A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

In this paper, we address the construction of a prior stochastic model for non-Gaussian deterministically-bounded positive-definite matrix-valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures. We first introduce the micromechanical framework and recall, in particular, Huet’s Partition Theorem. Based on the latter, we discuss the nature of hierarchical bounds and define, under some given assumptions, deterministic bounds for the apparent elasticity tensor. Having recourse to the Maximum Entropy Principle under the constraints defined by the available information, we then introduce two random matrix models. It is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically-bounded positive-definite matrix-valued random fields. Such a construction is presented and relies on a class of random fields previously defined. We finally exemplify the overall methodology considering an experimental database obtained from EBSD measurements and provide a simple numerical application.