1999 No 9

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The transverse strain response of cross-plied bre-reinforcedceramic-matrix composites

Eddy Vanswijgenhoven*, Martine Wevers, Omer Van Der BiestDepartment of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium

Received 11 June 1998; received in revised form 28 September 1998; accepted 5 November 1998

Abstract

A micromechanical model for the transverse strain response of cross-plied bre-reinforced ceramic-matrix composites has beendeveloped. The model uses dierent unit-cells to describe the composite material and takes into account all damage developing

during tensile testing. The followed approach has been assessed by comparing the simulated and experimentally observed response

of three dierent SiCf/CAS composites. Theory and experiment are in excellent agreement and a parametric study shows the limited

impact of the simplifying assumptions made. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: A. Ceramic-matrix composites (CMCs); B. Stress/strain curves; C. Computational simulation; C. Stress transfer; Transverse strain

1. Introduction

Fibre-reinforced ceramic-matrix composites (CMCs)

are of technological interest as lightweight materials forhigh-temperature environments. The mechanical response

of CMCs has received considerable attention, both from

an experimental and a theoretical point of view. The

longitudinal strain response is now reasonably well-

understood (see for example Ref. [1]).

The transverse strain response (i.e. in directions per-

pendicular to the applied load) of CMCs is less well

documented and understood. Matrix cracking and

interface debonding in unidirectional CMCs generally

results in a very distinct increase in transverse strain, 4tr[211]. The eect of transverse cracking of the 90 plies in

crossplied CMCs on the 4tr response is not so pronounced

[2,5,6,9]; damage development in the 0-plies on the

other hand results in a distinct increase in 4tr [2,5,6,9].

Fig. 1 compares the 4tr response of unidirectional and

cross-plied Tyranno F bre-reinforced barium magne-

sium aluminosilicate (SiCf/BMAS) [12]. The increase in

4tr typically occurs at lower longitudinal stresses, ', and

is less pronounced for cross-plied materials [2,6].

The transverse strain response of unidirectional

CMCs has been modelled in Refs. [8,10,11]. Their 4trresponse is governed by the Poisson contraction of the

bre and the matrix, the redistribution of mechanical

stress and the release of thermal strain upon damage

development, and the build-up of compressive radial

stresses at the bre/matrix interfaces due to the radialmismatch after sliding of bres with a certain roughness

[10,11]. Smith and Wood [13] and Han and Hahn [14]

proposed micromechanical models for the 4tr of cross-

plied polymeric-matrix composites (PMCs). Transverse

cracking in the 90 plies results in a redistribution of

longitudinal stresses which causes an increase in trans-

verse strain and a decrease in Poisson's ratio. These

models, developed for PMCs, can not be used for

CMCs however because they do not take into account

matrix cracking and interface debonding in the 0-plies

of CMCs.

In this paper a micromechanical model for the 4trresponse of crossplied CMCs is proposed. The model

focuses on the prediction of the overall over-the-width

transverse strain (i.e. the one that is generally measured

using strain gauges attached to the surface) during ten-

sile loading. It is the rst to describe 4tr throughout an

entire test, taking into account all relevant damage

mechanisms. The paper itself is organised as follows. In

Section 2 the model is described. This is followed by a

comparison between theory and experiment for three

dierent crossplied Nicalon bre-reinforced calcium-

aluminosilicate matrix composites (SiCf/CAS). Finally

some of the simplifying model assumptions are discussed.

Composites Science and Technology 59 (1999) 14691481

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved

PII: S0266-3538(98)00186-9

* Corresponding author.

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2. Model description

2.1. General assumptions

The rst step in the development of the model is the

identication, the simplication, and the description of

the dierent microstructural features and micro-

mechanical mechanisms governing the 4tr response. The

0 and 90 plies of the cross-plied CMC are assumed to

consist of a volume fraction Vf of homogeneously dis-

tributed, perfectly aligned bres [10,11]. The bres have

a unique radius R, are separated from the matrix by an

innitely thin bre-matrix interface, and have a surface

roughness characterised by a wavelength and anamplitude A A `` R [10,11]. The micromechanicalmechanisms taken into account by the model are:

1. the Poisson contraction of the dierent con-

stituents,

2. the redistribution of mechanical stress and the

release of thermal strain due to transverse cracking

in the 90 plies and matrix cracking, interface

debonding,and interface sliding in the 0 plies, and

3. the changes in radial interfacial mismatch as the

rough bres slide in the matrix.

