Mathematical Elasticity: General Preface1

Philippe G. Ciarlet

This book, which comprises three volumes, is intended to be both a thorough introduction to contemporary research in elasticity and a working textbook at the graduate level for courses in pure or applied mathematics or in continuum mechanics.

During the past decades, elasticity has become the object of a considerable renewed interest, both in its physical foundations and in its mathematical theory. One reason behind this recent attention is that it has been increasingly acknowledged that the classical linear equations of elasticity, whose mathematical theory is now firmly established, have a limited range of applicability, outside of which they should be replaced by the genuine nonlinear equations that they in effect approximate.

Another reason, similar in its principle, is that the validity of the classical lower-dimensional equations, such as the two-dimensional von Kármán equations for nonlinearly elastic plates or the twodimensional Koiter equations for linearly elastic shells, is no longer left unquestioned. A need has been felt for a better assessment of their relation to the corresponding three-dimensional equations that they are supposed to “replace”.

Thanks to the ever increasing power of available computers, sophisticated mathematical models that were previously intractable by approximate methods are now amenable to numerical simulations. This is one more reason why these models should be established on firm grounds.

This book illustrates at length these recent trends, as shown by the main topics covered:

- A thorough description, with a pervading emphasis on the nonlinear aspects, of the two existing mathematical models of threedimensional elasticity, either as a boundary value problem consisting of a system of three quasilinear partial differential equations of the second order together with specific boundary conditions, or as a minimization problem for the associated energy over an ad hoc set of admissible deformations (Vol. I, Part A);

- A mathematical analysis of these models, comprising in particular complete proofs of all the available existence results, relying either on the implicit function theorem, or on the direct methods of the calculus of variations (Vol. I, Part B);

- A mathematical justification of the well-known two-dimensional linear Kirchhoff-Love theory of plates, by means of convergence theorems in H1 as the thickness of the plate approaches zero (Vol. II, Part B);

- Similar justifications of mathematical models of junctions in linearly elastic multi-structures and of linearly elastic shallow shells (Vol. II, Part A);

- A systematic derivation of two-dimensional plate models from nonlinear three-dimensional elasticity by means of the method of formal asymptotic expansions, which includes a justification of well-known plate models, such as the nonlinear Kirchhoff-Love theory and the von Kármán equations (Vol. II, Part B);

- A description of the large deformation, frame-indifferent, nonlinear membrane and flexural theories recently obtained by formal asymptotic expansions, the former being justified by a convergence theorem (Vol. II, Part B);

- A mathematical analysis of the two-dimensional plate equations, which includes in particular a review of the existence and regularity theorems in the nonlinear case, and an introduction to bifurcation theory (Vol. II, Part B);

- A mathematical justification by means of convergence theorems, in H1 or L2, of the two-dimensional flexural, membrane, and Koiter equations of a linearly elastic shell (Vol. III, Part A);

- A systematic derivation of the two-dimensional membrane and flexural equations of a nonlinearly elastic shell by means of the method of formal asymptotic expansions, the former being again justified by a convergence theorem (Vol. III, Part B).

Although the emphasis is definitely on the mathematical side, every effort has been made to keep the prerequisites, whether from mathematics or continuum mechanics, to a minimum, notably by making the book as largely self-contained as possible. The reading of the book only presupposes some familiarity with basic topics from analysis and functional analysis.

Naturally, frequent references are made to Vol. I in Vol. II, and to Vols. I and II in Vol. III. However, I have also tried to render each volume as self-contained as possible. In particular, all relevant notions from three-dimensional elasticity are (at least briefly) recalled wherever they are needed in Vols. II and III.

References are also made to Vol. I regarding various mathematical notions (properties of domains in n, differential calculus in normed vector spaces, the Rellich-Kondrasov theorem, weak lower semi-continuity, etc.). This is a mere convenience, reflecting that I also regard the three volumes as forming the same whole. I am otherwise well aware that Vol. I is neither a. text on analysis nor on functional analysis. Any reader interested in a deeper understanding of such notions should consult the more standard texts referred to in Vol. I.

Each volume is divided into consecutively numbered chapters. Chapter m contains an introduction, several sections numbered Sect. m.1, Sect. m.2, etc., and is concluded by a set of exercises. Within Sect.m,n theorems are consecutively numbered, as Thm. m.n − 1, Thm. m.n − 2, etc., and figures are likewise consecutively numbered, as Fig. m.n − 1, Fig. m.n − 2, etc. Remarks and formulas are not numbered. The end of the proof of a theorem, or the end of a remark,<pg > x"/>is indicated by the symbol ■ in the right margin. In Chapter m, exercises are numbered as Ex. m.1, Ex. m.2, etc.

All the important results are stated in the form of theorems (there are no lemmas, propositions, or corollaries), which therefore represent the core of the text. At the other extreme, the remarks are intended to point out some interpretations, extensions, counter-examples, relations with other results, that in principle can be skipped during a first reading; yet, they could be helpful for a better understanding of the material. When a term is defined, it is set in boldface if it is deemed important, or in italics otherwise. Terms that are only given a loose or intuitive meaning are put between quotation marks.

Special attention has been given to the notation, which so often has a distractive and depressing effect in a first encounter with elasticity. In particular, each volume begins with special sections, which the reader is urged to consult first, about the notations and the rules that have guided their choice. The same sections also review the main definitions and formulas that will be used throughout the text.

Complete proofs are generally given. In particular, whenever a mathematical result is of particular significance in elasticity, its proof has been included. More standard mathematical prerequisites are presented (usually without proofs) in special starred sections, scattered throughout the book according to the local needs. The proofs of some advanced, or more specialized, topics, are sometimes only sketched, notably in order to keep the length of each volume within reasonable limits; in this case, ad hoc references are always provided. These topics are assembled in special sections marked with the symbolb, usually at the end of a chapter.

Exercises of varying difficulty are included at the end of each chapter. Some are straightforward applications of, or complements to, the text; others, which are more challenging, are usually provided with hints or references.

This book would have never seen the light, had not I had the good fortune of having met, and worked with, many exceptional students and colleagues, who helped me over the past two decades decipher the arcane subtleties of mathematical elasticity; their names are listed in the preface to each volume. To all of them, my heartfelt thanks!

I am also particularly indebted to Arjen Sevenster, whose constant interest and understanding were an invaluable help in this seemingly endless enterprise!

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