Table of Contents

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 order.

1. On the control of coefficients in partial differential equations by F. Murat and L. Tartar. This is a translation of a paper published in Lecture Notes in Economics and Mathematical Systems 107 (Control Theory, Numerical Methods, and Computer System Modelling), Springer-Verlag, 1975, 420-426. The translation was arranged by Doina Cioranescu with cooperation from the authors. The article is reproduced with permission from Springer-Verlag. It 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).
2. Estimation of homogenized coefficients by L. Tartar. This is a translation of a paper published in Lecture Notes in Mathematics 704 (Computing 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 applications 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 which have circulated in mimeographed form since 1978, based on lectures given by F. Murat at the University of Algiers. The translation 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 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 [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 Differential 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 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 [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].
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 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 [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.
6. Calculus of variations and homogenization by F. Murat and L. Tartar. 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 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 [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 chapter 5 of this book. 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 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.
8. Microstructures of composites of extremal rigidity and exact bounds on the associated energy density by L.V. Gibiansky and A.V. Cherkaev. This is a translation of the Russian article first circulated as Ioffe Physico-Technical Institute Publication 1115, Leningrad, 1987. The translation was done by Yury Gonorovsky with cooperation from the authors. 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.

We considered, at one point, preparing a comprehensive review of mathematical work related to the articles in this volume. But this seemed a Herculean task, so we abandoned it. One can take pleasure in the idea that the subject has become difficult to review: it means the foundation represented by this book is occupied by a rich and multifaceted scientific edifice. The references given above should permit interested readers to explore the literature on their own.

We express our gratitude to the many people who have helped assemble this volume:

-- the authors, who devoted much energy to proof-reading and improving the translations;

-- the translators, who hopefully found their tasks interesting, because they certainly didn't get paid very much; and

-- Edwin Beschler and his colleagues at Birkhauser-Boston, who have shown unlimited patience as the target date for completion of the volume was reset time and again.

The subject is far from completion. Much remains to be understood, in many different directions, concerning effective moduli, structural optimization, multiwell variational problems, and coherent phase transitions. Some specific areas of current interest include bounds for multicomponent composites; bone remodelling; polycrystal plasticity, and ``practical'' suboptimal design. We hope this volume will accelerate progress by helping fill in the foundations of the field.

