*Published Paper*

**Inserted:** 6 dec 2000

**Last Updated:** 10 dec 2003

**Journal:** J. Convex Anal.

**Volume:** 9

**Number:** 2

**Pages:** 1-24

**Year:** 2002

**Abstract:**

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.