A comparison of two formulations for effective relations for waves in a composite

Effective constitutive relations for waves in composites with random microstructure were proposed by Willis (2009) as relations between ensemble averages of stress and momentum, and “effective” strain and velocity which were related to a weighted ensemble average of displacement, and results of an example one-dimensional calculation were presented, explicitly demonstrating the possibility of coupling between mean stress and effective velocity, and mean momentum density and effective strain, even in the long-wavelength (or homogenization) limit. Relations of this type have recently been recognized to be inevitably non-unique, and a quite general prescription for defining unique relations has been advanced (Willis, 2011). The present work compares and contrasts the effective relations obtained by either formulation, for the example considered in 2009. The work of 2009 is generalized to the extent that the constituent materials are taken to have some dissipation. It emerges explicitly that an “effective elastic constant” obtained by the method of 2009 can display an apparent energy gain rather than loss. This is not the only term that contributes, however, and it is shown that the effective material remains dissipative, as it should. It is also confirmed, both theoretically and in the computation, that either formulation leads to exactly the same mean stress and momentum density, and to the same dissipation.

► Effective relations are calculated explicitly for a laminate with complex moduli.
► Intrinsic non-uniqueness of effective relations is clarified in relation to the example.
► Requirements of causality and positive dissipation are discussed and illustrated.