Topics in mathematical modelling..,
Cherkaev and Kohn eds, Intro.
Topics in mathematical modelling of composite materials
Andrej Cherkaev and Robert Kohn editors, Birkhauser, 1997
Table of Contents
From the Introduction
by Andrej Cherkaev and Robert Kohn
The past 20 years have witnessed a renaissance of theoretical work
on the macroscopic behavior of microscopically heterogeneous
materials. This activity brings together a number of related
themes, including: (1) the use of weak convergence as a rigorous
yet general language for the discussion of macroscopic behavior;
(2) interest in new types of questions, particularly the
``G-closure problem,'' motivated in large part by applications
of optimal control theory to structural optimization;
(3) the introduction of new methods for
bounding effective moduli, including one based on ``compensated
compactness''; and (4) the identification of deep links between
the analysis of microstructures and the multidimensional calculus
of variations. This work has implications for many physical problems
involving optimal design, composite materials, and coherent phase
transitions. As a result it has received attention and support from
numerous scientific communities -- including engineering,
materials science, and physics as well as mathematics.
There is by now an extensive literature in this area. But for
various reasons certain fundamental papers were never properly
published, circulating instead as mimeographed notes or preprints.
Other work appeared in poorly distributed conference proceedings
volumes. Still other work was published in standard books or
journals, but written in Russian or French. The net effect is a
sort of ``gap'' in the literature, which has made the subject
unnecessarily difficult for newcomers to penetrate.
The present book aims to help fill this gap by assembling a
coherent selection of this work in a single, readily accessible
volume, in English translation. We do not claim that these articles
represent the last word -- or the first word -- on their respective
topics. But we do believe they represent fundamental work, well
worth reading and studying today. They form the foundation upon
which subsequent progress has been built.
The decision what to include in a volume such as this is difficult
and necessarily somewhat arbitrary. We have restricted ourselves
to work originally written in Russian or French, by a handful of
authors with different but related viewpoints. It would have been
easy to add other fundamental work. We believe, however, that our
choice has a certain coherence. This book will interest
scientists working in the area, and those who wish to enter it.
The book contains papers we want our Ph.D. students
to study, to which they have not until now had ready access.
We now list the chapters in this book, and comment briefly on each
one. They are presented, here and in the book, in chronological
- On the control of coefficients in partial differential
equations by F. Murat and L. Tartar. The article represents some of the earliest work
recognizing the ill-posedness of optimal control problems when the
``control'' is the coefficient of a PDE. Other early work of a
similar type is described in the review article by Lurie and
Cherkaev (see chapter 7 of the present book).
- Estimation of homogenized coefficients by L. Tartar.
This is one of
the earliest applications of weak convergence as a tool for
bounding the effective moduli of composite materials.
- H-Convergence by F. Murat and L. Tartar. The theory of H-convergence provides
a mathematical framework for analysis of composites in complete
generality, without any need for geometrical hypotheses such as
periodicity or randomness. When specialized to the self-adjoint
case it becomes equivalent to G-convergence. Treatments of
G-convergence can be found elsewhere, including the books of Jikov,
Kozlov, and Oleinik  and Dal Maso . However the treatment by Murat and Tartar has
the advantage of being self-contained, elegant, compact, and quite
general. As a result it remains, in our opinion, the best
exposition of this basic material.
- A strange term coming from nowhere
by F. Murat and D.Cioranescu.
The focus of this work is somewhat different from the other
chapters of this book. Attention is still on the macroscopic
consequences of microstructures, and weak convergence still plays
a fundamental role, however in this work the fine-scale boundary
condition is of Dirichlet rather than Neumann or transmission type.
There has been a lot of work on problems with similar boundary
conditions but more general geometry, e.g. Dal Maso, G. and Garroni ,
and to problems involving Stokes flow, e.g. Allaire  and
Hornung . For work on structural optimization in
problems of this type see Butazzo, G. and Dal Maso  and Sverak .
- Design of composite plates of extremal rigidity by L.
Gibiansky and A. Cherkaev. This work provides an early
application of homogenization to a problem of optimal design. Most
prior work dealt with second-order scalar problems such as thermal
conduction; this article deals instead with plate theory (and, by
isomorphism, 2D elasticity). For subsequent related work see
Kohn and Strang 
Allaire and Kohn [9,10] and
especially the book of Bendsoe 
and the review paper by Rozvany, Bendsoe, and Kirsch
which have extensive bibliographies.
- Calculus of variations and homogenization by F. Murat
and L. Tartar. This work presents a very complete
treatment of optimal design problems in the setting of scalar
second-order problems, and structures made from two isotropic
materials. Such a treatment was made possible by the solution of
the associated ``G-closure problem'' a few years before. The
exposition of Murat and Tartar emphasizes the role of optimality
conditions. For related work we refer once again to the book of
Bendsoe , and also the article of Kohn and
- Effective characteristics of composite materials and
the optimal design of structural elements by K.A. Lurie and A. V.
Cherkaev. The paper presents a
comprehensive review of work by Russian community on homogenization
methods applied to structural optimization and can be viewed as
a theoretical introduction to optimal design problems
illustrated by a number of examples. The approach developed here is
strongly influenced by advances in control theory
(see the book by Lurie ) as well as by
practical optimization problems.
The paper is supplemented by an Appendix describing early (1972)
progress by Lurie and Simkina. That work in Russia was approximately
contemporary with work by Murat and Tartar on similar issues in France in
the early 1970s, including the first chapter of this book.
- Microstructures of composites of extremal rigidity and
exact bounds on the associated energy density by L.V. Gibiansky
and A.V. Cherkaev. This work is a straight continuation of the
problem which is discussed above, in chapter 5 of this book.
The bounds considered in this chapter by Gibiansky and Cherkaev
concern the rigidity or
compliance of a two-component elastic composite in three space
dimensions; however, the paper reflects a subtle shift of emphasis.
The mathematical community gradually realized during the
1980s that bounds on effective moduli are of broad interest in
mechanics, beyond their value for relaxing problems of structural
optimization. Here the translation method is applied
for proving such bounds -- based on the use
of lower semicontinuous quadratic forms.
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