Advances in Mathematical Modelling of Composite Materials

ADVANCES IN MATHEMATICAL MODELLING OF COMPOSITE MATERIALS

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Series on Advances in Mathematics for Applied Sciences - Vol. 15

ADVANCES IN MATHEMATICAL MODELLING OF COMPOSITE MATERIALS Editor

Konstantin Z. Markov

Faculty of Mathematics and Informatics University of Sofia Bulgaria

118* World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite I B , 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

ADVANCES IN M A T H E M A T I C A L M O D E L L I N G O F COMPOSITE MATERIALS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.

ISBN 981-02-1644-0

Printed in Singapore by Utopia Press.

V

Foreword

I n t h e past decades the t h o e r y o f microinhomogeneous and composite materials became one o f t h e m o s t a t t r a c t i v e topics i n mechanics o f solids. T h e r e are at least t w o m a i n reasons for the great interest i n t h i s field. T h e first lies i n t h e challenge t o describe a n d t o m a t h e m a t i c a l l y m o d e l , i n a comprehensive and c o m p u t a b l e way, the i n t e r c o n n e c t i o n between the m i c r o - and macroproperties and, i n p a r t i c u l a r , t h e m u l t i p a r t i c l e s interactions t h a t decisively determine t h e macroscopic mechanical behaviour o f composites o f m a t r i x t y p e . Moreover, t h e i n t e r n a l s t r u c t u r e o f m a n y composites is r a n d o m i n itself w h i c h leads t o great and well acknowledged a d d i t i o n a l t h e o r e t i c a l difficulties. T h e second reason is concerned w i t h t h e a p p l i c a t i o n s o f composite materials as s t r u c t u r a l elements w i t h o u t s t a n d i n g features. Such applications also s t i m u l a t e , i n t u r n , t h e need o f p r o p e r m e c h a n o - m a t h e m a t i c a l models o f composites w h i c h are t o adequately p r e d i c t t h e whole range ( f r o m l i n e a r l y elastic t o failure) o f their response u n der l o a d i n g , m a k i n g use o f the available i n f o r m a t i o n (at the same t i m e very r e s t r i c t e d , as a r u l e ) a b o u t t h e i r i n t e r n a l s t r u c t u r e . T h e m a t h e m a t i c a l models proposed s h o u l d thus p r o v i d e a firm basis for a reliable p r e d i c t i o n o f real composite m a t e r i a l s performance; as a first step i n checking t h e i r a p p l i c a b i l i t y is the c o m p a r i s o n o f the t h e o r e t i c a l findings w i t h the e x i s t i n g e x p e r i m e n t a l data. T h i s v o l u m e collects six c o n t r i b u t i o n s o f specialists w i t h different backgrounds, a c t i v e l y w o r k i n g on m a t h e m a t i c a l m o d e l l i n g o f the s t r u c t u r e

and

t h e r m o - m e c h a n i c a l behaviour o f composite materials. T h e c o n t r i b u t i o n s represent comprehensive i n the

field,

a n d detailed reviews of the a u t h o r s ' current research

together w i t h discussions and comparisons w i t h relevant exist-

i n g models, approaches a n d e x p e r i m e n t a l data. Various aspects o f composite m a t e r i a l s m o d e l l i n g are addressed; t o m e n t i o n o n l y a few, t h e y include: — A p p r o x i m a t e schemes o f the t y p e o f effective field, self-consistency, generalized r u l e o f m i x t u r e s , u n i t cell models, etc., w h i c h lead eventually t o simple a n d reasonable a n a l y t i c a l p r e d i c t i o n s o f the elastic a n d inelastic response o f the composites; — R i g o r o u s estimates for the overall properties o f the composites

under

a l i m i t e d a m o u n t o f s t a t i s t i c a l i n f o r m a t i o n , b o t h i n the linear and nonlinear ranges; — S t r u c t u r a l models o f r a n d o m n a t u r e p r e d i c t i n g the failure o f composites a n d t a k i n g i n t o account t h e i r real i n t e r n a l c o n s t i t u t i o n . M o r e specifically, t h e article of S. K a n a u n a n d V . L e v i n is devoted t o the

vi

i n v e s t i g a t i o n o f the macroscopic behaviour o f composite m a t e r i a l s o f m a t r i x type.

