Anti-plane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials

Abstract

The purpose of this research is to investigate the mechanical response in anti-plane shear of a class of incompressible isotropic hyperelastic materials for which the strain-energy density depends only on the first invariant of the strain tensor. Our concern is with the subclass of these materials that exhibits hardening at large deformations. In the molecular theory of elasticity, these models are called non-Gaussian, since they introduce a distribution function which is not Gaussian for the end-to-end distance of the polymeric chain. Two classes of these materials are considered, namely those with limiting chain extensibility and power-law models. The governing partial differential equation of equilibrium in anti-plane shear (a single second-order quasilinear partial differential equation) is obtained for specific constitutive models of the above type. Some solutions are derived using group symmetry reduction methods. Applications to crack problems and spatial decay of end effects are described. The results are applicable to rubber-like and biological materials.