Incremental equations and propagation of small amplitude waves for isotropic incompressible elastic bodies. The case the Hencky strain tensor is a function of the Cauchy stress tensor

Abstract

For a relatively new class of constitutive equation for incompressible elastic bodies, wherein the Hencky strain tensor is a function of the Cauchy stress tensor, incremental equations are obtained, where it is assumed the presence of an initial time-independent stress that causes large deformations, to which a small time-dependent stress tensor is added. That stress tensor is assumed to cause a small (incremental) time-dependent deformation. For the case of infinite media the incremental equations are solved assuming travelling waves, and the speed of such waves is obtained for different problems considering initial homogeneous distributions of stresses and strains. Some numerical calculations of the speed of such waves are presented, considering a particular constitutive equation for rubber published recently in the literature. The speed of such small amplitude waves are compared with the predictions of the classical theory of nonlinear elasticity for similar problems.