Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory

A stochastic constitutive theory is proposed in this work to propagate microstructure uncertainties in computational multiscale continuum models to bulk multiresolution material behavior. Ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized in detail, and this research transmits these uncertainties to coarser material resolutions by introducing a stochastic constitutive theory deduced from volume element simulations. To implement the stochastic upscaling process, two advanced uncertainty quantification methods are examined: statistical copula functions and random process polynomial chaos expansion. Both methods confront the mathematical difficulty in randomizing constitutive laws by capturing the marked correlation among constitutive parameters seen in complex materials, thus the results proffer a more accurate probabilistic estimation of constitutive material behavior. The contribution of this work is twofold: uncertainty is propagated from heterogeneous material "structure" to material "property" via the stochastic constitutive theory, and rigorous, data-driven mathematics are formalized to represent complicated dependence structures in multivariate statistical distributions. To the authors' knowledge, this is the first work in multiresolution mechanics that presents an approach to computationally derive correlation functions from numerical experiments, as opposed, for instance, to assuming one a priori. The method put forth in this research, though quite general, is applied to a mathematical example and plastic, high strength steel alloy for demonstration. Results include stochastic constitutive curve confidence intervals for the material stress-strain response and qualitative comparisons of the two stochastic methods detailed herein.