Testing Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations

TL;DR

This paper investigates the behavior of Hooke-like isotropic hyper-/hypo-elastic models under finite simple shear deformations, revealing stress tensor symmetries and stress responses for different deformation types and material models.

Contribution

It provides a comprehensive analysis of stress responses in various isotropic elastic and hypoelastic models under finite simple shear, highlighting symmetries and specific stress behaviors.

Findings

Cauchy stress components are equal under LFSS and rotated RFSS deformations.

LFSS and RFSS lead to pure shear stresses in Hill's hyperelastic models.

Certain hypoelastic models exhibit stress behavior similar to hyperelastic models under shear.

Abstract

We test some Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations (cf., Thiel et al. Int. J. Non-linear Mech. 112: 57--72, 2019) and show that (1) the components of the Cauchy stress tensor for any Cauchy/Green isotropic elastic material under left finite simple shear (LFSS) deformation are equal to the components of the rotated Cauchy stress tensor for the same material under right finite simple shear (RFSS) deformation; (2) for any Hill's linear isotropic hyperelastic material model based on a symmetrically physical (SP) strain measure, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (3) for any two-power Ogden's isotropic hyperelastic material model based on a SP strain function, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (4) for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates under LFSS deformation, the behavior of the Cauchy stress tensor components as a function of the shear parameter is qualitatively similar to that for the same materials under simple shear deformation. In addition, we confirm the results of Lin (Lin R.C. ZAMP, 75: 191, 2024) showing that for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates without initial stresses under RFSS deformation, the Cauchy stress tensor components coincide with those for the Hencky isotropic hyperelastic material.