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CISM COURSES AND LECTURES Series Editors: The Rectors Sandor Kaliszky - Budapest Mahir Sayir - Zurich Wilhelm Schneider - Wien The Secretary General Bernhard Schrefler - Padua Former Secretary General Giovanni Bianchi - Milan Executive Editor Carlo Tasso - Udine The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences. INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES -No. 430 MECHANICS OF RANDOM AND MULTISCALE MICROSTRUCTURES EDITED BY DOMINIQUE JEULIN ECOLE DES MINES DE PARIS MARTIN OSTOJA-STARZEWSKI MCGILL UNIVERSITY ~ Springer-Verlag Wien GmbH This volume contains 82 illustrations This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2001 by Springer-Verlag Wien Originally published by Springer-Verlag Wien New York in 2001 SPIN 10862933 In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been reproduced in their original forms. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader. ISBN 978-3-211-83684-2 ISBN 978-3-7091-2780-3 (eBook) DOI 10.1007/978-3-7091-2780-3 PREFACE Many solid materials - composites, granular media, metals, biomaterials, etc. - display microstructures of several length scales, oftentimes accompanied by a non-deterministic disorder. A better understanding and prediction of the resulting multiscale and random nature (~f such materials mesoscopic and/or macroscopic properties requires a modeling approach based on a combination of probabilistic concepts with methods of mechanics. The present course was organized to provide an introduction to this subject through a self-contained series ~f lectures by five mechanicians, materials scientists and mathematicians working in (micro)mechanics of random media. First, motivated by a review of advanced experimental techniques for the microstructure description, and by typical results involving fluctuations present in plasticity, damage and fracture phenomena, basic tools of applied probability and random processes have been introduced. At the second stage, mechanics of random media was presented from the standpoint of approximate solutions of partial differential equations with random coefficients. For example, static elasticity problems in random media were studied by means of perturbation expansion of the random stress and strain .fields, while bounds on elastic moduli were derived from variational principles. This approach was illustrated by optimal microstructures. Next, dynamic problems of scattering, homogenization, localization, and wavefront propagation in connection with waves in random media were addressed. The fourth area of focus was the description and Monte Carlo type simulations of random continuous or discrete media, and, on that basis, the development and application of techniques of computational micromechanics: finite and boundary elements, spring networks, molecular dynamics and lattice-gas. Finally, gi1·en the importance of reliability problems in a multitude of engineering applications, several fracture statistics models (brittle, ductile, fatigue), worked out _fi-mn a probabilistic approach, concluded the course. We acknowledge the commitment of Professors M.J. Beran, A. Pineau, and J.R. Willis in making the course possible. Their lectures and ours resulted in five chapters collected in this book. The course itself was enriched by several evening discussion sessions, where some participants presented their ongoing research projects. All in all, 30 persons from universities, government research establishments and industry in Europe, Israel, and North America attended the course. Last but not least, our thanks go to the rectors and staff of CISM. who made this event possible and helped with all the arrangements. Dominique Jeulin Martin Ostoja-Starzewski CONTENTS Page Preface Statistical Continuum Mechanics: an Introduction by M.J. Beran ................................................................................... .................................................................. l Random Structure Models for Homogenization and Fracture Statistics by D. Jeulin ........................................................................................................................................................ 33 Mechanics of Random Materials: Stochastics, Scale Effects, and Computation by M. Ostoja-Starzewski........................................................................ . .. .......................................... 93 The Randomness of Fatigue and Fracture Behaviour in Metallic Materials and Mechanical Structures by A. Pineau ................................................... . .................................. 163 Lectures on Mechanics of Random Media by J.R. Willis................................................ . .. . .. . .. ... .. ....................... 221 STATISTICAL CONTINUUM MECHANICS AN INTRODUCTION M.J. Beran Tel Aviv University, Tel Aviv, Israel ABSTRACT In these sections we outline some basic ideas underlying the solution of statisti cal problems in conductivity, acoustics and elasticity. The problems we consider are both static and dynamic and the statistical nature of the problems stems from the fact that coefficients like the heat conductivity or sound speed are random functions of position. We first point out that simple averaging procedures are inadequate to determine effective properties of a medium except in the limit of small pertur bations. In general they lead to an infinite hierarchy of statistical equations. For large variations of the random coefficients different teclmiques are required to obtain useful information. In the static problem variational principles are used extensively to find bounds on effective constants. However, we also discuss very approximate techniques like the self-consistent scheme. In the dynamic case we consider three techniques u .'