Homogenization via unfolding in domains separated by the thin layer of the thin beams

TL;DR

This paper investigates the asymptotic behavior of a thin heterogeneous layer composed of beams as its geometric parameters shrink, deriving the limit problem and transmission conditions using unfolding operators and displacement decomposition.

Contribution

It introduces a novel unfolding approach for thin beam layers and derives the effective limit problem with interface transmission conditions.

Findings

Derived the limit behavior of thin beam layers as parameters tend to zero.

Established transmission conditions across the interface in the homogenized limit.

Developed new unfolding operators tailored for different parts of the displacement decomposition.

Abstract

We consider a thin heterogeneous layer consisted of the thin beams (of radius $r$) and we study the limit behaviour of this problem as the periodicity $\varepsilon$, the thickness $\delta$ and the radius $r$ of the beams tend to zero. The decomposition of the displacement field in the beams developed in [Griso, Decompositions of displacements of thin structures, 2008] is used, which allows to obtain a priori estimates. Two types of the unfolding operators are introduced to deal with the different parts of the decomposition. In conclusion we obtain the limit problem together with the transmission conditions across the interface.