Analysis of Hooke-like isotropic hypoelasticity models in view of applications in FE formulations

Abstract

This paper presents an analysis of the constitutive relations of Hooke-like isotropic hypoelastic material models in Lagrangian and Eulerian forms generated using corotational stress rates with associated spin tensors from the family of material spin tensors. Explicit expressions were obtained for the Lagrangian and Eulerian tangent stiffness tensors for the hypoelastic materials considered. The main result of this study is a proof that these fourth-order tensors have full symmetry only for material models generated using two corotational stress rates: the Zaremba–Jaumann and the logarithmic ones. In the latter case, the Hooke-like isotropic hypoelastic material is simultaneously the Hencky isotropic hyperelastic material. For the material models considered, basis-free expressions for the material and spatial tangent stiffness tensors are obtained that can be implemented in FE codes. In particular, new basis-free expressions are derived for the tangent stiffness (elasticity) tensors for the Hencky isotropic hyperelastic material model.