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.

John R. Willis ^{1}

@article{CRMECA_2012__340_4-5_181_0, author = {John R. Willis}, title = {The construction of effective relations for waves in a composite}, journal = {Comptes Rendus. M\'ecanique}, pages = {181--192}, publisher = {Elsevier}, volume = {340}, number = {4-5}, year = {2012}, doi = {10.1016/j.crme.2012.02.001}, language = {en}, }

John R. Willis. The construction of effective relations for waves in a composite. Comptes Rendus. Mécanique, Volume 340 (2012) no. 4-5, pp. 181-192. doi : 10.1016/j.crme.2012.02.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.02.001/

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