Journal of Elasticity, vol. 3, no 3, September 1973 Noordhoff International Publishing - Leyden Printed in the Netherlands

Book review S. Fliigge, Handbueh der Physik (Encyclopedia of Physics). Gruppe 3: Mechanisches und thermisches Verhalten der Materie. Band 6a, Teil 2.

Festk~rpermechanik 2 (Mechanics of Solids 2) Edited by C. Truesdell, The Johns Hopkins University, Md., USA. Springer-Verlag, Berlin/Heidelberg/New York, 1972. 25 Fig. X, 745. Cloth D M 288.--; US $ 91.30. The subjects treated here have their roots in elasticity theory in the classical sense, but the style and content o f the treatment are thoroughly modern. This is not to say that the authors have merely dressed classical subject matter in the raiment o f modern notation and terminology. Nor do the authors merely collect and survey the researches of the past few decades into the foundations of certain well defined areas of solid mechanics. The articles in this volume are complete, tightly knit developments o f the underlying theory of each author's subject. What strikes the reader is the realization that virtually all the treatment of this underlying theory is taken from contemporary research literature. The lead article, nearly 300 pages by Professor Morton E. Gurtin entitled "The Linear Theory of Elasticity", is a careful and thorough exposition of the foundations of the theory. The development is logical and selfcontained with the uniformly high level of rigor consistent with modern taste in mathematical continuum physics. After a brief introductory chapter and another on mathematical preliminaries, the work is organized into three main chapters: Formulation of the linear theory, Elastostatics, and Elastodynamics. The reader expecting to find clean well motivated treatments of the classical problems of elasticity, or an extensive development o f methods for solving plane and three dimensional problems, will have to look elsewhere. What he will find here is a development of the basic theory which incorporates the significant results of relatively recent research on fundamental questions of the theory itself. Professor Gurtin has managed to organize the development in a natural and unforced manner without sacrificing continuity, so that most of the discussion is devoted to topics which are not well treated in conventional books. The following article by Professor Donald E. Carlson is entitled "Linear Thermoelasticity" and should properly be considered an extension of the first. A successful effort was made to achieve continuity in style and type of content so that the preceding general comments apply to this work as well. Furthermore, this permitted an unusually wide coverage in less than 50 pages, including a concise chapter setting the foundations of linear thermoelasticity very neatly in the context of general continuttm thermodynamics. Two short articles (about 40 pages each) by Professor Gaetano Fichera are entitled "Existence Theorems in Elasticity" and "Boundary Value Problems of Elasticity with Unilateral Constraints". These are remarkable in the depth of coverage and completeness of discussion achieved in so brief a treatment. The material constitutes a substantial portion of a modern course in the theory and treatment of partial differential equations. Of necessity the style is compact and the text densely written so that the work would be of questionable utility to a reader not already possessing some familiarity with functional analysis techniques in differential equations. Nevertheless these articles constitute a self-contained and fairly complete treatment of existence theorems for large classes o f problems arising in elasticity theory. The pair of articles closes with a brief historical sketch o f existence theory for elasticity problems. During the past decade the foundations of shell theory have been recast on an alternative basis. A significant literature has emerged in which the shell is regarded as a two dimensional Cosserat continuum in contrast to the conventional view of a shell as a three dimensional simple continuum whose thickness is relatively small. In just over 200 pages entitled "The Theory of Shells and Plates" Professor P. M. Naghdi has very skillfully interwoven this direct approach and the conventional one into a complete dual development of the theory indicating contrasts and correspondences as the development proceeds. The pace o f this article is somewhat more deliberate than that of the other articles in this volume, but this is welcome and results in a more readable text. As with the preceding articles the treatment is logical, consistent and complete with a very large part of the content reflecting the findings of relatively recent research. The final article by Professor Stuart S. Antman is entitled "The Theory of Rods". The treatment is abstract and reflects the infiuence that modern mathematical researches in continuum mechanics have had on the foundations o f special theories o f solid mechanics. A very general mathematical structure is introduced to represent rod approximations, and rational procedures for systematically constructing rod theories are developed.

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A brief discussion of the direct approach, regarding the rod as a Cosserat curve, is also given. The article closes with a chapter on problems for nonlinearly elastic rods in which a major effort is the proof of a global existence theorem. Although the volume is a collection o f individual articles, a common theme unites them. The articles are concerned solely with the development and elucidation of theory. Applications and solution methods are absent and even examples are scarce. But the theoretical questions confronted and their resolution pretty well survey contemporary research efforts into the foundations o f particular disciplines in solid mechanics. In this connection it should be noted that this book is only one part, the second, of the encyclopedia volume on solid mechanics. Due for publication by the time this review appears are part 1 on experimental foundations of solid mechanics, and part 3 containing articles on linear viscoelasticity, plasticity, elastic stability, and growth and decay of waves. The contents of this book argue against its permanence, since the lesson seems to be that current research and evolving taste in method and style will continue to change how we view the foundations of theory. Yet the unique value of this volume is that it stands as a relatively complete statement o f the current state of these foundations. Finally, it would be difficult not to comment on the price of this book which must be regarded as high by any reasonable standard. The publisher has done his usual excellent productior~ job for this series. The variety o f symbols and type permit the authors to construct notation schemes which are explicit and clear, minimizing ambiguity without becoming excessively cumbersome. Nevertheless, because o f its price, it is hard to imagine that the book will achieve the wide distribution it deserves outside of large libraries. Morris Stern Department of Aerospace Engineering and Engineering Meohanies The University of Texas at Austin, USA