On the role of large elastic deformation in fracture initiation -- the cases of plane stress and plane strain

Abstract

In recent years there has been an increase in interest in the problems of strength and deformation of highly elastic materials. A number of simpler cases which have been solved indicate some markedly different characteristics from their counterparts in infinitesimal elasticity. References 1 and 2 discuss these cases in some detail. It seems natural therefore to ask whether in the problem areas of fracture in which the application of the infinitesimal theory of elasticity has met with some success in brittle elastic materials an equally marked difference in behavior would result if the possibility of large strains were included in the analyses. The question is more easily asked than answered since the inclusion of the possibility of large strains often complicates the problem mathematically many fold. Only recently, Williams and Schapery considered large strains in the instability analysis of the spherical cavity. One of their conclusions is that the strain at the cavity increases without bound as the cavity becomes vanishingly small, at a finite critical tensile loading at some large radius. The results of Williams and Schapery thus reinforce the suspicion held by many that large strain would play an important role in the study of fracture mechanics of highly elastic materials. The present paper is devoted to the analogous problem of a plane sheet containing a circular hole while at some large radius an axially symmetric tensile loading p per unit length of the stretched boundary is applied. Notations of references 2 and 3 will be generally adopted and, their explanations will be provided whenever necessary for clarity.