Using transverse isotropy to model arbitrary deformation-induced anisotropy

A unifying framework is developed for the analysis of brittle materials. Heretofore diverse classes of models result from different choices for unspecified coefficient and distribution functions in the unified theory. Material response is described in terms of expectation integrals of transverse symmetry tensors. First, a canonical body containing cracks of all the same orientation is argued to possess macroscopic transverse isotropy. An orthogonal basis for the linear subspace consisting of all double-symmetric transversely-isotropic fourth-order tensors associated with a given material vector is introduced and applied to deduce the explicit functional dependence of the compliance of such contrived materials on the shared crack orientation. A principle of superposition of strain rates is used to write the compliance for a more realistic material consisting of cracks of random size and orientation as an expectation integral of the transverse compliance for each orientation times the joint distribution function for the size and orientation. Utilizing an evolving (initially exponential) size- dependence in the joint distribution, the general theory gives unprecedented agreement with measurements of the dynamic response of alumina to impact loading, especially upon release where the calculations predict the development of considerable deformation- induced anisotropy, challenging the conventional notion of shocks as isotropic phenomena