On compressible materials capable of sustaining axisymmetric shear deformations. Part 3: Helical shear of isotropic hyperelastic materials

A helical shear deformation is a composition of non-universal, axisymmetric, anti-plane shear and rotational shear deformations, shear states that are separately controllable only in special kinds of compressible and incompressible, homogeneous and isotropic hyperelastic materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the anti-plane and rotational shear displacement functions. For compressible materials, however, these shear deformations may not be separately possible in the same specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. We have recently presented simple algebraic conditions necessary and sufficient in order that both anti-plane shear and rotational shear deformations may be separately possible in the same material subclass. In this paper, under the same physical condition that the shear response function be positive, we present an essentially algebraic condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining controllable, helical shear deformations. It is then proved that helical shear deformations are possible in a specified hyperelastic material if and only if that material can separately sustain both axisymmetric, anti-plane shear and rotational shear deformations. The simplicity of the result in applications is illustrated in a few examples.