.i ed in different-type problems. We show how to treat the problem of scattering by a dilute collection of discrete scatterers, propagation of acoustic waves through a medium like the ocean and forward and backward scattering of acoustic waves by a one-dimensional random medium. 1 Basic statistical formulation of the static conductivity and elasticity problems In this lecture we shall present the statistical formulation of the static conduc tivity and elasticity problems using the concepts developed in the previous lectures. The basic governing equation for the static conductivity problem is Qi,i = 0, Qi = Ki;(x)&T' /ox; (1.1) Here Ki;(x) is the spatially variable thermal conductivity tensor, Qi(x) is the heat flux and T(x) is the temperature. For the elastic problem we have = = TijJ 0, Tij Cijkl(x)ekl (1.2) 2 M.J. Beran Here Cijkl(x) is the stiffness tensor, Ti;(x) is the stress tensor and ekl(x) is the strain tensor. The functions Ki;(x) and CiJkl(x) are random functions of position and as a con sequence of this the flmctions Qi(x), T(x), Ti;(x) and ekz(x) are also random functions of position. We shall assume that the boundary conditions are non-random and the same as the case in which the conductivity and stiffness tensors were independent of position. The statistical formulation of the conductivity and elastic problems are very similar and we shall present a detailed discussion of only the conductivity prob lem. At the conclusion we shall present the final results for the elasticity problem. Further, to simplify the presentation we shall consider the special case of a locally isotropic conductivity so that (1.3) and qi(x) = K(x)8T(x)f8xi (1.4) \Ve assume that all the statistical properties of K(x) are known. That is, for example, we know the mean, (K(x)), the mean square, (K2(x)), and in general P (K(x)), the one-point probability density function. However, this is just the 1 simplest statistical information. In addition, we a.'3Sume we know all the two-point moments of the form (Km{x)Kn(x)) and in general all the -n.-point moments. A complete description is given by the n-point probability density function in the limit that n approaches infinity. 1.1 Effective conductivity The quantity of perhaps most interest in considering the effect of the statistical properties of the microstructure is the effective conductivity. In many problems Eq. (1.4) may be replaced by (1.5) where K*(x) is termed the effect·ive conductitrity. (The quantity will be defined more precisely subsequently). The important thing to note at this point is that from a rigorous point of view K* depends on the n-point probability density function in the limit that n approaches infinity. As a practical matter this information is never available and so K must be approximated using more limited statistical information. For example, if K(x) has only small fluctuations about a mean value (K(x)), the effective constant K* may be fmmd using only (K1(x)K2(x)). However, it cannot be stressed too strongly that good approximations for K* may require a considerable amount of statistical information and in principle an infinite ammmt of statistical information is necessary for an exact determination. Statistical Continuum Mechanics 3 1.2 Statistical hierarchy of equations From a statistical point of view the quantities that we wish to determine from the solution of Eq. (1.1) are quantities like (T(x)}, (8T(x)/8xi}, (qi(x)} and the higher n-point moments and probability density functions associated with T(x), 8T(x)f8xi and Qi(x). It is instructive to begin by attempting to find an equation governing (T(x)} from Eqs. (1.1) and (1.4). We have the basic equation 8[K(x)OT(x)f8xi]/8xi = 0 (1.6) or in expanded form If we now take the ensemble average of Eq. (1.7) we find (1.8) Thus in place of having an equation governing the quantity (T(x)) we have an equation that contains the unknown quantity The quantity is unknown since we know nothing about the correlation between K(x) and T(x). It might be hoped that it is po..'!sihle to get an equation governing the quantity in Eq. (1.9) and have a determinate pair of equations. However, this attempt leads to the inclusion of additional unknown quantities. For example, we can find an equation governing the quantity (Ki(xi)81'(x)/8x2i) by multiplying Eq. (1.7) by Ki(x2) and averaging. We note that the expression in Eq. (1.9) is a special c~e of this expression when x2 = x1. We find then The pair of equations Eqs. (1.10) and (1.8) now contain the additional unknown term (1.11) By this procedure we can obtain an infinite hierarchy of equations which can only be solved exactly in the limit of n approaching infinity. It is possible to derive a functional equation governing T(x) and K(x) (see [3]) but we shall not consider this approach here. In this and other statistical theories considerable effort is devoted to terminating the hierarchy and obtaining a finite set of solvable equations.