Search

Abstract: 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. [Copyright &y& Elsevier]

Definitions of 'effective fields' for a randomly inhomogeneous material are offered, which guarantee automatic satisfaction of the equations of motion. The important case of a medium with periodic microstructure is included. In this special case, the definitions are completely explicit and can be applied without reference to random media. The presentation is mostly expressed in terms of electromagnetic waves. The reasoning is applicable also to other types of waves and its realization for elastodynamics is briefly outlined towards the end. Some of the effective fields are defined directly as ensemble averages, ensuring the exact satisfaction of the equations of motion, but the effective 'kinematic' fields to which they are related are defined more generally, as weighted averages. The main result of this work is an explicit formula for the tensor of effective properties. The important issue of uniqueness (or not) of the effective properties is explained and resolved. Self-adjointness of the original problem is not assumed. An attractive feature of the formulation is that self-adjointness at the local level implies self-adjointness at the level of the 'effective medium'. [ABSTRACT FROM AUTHOR]