Palestra: Homogeniztion of strongly heterogeneous elastic compositesg Application to phononic crystals modeling

Resumo

Phononic crystals are artificial crystals which mimic a crystalline atomic lattice, they are structured materials formed

of periodic micro-structures. Recently they have received growing interest since they may exhibit interesting properties

such as presence of band-gaps (a band-gap is a range of frequencies in which elastic or acoustic waves cannot propagate,

it is surrounded, above and below, by propagating states); hence they are good candidates for wave-guides or filters. For

example Vasseur and al. [1] have considered a two-dimensional binary solid-solid composite made of elastic arrays of

Duralumin cylindrical inclusions embedded in a resin epoxy matrix and they showed that measured transmission exhibit

absolute acoustic band gaps, see also [2]. In this talk we consider a three-dimensional composite material made of small

inclusions periodically embedded in an elastic matrix, the whole structure presents strong heterogeneities between its different

components. In the general framework of linearized elasticity we show that, when the size of the micro-structures

tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative mass density tensor. Hence

we are able to rigorously justify the existence of forbidden bands. In particular, we show how to compute these band gaps

and we illustrate the theoretical results with some numerical simulations. Some of our results are given in [3, 4].