Vibrations d'un composite élastique comportant des inclusions granulaires très lourdes : effets de mémoire

Bellieud, Michel

Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 807-812.

Soit Ω un ouvert borné de ${\mathbb{R}}^3$ et soient $({\rho }_{\epsilon })$, $({\mu }_{\epsilon })$, $({\lambda }_{\epsilon })$ des suites de fonctions périodiques sur Ω, de période ε, la fonction ${\rho }_{\epsilon }$ prenant des valeurs très grandes sur un ensemble de boules de rayon $r_{\epsilon }$ périodiquement distribuées de période ε ($r_{\epsilon }\ll \epsilon$). Nous étudions le comportement asymptotique ($\epsilon \to 0$) de l'équation :

$$\{\begin{array}{c}{\rho }_{\epsilon }\frac{{\partial }^2{\mathit{u}}_{\epsilon }}{\partial t^2}-\mathrm{div}({\mathit{\sigma }}_{\epsilon }({\mathit{u}}_{\epsilon }))={\rho }_{\epsilon }\mathit{f}\text{dans}\Omega \times (0,T)+\text{conditions aux limites},\hfill \\ {\mathit{\sigma }}_{\epsilon }({\mathit{u}}_{\epsilon })={\lambda }_{\epsilon }\mathrm{tr}(\mathit{e}({\mathit{u}}_{\epsilon }))\mathit{I}+2{\mu }_{\epsilon }\mathit{e}({\mathit{u}}_{\epsilon }),\mathit{e}({\mathit{u}}_{\epsilon })=\frac{1}{2}({\mathbf{\nabla }\mathit{u}}_{\epsilon }+{\mathbf{\nabla }}^T{\mathit{u}}_{\epsilon }),\hfill \end{array}$$

et nous trouvons une loi effective non locale déduite d'un système couplé d'équations hyperboliques.

Let Ω be a bounded open set of ${\mathbb{R}}^3$ and $({\rho }_{\epsilon })$, $({\mu }_{\epsilon })$, $({\lambda }_{\epsilon })$ be sequences of ε-periodic functions on Ω, the function ${\rho }_{\epsilon }$ taking very large values on a set of ε-periodically distributed balls of radius $r_{\epsilon }$ ($r_{\epsilon }\ll \epsilon$). We study the asymptotic behaviour as $\epsilon \to 0$ of the equation:

$$\{\begin{array}{c}{\rho }_{\epsilon }\frac{{\partial }^2{\mathit{u}}_{\epsilon }}{\partial t^2}-\mathrm{div}({\mathit{\sigma }}_{\epsilon }({\mathit{u}}_{\epsilon }))={\rho }_{\epsilon }\mathit{f}\text{in}\Omega \times (0,T)+\text{boundary conditions},\hfill \\ {\mathit{\sigma }}_{\epsilon }({\mathit{u}}_{\epsilon })={\lambda }_{\epsilon }\mathrm{tr}(\mathit{e}({\mathit{u}}_{\epsilon }))\mathit{I}+2{\mu }_{\epsilon }\mathit{e}({\mathit{u}}_{\epsilon }),\mathit{e}({\mathit{u}}_{\epsilon })=\frac{1}{2}({\mathbf{\nabla }\mathit{u}}_{\epsilon }+{\mathbf{\nabla }}^T{\mathit{u}}_{\epsilon }),\hfill \end{array}$$

and finally we find a non-local effective equation deduced from a homogenized system of hyperbolic equations.

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