Stochastic finite elements as a bridge between random material microstructure and global response

Abstract

This paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element. A consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window (or, a mesoscale finite element) in a stochastic finite element method (SFE). An estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles, which involve Dirichlet and Neumann boundary conditions on all the mesoscale finite elements, respectively. While in the classical case of a homogeneous material, these two bounds converge to each other as the finite element mesh becomes sufficiently fine, an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material. The proposed SFE method is illustrated with a numerical example of a sample two-phase Voronoi composite (with some 26 000 grains), where a reference solution taking into account the entire microstructure without any smearing out, is shown to fall between both energy bounds. A generalization to an ensemble response is straightforward.