Comparison of spherical and cubical statistical volume elements with respect to convergence, anisotropy, and localization behavior

The statistical volume element (SVE) technique is commonly used for the estimation of the effective properties of a micro-structured material. Mostly, cubical SVEs with periodic boundary conditions are employed, which result in a better convergence, compared to the uniform boundary conditions. In this work, the possibility of using spherical SVEs is discussed, since their use promises a reduction of the influence of the boundary, and thus a more efficient estimation of the effective material properties. We discuss the applicability of boundary conditions which are similar to the periodic boundary conditions to spherical SVEs. Then we assess the convergence (subject 1) of spherical and cubical SVEs to the effective material behavior for the uniform and periodic boundary conditions, focusing on the elastic and plastic properties of a macroscopically isotropic matrix-inclusion material. It is shown that the spherical SVEs perform indeed better than the cubical SVEs. Also, unlike the spherical SVEs, the cubical SVEs with periodic boundary conditions induce a spurious anisotropy (subject 2), which is quantified for the effective elastic properties. Finally, we examine the effect of the periodicity frame on the localization behavior (subject 3) of cubical SVE, since cubical SVE with periodic boundary conditions are commonly used to estimate macroscale material failure. It is demonstrated that the orientation of the periodicity frame affects the overall SVE response significantly. The latter is not observed for spherical SVE.