Compressible, isotropic hyperelastic materials capable of sustaining axisymmetric, antiplane shear deformations

Antiplane shear deformations are not controllable in every compressible or incompressible, homogeneous and isotropic elastic material. On the other hand, for both compressible and incompressible materials, Knowles has presented conditions on the strain energy function necessary and sufficient to determine whether a specified homogeneous and isotropic hyperelastic material having a monotone shear stress response function can sustain general, non-axisymmetric, antiplane sheer deformations. For compressible materials, he finds two conditions. Under a weaker condition on the shear response function, Jiang and Beatty derived a single necessary and sufficient condition on the strain energy function in order that the material may support axisymmetric, antiplane shear deformations; and they proved that while Knowles's conditions for general antiplane sheer deformations suffice, they are not necessary for a hyperelastic material to sustain axisymmetric, antiplane shear deformations. Here Ne present a somewhat easier proof of the theorem. The simplicity of the result in applications is illustrated in some examples.