Determining the Size of RVE for Nonlinear Random Composites in an Incremental Computational Homogenization Framework

Abstract

In this paper, we address the issue of determining the size of a representative volume element (RVE) in the case of nonlinear random composites with either elastoplastic or elasto-viscoplastic phases. In such a case, the general form of the effective constitutive behavior is not known in advance and the response must be evaluated either by direct numerical computations on the RVE, either by an appropriate approximation scheme. Previous methodologies for determining the size of RVE usually rely on analyzing the convergence of the RVE response computed numerically with respect to its size. In the present work, we analyze the convergence of parameters related to an incremental homogenization scheme, with respect to (i) the size of the RVE and (ii) to statistical convergence related to microstructure real-izations. For that purpose, we combine an incremental homogenization method with a statistical convergence analysis of parameters related to the matrix phase only. The advantage is that the range of parameters to be identified is much narrower than for a general empirical constitutive law. Once identified and the convergence analysis performed with respect to both size of RVE and statistical realizations, the macroscopic constitutive law can be readily used for structure calculations. We illustrate the methodology by analyzing two-dimensional microstructures with randomly distributed cylindrical elastic rigid fibers, embedded in a elastoplastic or elasto-viscoplastic matrix. For these materials, the existence of an RVE is demonstrated for sizes of RVE corresponding to 17 − 18 and 14 − 15 times the diameter of the inclusions, respectively.