The cracks are assumed equidistant and the relation

between the transverse crack density &90 in the 90 plies

and the matrix crack density &0 in the 0 plies and the

applied stress ' is assumed to be given by a three-para-

meter Weibull function [10,11,15]:

& ' &st 1 exp ' 'm

'norm m

X I

&st is the saturation crack density, 'm the stress at

which the rst cracks form, 'norm a normalising para-

meter, and m is the Weibull modulus. In the case of

CMCs matrix cracks in the 0 plies form at higher

stresses than transverse cracks in the 90 plies [1,5,12].

Thus it is assumed that 'mY90 ` 'mY0. In a rst

approximation bre failure in the 0 plies is neglected

and interfacial sliding is assumed to occur against a

constant interfacial sliding stress (. These latter simpli-

fying assumptions have been shown to involve small

errors, while more realistic assumptions (e.g. Coulombfriction) are possible at the expense of the simplicity of

the calculations [10]. Delamination cracking between

dierent plies is seldom observed during tensile loading

of crossplied CMCs [1,2,5,6,9,13] and is not considered

either.

The simplifying assumptions of the previous para-

graph allow to describe a crossplied CMC with a given

homogeneous microstructure and an evenly distributed

damage state by one compatible unit-cell. This

approach has also been successfully used to model the

4tr response of unidirectional CMCs [10,11]. In Section

2.2 it is explained how for ' ` 'mY90 the undamagedcomposite material is described by a unit-cell, consisting

of an intact 0 ply and an intact 90 ply. It is shown

that, although the external stress applied to the unit-cell

is uni-axial, its internal stress state is bi-axial because of

compatibility eects. For 'mY90 ` ' ` 'mY0 the compo-

site material is described by a unit-cell with a cracked

90 ply (Section 2.3). For 'mY0 ` ' a unit-cell with a

cracked 0 ply and a cracked 90 ply is used (Section

2.4). The 4tr response of the cracked 0 ply is, as a rst

approximation, modelled making abstraction of the bi-

axial stress state in the 0 ply. The use of this one-

dimensional, axi-symmetric approximation, developed

for the 4tr response of unidirectional CMCs [10,11], isjustied in the sense that the modes of cracking con-

sidered in the paper are not signicantly inuenced by

the presence of transverse stresses. Matrix and trans-

verse cracks are mode I cracks, while interface debond-

ing is mainly governed by mode II shear [16].

2.2. Intact composite

A large volume of undamaged crossplied CMC is

described by a unit-cell, consisting of one 0 ply with a

thickness t0 and one 90 ply with thickness t90 [13]. The

Fig. 1. Comparison between the monotonic transverse strain response

of unidirectional and crossplied Tyranno F silicon carbide bre-rein-

forced barium magnesium aluminosilicate (SiCf/BMAS). Matrix

cracking and interface debonding start at 350MPa in unidirectional

SiCf/BMAS [12]. The result is a pronounced increase in transverse

strain. The rst transverse crack forms in crossplied material at

75MPa [12]. The eect of transverse cracking on the transverse

response is not so pronounced. At 200 MPa matrix cracks form in the

0 plies of crossplied SiCf/BMAS, causing a strong increase in trans-

verse strain.

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appropriate way of choosing t0 and t90 is illustrated in

Fig. 2. For symmetric lay-ups with 0 and 90 plies with

one specic thickness, the choice of t0 and t90 is

straightforward. For composites of which the thickness

of one type of ply varies, an average value is used.

