Rheological models for large deformations of elastic–viscoplastic materials

Four different rheological models for large deformations of elastic–viscoplastic materials are considered. Each model is hyperelastic with the stress being determined by derivatives of a strain energy function. Model 1 is based on a nonlinear evolution equation for a unimodular elastic distortional deformation tensor and is considered to be the reference model since it includes the special case of a nonlinear isotropic elastic solid. Models 2–4 are based on simplified evolution equations which depend linearly on a deviatoric elastic strain tensor. Equations of this type have been used to model non-Newtonian fluids. The main objective of this paper is to examine the influence of approximations introduced to obtain these simplified evolution equations. Examples of steady-state simple shear and steady-state isochoric extension relative to a rotating coordinate system are used to study the response of the simplified models. In particular, it is shown that these simplified models predict the unphysical result that the shear stress decreases towards zero for large values of the rate of deformation in simple shear.