Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy-Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney-Rivlin model remains an open question which can be resolved only by large strain experiments.