Table Of ContentCISM 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
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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.