The longitudinal strain in the 0 ply 40 and the 90 ply

4

90

are equal and given by the longitudinal strain of theintact composite 4:

40 490 4 '

EpX P

The rule-of-mixtures value for the Young's modulus of

an undamaged crossplied composite Ep is only a frac-

tion of a percent lower than the value obtained using

laminated plate theory [13,17]:

Ecp

t0E0 t90E90

t0 t90 X Q

The Young's modulus of the 0 plies E0 is calculated

from the bre volume fraction Vf, the Young's moduli

of the bre and the matrix Ef and Em, and the Poisson

coecients of the bre and the matrix #f and #m [18]. A

similar procedure exists for the Young's modulus of the

90 plies E90 [18]. Because the present model neglects the

presence of a compliant brematrix interphase (the

bre and matrix are separated by an interface), the the-

oretically calculated value for E90 would be an over-

estimation. In order to avoid this E90 is experimentallymeasured.

If the residual stresses at the ply level are not taken

into account, the longitudinal stresses in the 0 ply '0

and in the 90 ply '90 are given by:

'0 'E0

Ep

'90 'E90

EpX

R

Residual stresses at the ply level can easily be taken intoaccount by the present model at the expense of the sim-

plicity of the equations. However, these residual stres-

ses, arising from the dierence in the thermal expansion

coecients in a direction along the bres 0 and in a

direction perpendicular to the bres 90 are generally

much smaller than the intra-ply residual stresses arising

from the dierence in thermal expansion coecient of

the bre f and the matrix m. Schapery derived the

following equations for the expansion coecients 0and 90 [19]:

0

mEm 1 Vf fEfVf

Em 1 Vf EfVf

90 1 #m m 1 Vf fVfX

S

In the case of CMCs f and m are typically quite close.

A small dierence in f and m generally implies that 0and 90 are close as well. Even for a dierence between

the test temperature and the stress free temperature

T 1000C, the residual stresses at the ply level incrossplied SiCf/CAS are only of the order of magnitude

of 10 MPa [20,21]. The more important residual stresses

arising from the dierence in thermal expansion of bre

and matrix are however taken into account.The longitudinal stresses in the 0 and 90 ply result

in a contraction in the directions perpendicular to the

applied stress '. The over-the-width deformation of the

dierent plies is not free. Compatibility demands that

the transverse strain is equal for the 0 and the 90 ply.

The overall over-the-width transverse strain 4tr is calcu-

lated as follows.

First the over-the-width transverse strains are calcu-

lated for the 0 plies 40tr and 90 plies 490tr , not taking into

account the presence of the other ply. These `free'

transverse strains are given by:

40tr #040 #0

'0

E0

490tr #90490 #90

'90

E90

T

The Poisson coecient of the 0 plies #0 is calculated

from the constituent properties of the composite [18].

#90 is an experimentally determined parameter, although

its value can be theoretically calculated [18]. In the fol-

lowing step the `free' transverse strain of the dierent

plies is made equal to the overall transverse strain 4trFig. 2. Appropriate choice of the ply-thickness t0 and t90 in the unit-

cells used to describe the intact composite.

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through the application of transverse stresses '0tr and

'90tr , with an average value equal to zero [10,11,13]:

4tr 40tr

'0trE90

490tr '90trE0

t0'0tr t90'

90tr 0X

U

This set of equations gives the overall over-the-width

transverse strain 4tr of the undamaged composite. The

corresponding value of the Poisson coecient of the

undamaged composite is very close to the one calculated

using laminated plate theory [17].

2.3. Composite with transverse cracks

The proposed model describes a composite with

transverse cracks in the 90 plies by a unit-cell with an

intact 0 ply and a 90 ply with one matrix crack (Fig. 3).

The length of the unit-cell equals 2s or 1/&90 with &90 thecrack density in the 90 plies.

The formation of a transverse crack in the 90 ply

leads to a redistribution of the stresses in the vicinity of

the crack. The altered stress distributions are commonly

computed through a shear-lag analysis [13,14,2224]. If

it is assumed that the longitudinal stresses in the 0 ply

'0 x and the 90 ply '90 x are constant over thetransverse cross-sections (i.e. their values only depend

on the longitudinal distance from the transverse crack

x), the longitudinal stress in the 90 ply after transverse

cracking is given by [13]:

