Homogenization of a thin linear elastic plate reinforced with a periodic mosaic of small rigid plates

TL;DR

This paper develops a homogenized limit model for thin elastic plates reinforced with a periodic mosaic of small rigid plates, revealing a decoupled 2D behavior with specific deformation modes.

Contribution

It introduces a combined homogenization and dimension reduction approach for complex rigid-elastic structures, providing new Korn-type inequalities and a detailed characterization of the limit deformation fields.

Findings

Limit model shows bending as sum of functions of single variables.

In the 2D limit, the structure behaves as two decoupled plates with three degrees of freedom each.

Correctors are explicitly constructed within the periodicity cell.

Abstract

In the framework of linearized elasticity, we study thin elastic composite plates with thickness $\delta$. The plates contain small, rigid rectangular plates distributed periodically along $\varepsilon$. Between two neighboring rigid plates is an elastic beam with thickness $\delta < \varepsilon/3 < 1$. Through a simultaneous process of homogenization and dimension reduction, we obtain the limit model. Our analysis yields Korn-type inequalities adapted to the rigid-elastic geometry of the structure and provides a precise characterization of the limit deformation and displacement fields. In the $2$D limit problem, the bending is the sum of two functions, each depending on only one variable. This is due to the fact that the mixed derivatives of the outer-plane displacement vanish. Finally, the limiting 2D problem is two decoupled plates or strips, each one with just three degrees of freedom: shear along the strip axis, the cross-contraction (-extension), and the cross-bending. The corresponding correctors are defined in the same way in the periodicity cell. In the linearized setting, all the correctors are decomposed.