A note on the homogenization of incommensurate thin films

TL;DR

This paper extends homogenization results for thin films to incommensurate cases using a geometric almost-periodicity approach, broadening the understanding of material behavior in complex layered structures.

Contribution

It introduces a novel homogenization method for incommensurate thin films using a geometric almost-periodicity argument, unlike previous periodicity-based approaches.

Findings

Established homogenization results for incommensurate thin films.

Demonstrated the applicability of cut-and-project techniques to homogenization.

Extended the theoretical framework beyond periodic media.

Abstract

Dimension-reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost-periodicity of the energies in the directions of the mid-plane of the film. In this note we consider thin films, obtained as sections of a periodic medium with a mid-plane that may be incommensurate; that is, not containing periods other than $0$. A geometric almost-periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.