Heterogeneous Thin Films: Combining Homogenization and Dimension Reduction with Directors

We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely, the film thickness and the period of oscillating microstructures, by means of $\Gamma$-convergence. On a technical level, this requires a subtle merging of homogenization tools, such as multiscale convergence methods, with dimension reduction techniques for functionals subject to differential constraints. One observes that the results depend critically on the relative magnitude between the two scales. Interestingly, this even regards the fundamental question of locality of the limit model and, in particular, leads to new findings also in the gradient case.