On the constitutive relations for anisotropic materials—Triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal systems

Abstract

We consider constitutive relations of the form W = W( B,…, C) where W is a scalar polynomial function of the components of B,…, C which is invariant under the group A of transformations describing the symmetry of the material considered. We make no restrictions as to the number or kind of quantities B,…, C appearing as arguments of W. We proceed by listing the elements of the integrity basis for the polynomial W( B,…, C) invariant under A which are multilinear in the components of B,…, C. The non-linear elements of the integrity basis are readily obtained once the multilinear elements are given. We give complete results for the cases where the material considered is a single crystal belonging to one of the triclinic, monoclinic, rhombic, tetragonal or hexagonal crystal systems (except for the crystal D 6 h ).