The Cauchy Relations in Linear Elasticity Theory

TL;DR

This paper explores the mathematical structure of the Cauchy relations in linear elasticity, revealing their group-theoretical basis through tensor decomposition.

Contribution

It introduces a group-theoretical perspective on the Cauchy relations by decomposing the elasticity tensor into irreducible components.

Findings

Decomposition of the elasticity tensor into two irreducible parts with 15 and 6 components.

Identification of the vanishing of the 6-component part as the Cauchy relations.

First recognition of the group-theoretical underpinning of the Cauchy relations.

Abstract

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The {\it vanishing} of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.