'90 x 'E90

Ep1 osh lx

osh ls

X V

The stress redistribution factor l is given by [13]:

l2

Gt t0 t90 Ep

t290t0E0E90X W

is a shear-lag parameter. Its value is equal to 3 if the

longitudinal displacement across the transverse ply is

parabolic [13]. Its value is 1 for a linear variation of

displacement. Gt is the shear modulus of the composite

material [13]. The longitudinal stress in the 0 ply '0 x is given by [13]:

'0 x 'E0

Ep

t90

t0'

E90

Ep1 osh lx

osh ls

X IH

The longitudinal strain of the composite 4 equals the

longitudinal strain of the continuous 0 ply 40:

4 40

s0

'0 x E0dx

s

'

Ep

t90t0

'E90Ep

1 tnh ls ls

E0

X II

As discussed earlier for undamaged crossplied compo-

sites the transverse strain of the 0 ply 40tr needs to be

equal to the transverse strain of the 90 ply 490tr . Trans-

verse cracking in the 90 plies does not change this.

Compatibility is again ensured by applying transverse

stresses, with an average value equal to zero. The over-

all over-the-width transverse strain 4tr is calculated asfollows. First the average longitudinal strain is calcu-

lated for the 0 and 90 plies. These average longitudinal

strains 40 and 490 are not necessarily equal since the 90

ply is cracked. The average longitudinal strain of the 0

ply 40, which is also the average longitudinal strain of

the composite, is given by Eq. (9). The average long-

itudinal strain of the 90 ply 490 is given by:

490 ' 1

tnh ls ls

Ep

X IP

Fig. 3. Schematic representation of (a) a unit-cell with no damage, (b)

a unit-cell with a cracked 90 ply and an intact 0 ply, and (c) a unit-

cell with a cracked 90 ply and a cracked 0 ply. The load is applied in

the horizontal direction.


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In the following step the transverse strains of the free 0

ply and 90 ply are calculated using Eq. (6). Finally the

overall over-the-width transverse strain 4tr of the com-

posite is obtained from Eq. (7).

2.4. Composite with transverse cracks and matrix cracks

Matrix cracking in the 0 plies complicates the analy-

sis further. Linear changes in the longitudinal stresses in

the bre 'f and matrix 'm, caused by matrix cracking in

the 0 plies [25,26], need to be superimposed on the

hyperbolic changes in the 0 and 90 ply stresses, arising

from transverse cracking in the 90 plies. To accomplish

this the following approach has been developed.

The unit-cell to describe the damaged composite

consists of a cracked 0 ply and a cracked 90 ply. Its

length equals 1/&90 and it contains one crack in the 90

ply and &0a&90 cracks in the 0 ply. It can be argued that

&0a&90 has no physical meaning. A unit-cell can for

example not have half a crack in the 0 ply. &0a&90should however be considered as an ad-hoc parameter

used in the calculation of the 4tr response where

&0a&90 ` 1 is perfectly possible.

At each stress level, the average longitudinal stresses

in the 90 ply '90 and in the 0 ply '0 are calculated:

'0 'E0

Ep

t90

t0'

E90

Ep1 tnh ls

ls

'90 'E90

Ep1 tnh ls

ls

X

IQ

These equations are an approximation. When damagedevelops in the 0 plies, the stiness deviates from E0and the stress redistribution between the dierent plies

is not calculated rigorously correct. However, it will be

demonstrated in Section 4 that the error is small.

Making abstraction of the bi-axial stress state in the

0 plies by assuming that interface debonding and bre

sliding only occurs under inuence of the average long-

itudinal stress '0, the length over which the brematrix

interfaces debond Ld '0

is given by [10,11,26,27]:

Ld '0

R

2(

'0

Vf

Ef

E0'0 SfYx 'f

X IR

R is the bre radius, ( the constant interfacial sliding

stress, SfYx the longitudinal residual stress in the bre,

and 'f the bre stress discontinuity at the debond

crack tip. 'f can be calculated from the elastic prop-

erties of bre and matrix and the interface debond

energy i [27]. For i % 0, 'f % 0. The thermal resi-dual stress state in the 0 ply plays an important role in

its 4tr response [10,11]. Most important are the long-

itudinal residual stresses in the bre and the matrix SfYxand SmYx, and the radial residual stress at the bre

matrix interface SiYr [10,11]. In this paper the residual

stresses are calculated for a cylinder of bre surrounded

by a cylindrical matrix shell. The equations used for the

calculation are given in Ref. [28] and are not repro-

duced here. For the calculations it is assumed that the

matrix surface is radial stress free and that SfYx and SmYx

outbalance each other. Strictly speaking, the latter can

only be the case for composites with no residual stressesat the ply level. However, because CMCs have typically

only small residual stresses at the ply level, it is expected

that this approximation does not introduce large errors.

