Uncertainty Quantification of Random Heterogeneous Media Using XFEM

Abstract

AbstractThe macroscopic mechanical properties of two-phase heterogeneous materials, consisting of random inclusions in a solid medium, are governed by the individual material properties of the inclusions and the medium, as well as the volume fraction and spatial distribution of the inclusions. In design and analysis using such materials, the macroscopic material properties are often used. The characterization of the macroscopic properties based on the accurate micro-structure modeling may often be computationally expensive. Hence, homogenization of representative micro-structures is often used to get these macroscopic properties. In this paper too we adopt this approach for a medium with random elliptical inclusions. Monte Carlo (MC) simulations are used to obtain different micro-structure realizations, enabling the statistical modeling of macroscopic material properties. The material is modeled using the extended finite element method (XFEM), where the inclusions are modeled independent of the finite element mesh using the level set method, thereby reducing the computational cost involved in the MC simulations. The effective elastic modulus and Poisson’s ratio of the heterogeneous material are obtained using Hill’s averaging theorem, following which the failure stress is obtained using the computed homogenized elastic properties. The excellent synergy between XFEM and MC simulations gives the statistical characteristics of these effective material properties, including the variation in their estimates induced by the random micro-structure. It is shown that the uncertainty in the homogenized failure stress is much higher than the uncertainty in the elastic properties.KeywordsXFEMTwo-phase heterogeneous materialMonte Carlo simulationHomogenizationElastic propertiesFailure stress