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

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.