#z(1 - 6) 2
2Ea =~-~ - 2~-~ 0",~o~ E~(1 - d'H+(o22))
0~2~ Ya' 8[Edl,, d o~ H+(oee )
+ = ad' = - d')
G]~z(1 - d)
where lEd] denotes the mean value of the strain
energy throughout the thickness of the ply. Associated
forces Yd and Y~ are analogous to energy-release
3 2 rates. They govern damage development, just as the
energy-release rate governs crack propagation.
The model distinguishes two ply-degradation
mechanisms that contribute to damage development:
Fig. 1. Elementary ply. (i) matrix micro-cracking and (ii) fibre~matrix
 Modelling of mechanical compression behaviour 37

debonding. After the experimental results, two This leads to:

quantities are introduced in order to describe this . .
0 1 2 0 1 2 + a2022022
development:
5 = ~R
Y(t) = sup(X/Yd(r) + br~(~)) (R + n,,)
g<t
iff =0 andf =0; otherwise/5 = 0
¥'(t) = sup(~)
X;<Zt
The model assumes that no plastic yield exists in the
The damage-development laws are very simple and fibre direction (~lt = 0). Furthermore, in the expres-
are written as follows: sion of the function f, we assume that the stress 0"tt
has no effect on plasticity development. Squaring
d - ( Y - YI) + if d < 1 and Y' < Y~; otherwise d = 1 previous equations and summing them, we obtain an
Y~
expression for/5 in terms of the effective plastic strain
d' = ( Y - Y ; ~ ) + i f d ' < l a n d Y ' < ' otherwise
Y~; " d'= 1 rate:
rc'
/5 = ~/4~lO~+ a 2 ~
(x)+=x ifx>0; otherwise ( x ) + = 0
Two damage mechanisms are present. The progressive
2.1.4 Identification
one depends on the parameters Y¢, Yc', Y~'~and b, The identification of the different materials param-
which are material characteristics. The brittle-damage eters that have been introduced requires a minimum
threshold, Y~, determines the brittle behaviour of the of three tension tests. In order to measure the damage
fibre/matrix interface in transverse tension. variable and the inelastic strains it is necessary to load
2.1.3 Inelastic strain and to unload. The stacking sequences which give the
In order to model the inelastic strains induced by best results are the following:
damage, a plasticity model is built up based on the
following effective quantities: [0,90].~ identification of E':, v~z and of the
ultimate strain in the fibre direction;
19"11 0"22 [+45].~ identification of G~2, Yo, Yc and R; and
[+67,5],s identification of E~, Y[~, Y~, b, a 2, and
effective stress: 0 = (1 - d')H+(0"22) Y~.
0"1~2
A test on a unidirectional [0],s laminate, more difficult
1-d
to perform correctly than tension tests on [0,90],s

effective plastic strain rate: }P = { i~l

~2(1 - d'H+(o~2))
~,~2(~ - a)
laminate, is better for the identification of ¢~2.

2.2 Extension of the previous model

which are associated with plasticity dissipation ~e: From the authors' bending tests 9"t2 on carbon/epoxy
~ = Tr[a~ p] = Tr[O~Ol laminates (T300/914 and IM6/914), it appears that the
Young's modulus for compression in the fibre
~ e hardening is assumed to be isotropic (i.e. depends direction, initially very close to that for tension,
on a scalar variable p), and the elastic domain is decreases with the compression load. Furthermore,
defined by the function f such that: this non-linearity involves neither inelasticity (no
f = ~0~2 + a20~2 - - R ( p ) - Ro elastic strains after unloading) nor damage in the
mechanical sense (after one loading cycle, up to 90%
where the threshold R is a function of the of the ultimate strain, the Young's modulus remains
accumulated plastic strain p; p ~ R ( p ) is a material unchanged) and a quite satisfactory description of the
characteristic function; and a 2 is a material charac- Young's modulus change in the fibre direction is the
teristic constant. following:
The yield conditions are then written:
. Ei(~711) "~--E(~ -~- Cirri l H - - ( ~ I I)
g~ = 0
where a~ is a material parameter which governs the
~2 = a2 622
loss of rigidity in compression. With such a scheme,
R + Ro p
the tension behaviour remains elastic, linear and
• 012 brittle; in compression the behaviour is elastic,
~2
2(R + Ro) p non-linear and brittle. Taking the previous expression
38 O. A llix, P. Ladevbze, E. Vittecoq

into account, the free energy expression becomes: Additionaltighteningsupport

A-A
2Ed = E~e~+ ~ e,,H3 -(e,,) + 2vt]eE~(l - d'H+(e~))

