Large elastic deformations of isotropic materials. I. Fundamental concepts

Abstract

The mathematical theory of small elastic deformations has been developed to a high degree of sophistication on certain fundamental assumptions regarding the stress-strain relationships which are obeyed by the materials considered. The relationships taken are, in effect, a generalization of Hooke’s law— ut tensio, sic vis . The justification for these assumptions lies in the widespread agreement of experiment with the predictions of the theory and in the interpretation of the elastic behaviour of the materials in terms of their known structure. The same factors have contributed to our appreciation of the limitations of these assumptions. The principal problems, which the theory seeks to solve, are the determination of the deformation which a body undergoes and the distribution of stresses in it, when certain forces are applied to it, and when certain points of the body are subjected to specified displacements. These problems are always dealt with on the assumption that the generalization of Hooke’s law is obeyed by the material of the body and that the deformation is small, i.e. the change of length, in any linear element in the material, is small compared with the length of the element in the undeformed state. Apart from the fact that the generalization of Hooke’s law is obeyed accurately by a very wide range of materials, under a considerable variety of stress and strain conditions, it has the further advantage that it leads to a mathematically tractable theory.