On the derivation of homogenized bending plate model

TL;DR

This paper derives a homogenized bending plate model by combining homogenization and dimension reduction techniques, focusing on thin elastic plates with oscillating energy densities at specific scales.

Contribution

It introduces a novel derivation of the $ ext{Gamma}$-limit for thin plates with oscillatory energy densities, integrating homogenization with dimension reduction.

Findings

Derived the $ ext{Gamma}$-limit for plates with oscillating energy densities.

Established the relation between oscillation scales and plate thickness.

Provided a rigorous mathematical framework for nonlinear bending theory.

Abstract

We derive, via simultaneous homogenization and dimension reduction, the $\Gamma$-limit for thin elastic plates of thickness $h$ whose energy density oscillates on a scale $\eh$ such that $ \eh^2 \ll h\ll \eh$. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates.