Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

Abstract

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.