12 - Modeling non-Gaussian random fields of material properties in multiscale mechanics of materials

Abstract

The proper representation of random physical quantities and system parameter uncertainties is a key ingredient of predictive science. This modeling aspect must ensure, in particular, that all samples drawn from the stochastic model satisfy the requirements raised by the mathematical analysis of the associated boundary value problem. In addition, the model is typically required to mimic physics-based constraints, ranging from the prescription of material symmetries to the multiscale-inferred definition of anisotropic statistical fluctuations. In this chapter, we present a methodology that allows non-Gaussian, tensor-valued, and vector-valued random fields to be modeled within an information-theoretic framework. The overall framework is first presented and illustrated on a simple application. The approach is subsequently developed to model tensor-valued random coefficients for linear elliptic operators. This case necessitates a specific treatment of symmetry constraints, which is addressed in detail. The modeling of stochastic nonlinear constitutive laws is finally discussed, and results devoted to the randomization of spatially dependent, Ogden-type strain energy functions are provided.