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Journal Articles Mathematical Models and Methods in Applied Sciences Year : 2007

The Neumann sieve problem and dimensional reduction: a multiscale approach

Abstract

We perform a multiscale analysis for the elastic energy of a $n$-dimensional bilayer thin film of thickness $2\delta$ whose layers are connected through an $\epsilon$-periodically distributed contact zone. Describing the contact zone as a union of $(n-1)$-dimensional balls of radius $r\ll \epsilon$ (the holes of the sieve) and assuming that $\delta \ll \epsilon$, we show that the asymptotic memory of the sieve (as $\epsilon \to 0$) is witnessed by the presence of an extra interfacial energy term. Moreover we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of $\delta$ and $r$. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.

Dates and versions

hal-00265690 , version 1 (19-03-2008)

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Nadia Ansini, Jean-François Babadjian, Caterina Ida Zeppieri. The Neumann sieve problem and dimensional reduction: a multiscale approach. Mathematical Models and Methods in Applied Sciences, 2007, 17 (5), pp.681-735. ⟨10.1142/S0218202507002078⟩. ⟨hal-00265690⟩

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