The longitudinal strain of the composite 4 equals the

longitudinal strain of the bres in the cracked 0 ply.

For Ld '0

` 1a&0 the interfaces are not totally debon-

ded and the longitudinal strain 4 is given by [10]:

4 Ld '

0 '0Vf (RLd '0

Ef

12&0

Ld '0

'0

E0

12&0

X IS

For Ld '0

b 1a&0 total interface debonding has occur-red and 4 is given by [10]:

4

'0

Vf (

R1

2&0

Ef

X IT

Fig. 4. Schematic representation of the algorithm used to calculate the

overall transverse strain response of crossplied CMCs. At a given

longitudinal stress level ', the crack densities in the 90 and 0 plies are

calculated using Eq. (1). The next step is the calculation of the average

longitudinal stresses '0 and '90. &0, &90, '0 and '90 are then used to

calculate the `free' transverse strains 40tr and 490tr . Eq. (7) nally gives

the overall transverse strain 4tr.


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The average longitudinal stress '0 and the matrix crack

density &0 of the 0 ply are used to calculate the average

transverse strain in the 0 plies, as explained in Refs.

[10,11]. An elastic analysis reveals that the 4tr response

of a 0 ply depends on whether the bre and the matrix

are in radial contact. Thus it rst is checked if the bre

and the matrix are in radial contact along the debondedinterface [11]:

R '0Y xH

5 '0Y xH

X IU

R is the radial mismatch caused by sliding of a rough

bre, the debond crack opening, and xH the distance

from a matrix crack in the 0 ply. The radial mismatch

R depends on the relative longitudinal displacement

between the bre and the matrix u. For bres whose

surface roughness is random and small compared to R,

R '0Y xH

is modelled by [10,11,29]:

R '0Y xH A j u '0Y xH

j

a2for j u '0Y xH

j 4a2

R '0Y xH

A for j u '0Y xH

j 5a2X

IV

A is the characteristic roughness amplitude of the bres

and the characteristic roughness wavelength. The

debond crack opening is given by:

'0Y xH

m f T #f'f '

0Y xH

Ef #m

'm '0Y xH

Em

X

IW

Note that for the calculation of 40tr the distance from a

matrix crack in the 0 ply xH is used (Fig. 3). Because the

overall longitudinal stress in the cracked 0

ply isassumed to be constant and to be given by Eq. (13),

40tr can be calculated by letting xH vary between 0 and

1/(2&0). If the bre and the matrix are not in radial

contact 40tr '0Y xH

is given by [11]:

40tr '0Y xH

#m

'm '0Y xH

Em

2Vf

1 Vf

SiYr

Em #m

SmYx

EmX

PH

SiYr is the radial residual stress at the brematrix inter-

face. SmYx is the longitudinal residual stress in the matrix.

For radial contact between the bre and the matrix the

radial stress at the interface 'iYr s0

Y x

is given by [10]:'iYr '

0Y xH

R '0 YxH

R #f

Ef'f '

0Y xH

#mEm

'm '0Y xH

m f T

1Vf1Vf

#m

Em 1#f

Ef

PI

Fig. 5. Experimentally observed relationships between the longitudinal stress ', the longitudinal strain 4, and the transverse strain 4tr for the dif-

ferent crossplied SiCf/CAS composites investigated by Karandikar and Chou [6].


