Universal States of Finite Anti-Plane Shear: Ericksen's Problem in Miniature

Abstract

A solid body which in its undeformed state occupies a cylindrical region is said to undergo a deformation corresponding to anti-plane shear if each particle of the body is displaced parallel to the generators of the cylinder by an amount which is independent of the axial position of the particle. The displacement vector field thus has a nonvanishing component u only in the axial direction, and u is a function of position on a cross-section D of the cylinder. Problems involving such deformations are ordinarily simpler than those in which the displacement vector has a more elaborate character, and for this reason they often serve usefully as pilot problems for the analysis of qualitative effects, especially when nonlinearity is involved. (Examples may be found in [1] and in the references cited in [2].) The present paper is intended to illustrate anti-plane shear in its role as exemplar in the setting of finite elasticity theory and with particular reference to an issue which has come to be called Ericksen' s problem.