  The Cauchy relations distinguish between rari- and multi-constant linear
elasticity theories. These relations are treated in this paper in a form that
is invariant under two groups of transformations: indices permutation and
general linear transformations of the basis. The irreducible decomposition
induced by the permutation group is outlined. The Cauchy relations are then
formulated as a requirement of nullification of an invariant subspace. A
successive decomposition under rotation group allows to define the partial
Cauchy relations and two types of elastic materials. We explore several
applications of the full and partial Cauchy relations in physics of materials.
The structure's deviation from the basic physical assumptions of Cauchy's model
is defined in an invariant form. The Cauchy and non-Cauchy contributions to
Hooke's law and elasticity energy are explained. We identify wave velocities
and polarization vectors that are independent of the non-Cauchy part for
acoustic wave propagation. Several bounds are derived for the elasticity
invariant parameters.