Andrej V. Cherkaev
Robert V. Kohn

1. Jikov, V.V., Kozlov, S.M., and Oleinik, O.A. (1994) Homogenization of Differential Operators and Integral Functionals. Springer-Verlag.
2. Dal Maso, G. (1993) An Introduction to $\Gamma$-convergence. Birkhauser, 1993.
3. Dal Maso, G. and Garroni, A. (1994) New results on the asymptotic behavior of Dirichlet problems in perforated domains. Math. Meth. Appl. Sci. 3, 373-407.
4. Allaire, G. (1991) Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Rat. Mech. Anal. 113, 209-259.
5. Hornung, U. et al. (in press) Homogenization and Porous Media. Springer-Verlag.
6. Butazzo, G. and Dal Maso, G. (1991) Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23, 17-49.
7. Sverak, V. (1993) On optimal shape design. J. Math. Pures Appl. 72 (1993) 537-551.
8. Kohn, R.V. and Strang, G. (1986) Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39 pp. 113 - 137, 139 - 182, 353 - 377.
9. Allaire, G. and Kohn, R.V. (1993) Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Quart. Appl. Math. 51, 675-699.
10. Allaire, G. and Kohn, R.V. (1993) Optimal design for minimum weight and compliance in plane stress, Euro. J. Mechanics A/Solids 12, 839-878.
11. Bendsoe, M. P. (1995) Optimization of structural topology, shape, and material. Springer. Berlin ; NY
12. Rozvany G.I.N., Bendsoe M.P., and Kirsch U. (1994) Layout optimization of structures. Appl. Mech. Rev., vol. 48, No 2, pp. 41-119; Addendum: Appl. Mech. Rev., Vol. 49, No 1, p. 54.
13. Lurie, K. (1993). Applied Optimal Control, Plenum, New York.
14. Lurie, K.A. and Cherkaev, A. V. (1984) Exact estimates of conductivity of mixtures composed of two isotropical media taken in prescribed proportion, Proceedings of Royal Soc. of Edinburgh, {\bf A}, 99(P1-2), 71-87. First version: Phys.-Tech. Inst. Acad. Sci. USSR, Preprint 783, 1982, (in Russian).
15. Tartar, L. (1985) Estimations fines des coefficients homogeneises, Ennio de Giorgi's Colloquium, P.Kree,ed., Pitman Research Notes in Math. 125 , 168-187.
16. Lurie, K. and Cherkaev A. (1986) Exact estimates of conductivity of a binary mixture of isotropic compounds. Proceed. Roy. Soc. Edinburgh, sect.A, 1986, 104(P1-2), p.21-38. First version: Phys.-Tech. Inst. Acad. Sci. USSR, Preprint N.894, 1984 (in Russian).
17. Francfort, G. and Murat, F. (1991) Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal., 94, pp.301-307.
18. Gibiansky, L.V., and Milton, G. W. (1993) On the effective viscoelastic moduli of two-phase media: I. Rigorous bounds on the complex bulk modulus, Proc. R. Soc. Lond. A , {\bf 440}, 163-188.
19. Cherkaev A.V., and Gibiansky, L.V. (1992) The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites, Proceedings of Royal Society of Edinburgh 122A, pp.93-125.
20. Cherkaev, A.V. and Gibiansky, L.V. (1993) Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite, J. Mech. Phys. Solids {\bf 41}, 937-980.
21. Milton, G.W. (1990) On characterizing the set of possible effective tensors of composites: the variational method and the translation method. Comm. Pure Appl. Math. 43, 63-125.
22. Milton, G.W. (1990b) A brief review of the translation method for bounding effective elastic tensors of composites, Continuum Models and Discrete Systems, ed. by G. A. Maugin, vol. 1 , 60-74.
23. Milton, G.W. Effective moduli of composites: exact results and bounds. In preparation
24. Gibiansky, L.V. and S. Torquato, S. 1996 Rigorous link between the conductivity and elastic moduli of fiber-reinforced composite materials. Trans. Roy. Soc. London {\bf A} {\bf 452}, 253-283.
25. Nesi, V. and Milton, G.W. (1991). Polycrystalline configurations that maximize electrical resistivity. J. Mech. Phys. Solids 39, 525-542.
26. M. Avellaneda, A.V. Cherkaev, L.V. Gibiansky, G.W. Milton, M. Rudelson (1996) A complete characterization of the possible bulk and shear moduli of planar polycrystals. J. Mech. Phys. Solids, in press.
27. Milton, G.W. and Kohn, R.V. (1988) Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids , vol. 36, (1988), 597-629.
28. Kohn, R.V. and Lipton, R. Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials, Arch. Rational Mech. Anal. , 102, (1988), 331-350.
29. Allaire, G. and Kohn, R.V. (1993) Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, Quart. Appl. Math. 51, 643-674.
30. Allaire, G., Bonnetier, E., Francfort, G., and Jouve, F. (1996) Shape optimization by the homogenization method, Numer. Math., in press.
31. Cherkaev, A.V. and Palais, R. (1995) Optimal design of three-dimensional elastic structures. in: Proceedings of the First World Congress of Structural and Multidisciplinary Optimization, Olhoff, N. and Rozvany, G. eds., Pergamon , 201-206
32. Ball, J.M. and James, R.D. (1987) Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100, 13-52.
33. Ball, J.M. and James, R.D. (1992) Proposed experimental tests of a theory of fine microstructure and the two well problem, Phil. Trans. Roy. Soc. London 338A, 389-450.
34. Bhattacharya, K. and Kohn, R.V. (1996) Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials, Arch. Rat. Mech. Anal., in press.
35. Christensen, R.M. 1979 Mechanics of Composite Materials (Wiley-Interscience, New-York).
36. Nemat-Nasser, S. and Hori, M. 1993 Micromechanics: Overall Properties of Heterogeneous Materials , North Holland.