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

This paper presents a finite element method aimed at the introduction of mi crostructural material randomness below the level of a single finite elemen t. A consideration of dependence of effective moduli on scare and on the de fining boundary conditions leads to an identification of a finite element a s a mesoscale window (or, a mesoscale finite element) in a stochastic finit e element method (SFE). An estimation of the global response can be obtaine d through bounds stemming from minimum potential energy and complementary e nergy principles, which involve Dirichlet and Neumann boundary conditions o n all the mesoscale finite elements, respectively. While in the classical c ase of a homogeneous material, these two bounds converge to each other as t he finite element mesh becomes sufficiently fine, an optimal mesoscale with respect to the difference between both bounds may exist in the case of a h eterogeneous material. The proposed SFE method is illustrated with a numeri cal example of a sample two-phase Voronoi composite (with some 26 000 grain s), where a reference solution taking into account the entire microstructur e without any smearing out, is shown to fall between both energy bounds. A generalization to an ensemble response is straightforward. (C) 1999 Elsevie r Science S.A. All rights reserved.