Fuzzy‐stochastic FEM–based homogenization framework for materials with polymorphic uncertainties in the microstructure

Abstract

SummaryUncertainties in the macroscopic response of heterogeneous materials result from two sources: the natural variability in the microstructure's geometry and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function. The second type of uncertainty is denoted epistemic uncertainty. This kind of uncertainty cannot be described using probabilistic methods. Models considering both sources of uncertainties are called polymorphic. In the case of polymorphic uncertainties, some combination of stochastic methods and fuzzy arithmetic should be used. Thus, in the current work, we examine a fuzzy‐stochastic finite element method–based homogenization framework for materials with random inclusion sizes. We analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element. The stochastic finite element method is used to obtain the material response in the case of random inclusion radii. Due to unavoidable noise in experimental data, an insufficient number of samples, and limited accuracy of the fitting procedure, the radii distribution density cannot be obtained exactly; thus, it is described in terms of fuzzy location and scale parameters. The influence of fuzzy input on the homogenized stress measures is analyzed.