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The free transverse strain along a debond crack with

radial contact between bre and matrix is given

4trY0 '0Y x

is given by [10]:

40tr '0Y xH

2

Vf

1 Vf

'iYr '0Y xH

Em

#m'm '

0Y xH Em

2Vf

1 Vf

SiYr

Em #m

SmYx

EmX

PP

The average transverse strain of the 0 ply is calculated

by averaging the local transverse strains in the 0 ply as

explained in Refs. [10,11]. The relation between the average

longitudinal stress '0 and the average free transverse

strain 40tr is no longer linear [10,11]. Damage develop-

ment typically results in an increase in 40tr (Refs. [10,11]).

The transverse strain of the 90 ply is still given by Eq. (6).

The overall-over-the width transverse strain 4tr of the

composite is found by solving Eq. (7). The algorithm usedto calculate the overall 4tr response of crossplied compo-

sites is schematically represented in Fig. 4. For each stress

level 'the crack densities &90 and &0 are calculated. Eq. (13)

subsequently gives the average longitudinal stresses '90

and '0. These values allow to calculate the longitudinal

Fig. 6. Comparison between the experimentally observed evolutions of the transverse crack density &90 in the 90 plies and the matrix crack density

&0 in the 0 plies with longitudinal strain 4 and the ones used for the simulation of the response of the dierent SiC f/CAS composites investigated by

Karandikar and Chou [6].


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strain 4 and the free transverse strains 490tr and 40tr. For

the undamaged plies and the cracked 90 ply the free

transverse strain is given by Eqs. (6), (11), and (12). To

calculate the free transverse strain of the cracked 0 ply

the procedure, developed for the 4tr of unidirectional

CMCs [10,11], is followed. Eq. (7) nally gives the

transverse strain of the crossplied composite.

3. Comparison between experiment and theory

Karandikar and Chou [6] investigated the monotonic

over-the-width transverse strain response of three dif-

ferent crossplied SiCf/CAS composites. The laminate

lay-up of the investigated CMCs was equal to (03,90,03)

for composite 1, (03

,903

,03

) for composite 2 and(0,90,0,90,0,90,0,90,0) for composite 3. Fig. 5 gives the

experimentally observed relationships between the

Fig. 7. Experimentally observed evolution of the secant Poisson's

ratio #se with longitudinal strain 4 for the dierent SiCf/CAS compo-

sites investigated by Karandikar and Chou [6].

Table 1

Constituent properties used for the theoretical simulation of the

transverse response of the 0 plies of the SiCf/CAS composites

investigated by Karandikar and Chou [6]

Basic constituent properties

Vf 0.35

Ef (GPa) 195

Em (GPa) 98

#f 0.15

#m 0.30

R (mm) 7.5

( (MPa) 12'f (MPa) 0

A (nm) 20

(nm) 500

f (106 K1) 3

m (106 K1) 5

T (K) 1000

Derived properties

E0 (GPa) 132.0

#0 0.245

SiYr (MPa) 92SfYx (MPa) 232SmYx (MPa) 125

Table 2

Constituent properties used for the theoretical simulation of the

crossplied SiCf/CAS composites investigated by Karandikar and Chou

[6]. The rst number refers to the (03,90,03) composite, the second to

the (03,903,03) lay-up and the third to (0,90,0,90,0,90,0,90,0)

Basic constituent properties

t0 (mm) 0.517

0.525

0.091

t90 (mm) 0.0810.263

0.091

1

1

1

E90 (GPa) 102

#90 0.223

Gt (GPa) 48

&90Yst (mm1) 6

2.5

4.2

'90Ym (MPa) 64

30

42

'90Ynorm (MPa) 130100

100

&0Yst (mm1) 8

8

9

'0Ym (MPa) 112

78

82

'0Ynorm (MPa) 120

100

70

m90 3

4

3

m0 33

4

Derived properties

Ep (GPa) 127.7

122.0

117.0

#p 0.241

0.241

0.238

l (mm1) 8.45

3.08

10.0


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longitudinal stress ', the longitudinal strain 4, and the

transverse strain 4tr. Fig. 6 gives the experimentally

observed evolution of the crack density in the 90 and

the 0 plies for the dierent composites. The 4tr response

was quantitatively characterised by the secant Poisson's

ratio #se, dened as the negative of the ratio of the

instantaneous 4tr divided by the instantaneous 4. Fig. 7gives the evolution of #se with 4 for the three dierent

composites.

