The Neumann sieve problem and dimensional reduction: a multiscale approach
Mathematical Models and Methods in Applied Sciences, 2007•World Scientific
We perform a multiscale analysis for the elastic energy of a n-dimensional bilayer thin film of
thickness 2δ whose layers are connected through an ε-periodically distributed contact zone.
Describing the contact zone as a union of (n-1)-dimensional balls of radius r≪ ε (the holes of
the sieve) and assuming that δ≪ ε, we show that the asymptotic memory of the sieve (as ε→
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 δ and r. We …
thickness 2δ whose layers are connected through an ε-periodically distributed contact zone.
Describing the contact zone as a union of (n-1)-dimensional balls of radius r≪ ε (the holes of
the sieve) and assuming that δ≪ ε, we show that the asymptotic memory of the sieve (as ε→
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 δ and r. We …
We perform a multiscale analysis for the elastic energy of a n-dimensional bilayer thin film of thickness 2δ whose layers are connected through an ε-periodically distributed contact zone. Describing the contact zone as a union of (n - 1)-dimensional balls of radius r ≪ ε (the holes of the sieve) and assuming that δ ≪ ε, we show that the asymptotic memory of the sieve (as ε → 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 δ and r. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.
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