Extension and torsion of incompressible non-linearly elastic solid circular cylinders

The problem of torsion superimposed on axial extension of a solid circular cylinder composed of an incompressible isotropic hyperelastic material has been extensively investigated in the literature on non-linear elasticity following the pioneering theoretical investigations of Rivlin. The classical results of Rivlin were given in terms of a general strain-energy density that depends on the principal invariants of the Cauchy-Green deformation tensor. Here we reformulate these results in terms of a general strain-energy density that depends on alternative invariants, namely the invariants of the stretch tensor. Such an approach was carried out in a paper in this journal by Rivlin for the special case of pure torsion. As was remarked there, this is a relatively straightforward procedure and is a viable alternative to development of the theory from first principles in terms of the invariants of the stretch tensor. As an illustrative example, we consider the well-known Varga model for incompressible materials. For this material model, it is shown that the stretched circular cylinder always tends to further elongate on twisting. The special case of pure torsion is also briefly considered. The resultant axial force necessary to maintain pure torsion is compressive for the Varga model. In the absence of such a force, the bar tends to elongate on twisting, reflecting the celebrated Poynting effect.