Synopses & Reviews
The object of this book is the systematic exposition of recent work of the author's on boundary value problems of finite elasticity. These results concern an n-dimensional generalization of the three-dimensional elasticity which, aside from leading to a great many interesting mathematical situations, often shed light on certain aspects of the three-dimensional case. The book begins with a brief introduction to some general concepts, in order to show how the boundary value problems studied in the text arise. This is followed by the development of some technical material needed in the rest of the book. Subsequent chapters are devoted to obtaining theorems of existence, uniqueness and analytic dependence on the load, near special deformations for boundary value problems of place, and traction in finite elastostatics.
Synopsis
In this book I present, in a systematic form, some local theorems on existence, uniqueness, and analytic dependence on the load, which I have recently obtained for some types of boundary value problems of finite elasticity. Actually, these results concern an n-dimensional (n 1) formal generalization of three-dimensional elasticity. Such a generalization, be- sides being quite spontaneous, allows us to consider a great many inter- esting mathematical situations, and sometimes allows us to clarify certain aspects of the three-dimensional case. Part of the matter presented is unpublished; other arguments have been only partially published and in lesser generality. Note that I concentrate on simultaneous local existence and uniqueness; thus, I do not deal with the more general theory of exis- tence. Moreover, I restrict my discussion to compressible elastic bodies and I do not treat unilateral problems. The clever use of the inverse function theorem in finite elasticity made by STOPPELLI 1954, 1957a, 1957b], in order to obtain local existence and uniqueness for the traction problem in hyperelasticity under dead loads, inspired many of the ideas which led to this monograph. Chapter I aims to give a very brief introduction to some general concepts in the mathematical theory of elasticity, in order to show how the boundary value problems studied in the sequel arise. Chapter II is very technical; it supplies the framework for all sub- sequent developments.
Table of Contents
Contents: A Brief Introduction to Some General Concepts in Elasticity.- Composition Operators in Sobolev and Schauder Spaces. Theorems on Continuity, Differentiability and Analyticity.- Dirichlet and Neumann Boundary Problems in Linearized Elastostatics. Existence, Uniqueness and Regularity.- Boundary Problems of Place in Finite Elastostatics.- Boundary Problems of Traction in Finite Elastostatics. An Abstact Method. The Special Case of Dead Loads.- Boundary Problems of Pressure Type in Finite Elastostatics.- Appendices.- Bibliography.- Index of Notations.- Index.