A stochastic multiscale approach to deal with the homogenization of random nonlinear heterogeneous materials defined in high dimensional parameters space

Abstract : we address the construction of random macroscopic constitutive laws by homogenization of random nonlinear heterogeneous materials for which a high number of random parameters is needed to characterize the uncertainties at the microscopic scale. We base our new approach on a non-concurrent multiscale method recently proposed for computing the homogenization of nonlinear heterogeneous materials [1,2] which still represents a difficult task. This technique, based on a numerical construction of the strain density function associated with a microstructure, allows to encounter this difficulty in a deterministic framework. However, in order to obtain an efficient mechanical model, one must take into account the different sources of uncertainties. In this work, we propose to extend this method to the stochastic framework and we therefore consider random microstructures. We focus on hyperelastic materials made of reinforced rigid fibers characterized by random geometrical parameters.