Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.