Calculus of Variations and Geometric Measure Theory
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G. Bouchitté - I. Fragalà

Homogenization of elastic thin structures: a measure-fattening approach

created on 06 Dec 2000
modified on 10 Dec 2003


Published Paper

Inserted: 6 dec 2000
Last Updated: 10 dec 2003

Journal: J. Convex Anal.
Volume: 9
Number: 2
Pages: 1-24
Year: 2002


We study the homogenization of vector problems on thin periodic structures in $R^n$. The analysis is carried out within the same measure framework that we previously enforced for scalar problems, namely each periodic, low-dimensional structure is identified with the overlying positive Radon measure $\mu$. Thus, we deal with a sequence of measures $\{ \mu _\epsilon\}$, whose periodicity cell has size $\epsilon$ converging to zero, and our aim is to identify the limit, in the variational sense of $\Gamma$-convergence, of the elastic energies associated to $\mu_\epsilon$. 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 ({\it i.e.\ }$\delta\gg \epsilon$), and, for all $\delta>0$, $\mu _\delta$ satisfy suitable {\sl fatness} conditions, which generalize the {\sl connectedness} hypotheses needed in the scalar case. We conclude by pointing out some related problems and conjectures.

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