Cylindrical axial shear generated by an applied Piola–Kirchhoff stress

Abstract

A novel approach to the classical problem of axial shear of isotropic incompressible non-linearly elastic materials is proposed here. It is assumed that only the axial first Piola–Kirchhoff shear stress components are not identically zero, instead of the usual semi-inverse assumption on the displacement field of a typical particle. The form of the displacement consistent with this stress formulation is then obtained, assuming that the so-called Empirical Inequalities hold. The classical displacement formulation of axial shear is derived for the class of generalised neo-Hookean materials. The absence of a normal stress effect is noted. The difficulties in solving the corresponding problem in the context of Cauchy stress are highlighted.