Some homogenization results for viscoelastic bodies at fixed frequency.

Pierre Seppecher
Université de Toulon et du Var

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.

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