Civil Engineering Association

Product Details: Hardcover
Publisher: Oxford; 1st edition (1960)
ASIN: B000XHQ0KY

CONTENTS
I. THE STRAIN ENERGY FUNCTION
II. GENERAL SOLUTIONS OF PROBLEMS
III. FINITE PLANE STRAIN
IV. THEORY OF ELASTIC MEMBRANES
V. THEORY OF SUCCESSIVE APPROXIMATIONS
VI. APPROXIMATE SOLUTION OF TWO-DIMENSIONAL PROBLEMS
VII. REINFORCEMENT BY INEXTENSIBLE CORDS
VIII. THERMODYNAMICS OF DEFORMATION
IX. STABILITY
X. EXPERIMENTAL APPLICATIONS
XI. RHEOLOGICAL EQUATIONS OF STATE
APPENDIX. REDUCTION OF A MATRIX POLYNOMIAL OF TWO MATRICES
AUTHOR INDEX
SUBJECT INDEX



PREFACE
THE rapid development of the theory of large elastic deformations during the past decade was motivated in the first instance by the increasing importance of rubber in industrial applications and the need to get a clear picture of its mechanical properties. For this reason, the early work in the field was concentrated largely on ideally elastic, isotropic, incompressible materials with vulcanized rubber as a typical example of a substance of this kind. The initial success in solving a number of simple problems led to corresponding investigations for compressible and aeolotropic bodies, while the difficulty of solving all but the simplest problems in complete generality has led to the development of approximation procedures.
The general theory of elasticity for finite deformations has been given by Green and Zerna in the book Theoretical Elasticity (subsequently referred to as T.E.) published by the Clarendon Press (1954). This theory is presented in compact form with the aid of tensor notation and the results are applied to solve a numbe of special problems, mainly for isotropic incompressible materials. In the present book attention is concentrated on subsequent development. A summary of the essential basic formulae of the finite theory is given in Chapter I, again using the tensor notation of T.E.; for proofs of these formulae and an exposition of the elementary tensor analysis required the reader is referred to the earlier book. Chapter I then proceeds to an examination of the form of the strain energy function for the basic crystal classes together with the development of the stress-strain relations for orthotropic and transversely isotropic materials. Curvilinear aeolotropy and materials subject to constraints are also examined.
Chapter II contains some of the exact solutions of the finite theory, mainly for aeolotropic bodies. The earlier part of this chapter is concerned with cylindrically symmetrical problems and the flexure problem in which the results apply for a perfectly general form of strain energy. In later sections solutions are derived using the restricted Mooney form of strain energy for rubber-like materials.
In Chapter III the theory of plane strain is developed using tensor notation initially and specializing subsequently to give a complex variable formulation. A number of special problems are examined using this theory. Chapter IV deals with plane stress and the membrane theory of thin shells. In applications attention is confined to axially symmetrical problems which involve only one independent variable. For some of these problems, the membrane equations can be solved analytically; for others, simple numerical methods of integration are available.
A method of successive approximation is developed in Chapter V and illustrated by simple examples. Chapter VI deals with the application of this approximation method to two-dimensional problems, attention being focused largely upon plane strain. The equations derived resemble those of the classical infinitesimal theory, and a complex variable formulation permits the use of the powerful techniques evolved for the classical theory by Muskhelishvili.
The reinforcement of elastic materials by systems of thin flexible inextensible cords is considered in Chapter VII, the main problems examined being either two-dimensional or those possessing cylindrical symmetry. Such problems arise in many industrial applications of rubber. Chapters VIII and IX are concerned with the theories of thermoelasticity and elastic stability respectively for finite deformations. A description of some of the more important physical experiments which have been carried out on vulcanized rubber is contained in Chapter X. These experiments illustrate how a completely general theory, such as that developed in the earlier chapters, may be applied to evaluate the mechanical properties of real materials.
In the last chapter of the book an account is given of some of the recent developments in non-linear continuum mechanics. The theory may be regarded as a natural extension of that of finite elasticity when the assumption of ideal elasticity is relaxed. Owing to the rapid advances now being made in this field attention is confined to a consideration of the kinematics of deformation and some of the simpler forms of the stress deformation relations.
Many of the developments described in this book have been due to the initiative of Professor R. S. Rivlin, and the debt which the subject owes him is apparent from the references in the text. The authors record with pleasure their own indebtedness to him for many stimulating discussions and contacts during the past few years. Acknowledgement should also be made of the encouragement and financial support given to a number of workers in the field by the British Rubber Producers' Research Association with whom Professor Rivlin and one of the present authors (J. E. A.) were at one time associated.

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