The material parameters used to simulate the 4trresponse of the dierent composites are given in Tables 1

and 2. The rst table gives the parameters needed for

the simulation of the 0 plies [6,10,11]. These parameters

are the same for all composites under investigation. The

second table gives the parameters needed for the simu-

lation of the entire composite. These vary with compo-

site lay-up. R, Vf, Ef, Em, #m, (, Gt, E90, and #90 are

taken from Ref. [6]. #f, f, m, T, and 'f are taken

from Refs. [7,10,30]. The Nicalon bre roughness has

been characterised using atomic force microscopy[31,32] and the values for A and have been used to

simulate the response of unidirectional SiCf/CAS suc-

cessfully [10]. E0, #0, SfYx, SmYx and S

iYr are calculated

from the constituent properties of the 0 ply [18,28]. Ep,

#p, and l are calculated from the basic constituent

properties of Tables 1 and 2. For the simulation it was

Fig. 8. Comparison between the experimentally observed and theoretically predicted evolution of the secant Poisson coecient #se of the three

dierent composites investigated by Karandikar and Chou [6] for the constituent properties of Tables 1 and 2.


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assumed that the relation between the crack density &

and the applied stress ' is given by a three-parameter

Weibull function. The experimentally observed evolu-

tion of the crack densities in the 90 and 0 plies is

compared to these used for the simulation in Fig. 6.

The experimentally observed and theoretically predicted

evolutions of the secant Poisson's ratio #se are com-pared in Fig. 8. The main features of the experimentally

observed response are reproduced by the model. For all

three dierent composites the decrease in #se, the long-

itudinal strain range over which #se decreases, and the

nal value of #se are accurately predicted.

The model not only predicts the experimentally

observed response. It also provides explanations for the

transverse strain response. In this section the behaviour

of the (03,903,03) composite is explained. The response of

the other composites can be explained in a similar manner.Fig. 9(a) shows the evolution of the average longitudinal

stresses in the 90 plies '90 and the 0 plies '0 with

applied longitudinal strain 4. After the development of

Fig. 9. Explanation of the transverse strain response of (03,903,03) SiCf/CAS composite. (a) Theoretical evolution of the longitudinal stresses in the 0

plies '0 and the 90 plies '90 with applied longitudinal strain 4. (b) Theoretical evolution of the free transverse strain of the 0 plies 40tr and the 90 plies

490tr with applied longitudinal stress '. (c) Theoretical evolution of the average radial mismatch in the 0 pliesR with applied longitudinal strain 4. (d)

Comparison between the experimentally observed and theoretically predicted relation between longitudinal stress ' and transverse strain 4tr.


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damage in the 90 plies '90 becomes approximately con-

stant. '0 still increases as the intact bres in the 0 plies

can take extra load. The `free' transverse strains of the

90 plies 490tr and 0 plies 40tr are compared in Fig. 9(b).

After damage development the `free' transverse strain of

the 90 ply remains negative and approximately con-

stant. The `free' transverse strain of the 0

ply increasesto a positive value, because of an increase in average

radial mismatch R between the bre and matrix

[Fig. 9(c)] [10,11]. Combining the `free' responses of the

90 and the 0 plies into an average transverse response

gives the response of the crossplied composite [Fig. 9(d)].

The theoretical prediction is in good agreement with the

experimentally observed response. The small `hill' in the

simulated evolution of #se (Fig. 8) is caused by the

simultaneous, total debonding of all brematrix inter-

faces in the simulation. In reality matrix cracks are not

perfectly equidistant and this does not occur [1,15,33].

Instead total interface debonding occurs over a stress

range, causing the `hill' to disappear. Theoretically this canbe incorporated by working with distributions of matrix

cracks spacings, as explained in Ref. [10]. This latter

falls, however, outside the scope of the present paper.

4. Discussion

The simplifying assumptions made to develop the

model for the 4tr response of a damage 0 ply have been

discussed in Refs. [10,11]. In this section only the

assumptions used to model the 4tr response of crossplied

material are discussed.

