  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.
