Our aim is to characterize the set of all materials which can be obtained by homogenizing high contrasted viscoelastic bodies. We use a variational framework. Indeed lossy elastodynamic problems can be seen as the search of saddle points of convex-concave functionnals (see Milton & al. Proc. R. Soc. A 465, 367-396, 2009). A suitable notion of convergence of convex-concave functions and associated saddle points is the epi-hypo-convergence (see Attouch & al., Trans. Amer. Math. Soc. 280, 1-44, 1983.)
We follow, for elastodynamics at fixed frequency, a scheme which has proved to be efficient in the elastostatic case. In this scheme different steps are necessary. (i) A first homogenization result shows that materials exhibiting simple non local interactions can be obtained. These interactions are simple as they are two-points interactions with a fixed range and direction. (ii) An addivity property allows to reach multiple interactions : truss-like interactions. (iii) A second homogenization result proves that some nodes of the previous trusses can be set free. Hence the truss-like interactions become mechanisms. (iv) the possible responses of such mechanisms are characterized. This step has already been studied (Milton & al. Proc. R. Soc. Lond. A, 464, 967–986, 2008) at a fixed frequency and, very recently, the dependence with respect to the frequency has been investigated (Vasquez & al., arXiv:0911.1501v1). (v) The last step is the approximation of any realizable functional by mechanisms using regularization and discretization procedures.
In this contribution some crucial homogenization steps will be presented.