A Note on Anti-plane Shear for Compressible Materials in Finite Elastostatics

This note gives a necessary and sufficient condition that a compressible, isotropic elastic material should admit non-trivial states of finite anti-plane shear. One of the simplest classes of deformations of solids is that of anti-plane shear, in which each particle of a cylindrical body undergoes a displacement parallel to the generators of the cylinder and independent of the axial position of the particle. Problems involving deformations of this kind are often helpful in the study of qualitative effects whose analogues in more elaborate deformations such as plane strain may be much Jess accessible because of technical complexities. Because of their utility in this respect, anti-plane shear fields have proved to be especially instructive in crack problems of the type arising in fracture mechanics. (Examples may be found in the references given in [6]). In the linearized theory of elasticity, non-trivial equilibrium fields of anti-plane shear are possible in the absence of body forces for any homogeneous, isotropic material. In contrast, the exact theory of anti-plane shear in finite elasticity is marred slightly by the fact that not all such materials have this property. This inference is implicit in the work of Adkins [I] for incompressible elastic materials. (See Section 2.20 of [4] for a summary of the analysis in [!].) Much of his study, which appears to have been the first detailed investigation of anti-plane shear in finite elastostatics, is limited to the so-called Mooney material which does admit non-trivial states of anti-plane shear and which is of especial interest in the theory of rubber elasticity. In [6] a necessary and sufficient condition that an incompressible, isotropic elastic material should sustain non-trivial states of finite anti-plane shear was given in terms of the strain energy density characteristic of the material. In this note an analogous condition is established for compressible materials. © Copyright Australian Mathematical Society 1977 Copyright. Apart from any fair dealing for scholarly purposes as permitted under the Copyright Act, no part of this JOURNAL may be reproduced by .any process without written permission from the Treasurer of the Australian Mathematical Society.