From statics of composites to acoustic metamaterials

The Hashin-Shtrikman formalism for the static response of a composite is first summarized, in a form applicable to anisotropic media and arbitrary two-point correlations. Its extension to dynamic response is then explained and in particular the extra coupling, for stress with velocity as well as strain, and momentum density with strain as well as velocity that inevitably follows, is highlighted. The more recent dynamical formulations which incorporate extra flexibility through the use of a weighted mean displacement also receive mention. The article is concluded with an explicit analysis of a simple model random composite and some lessons are drawn from the solution, which exposes at least one clear gap in existing knowledge and requires further research. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.