Homogenization of bending theory for plates; the case of elastic laminates

TL;DR

This paper investigates how oscillations in material properties affect the bending behavior of elastic plates, showing that in some cases these oscillations do not influence the effective behavior, while in others they do.

Contribution

It demonstrates, using $ extGamma$-convergence, that oscillations in the thickness direction do not affect the homogenized model unless coupled with in-plane periodic oscillations.

Findings

Oscillations in the thickness direction alone do not influence the effective behavior.

Periodic in-plane and thickness oscillations lead to homogenization effects.

The analysis is based on $ extGamma$-convergence techniques.

Abstract

In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of thickness. We study two different cases. First, we show, starting from 3D elasticity, by means of $\Gamma$-convergence and under general (not necessarily periodic) assumption, that the effective behavior of the limit is not influenced by oscillations in the direction of thickness. In the second case, we study periodic in-plane oscillations of the energy density coupled with periodic oscillations in the direction of thickness. In contrast to the first case we show that there are homogenization effects coming also from the oscillations in the direction of thickness.