Topics in the Mathematical Modelling of Composite Materials

327 Pages·2018·25.119 MB·English

by  Andrej V. Cherkaev, Robert Kohn

Modern Birkhäuser Classics
Andrej V. Cherkaev
Robert Kohn
Editors
Topics in the
Mathematical
Modelling of
Composite Materials
Modern Birkhäuser Classics

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Topics in the
Mathematical
Modelling
of Composite
Materials
Andrej V. Cherkaev
Robert Kohn
Editors
Reprint of the 1997 Edition
Editors
Andrej V. Cherkaev Robert Kohn
Department of Mathematics Courant Mathematical Institute
University of Utah New York University
Salt Lake City, Utah, USA New York, USA

ISSN 2197-1803 ISSN2197 -1811 (electronic)
Modern Birkhäuser Classics
ISBN 978-3-319-97183-4 ISBN 978-3-319-97184-1 (eBook)
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Topics in the Mathematical
Modelling of Composite Materials
Andrej Cherkaev
RobertKohn
Editors
Springer Science+Business Media, LLC
Andrej Cherkaev Robert Kohn
Department of Mathematics Courant Mathematical Institute
University of Utah New York University
Salt Lake City, UT 84112 New York, NY 10012
Library of Congress Cataloging-in-Publication Data
Topics in the mathematical modelling of composite materials / Andrej
Cherkaev, Robert Kohn, editors.
p. cm. --(Progress in nonlinear differential equations and
their applications ; v. 31)
Includes bibliographical references.
ISBN 978-1-4612-7390-5 ISBN 978-1-4612-2032-9 (eBook)
DOI 10.1007/978-1-4612-2032-9
1. Composite materials--Mathematical models. 1. Cherkaev,
Andrej. 1950- . II. Kohn, Robert V. III. Series.
TA418.9.C6T676 1997
620.1'18'015118--dc21 97-180
CIP
Printed on acid-free paper
© Springer Science+Business Media New York 1997
Originally published by Birkhăuser Boston in 1997
Softcover reprint ofthe hardcover lst edition 1997
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ISBN 978-1-4612-7390-5
Reformatted and typeset from disks by Texniques, Inc., Boston, MA.
987 6 5 4 3 2 1
Contents
Introduction
Andrej V. Cherkaev and Robert V. Kohn .............................. vii
1. On the Control of Coefficients in Partial Differential Equations
Franr;ois Murat and Luc Tartar ........................................ 1
2. Estimations of Homogenized Coefficients
Luc Tartar ............................................................ 9
3. H-Convergence
Franr;ois Murat and Luc Tartar ....................................... 21
4. A Strange Term Coming from Nowhere
Doina Cioranescu and Franr;ois Murat ................................ 45
5. Design of Composite Plates of Extremal Rigidity
L. V. Gibiansky and A. V. Cherkaev .................................... 95
6. Calculus of Variations and Homogenization
Franr;ois Murat and Luc Tartar ...................................... 139
7. Effective Characteristics of Composite Materials
and the Optimal Design of Structural Elements
K.A. Lurie and A. V. Cherkaev ...................................... 175
Appendix: Local distribution of a MHD Channel
in the case of optimal distribution of the
resistivity of the working medium
K. A. Lurie and T. Y. Simkina ..................................... 259
8. Microstructures of Composites of Extremal Rigidity
and Exact Bounds on the Associated Energy Density
L. V. Gibiansky and A. V. Cherkaev .................................. 273
vii
Introduction
Andrej V. Cherkaev and Robert V. Kohn
In the past twenty years we have witnessed a renaissance of theoretical
work on the macroscopic behavior of microscopically heterogeneous mate
rials. 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 "com
pensated 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, circu
lating 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 trans
lation. 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 of 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.
ix
viii A. V. Cherkaev and R. V. Kohn
1. On the control of coefficients in partial differential equations by F. Mu
rat and 1. Tartar. This is a translation of a paper published in Lec
ture Notes in Economics and Mathematical Systems 107 (Control The
ory, Numerical Methods, and Computer System Modelling), Springer
Verlag, 1975, 420-426. The translation was arranged by Doina Cio
ranescu with cooperation from the authors. The article is reproduced
with permission from Springer-Verlag. It represents some of the ear
liest 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 the paper by Lurie and Cherkaev, this volume).
2. Estimation of homogenized coefficients by L. Tartar. This is a transla
tion of a paper published in Lecture Notes in Mathematics 704 (Com
puting Methods in Applied Sciences and Engineering), Springer-Verlag,
1977, 364-373. The translation was arranged by Doina Cioranescu
with cooperation from the authors. It is reproduced in this form with
the permission of Springer-Verlag. This is one of the earliest applica
tions of weak convergence as a tool for bounding the effective moduli
of composite materials.
3. H-Convergence by F. Murat and L. Tartar. This is a translation of
notes that have circulated in mimeographed form since 1978, based on
lectures given by F. Murat at the University of Algiers. The transla
tion was done by Gilles Francfort with cooperation from the authors.
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 spe
cialized to the self-adjoint case it becomes equivalent to G-convergence.
Treatments of G-convergence can be found elsewhere, including the
books of Jikov et al. [1] and Dal Maso [2]. 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.
4. A strange term coming from nowhere by F. Murat and D. Cioranescu.
This is a translation of a paper published in Nonlinear Partial Differ
ential Equations and their Applications - College de France Seminar,
H. Brezis, J.-L. Lions, and D. Cioranescu eds., Pitman, 1982, volume 2
pp. 98-138 and volume 3, pp. 154-178. The translation was arranged
by D. Cioranescu with cooperation from F. Murat. It is reproduced
in this form with the permission of Pitman. 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
Introduction ix
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, for example,
Dal Maso and Garroni [3], and on problems involving Stokes flow, for
example, Allaire [4] and Hornung et al. [5]. For work on structural
optimization in problems of this type see Buttazzo and Dal Maso [6]
and Sverak [7].
5. Design of composite plates of extremal rigidity by L. Gibiansky and
A. Cherkaev. This is a translation of a Russian article first circulated
as Ioffe Physico-Technical Institute Publication 914, Leningrad, 1984.
The translation was done by Yury Gonorovsky and Leonid Gibiansky.
This work provides an early application of homogenization to a prob
lem 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 re
lated work see Kohn and Strang [8], Allaire and Kohn [9, 10], and
especially the book of Bendsoe [11] and the review paper by Rozvany
et al. [12] which have extensive bibliographies.
6. Calculus of variations and homogenization by F. Murat and L. Tar
tar. This is a translation of a paper published in Les Methodes de
l'Homogeneisation: Theorie et Applications en Physique, Eyrolles, 1985,
pp. 319-369. The translation was done by Anca-Maria Toader with
cooperation from the authors. This work is reproduced in this form
with the permission of Eyrolles. It presents a very complete treat
ment 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 "0-
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 by Bendsoe [11], and also the article
of Kohn and Strang [8].
7. Effective characteristics of composite materials and the optimal design
of structural elements by K.A. Lurie and A. V. Cherkaev. This is a
translation of the Russian article published in Uspekhi Mekhaniki 9:2,
1986,3-81. However a portion of the Russian version has been omitted
to avoid overlap with the paper by Gibiansky and Cherkaev, (Chap
ter 5 this volume.) The translation was done by Natalia Alexeeva
with cooperation from the authors. The article is reproduced in this
form with the permission of Uspekhi Mekhaniki. The paper presents a
comprehensive review of work by the Russian community on homoge
nization methods applied to structural optimization and can be viewed
as a theoretical introduction to optimal design problems illustrated by