Homogenization of Elastic Thin Structures: a Measure-Fattening Approach

\def\ren{{\mathbb R}^n} We study the homogenization of vector problems on thin periodic structures in $\ren$. The analysis is carried out within the same measure framework that we previously published for scalar problems [see "Homogenization of thin structures by two-scale method with respect to measures", SIAM J. Math. Analysis 32 (2001) 1198--1226], namely each periodic, low-dimensional structure is identified with the overlying positive Radon measure $\mu$. Thus, we deal with a sequence of measures $\{\mu_\varepsilon\}$, whose periodicity cell has size $\varepsilon$ converging to zero, and our aim is to identify the limit, in the variational sense of $\Gamma$-convergence, of the elastic energies associated to $\{\mu_\varepsilon\}$. We show that the explicit formula for such homogenized functional can be obtained combining the application of a two-scale method with respect to measures, and a fattening approach; actually, it turns out to be crucial approximating $\mu$ by a sequence of measures $\{\mu_\delta\}$, where $\delta$ is an auxiliary, infinitesimal parameter, associated to the thickness of the structure. In particular, our main result is proved under the assumption that the structure is asymptotically not too thin (i.e.\ $\delta \gg \varepsilon$), and, for all $\delta>0$, $\mu_\delta$ satisfy suitable {\it fatness} conditions, which generalize the {\it connectedness} hypotheses needed in the scalar case. We conclude by pointing out some related problems and conjectures.