Enabling integrated material and product design under uncertainty through stochastic constitutive relations

This paper presents a computational framework that mathematically propagates material microstructure uncertainties to coarser system resolutions for use in multiscale design frameworks. The computational framework uses a homogenized stochastic constitutive relation that links microstructure uncertainty with stochastic material properties. The stochastic constitutive relation formulated in this work serves as the critical link between the material and product domains in integrated material and product design. Ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized, and stochastic cell averaging is achieved by two advanced uncertainty quantification methods: random process polynomial chaos expansion and statistical copula functions. Both methods confront the mathematical difficulty in randomizing constitutive law parameters by capturing the marked correlation among them often seen in complex materials, thus the results proffer a more accurate probabilistic estimation of constitutive material behavior. The method put forth in this research, though quite general, is applied to a plastic, high strength steel alloy for demonstration.