Symmetry classes of linear elasticity induced by fabric tensors

The relationship between microstructural fabric and the symmetry class of the homogenized elasticity tensor is investigated within a rigorous measure-theoretic framework where fabric tensors are defined as tensorial moments of a positive measure on the unit sphere, and constitutive mappings are assumed to be isotropic. Within such a context, it is shown that isotropic mappings depending on the sole second-order fabric moment cannot generate effective elasticity with symmetry lower than orthotropy; such a limitation, previously acknowledged as a practical bound of second-order models, is here established as a structural theorem. In order to overcome such a limitation, it is proved that inclusion of the fourth-order moment is mandatory. In particular, under a non-degeneracy condition generically typical of isotropic mappings, triclinic effective elasticity is equivalent to absence of rotational invariances for the fouth-order fabric tensor. Accordingly, a complete spectral classification of the eight classical symmetry classes of linear elasticity is provided in terms of the joint spectral structure of the second- and fourth-order fabric tensors. It is further proved that triclinic symmetry is the generic condition in the space of completely symmetric fourth-order tensors, so that higher symmetry classes are exceptional rather than typical. The developed theoretical framework is validated numerically on eight three-dimensional lattice truss models designed to approximate prescribed symmetry classes, thus confirming that the fourth-order fabric tensor governs the symmetry of the effective elastic response both qualitatively and quantitatively.