Effective constitutive relations for waves in composites and metamaterials

Abstract

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’.