On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

TL;DR

This paper explores the relationship between the Cauchy stress tensor and the logarithm of the left Cauchy-Green strain tensor in isotropic finite hyperelasticity, providing a new proof based on tensor exponential derivatives.

Contribution

It introduces a concise proof connecting the Cauchy stress tensor and strain tensor logarithm via a potential, using an explicit tensor exponential derivative formula.

Findings

Established a link between the Cauchy stress tensor and the strain tensor logarithm.

Provided a new proof method based on explicit tensor exponential derivatives.

Clarified the potential-driven constitutive law in hyperelasticity.

Abstract

If the constitutive law linking the second Piola-Kirchhoff stress tensor and the right Cauchy-Green strain tensor derives from a potential, then the Cauchy stress tensor and the logarithm of the left Cauchy-Green strain tensor are linked by a related potential. We give a new and concise proof which is based on an explicit formula expressing the derivative of the exponential of a tensor.