Table Of ContentADVANCES 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
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ADVANCES IN MATHEMATICAL MODELLING OF
COMPOSITE MATERIALS
Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd.
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V
Foreword
In the past decades the thoery of microinhomogeneous and composite ma
terials became one of the most attractive topics in mechanics of solids. There
are at least two main reasons for the great interest in this field. The first lies in
the challenge to describe and to mathematically model, in a comprehensive and
computable way, the interconnection between the micro- and macroproperties
and, in particular, the multiparticles interactions that decisively determine the
macroscopic mechanical behaviour of composites of matrix type. Moreover, the
internal structure of many composites is random in itself which leads to great
and well acknowledged additional theoretical difficulties. The second reason is
concerned with the applications of composite materials as structural elements
with outstanding features. Such applications also stimulate, in turn, the need
of proper mechano-mathematical models of composites which are to adequately
predict the whole range (from linearly elastic to failure) of their response un
der loading, making use of the available information (at the same time very
restricted, as a rule) about their internal structure. The mathematical models
proposed should thus provide a firm basis for a reliable prediction of real com
posite materials performance; as a first step in checking their applicability is
the comparison of the theoretical findings with the existing experimental data.
This volume collects six contributions of specialists with different back
grounds, actively working on mathematical modelling of the structure and
thermo-mechanical behaviour of composite materials. The contributions rep
resent comprehensive and detailed reviews of the authors' current research
in the field, together with discussions and comparisons with relevant exist
ing models, approaches and experimental data. Various aspects of composite
materials modelling are addressed; to mention only a few, they include:
— Approximate schemes of the type of effective field, self-consistency, gen
eralized rule of mixtures, unit cell models, etc., which lead eventually to simple
and reasonable analytical predictions of the elastic and inelastic response of
the composites;
— Rigorous estimates for the overall properties of the composites under
a limited amount of statistical information, both in the linear and nonlinear
ranges;
— Structural models of random nature predicting the failure of composites
and taking into account their real internal constitution.
More specifically, the article of S. Kanaun and V. Levin is devoted to the
vi
investigation of the macroscopic behaviour of composite materials of matrix
type. A generalized version, with various implementations, of the effective
(self-consistent) field method is developed and reviewed in depth. Explicit
formulae for the overall elastic, thermoelastic and dynamic properties of such
composites are derived and compared with experimental data.
In the paper of K. Markov and K. Zvyatkov a general method of placing
variational bounds on the effective properties of random heterogeneous solids
is exposed in detail. It is shown how the earlier variational procedures due to
Hashin and Shtrikman, Milton and Phan-Thien, and Willis find natural places
in the proposed general scheme. Special attention is paid to dispersions of
spheres.
In the contribution of D. R. S. Talbot, the generalization of the Hashin-
Shtrikman variational principles to nonlinear problems is reviewed. A new
methodology is described which relies on a nonlinear comparison material and
which has the potential of producing two new (upper and lower) bounds. The
methodology is outlined in the context of a nonlinear dielectric problem and
some results for a periodic composite are described.
The paper of K. Herrmann and I. Mihovsky presents an unified approach
to the inelastic deformation and failure of composites comprising a ductile
matrix and parallelly aligned stiff fibres. The approach reduces a large variety
of real inelastic and failure problems for such composites to particular cases
of a certain general mathematical plasticity problem. The results derived are
shown to compare favourably with existing experimental data and theoretical
estimates.
The contribution of 0. Pedersen and B. Johannesson discusses numerous
recent applications of the effective medium theory, cell models and an extended
version of the rule of mixtures in modelling thermoelastic and plastic behaviour
of fibrous composites. The advantage of the new computer software, capable of
symbolic computations, is convincingly demonstrated on practically important
types of such composites.
The paper of D. Jeulin is concerned with a general probabilistic approach
to relate the microstructure with the overall properties. The method predicts
scale effects in the fluctuations of these properties; it allows also to specify the
probability of fracture of the composites under realistic loading conditions.
It is hoped that this volume will contribute to new insights and better
understanding of the composite materials behaviour in connection with their
internal structure. Hence it could be expected to further the research activity
in this interesting and important, both from the theoretical and practical points
of view, field.
vii
The support of the Bulgarian Ministry of Education and Science (under
Grant No. MM 26-91), in preparing the final camera-ready manuscripts of
some of the papers of this volume, is gratefully acknowledged.
Sofia, November 24th, 1993 Konstantin Markov
Editor
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ix
Contents
Foreword v
Effective Field Method in Mechanics of Matrix Composite Materials 1
S. K. Kanaun and V. M. Levin
Functional Series and Hashin-Shtrikman Type Bounds on the
Effective Properties of Random Media 59
K. Z. Markov and K. D. Zvyatkov
Bounds for the Effective Properties of Nonlinear Composite Materials 107
D. R. S. Talbot
On the Modelling of the Inelastic Thermomechanical Behaviour and
the Failure of Fibre-Reinforced Composites — A Unified Approach 141
K. P. Herrmann and I. M. Mihovsky
Modelling of Elastic and Inelastic Behaviour of Composites 193
O. B. Pedersen and B. Johannesson
Random Structure Models for Composite Media and Fracture Statistics 239
D. Jeulin
1
EFFECTIVE FIELD METHOD IN MECHANICS
OF MATRIX COMPOSITE 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. LEVIN
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 be
haviour 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 propaga
tion 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
In the last ten years the theory of microinhomogeneous and composite mate
rials became an attractive topic in mechanics and physics of solids. Acute interest in
investigation and modelling of physical and mechanical properties of composite mate
rials is inherently connected with the constantly increasing area of their applications.
In many respects composites proved to be superior to the known homogeneous mate
rials; first of all they have superior physical and mechanical properties, secondly, it is
possible to design the composite structure and to create materials with the prescribed