~ +
X elle22 E~(1 - d ,H +(eee))eee
~2
+ 4G'?~(1 - d ) e ~
Tighteningspring~
This expression is no longer convex when e~ is
smaller than -E~/2~. This value is larger than the
experimental ultimate strain is compression. Fig. 3. Gripping.
Nevertheless, it is preferable in order to ensure the
stability of numerical algorithm using this model, to loading of the specimen is made by freely rotating
slightly modify this expression. Thus, for e ~ < plane supports, which remain parallel to the specimen
-E~/2~, the strain energy becomes: (Fig. 3). The maximum strain, measured using gauges
in a non-disturbed area, is then close to the ultimate
e'; [ e,, + ~e';j l + 2v'~E~(1 -
2E~ = - 2~ d'u+(e~))~,,~
strain.
In a compression test, in which the stroke length is
+E~(1 - ~ 'n + ( ~ )~) ~ '~2 + 46't~(1 - d ) ~ often less than a tenth of millimetre, any sudden pull
at the end of loading leads to specimen failure. The
set-up has 10 elastic grips; two compensation bars,
3 IDENTIFICATION which insure the distribution of the load prescribed by
3.1 Description of the bending set-up the tension machine on which the set-up is fixed; and
A special bending set-up has been developed t2 in a narrow grip on each of the four heads to annul the
order to characterise the compression behaviour. The tension in the specimen. The rotation of the support is
principles of this set-up are presented in Fig. 2. allowed by using thin crossing plates. The dimensions
The loading of the specimen can induce parasitic of the specimen are the following: 50mm for the
deformations, such as tension-compression, shear or distance between the two central supports, 30 mm for
torsion. The set-up includes some internal degrees of the width and 3 mm for the thickness (24 plies). The
freedom to prevent tension-compression loads. In available area for the measurement is larger than
addition, the use of three strain gauges on each side 10 mm × 10 mm.
(oriented at 0°, 45 ° and 90°) allows the determination It is recommended to use a [0,90],s for the
of shear and torsion effects encountered by the determination of a.
specimen during the test.
The geometry of the support produces excess stress 3.2 Validation of the set-up
near the supports. In order to minimise them, the Figure 4 shows the results which were obtained on a
T300/914 [0,9016s specimen equipped with eight
longitudinal gauges (four on each side, distributed in
the central area). The dispersion remained very low.
The shear strain in the plane remained lower than
0.03% on each side, and its character cannot be
] reproduced from one test to another. The parasitic
torsion and out-of-plane shear loadings prescribed by
the set-up are negligible.

/
-!)/
L°ad(N*102) /

'-..,,,~,.~,~.,~,.~
-1.0 -0.5 0.0 0.5 1.0
Fig. 2. Bending setup. Fig. 4. Validation test.
 Modelling of mechanical compression behaviour 39

Load(N *102)

12.

8.
Longitudinal strain Longitudinal stress

4. Fig. 6. Longitudinal strain and stress distributions.

0. , , ~L°ngitudinalswain(%)
normal force at the middle of the specimen. Then:
o.o o'.~ 0:8

Fig. 5. Test on a T300/914 [0, 90],s.

