The construction of effective relations for waves in a composite

Implications of a recent general formulation for the effective dynamic response of a composite, in which “effective displacement” is defined as a weighted average, which could, for instance, be an average over the matrix material, are developed. A general formula is already known [J.R. Willis, Effective constitutive relations for waves in composites and metamaterials, Proc. R. Soc. A 467 (2011) 1865–1879], but it is expressed in terms of the Greenʼs function of the actual composite. A corresponding formula, expressed relative to a comparison medium, is developed here. The property of self-adjointness of the problem for the actual medium is transmitted to the corresponding problem for the “effective medium”. This permits, in the case of self-adjointness, variational characterizations of the effective response, both directly and in a formulation of “Hashin–Shtrikman” type relative to a comparison medium. The exposition is for waves in a viscoelastic composite but it applies equally to other physical examples, including electromagnetic waves.