Irreducible decompositions of the elasticity tensor under the linear and orthogonal groups and their physical consequences

TL;DR

This paper provides a detailed mathematical decomposition of the elasticity tensor in linear elasticity, revealing its physical implications such as the Cauchy relations and wave behavior.

Contribution

It introduces a new irreducible decomposition of the elasticity tensor under both linear and orthogonal groups, clarifying its physical significance.

Findings

Cauchy relations correspond to A=0

Longitudinal waves relate to S, transverse waves to A

Finer tensor decompositions clarify physical properties

Abstract

We study properties of the fourth rank elasticity tensor C within linear elasticity theory. First C is irreducibly decomposed under the linear group into a "Cauchy piece" S (with 15 independent components) and a "non-Cauchy piece" A (with 6 independent components). Subsequently, we turn to the physically relevant orthogonal group, thereby using the metric. We find the finer decomposition of S into pieces with 9+5+1 and of A into those with 5+1 independent components. Some reducible decompositions, discussed earlier by numerous authors, are shown to be inconsistent. --- Several physical consequences are discussed. The Cauchy relations are shown to correspond to A=0. Longitudinal and transverse sound waves are basically related by S and A, respectively.