Heterogeneous thin films: Combining homogenization and dimension reduction with directors

TL;DR

This paper investigates the asymptotic behavior of multiscale integral functionals in thin films with microstructures, using $ ext{Gamma}$-convergence, revealing scale-dependent effects on the limit models and locality properties.

Contribution

It combines homogenization and dimension reduction techniques for differential-constrained functionals, providing new insights into scale interactions and locality in thin film models.

Findings

Results depend critically on the relative scales of thickness and microstructure period.

New findings on the locality of the limit model in the gradient case.

Methodology merges homogenization tools with dimension reduction techniques.

Abstract

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 subtile 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.