A generalized version, w i t h various i m p l e m e n t a t i o n s , o f the effective

(self-consistent) field m e t h o d is developed a n d reviewed i n d e p t h .

Explicit

formulae for the overall elastic, thermoelastic a n d d y n a m i c p r o p e r t i e s o f such composites are derived and compared w i t h e x p e r i m e n t a l d a t a . I n the paper o f K . M a r k o v and K . Z v y a t k o v a general m e t h o d o f p l a c i n g v a r i a t i o n a l b o u n d s on the effective properties o f r a n d o m heterogeneous solids is exposed i n d e t a i l . I t is shown how the earlier v a r i a t i o n a l procedures due t o Hashin a n d S h t r i k m a n , M i l t o n a n d P h a n - T h i e n , and W i l l i s find n a t u r a l places i n the proposed general scheme.

Special a t t e n t i o n is p a i d t o dispersions o f

spheres. I n the c o n t r i b u t i o n o f D . R. S. T a l b o t , the generalization of the H a s h i n S h t r i k m a n v a r i a t i o n a l principles t o nonlinear problems is reviewed.

A new

m e t h o d o l o g y is described w h i c h relies on a nonlinear c o m p a r i s o n m a t e r i a l and w h i c h has the p o t e n t i a l o f p r o d u c i n g t w o new ( u p p e r a n d l o w e r ) b o u n d s . T h e m e t h o d o l o g y is o u t l i n e d i n the c o n t e x t o f a nonlinear dielectric p r o b l e m a n d some results for a periodic composite are described. T h e paper o f K . H e r r m a n n and I . M i h o v s k y presents an unified approach to the inelastic d e f o r m a t i o n and failure o f composites c o m p r i s i n g a d u c t i l e m a t r i x a n d parallelly aligned stiff fibres. T h e approach reduces a large variety of real inelastic and failure problems for such composites t o p a r t i c u l a r cases o f a c e r t a i n general m a t h e m a t i c a l p l a s t i c i t y p r o b l e m . T h e results d e r i v e d are shown to compare favourably w i t h e x i s t i n g e x p e r i m e n t a l d a t a a n d t h e o r e t i c a l estimates. T h e c o n t r i b u t i o n o f 0 . Pedersen and B . Johannesson discusses numerous recent applications o f the effective m e d i u m theory, cell models and an e x t e n d e d version o f the rule o f m i x t u r e s i n m o d e l l i n g thermoelastic a n d plastic b e h a v i o u r o f fibrous composites. T h e advantage o f the new c o m p u t e r software, capable o f symbolic c o m p u t a t i o n s , is c o n v i n c i n g l y d e m o n s t r a t e d o n p r a c t i c a l l y i m p o r t a n t types o f such composites. T h e paper of D . J e u l i n is concerned w i t h a general p r o b a b i l i s t i c approach t o relate the m i c r o s t r u c t u r e w i t h the overall properties. T h e m e t h o d p r e d i c t s scale effects i n the

fluctuations

o f these properties; i t allows also t o specify t h e

p r o b a b i l i t y o f fracture o f the composites under realistic l o a d i n g c o n d i t i o n s . I t is hoped t h a t this volume w i l l c o n t r i b u t e t o new insights a n d b e t t e r u n d e r s t a n d i n g o f the composite materials behaviour i n connection w i t h t h e i r i n t e r n a l s t r u c t u r e . Hence i t could be expected t o further the research a c t i v i t y in t h i s interesting and i m p o r t a n t , b o t h f r o m the t h e o r e t i c a l and p r a c t i c a l p o i n t s of view,

field.

vii

T h e s u p p o r t o f the B u l g a r i a n M i n i s t r y o f E d u c a t i o n and Science (under G r a n t N o . M M 26-91), i n p r e p a r i n g the final camera-ready manuscripts o f some o f the papers o f t h i s v o l u m e , is gratefully acknowledged.

Sofia, November 2 4 t h , 1993

Konstantin

Markov Editor

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ix

Contents

Foreword

v

Effective F i e l d M e t h o d i n Mechanics o f M a t r i x C o m p o s i t e M a t e r i a l s

1

S. K. Kanaun

and V. M.

