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