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

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... ve effects whose analogues in more elaborate deformations such as plane strain may be much less 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 [1] for incompressible elastic materials. (See Section 2.20 of [4] for a summary of the analysis in [1].) 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.