Levin

F u n c t i o n a l Series a n d H a s h i n - S h t r i k m a n T y p e Bounds o n the Effective P r o p e r t i e s o f R a n d o m M e d i a K. Z. Markov

and K. D.

B o u n d s for t h e Effective Properties o f N o n l i n e a r C o m p o s i t e M a t e r i a l s D. R. S.

59

Zvyatkov 107

Talbot

O n t h e M o d e l l i n g o f the Inelastic T h e r m o m e c h a n i c a l Behaviour and the Failure o f Fibre-Reinforced Composites — A Unified A p p r o a c h K. P. Herrmann

and I . M.

M o d e l l i n g o f Elastic a n d Inelastic Behaviour of Composites O. B. Pedersen

and B.

Jeulin

193

Johannesson

R a n d o m S t r u c t u r e M o d e l s for C o m p o s i t e M e d i a a n d Fracture Statistics D.

141

Mihovsky

239

1

EFFECTIVE FIELD METHOD OF M A T R I X C O M P O S I T E

I N MECHANICS MATERIALS

S. K . KANAUN Institute Tecnologico y de Esiudios Superiores de Monterrey, Campus Estado de Mexico, Apartado postal 214, 53100 Atizapdn de Zaragoza, Edo. de Mexico, Mexico and V. M. L E V I N Department cf Applied Mechanics, Petrosavodsk State University, Lenin pr. 33, Petrosavodsk 185640, Russia

ABSTRACT The article is devoted to the investigation of elastic, thermoelastic and dynamic behaviour of the important class of the so-called matrix composites. They represent a homogeneous matrix containing a random array of filling particles. A generalized version of the effective (self-consistent) field method is developed and its various implementations are reviewed in depth. The basic idea of the method is to treat every inclusion as isolated, but undergoing an effective external field, generated by the surrounding inclusions. Here, in distinction to the traditional forms of the effective field method, this field is considered, however, to be random and special techniques are proposed for calculating its statistical moments. The performance of the method is illustrated for composites, containing inclusions of various shapes (spheres, ellipsoids, long and short fibers, cracks, etc.). Explicit formulae for the overall elastic and thermoelastic properties of such composites are given and then compared to experimental data. The dynamic case (monochromatic wave propagation through the composite) is addressed as well. In the long wave approximation the propagation velocities and attenuation factors of the elastic waves in composites are derived and analyzed.

1.

Introduction I n the last ten years the theory of microinhomogeneous and composite mate-

rials became an attractive topic i n mechanics and physics of solids. Acute interest i n investigation and modelling of physical and mechanical properties of composite materials is inherently connected w i t h the constantly increasing area of their applications. I n many respects composites proved to be superior to the known homogeneous materials; first of a l l they have superior physical and mechanical properties, secondly, i t is possible to design the composite structure and to create materials w i t h the prescribed

2 i n advance properties optimal for the operation conditions of the whole structure. Comprehensive investigations of the influence of microstructure on the whole specter of physical and mechanical properties of composite materials are necessary for the solution of the latter problem. This article is devoted to the investigation of elastic and thermoelastic deformations of an important class of composite materials, namely, the so-called m a t r i x composites. The microstructure of such composites usually has the following specifics: it consists of a homogeneous matrix and a set of filling particles (i.e., inclusions) uniformly distributed i n the matrix. Examples of such composites are: composites reinforced by particles of different forms and shapes, unidirectional fiber composites, laminate and hybrid composites filled w i t h different types of fibers or/and inclusions. B y an appropriate choice of the components composite materials w i t h high shock-resistance properties, controlled dumping, high strength and stiffness and low thermal expansion can be created. As a rule microstructure of real composite material is stochastic: random parameters characterize shapes, sizes and physical properties of inclusions. Distribution of the inclusions i n the volume of the matrix is also random. As a result strain and stress fields i n these composites even under the deterministic external loading are stochastic. A n important problem i n mechanics of composite materials is estimation of mean values (mathematical expectations) of these fields under the deterministic external loading (i.e., the homogenization problem). The solution of this problem allows to find the effective properties of composite material and to substitute the so obtained homogeneous material, yielding known deterministic law of deformation, i n the failure analysis of structures made of the composite. Note that matrix composites may be reinforced by regular (periodical) as well as random systems of inclusions. I n the former case the homogenization problem can be solved w i t h any degree of accuracy i f advanced computer techniques is u s e d . But in case of stochastic composites this problem can be solved only approximately. The main difficulty is i n the description of many particle interactions between randomly placed inclusions. There is a group of methods i n theoretical physics known as self-consistent schemes which allow to solve effectively the many-particle problem. These schemes reduce the many-particle problem to a one particle problem and, thus, provide the opportunity for the effective solution of homogenization problem. The ideas of these methods can be also applied to the solution of homogenization problem i n the mechanics of microinhomogeneous media. The first attempts to solve homogenization problem w i t h i n the framework of self-consistent schemes were based on the following assumption: every inclusion was considered as isolated and embedded i n a homogeneous medium w i t h the effective properties of the whole composite structure. External field applied to this inclusion was considered as equal to the external field acting on the whole composite structure. This version of self-consistent schemes may be called the method of effective medium. Application of this method and some of its modifications to the solution 1 - 3

