Families of Hooke-like isotropic hyperelastic material models and their rate formulations

We introduce a new (HHE) family of Hooke-like isotropic hyperelastic material models associated with the Doyle–Ericksen family of strain tensors with Eulerian forms of constitutive relations (CRs). This family contains the well-known Hencky and Simo–Pister (modified neo-Hookean) isotropic hyperelastic material models. The main feature of the new family of material models is that the rate counterparts of CRs for material models from this family are simultaneously CRs for Hooke-like isotropic hypoelastic-type material models based on Eulerian continuous strain-consistent convective stress rates. Models from the new family extend the only previously known Hooke-like isotropic hypoelastic material model based on the corotational logarithmic stress rate with Hooke-like hyperelastic counterpart to an infinite number of such models based on non-corotational stress rates. In addition, we develop unified Eulerian forms of CRs and specific potential strain energies for the known families of Hill (HLIH) and Kellermann–Attard (K–A) Hooke-like isotropic hyperelastic material models. For all three families (HHE, HLIH, and K–A) of material models, new explicit basis-free (eigenprojection based) expressions for the fourth-order elasticity tensors with full symmetry are obtained. Expressions for the components of the Cauchy stress tensor versus axial stretch in the simple elongation problem are derived, and their plots for integer values of the parameter \(n=\pm 2,\pm 1, 0\) generating material models from the families considered are constructed. From an analysis of these plots, it is concluded that the Simo–Pister isotropic hyperelastic material model is the best model among those considered.