Scaling Functions in Spatially Random Composites

We review the key issues involved in scaling and homogenization of random composite materials. In the first place, this involves a Hill-Mandel condition in the setting of stochastic micromechanics. Within this framework, we introduce the concept of a scaling function that describes "finite-size scaling"of thermally conducting or elastic crystalline aggregates. While the finite size is represented by the mesoscale, the scaling function depends on an appropriate measure quantifying the single-crystal anisotropy. Based on the scaling function, we construct a material scaling diagram, from which one can assess the scaling trend from a statistical volume element (SVE) to a representative volume element (RVE) for many different materials. We demonstrate these concepts with the scaling of the fourth-rank elasticity and the second-rank thermal conductivity tensors. We also briefly discuss the trends in approaching the RVE for linear/nonlinear (thermo)elasticity, plasticity, and Darcy permeability.