3 of homogenization problem for the microinhomogeneous media of various structures was considered i n a number of a r t i c l e s . The method developed i n this article is a generalized version of the effective (self-consistent) field method. I n this method every inclusion is considered as an isolated one i n the homogeneous medium—the matrix of the composite. The presence of surrounding inclusions is taken into account by introducing an effective external field acting on this inclusion. Here, i n distinction to the traditional form of the effective field m e t h o d , the effective field is considered to be random and special technique is being developed for calculating its statistical moments. I n principle this method allows to calculate statistical moments of arbitrary order of elastic and thermoelastic fields i n m a t r i x composites w i t h various types of reinforcing elements and obeying different laws of statistical distribution in the volume of the m a t r i x . The solution of the problem for an isolated inclusion embedded i n homogeneous medium (i.e., the one-particle problem) represents an important stage i n the realization of the method. I t should be mentioned that the success i n realization of any self-consistent scheme always depends on the degree of sophistication of algorithms for the solution of one-particle problems. The above mentioned considerations determine the plan of the article. Its first part (Section 2) is devoted to the solution of three-dimensional problems of elasticity and thermoelasticity for a homogeneous medium w i t h an isolated inclusion. The list of these problems contains the main types of reinforcing elements usually used as fillers in m a t r i x composites: quasispherical particles, t h i n stiff flakes, short axisymmetrical fibers, long unidirectional fibers, t h i n compliant inclusions and cracks. I n Section 3 the general scheme for the solution of homogenization problem on the basis of effective field method is presented. The method is developed here for elasto and thermoelastostatic problems. The effective elastic moduli and coefficients of thermal expansion for composites w i t h above mentioned fillers are obtained. I t is shown that i n the general case the operator connecting mathematical expectations of stress and strain fields i n composite materials is nonlocal. I f an external field is sufficiently smooth this operator can be approximated by a differential operator of finite order. I t turns out that the equation satisfied by the average displacement vector coincides w i t h the known version of the couple stress theory of elasticity. Section 4 is devoted to the solution of the homogenization problem i n dynamic case. We consider monochromatic wave propagation through composite materials. We obtain the averaged equation of motion of this medium (the effective wave operator) i n the long wave approximation. I t is shown that this operator describes wave propagation i n some homogeneous medium w i t h spatial and temporal dispersion. The Green tensor of the effective wave operator and the expressions for the propagation 4-8

9 - 1 1

4

velocities and attenuation factors of elastic wave i n composites are analyzed. 2. E q u i l i b r i u m o f H o m o g e n e o u s E l a s t i c M e d i u m W i t h a S i n g l e I n c l u s i o n 2.1.

Integral Equations for the Strain and Stress

Fields

Let us consider an unbounded elastic medium described by the elastic moduli tensor C(x), which is representable i n the form C(x)

= C + C (x), 0

Ci{x) = C V{x),

1

C i = C - c V

1

(2-1)

Here V(x) is the characteristic function of a region V occupied by the inclusion w i t h the elastic moduli C, Co is a constant tensor of the elastic moduli of the medium. The equations of static elasticity in displacements are w r i t t e n i n the form ViC {x)V u,(x) ijkl

= -q,(x).

k

(2.2)

The quantity q denotes an external force, decreasing sufficiently rapidly as \x\ —* so. For this equation to have a unique solution, certain conditions at infinity must be added. It can be shown that the strain field e.v, = V(,Uj) in the medium w i t h an inhomogeneity satisfies an integral equation which is completely equivalent to Eq. (2.2) 12

