3R27. Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics. - GW Milton (Dept of Math, Univ of Utah, Salt Lake City UT). Cambridge UP, Cambridge, UK. 2002. 719 pp. ISBN 0-521-78125-6. $80.00.

AT Sawicki (Inst of Hydro-Eng, Koscierska 7, Gdansk-Oliwa, 80-953, Poland).

This is a book about the mathematical world of composites, where the electrical, thermal, magnetic, thermoelectric, mechanical, piezoelectric, poroelastic, and electromagnetic properties of these materials are described in detail. It is rather unusual to cover such a broad spectrum of difficult problems in a single volume, since most other books on composite materials are restricted to particular aspects of their behavior as, for example, mechanical properties, or even particular geometries as fiber-reinforced composites, etc.

For applied mathematicians, the theory of composites is the study of partial differential equations with rapid oscillations in their coefficients. These equations have similar mathematical structure for different physical phenomena, as those mentioned above, which enables unified treatment of various problems. Obviously, such an approach restricts applicability of various theories presented only to certain classes of materials and their properties, but the author is aware of it. For example, important problems dealing with plasticity and strength of composites, etc, are not analyzed, but respective references to other sources of information are provided.

In this book, the author presents the classical approach, where the effective (or homogenized) equations describe composites’ properties at the macroscopic level. These properties are related to the microstructure of composites and respective properties of the constituents. The book consists of 31 chapters, each containing several sections (from 3 to 14), and an extensive list of references. The first two chapters are of an introductory character. Chapters 3–9 deal with the exact results for effective moduli, and Chapter 10 discusses some approximations for estimating these moduli. In Chapter 11, some wave propagation problems are considered. Chapters 12–18 cover the general theory concerning effective tensors, including important variational principles. Chapters 19 and 20 provide some information on the so-called theory of Y-tensors which parallels that of effective tensors. Chapters 21–26 are devoted to variational methods for bounding effective tensors. Chapters 27–29 deal with the analytical properties of the effective tensors. The last two chapters discuss the set of effective tensors and bounding of effective moduli as a quasiconvexification problem.

In this reviewer’s opinion, this book is written mainly for applied mathematicians working in the field of composites. Engineers and other specialists may find it difficult to follow. Engineers would expect more practically oriented theoretical guidelines and experimental validations of sophisticated theories. But we cannot expect such definite treatment from a single textbook. This book shows the deep erudition of the author and is well organized, written with precision, and nicely edited. Particular parts have been discussed with many professionals, including well-known barons of the science of composites, for example, Zvi Hashin or John Willis, who, together with the author’s outstanding achievements in the field of mathematical analysis of composites, guarantee a high standard. Theory of Composites deserves its proper place in each library of applied mathematics and mechanics of materials, as it is an excellent addition to the existing literature on composites.