It is not my intention to present a treatise of elasticity in the follow ing pages. The size of the volume would not permit it, and, on the other hand, there are already excellent treatises. Instead, my aim is to develop some subjects not considered in the best known treatises of elasticity but nevertheless basic, either from the physical or the analytical point of view, if one is to establish a complete theory of elasticity. The material presented here is taken from original papers, generally very recent, and concerning, often, open questions still being studied by mathematicians. Most of the problems are from the theory of finite deformations [non-linear theory], but a part of this book concerns the theory of small deformations [linear theory], partly for its interest in many practical questions and partly because the analytical study of the theory of finite strain may be based on the infinitesimal one.

Elasticity theory is a classical discipline. The mathematical theory of elasticity in mechanics, especially the linearized theory, is quite mature, and is one of the foundations of several engineering sciences. In the last twenty years, there has been significant progress in several areas closely related to this classical field, this applies in particular to the following two areas. First, progress has been made in numerical methods, especially the development of the finite element method. The finite element method, which was independently created and developed in different ways by sci entists both in China and in the West, is a kind of systematic and modern numerical method for solving partial differential equations, especially el liptic equations. Experience has shown that the finite element method is efficient enough to solve problems in an extremely wide range of applica tions of elastic mechanics. In particular, the finite element method is very suitable for highly complicated problems. One of the authors (Feng) of this book had the good fortune to participate in the work of creating and establishing the theoretical basis of the finite element method. He thought in the early sixties that the method could be used to solve computational problems of solid mechanics by computers. Later practice justified and still continues to justify this point of view. The authors believe that it is now time to include the finite element method as an important part of the content of a textbook of modern elastic mechanics.

The book acquaints the reader with the basic concepts and relations of elasticity and plasticity, and also with the contemporary state of the theory, covering such aspects as the nonlinear models of elasto-plastic bodies and of large deflections of plates, unilateral boundary value problems, variational principles, the finite element method, and so on.

This book provides the general reader with an introduction to mathematical elasticity, by means of general concepts in classic mechanics, and models for elastic springs, strings, rods, beams and membranes. Functional analysis is also used to explore more general boundary value problems for three-dimensional elastic bodies, where the reader is provided, for each problem considered, a description of the deformation; the equilibrium in terms of stresses; the constitutive equation; the equilibrium equation in terms of displacements; formulation of boundary value problems; and variational principles, generalized solutions and conditions for solvability.Introduction to Mathematical Elasticity will also be of essential reference to engineers specializing in elasticity, and to mathematicians working on abstract formulations of the related boundary value problems.

This volume is a thorough introduction to contemporary research in elasticity, and may be used as a working textbook at the graduate level for courses in pure or applied mathematics or in continuum mechanics. It provides a thorough description (with emphasis on the nonlinear aspects) of the two competing mathematical models of three-dimensional elasticity, together with a mathematical analysis of these models. The book is as self-contained as possible.

TO THE FIRST ENGLISH EDITION. In preparing this translation, I have taken the liberty of including footnotes in the main text or inserting them in small type at the appropriate places. I have also corrected minor misprints without special mention .. The Chapters and Sections of the original text have been called Parts and Chapters respectively, where the latter have been numbered consecutively. The subject index was not contained in the Russian original and the authors' index represents an extension of the original list of references. In this way the reader should be able to find quickly the pages on which anyone reference is discussed. The transliteration problem has been overcome by printing the names of Russian authors and journals also in Russian type. While preparing this translation in the first place for my own informa tion, the knowledge that it would also become accessible to a large circle of readers has made the effort doubly worthwhile. I feel sure that the reader will share with me in my admiration for the simplicity and lucidity of presentation.