' :3
~h/2
Mf bj_h/2 e(y)y dy
=

~h/2
For the majority of the specimens, failure occurred N = bj_h/z tr(y) dy
on the compression side under the support. T h e r e
were, however, some specimens which failed between In order precisely to determine Mf and N, the
the supports. This shows that the additional stresses thickness of each ply is measured, as follows.
induced by the set-up are comparable to those For a perfect test, Mt is equal to the prescribed
induced by material imperfections. moment Mt, and N is equal to zero. For each
measurement point i e { 1, 2 . . . . . m,) ~ and E~ are
obtained by minimising the quantities:
3.3 R e s u l t s
Figure 5 gives results obtained on a [0,9016s T300/914 f = ~ {kh2[Ni]2 + (1 - k)[M~ - M~]2}
laminate. The upper curve corresponds to the tension i=t
side, the lower curve to the compression side (in that
where k is a coefficient introduced so that the
case, there is a change of sign of the strain).
maximum stress induced by the normal force and by
The validation of the model is made by using the
the moment are of the same order of magnitude. For
classical assumption that the longitudinal strain is
example, k is taken equal to 0.03 for a unidirectional
linear throughout the thickness. This hypothesis is
laminate or for a [0,9016s laminate, and equal to 0.1
correct when the thickness is sufficiently low
for a [06,906]s laminate.
compared with the length of the specimen (Fig. 6).
The quality of the indentification can be controlled
The Young's modulus, which connects the lon-
by introducing the two following error indicators:
gitudinal strain and stress, is then:

for the 0 ° layers in tension: E~~ IMp-

for the 90 ° layers: E2 i i~l
~an M~ ~ lk) m
for the 0° layers in _ _

compression: [E~ + a~e,,H-(e,~)let,

E2 is supposed to be constant and equal to E~. E] ~ is ~ = 1 ~1~

also constant and may be determined either by means M ~ h~
of a tension test on the same material or by the
bending test. The procedure for the identification of cr For the test on a T 3 ~ / 9 1 4 [0,90]~ laminate, one
is described hereafter. obtains results shown in Table 1. The retained values
Let Mf denote the bending moment and N the are 133 GPa for E~ and 25 G P a / % for ~. In the case

Table 1. Results of the test on T300-914 10, 90l~s

Material Stacking Test no. E~l o: r/~vc~° r/~ ~"

sequence (GPa) (GPa/%) (%) (%)
1 131-3 28.3 0-197 0-363
2 139.2 23-6 0-054 1 0-091
T300/914 [0, 9016s 3 129.6 24.0 0-088 2 0"103
4 129.9 25.5 0-210 0.324
40 O. Allix, P. Ladevbze, E. Vittecoq

of an 1M6/914 laminate, these values are 165 GPa and 16.¸

Lo~gitudi~ls~.,,a~ ,lO1)
4 0 G P a / % . The loss of rigidity for the IM6/914
laminate is thus greater than that for the T300/914 .......................~-°
laminate.
In the case of an T300/914, the ultimate
compression strain is about 1.3% and the secant ~..
compression modulus decreases more than 30GPa
between the beginning and the end of loading. The 0. ~
tangent modulus decreases from 133 to 70 GPa, which 0.0 0.4 0.8 1.2
P o ~ : S ~ f i o n ofa ~ g ~ ~ a ~914 [0,~]~
corresponds to a loss of rigidity greater than 50%. C~es : ~m~ion ~ s t on a ~ 9 1 4 [0,~]4s ( .... : Side A, ~ : Side B )