The rst concerns the stress redistribution caused by

transverse cracking in the 90 plies. In this paper a shear-

lag solution proposed by Smith and Wood [13] has been

used. Several other approaches do exist [14,2224]. Theyare however all similar in nature. '90 x increases withthe distance from the transverse crack x and the eect of

using other approaches on the predicted behaviour is

quite small. Fig. 10 for example compares the theore-

tical evolution of #se of composite 3 with 4 for dierent

values of the shear-lag parameter . Changing the

value between 1 and 3 (i.e. the extreme values suggested

in Ref. [13]) only slightly alters the model prediction.

The second assumption to be discussed is the use of

an average longitudinal stress '0 to calculate the trans-

verse strain of the 0 ply 40tr. '0 is calculated from Eq.

(11) assuming that the 0 ply is intact. This is however

no longer the case after matrix cracking. Eq. (11) over-estimates '0 by not taking into account the stiness

degradation of the cracked 0 ply. The eect of this

assumption can be assessed by a studying the eect of a

change in the value of E0, used in Eqs. (8) and (11), on

the theoretical 4tr response of a composite with cracks in

the 90 and 0 plies. Fig. 11 shows that the eect of a

decrease in E0 from 132 GPa, for an intact 0 ply, to

70 GPa, for a fully cracked 0 ply, is quite limited. A

decrease in Young's modulus of the damaged 0 ply

Fig. 10. Illustration of the eect of a change in the longitudinal stress

redistribution on the theoretical response of (0,90,0,90,0,90,0,90,0)

SiCf/CAS. For clarity, the curves have been shifted along the #se axis

by 0.02 and 0.04. Changing the stress redistribution parameter

between 1 and 3 has only a limited eect on the predicted 4tr response.

Fig. 11. Illustration of the eect of neglecting the stiness degradation

of the 0 ply on the theoretical response of (0,90,0,90,0,90,0,90,0) SiCf/

CAS in the presence of transverse cracks and matrix cracks. For

clarity, the curves have been shifted along the #sec axis by 0.02 and

0.04. Changing E0 between 132 and 70 GPa has only a limited eect on

the predicted 4tr response.


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lowers at the same time the Young's modulus of the

damaged composite. The result is that '0 is only slightly

decreased; '90 is slightly increased. Furthermore it is

theoretically possible to take into account the change in

0 ply stiness at the expense of more complex, iterative

calculations.

The third assumption concerns the use of an average'0 value to calculate 40tr. For cracked unidirectional

materials the transverse strain increases non-linearly

with the applied stress [10,11]. In general 40tr x will bemost positive close to a transverse crack in the 90 plies

because the stress in the 0 ply '0 x is highest there.Depending on the exact stress distributions in 90 and

0 plies the followed approach will slightly under-

estimate or overestimate the transverse strain of the

composite material. The eect of small errors in 40tr can

easily be assessed by comparing the response of a

crossplied composite with a slightly dierent response of

the 0 plies. This can for example be accomplished by

changing the bre roughness amplitude A, a constituentproperty with an important eect on the 4tr response of

unidirectional CMCs [10]. Fig. 12 shows that the gen-

eral characteristics of the transverse strain response do

not change if A is changed between 15 and 25 nm. The

90 ply constraints the dierences in 40tr response.

5. Conclusions

A micromechanical model describing the relationship

between longitudinal stress ' and transverse strain 4tr

during tensile testing of crossplied CMCs has been pre-

sented. The model describes crossplied CMCs by dier-

ent unit-cells, each representing a damage state

characteristic for the material. It takes into account the

Poisson contraction of the dierent constituents, the

redistribution of mechanical stress and the release of

thermal residual strain due to damage development, andthe build-up of radial mismatch stresses at the bre-

matrix interfaces in the 0 plies, due to radial interfacial

mismatch after debonding and sliding.

The validity of the proposed model has been assessed

by comparing the theoretically simulated and experi-

mentally observed response of three dierent crossplied

SiCf/CAS CMCs. Theory and experiment are in excel-

lent agreement for the dierent sets of experimentally

determined constituent properties. In the discussion

section the simplifying assumptions, used to develop the

model, are shown to involve only small errors.

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