£ij(x) + J Kij l(x

— x')Ciklmn{x')e n{x)

k

dx =

m

Kijki(x)

= -[V,V G (^)](i,)(H i

j i

£ ij, 0

(2.3)

)

Here the kernel K{x) is a generalized function of the degree -3. The properties of this function were investigated i n Ref. 12. I t is also convenient to introduce the tensor of elastic compliance of the medium w i t h the inhomogeneity B{x) B

= B + B {x), 0

l

= B - B

B {x)

1

0

,

1

= BiV(x), 1

B = C~\

Bo = C ~ . 0

(2.4)

The equation for the stress field a(x) follows now from Eq. (2.3) (sometimes the subscripts will be dropped for simplicity) o(x) where er = C e , 0

0

0

- J S(x

- a;')5i(a;')o-(i') dx' = i(jf>k)l is the unit tensor of the forth order. If the medium is isotropic, the tensors A and D have the symmetry of the ellipsoid and are defined by nine essential components. A l l these components can be expressed i n terms of standard elliptical integrals. The explicit forms of these tensors are given i n Ref. 12. 2.3. A Thin Isolated Inclusion

in a Homogeneous Elastic

Medium

Let us examine a again homogeneous elastic medium w i t h a tensor of elastic moduli Co, containing a single inclusion of moduli tensor C. The inclusion perfectly fits into the undeformed medium and occupies a finite volume V . We assume that one of the characteristic length parameter of this volume h is much smaller than the two others (of the order / ) . Thus the ratio 6\ = h/l is small. The external loading i n the medium is represented by body forces and stresses at infinity. The prime interest for applications are t h i n inclusions w i t h elastic moduli essentially different from the moduli of the medium. I n this case the ratio of characteristic moduli of elasticity of the inclusion and of the medium (O^Cg ) = S ) is either small (soft inclusions) or large (stiff inclusions). I t should be noted that the most valuable information about the stress fields i n the vicinity of t h i n inclusions is contained i n the main terms of the asymptotic 1

2

6 expansion of these fields over the parameters Si and S . I n order to construct these terms i t is necessary to find the limiting solution of the elastic problem under study when 0 (or 8 —> oo) and the ratio = 0) the homogenization problem is reduced to the construction of the operator C„. On the basis of Eq. (3.1) we can write the expression for the mean values of the strain and stress fields i n the form: ( e ( i ) ) = e -J 0

K{x

- x')(q(x')}

1

dx',

(ff(aej) = o - - | s ( x - x ' ) C - ( ( i ' ) ) d i ' . 0

0

9

(3.11)

For a spatially uniform set of inclusions £(x), o~(x) and q(x) are homogeneous random ergodic functions. Therefore, e.g., the average (q) is a constant tensor whose value can be found by spatial averaging of a typical fixed realization of the random function q(x), similarly to Eq. (3.6). Because of the linearity of the problem the strain field E(X) is represented by the external field E through the relation 0

£(x) = ( A £ ) ( x ) . 0

(3.12)

15

Here A is a certain linear operator. I f the field e is constant, A is the operator of multiplication by the function A ( x ) . This function is a certain fourth-rank tensor obtained from the solution of the many particle problem. After substituting the expression for e(x) i n the formula for q(x) i n Eq. (3.1) and averaging the result, we obtain 0

(q)=pPe ,

P={P ),

0

P = - [c (x)A(x)dx,

V

v

(3.13)

1

VJv

where p is the volume concentration of the inclusions p = (V(x)), v is the volume of a typical inclusion and the average (P ) is calculated from the ensemble distribution of the random variable P . I t is assumed henceforth that the average strain i n the composite (e) coincides w i t h the external field £n and does not depend on the properties and concentration of the inclusions ((e) = e ) . (A more general case w i l l be considered in Section 3.6.) This mean value is determined by the conditions at infinity. I n this case the effect of the operators K and S i n Eq. (3.1) on the constant tensors (q) and Co (q) is determined by Eq. (3.3). I t follows from Eqs. (3.11) to (3.13) that the expression for the average stresses i n the medium w i t h inclusions has the form v

v

0

1

(