Micromechanically based stochastic finite elements: length scales and anisotropy

The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the finite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elementsʹ stiffness matrices from the microstructures over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.