ABSTRACT
The problem of effective properties of material microstructures has received considerable, and ever-growing, attention over the past thirty years. By effective (or overall, macroscopic, global) is meant the response assuming existence of a representative volume element (RVE). The RVE in the case of disorder (i.e., lack of microstructural periodicity), implies that there must be some scale larger than the microscale (e.g., single heterogeneity size) to ensure a homogenization limit. Overall, most studies of effective properties simply assume that the RVE is attained and do not specify its size-scarce prescriptions of solid mechanics vaguely state that domains roughly 10 to 100 times larger than the heterogeneity should be taken. In the late 1980s work began on the determination of RVE in the sense of Hill (1963) and on continuum random fields serving as input into stochastic finite element methods. This then led 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 various factors, and displays certain tendencies. This chapter discusses that issue for linear elastic materials, thus setting the stage for nonlinear and/or inelastic materials as well as for continuum random field models and stochastic boundary value problems, topics dealt with in the subsequent chapters.
