Scaling to RVE in random media

The problem of effective properties of material microstructures has received considerable attention over the past half a century. By effective (or overall, macroscopic, global) is meant the response assuming the existence of a representative volume element (RVE) on which a homogeneous continuum is being set up. Since the efforts over the past quarter century have been shifting to the problem of the size of RVE, this chapter reviews the results and challenges in this broad field for a wide range of materials. For the most part, the approach employed to assess the scaling to the RVE is based on the Hill-Mandel macrohomogeneity condition. This leads to bounds that explicitly involve the size of a mesoscale domain-this domain also being called a statistical volume element (SVE)-relative to the microscale and the type of boundary conditions applied to this domain. In general, the trend to pass from the SVE to RVE depends on random geometry and mechanical properties of the microstructure, and displays certain, possibly universal tendencies. This chapter discusses that issue first for linear elastic materials, where a scaling function plays a key role to concisely grasp the SVE-to-RVE scaling. This sets the stage for treatment of nonlinear and or/inelastic random materials, including elasto-plastic, viscoelastic, permeable, and thermoelastic classes. This methodology can be extended to homogenization of random media by micropolar (Cosserat) rather than by classical (Cauchy) continua as well as to homogenization under stationary (standing wave) or transient (wavefront) loading conditions. The final topic treated in this chapter is the formulation of continuum mechanics accounting for the violations of second law of thermodynamics, which have been studied on a molecular level in statistical physics over the past two decades. We end with an overview of open directions and challenges of this research field.