Henky's elasticity model with the logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods

Abstract

Hencky's elasticity model is a finite strain elastic constitutive equation derived by replacing the infinitesimal strain measure in the classical strain-energy function of infinitesimal isotropic elasticity with Hencky's logarithmic strain measure. Anand [1,2] has demonstrated that, with only the two classical Lamé elastic constants measurable at infinitesimal strains, predictions of the just-mentioned simple model for a wide class of materials for moderately large deformations may be in better agreement with experimental data than other known finite elasticity models. The deformation modes considered in Anand's work are simple tension and compression, simple shear, and simple torsion and combined extension-torsion of solid cylinders, etc. Here, we indicate some remarkable properties of this Hencky model and, mainly, we investigate the large deformation responses of this model in torsion of cylindrical tubes and rods with free ends. It is pointed out that if in inelastic modeling, especially in modeling of metal plasticity, the widely-used hypoelastic formulation for the elastic rate of deformation is required to be exactly integrable to an elastic relation, as it should be, then the resulting elastic relation is just the Hencky model, and, further, this model is hyperelastic and the only possible one. In the main aspect, i.e. for the torsion of cylindrical tubes and rods with free ends, we derive explicit analytical solutions for the case of compressible small deformations and for the case of incompressible large deformations. The results derived show, in a clear and direct manner, second order effects, including the well-known Poynting effect regarding the length change in the axial direction. It is noticeable that, with only the material properties measurable at infinitesimal strains, the Hencky model can predict the just-mentioned second order effects, in particular the Poynting effect, and its predictions are in good accord with experiments reported in the literature.