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The Cauchy relations distinguish between rari- and multi-constant linearelasticity theories. These relations are treated in this paper in a form thatis invariant under two groups of transformations: indices permutation andgeneral linear transformations of the basis. The irreducible decompositioninduced by the permutation group is outlined. The Cauchy relations are thenformulated as a requirement of nullification of an invariant subspace. Asuccessive decomposition under rotation group allows to define the partialCauchy relations and two types of elastic materials. We explore severalapplications of the full and partial Cauchy relations in physics of materials.The structure's deviation from the basic physical assumptions of Cauchy's modelis defined in an invariant form. The Cauchy and non-Cauchy contributions toHooke's law and elasticity energy are explained. We identify wave velocitiesand polarization vectors that are independent of the non-Cauchy part foracoustic wave propagation. Several bounds are derived for the elasticityinvariant parameters.

Cauchy relations in linear elasticity: Algebraic and physics aspects