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ADVANCES IN MATHEMATICAL MODELLING OF COMPOSITE MATERIALS This page is intentionally left blank 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 IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH ADVANCES IN MATHEMATICAL MODELLING OF 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 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 This page is intentionally left blank 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