Nonlinear Cauchy Elasticity

TL;DR

This paper revisits nonlinear Cauchy elasticity, a less-explored form of elasticity where stresses relate directly to strains without an energy function, addressing its physical plausibility and recent applications.

Contribution

It provides a modern analysis of Cauchy elasticity, clarifying its theoretical foundations and relevance to active solids and other contemporary applications.

Findings

Cauchy elasticity allows non-zero work along closed strain paths.

It challenges the traditional energy-based framework of elasticity.

The paper discusses the physical plausibility of Cauchy materials.

Abstract

Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain-energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form.