Homogenization and Smoothing: A Unified View of Two Derivations of Effective Property Theories and Extensions

The response of a continuum characterized by two widely differing length scales, parameterized by the dimensionless ratio ε, is considered in the context of the composite materials problem. The development of a bulk property theory appropriate in the ε → 0 limit is examined, both from the perspective of the deterministic homogenization literature and the smoothing method associated with statistical continuum theory, and a unified framework is established. The extension of bulk property theories through the development of ordered expansions in powers of ε is discussed and specifically related to analogous treatments in linear-gas relaxation theory.