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Journal of Convex Analysis 09 (2002), No. 2, 339--362
Copyright Heldermann Verlag 2002

Homogenization of Elastic Thin Structures: a Measure-Fattening Approach

Guy Bouchitté
Dép. des Mathématiques, Université de Toulon, B. P. 132, 83957 La Garde, France

Ilaria Fragalà
Dip. di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy


\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.

Keywords: Thin structures, homogenization, two-scale convergence, periodic measures.

MSC: 35B40, 28A33; 74B05

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