On compressible materials capable of sustaining axisymmetric shear deformations. Part 2: rotational shear of isotropic hyperelastic materials

Abstract

It is well known that rotational shear deformations, though not universal, are controllable in specific kinds of compressible and incompressible materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the rotational shear displacement function. For compressible materials, however, rotational shear deformations may not be possible for a specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. Polignone and Horgan have recently derived two conditions in the form of nonlinear ordinary differential equations necessary for hyperelastic materials to sustain rotational shear deformations. In the present paper, under the condition that the shear response function be positive, we present a single, essentially algebraic, condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining rotational shear deformations by application of surface tractions alone. Several examples illustrate the simplicity of the result in applications. We have recently presented similar kinds of results on axisymmetric, anti-plane shear deformations. With the aid of an auxiliary necessary condition on the strain energy function, here we present a simple necessary and sufficient condition for which both anti-plane shear and rotational shear deformations are separately possible in the same material subclass. These algebraic conditions are illustrated in several examples.