A stochastic model for elasticity tensors exhibiting uncertainties on material symmetries

This study addresses the stochastic modeling of media whose elasticity tensor exhibits uncertainties on the material symmetry class to which it belongs. More specifically, we focus on the construction of a probabilistic model which allows realizations of random elasticity tensors to be simulated, under the constraint that the mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilistic model derived in [6] which allows the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. A new methodology and parameterization of the model are then defined. The proposed approach is exemplified considering the mean to transverse isotropy. The efficiency of the methodology is demonstrated by computing the mean distance of the random elasticity tensor to this material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemannian metric and the Euclidean projection, respectively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the nonparametric probabilistic approach (with anisotropic fluctuations) proposed in [12] is finally provided.