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Topics
in mathematical modelling of composite materials
Andrej Cherkaev and Robert Kohn editors,
Birkhauser, 1997
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Table of Contents

Introduction by A. Cherkaev and R. Kohn

On the control of coefficients in partial differential equations by
F. Murat and L. Tartar.

Estimation of homogenized coefficients by L. Tartar.

HConvergence by F. Murat and L. Tartar.

A strange term coming from nowhere by F. Murat and D.Cioranescu.

Design of composite plates of extremal rigidity by L. Gibiansky
and A. Cherkaev.

Calculus of variations and homogenization by F. Murat and L. Tartar.

Effective characteristics of composite materials and the optimal design
of structural elements by K.A. Lurie and A. V. Cherkaev.
Appendix by K.A. Lurie and T.Simkina.

Microstructures of composites of extremal rigidity and exact bounds
on the associated energy density by L.V. Gibiansky and A.V. Cherkaev.
For the annotation of the chapters see
the following:
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 ``Gclosure 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 order.

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 illposedness 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.

HConvergence by F. Murat and L. Tartar. The theory of Hconvergence
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 selfadjoint case it becomes equivalent
to Gconvergence. Treatments of Gconvergence can be found elsewhere, including
the books of Jikov, Kozlov, and Oleinik [1] and Dal Maso [2]. However the
treatment by Murat and Tartar has the advantage of being selfcontained,
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 finescale 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
[3], and to problems involving Stokes flow, e.g. Allaire [4] and Hornung
[5]. For work on structural optimization in problems of this type see Butazzo,
G. and Dal Maso [6] and Sverak [7].

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 secondorder
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 [8] Allaire and Kohn [9,10] and especially
the book of Bendsoe [11] and the review paper by Rozvany, Bendsoe, and
Kirsch [12] 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 secondorder problems, and structures made from
two isotropic materials. Such a treatment was made possible by the solution
of the associated ``Gclosure 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 [11], and also the article
of Kohn and Strang [8].

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 [13]) 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 twocomponent
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.
@Book{Cherkaev:1987:TMM,
TITLE = {Topics in the mathematical modelling of composite materials},
EDITOR = {Cherkaev, Andrej and Kohn, Robert},
PUBLISHER = {Birkh\"auser Boston Inc.},
ADDRESS = {Boston, MA},
YEAR = {1997},
PAGES = {xiv+317},
ISBN = {0817636625},
MRCLASS = {7302 (35B27 49J40 49Q10 73B27 73K20 73K40)},
MRNUMBER = {98i:73001},