Invariant elastic constants and eigentensors of orthorhombic, tetragonal, hexagonal and cubic crystalline media

The purpose of this paper is to present a simple and direct way of determining the eigenvalues and eigentensors, as well as their orientations, for all crystals of the orthorhombic, tetragonal, hexagonal and cubic symmetries, a procedure based on the spectral decomposition of the compliance and stiffness fourth-rank tensors. First, both the eigenvalues and the idempotent fourth-rank tensors are derived for the orthorhombic and tetragonal-7 symmetries. The latter decompose, respectively, the second-rank symmetric tensor spaces of orthorhombic and tetragonal-7 media into orthogonal subspaces, consisting of the stress and strain eigentensors, and split the elastic potential into distinct noninteracting strain-energy parts. Accordingly, the spectrum of the compliance tensor of the tetragonal-6 symmetry is evaluated, by reduction of the eigenvalues and eigentensors of either the orthorhombic or tetragonal-7 symmetry. These results are, then, applied in turn to each of the hexagonal and cubic crystal systems. In each case, the eigenvalues, the idempotent tensors and the stress and strain eigentensors are easily derived as particular cases of the results obtained for the tetragonal-6 symmetry. Furthermore, it is noted that the positivity of the eigenvalues for each symmetry is equivalent to the positive definiteness of the elastic potential and, thus, necessary and sufficient conditions are acquired, in terms of the compliance-tensor components, characteristic of each symmetry.