Material spatial randomness: From statistical to representative volume element

The material spatial randomness forces one to re-examine various basic concepts of continuum solid mechanics. In this paper we focus on the Representative Volume Element (RVE) that is commonly taken for granted in most of deterministic as well as in stochastic solid mechanics, although in the latter case it is called the Statistical Volume Element (SVE). The key issue is the scale over which homogenization is being carried out - it is called the mesoscale, separating the microscale (level of microheterogeneities) from the macroscale (level of RVE). As the mesoscale grows, the SVE tends to become the RVE. This occurs in terms of two hierarchies of bounds stemming from Dirichlet and Neumann boundary value problems on the mesoscale, respectively. Since generally there is no periodicity in real random media, the RVE can only be approached approximately on finite scales. We review results on this subject in the settings of linear elasticity, finite elasticity, plasticity, viscoelasticity, thermoelasticity, and permeability.