Stochastic modeling of anisotropy in multiscale analysis of heterogeneous materials: A comprehensive overview on random matrix approaches

Abstract The aim of this paper is to provide a general overview on random matrix ensembles for modeling stochastic elasticity tensors that exhibit uncertainties on material symmetries. Such an issue is of primal importance in many practical situations involving either a computational or experimental analysis on random heterogeneous materials (such as bones, reinforced composites, etc.). For this purpose, we first define a stochastic measure of anisotropy, the definition of which relies on the use of distances in the set of fourth-order elasticity tensors. We subsequently describe two random matrix ensembles that have been proposed within the framework of information theory and making use of a MaxEnt approach. In particular, we discuss the relevance of each of those with respect to constraints on the proposed anisotropy measure. It is shown that the capability of prescribing the mean distance to a given symmetry class depends, in view of the eigensystem-based characterization, on the behavior of the random eigenvalues. Finally, we propose a procedure allowing for the identification of the stochastic representation, should a set of experimental data be available. The approach, which is based on the use of the maximum likelihood principle, is exemplified in the case of experimental realizations that are almost transversely isotropic.