Fig. 8. Comparison between bending test and compression

test.
4 VERIFICATION
5 POSSIBLE EXPLANATION: THE L E E -
4.1 Bending tests on other stacking sequences HARRIS MODELS m3
Once the longitudinal behaviour is identified by a
bending test on [0,9016s specimens it is possible to 5.1 Deseriplion of the model
simulate the response of any stacking sequences in a The basic assumption of this modeP 3 is the
bending test. For example, Fig. 7 shows the misalignment of the fibre before any loading, For the
comparison between experiment and simulation in the sake of simplification, the study is limited to the plane
case of a [06,906]s laminate. case described in Fig. 9. The fibres are supposed to
belong to the plane [1, 2], and the defect of alignment
4.2 Tension tests is taken as follows:
Rudimentary tension tests (fixed cheeks for the ~X
gripping of the heads of the specimen) have been Yo = H0 sin -~-
carried out on specimens dedicated to bending (same
plate, same specimen dimensions). The displacement of the fibre is:
The measured longitudinal Young's modulus is very
U = u(x)x + v ( x ) y
closed to that obtained in the bending test (the
difference being lower than 5%). The longitudinal This model is independent of the position of the layer
maximum strain measured on the side in tension when inside the stacking sequence. To take into account this
failure occurred on the compression side was slightly parameter, or the influence of the orientation of the
greater than the ultimate strain obtained by means of adjacent layers, an approach similar to the one
the tension test. proposed in Ref. 17 can be used.
Two types of behaviour can be easily distinguished,
4.3 Compression test whether the fibres are in or our of phase. 14 In the first
Compression tests have been carried out on the same case, the load acting on the matrix is shear; in the
material by means of a set-up especially studied for second case it is tension-compression. The action of
composite laminates. 23 As shown in Fig. 8, there is a the matrix on the fibre may be schematised as follows:
good correlation between results obtained in bending
- - b y a linear density of force normal to the fibre:
and in compression. Because there is a significant
f = - Kfvy
dispersion in compression tests the comparison is
- - b y a linear density of couple: e = - K c v ' z
more difficult when using this kind of test.
For a proportion of fibre r/f = 0.6, the value of the
00., Bending mom~t (Nm) rigidities Kf and Kc are such that Kf >> K¢. Therefore,
only the in-phase case has been studied. The problem,
neglecting the tension-compression energy with regard
40..

20.

0. ~ ~ ~ , ~ ~ ~ , i
0.0 0,4 0.8 1.2
Curve : T¢-q rosult ~ 2 L x~
Points : Simulation (with 133 GPa and 25 GPa/%) 0l /" ~ ~ --
Initialposition
Fig. 7. Comparison between experiment and simulation:
T300/914 [06, 906]s. Fig. 9. Description of the fibre undulation.
 Modelling o f mechanical compression behaviour 41

to the bending energy, can be described by the 60.. l~iag mammtflqm) . .

following equation:~3
d4v _ d2v _ d~y0
e f t , - ~ + (t" - Kc) ~ + / " ~-T = 0 4 0 .

where Ef is the Young's modulus of the fibre; If is the

moment of inertia of the fibre; and P is the 20.
compression force acting on the fibre.
For an initial warping Yo= Ho sin :rx/L, one obtains:
0.
1 :rx 0.0 0.4 0.8 1.2
v(x) = ~ N, sin ~ Fig. 10. Comparison between the experiment and the
Lee-Harris model in the case of a [0~,90,]s T300-914
with laminate.
Y = Eflf :r2 + KcL 2
pL ~ loss of rigidity in compression. For the T300/914
Considering that the strain measurement takes into specimen, at failure, that is for O= 1250MPa, the
account many waves, the measured longitudinal secant modulus is:
- - - -

strain, g (given by a strain gauge), has the following Er = 0-76 E0

expression: - -

P ~2Ho2 2 Y - 1 where Eo is the initial longitudinal modulus of the ply.

= (v- In addition the small observed increase of rigidity in
tension is introduced, approximately 5% for the
where Sf is the fibre cross-section. The first term T300/914 specimen:
comes from the compression of the fibre. The second - -

