Commutativity and non-commutativity of limits in the nonlinear bending theory for prestrained microheterogeneous plates

TL;DR

This paper investigates how different limiting processes affect the derivation of nonlinear bending models for prestrained microheterogeneous plates, revealing conditions under which limits commute or not.

Contribution

It provides a detailed comparison of effective models obtained via simultaneous and consecutive limits in homogenization and dimension reduction, highlighting the impact of microstructure and thickness scales.

Findings

Limits commute when microstructure scale is much smaller than thickness.

Different limit models emerge when plate thickness is much smaller than microstructure scale.

The rate of convergence between models is analyzed in the regime where microstructure is smaller than thickness.

Abstract

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $\Gamma$-limits as the thickness $h\ll1$ and the size of the material microstructure $\e\ll1$ vanish. In the regime $\e\ll h$ we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $h\ll \e$.