Eo = 0.95r/fEf
one is computed by assuming that the length of the
fibre remains constant. It is thus the relative decrease For an amplitude, Ho comprises between 0.5 and
of the ply length due to the increase of the undulation 2/am, and for a wavelength, L, larger than 0.1 mm,
amplitude. This last term is in fact equal to: K~ is nearly equal to 0.095 N. It should be noted that
these values are in agreement with that deduced from
2~ £~ [ [ d ( Y ~ v ' l e - [ ~ ] 2 ] ~ the experimental characterisation of the undulation
proposed in Ref. 24. This set of parameters, being
and leads to the previous expression by linearisation. fixed, one can compare the model prediction with the
~ e average longitudinal stress is equal to: experiment results (Fig. 10). For this comparison, the
results of the experiment have already been presented
O = p ~f
in Fig. 7.
s, It is interesting to note that the sensitivity of the
~ u s , an expression for the secant modulus is: results with respect to the value of L and Ho is low.
This allows a unique identification of Kc for a wide
~=~= +~ class of defects. Furthermore, the behaviour is purely
e 4LZO~i)J elastic; the loading and unloading curves are identical.
with The comparison of the simulations corresponding to
Eflf~ 2 + KcL 2 the two models (Figs 7 and 10) shows a difference: the
Y = r/f ~rSfL2 decrease of rigidity is less progressive in the case of
the micro-mechanical model of Lee-Harris. The
5.2 Identification secant Young's moduli at failure are closed, but the
The number of parameters which are introduced is tangent ones are different, as shown in Fig. 11.
important. The characteristics of the fibre (r/f, Ef, Sf
and lr) are deduced from those given by the
manufacturer. For example, in the case of a T300/914 S6cantmoduli(GPa)
laminate:
r/f = 0.6; Ef = 225 GPa;
Sf = 44.2 ~m 2, If = 155/.tm 4
!/...
The identification itself concerns the rigidity K¢, L and -1400. 1600.
Ho which define the initial shape. These parameters Fig. 11. Comparison between the secant moduli obtained by
are obtained in order to reproduce the experimental the two models.
42 O. All&, P. LadeoOze, E. Vittecoq

6 CONCLUSION 1/I. Woolstencroft, D. H., Curtis, A. R. & Haresceugh, R.

The authors show that there is a significant loss of I., A comparison of test techniques used for evaluation
rididity in compression in the fibre direction. In this of unidirectional compressive strength of carbon
fibre-reinforced plastic. Composites, 12 11981) 275.-80.
p a p e r a model is proposed which allows us to take this
11. Adsit, N. R., Compression testing of graphite-epoxy. In
phenomenon into account in the structural computa- Compression Testing of Homogeneous Materials and
tion of composite structures submitted to complex Composites, ASTM STP 808. American Society of
loadings. The model is quite simple and should be Testing and Materials, Philadephia, PA, 1983, pp.
satisfactory for a wide class of continuous long-fibre 175-86.
12. Vittecoq, E., On the compression behaviour of
composites. It introduces only one additional scalar
carbon-epoxy laminates. Thesis, University of Paris,
p a r a m e t e r which can be precisely identified by means Paris 1991.
of a bending test on a [0 °, 90°],~ cross-ply laminate. 13. Lee, J. W. & Harris, C. E., A micromechanics model
The behaviour remains elastic and brittle in for the effective Young's modulus of a piecewise
compression but is strongly non-linear. The tangent isotropic laminate with wavy patterns. J. Comp. Mater.,
22 (1988) 717-41.
Young's modulus decrease is about 50% between the
14. Rosen, B. W., Mechanics of composite strengthening.
beginning and the end of the loading. In Fiber Composite Materials, American Society of
Although the loss of rigidity can be described by Metals, 1965, pp. 37-75.
this simple model, its physical explanation is still an 15. Piggott, M. R., A theoretical framework for compres-
open problem. The L e e - H a r r i s , micro-mechanical sive properties of aligned fibre composites. J. Mater.
Sci. 16 (1981) 2837-45.
model, which is developed and identified in this 16. Steif, P. S., A simple model for the compressive failure
paper, does however give, at least, a qualitative of weakly bonded fiber reinforced composites. J. Comp.
explanation of the loss of rigidity in compression of Mater., 22 (1988) 818-28.
continuous long-fibre composites. 17. Grandidier, J. C. & Potier-Ferry, M., Microflambage
des fibres dans un mat6riau composite ~t fibres longues.
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des Composites. Comptes-rendus des JNC5, ed. C. fragmentation en compression sur des syst~mes fibre de
Bathias & D. MenkEs. Piuralis, Paris, 1986, pp. 667-83. carbone/matrice organique. In Comptes-rendus des
2. Allix, O., Ladev~ze, P., Le Dantec, E. & Vittecoq E., huitidmes Journdes nationales sur les composites, ed. O.
Damage mechanics for composite laminates under Allix, J. P. Favre & P. Ladev~ze. AMAC, 1992, pp.
complex loading. In Yielding, Damage and Failure of 41-52.
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