Derivation of a homogenized von-Karman shell theory from 3D elasticity

TL;DR

This paper derives a homogenized von Kármán shell theory from 3D nonlinear elasticity, analyzing how different asymptotic regimes of shell thickness and material oscillations affect the resulting models.

Contribution

It provides a rigorous derivation of homogenized shell models considering two small parameters and identifies different asymptotic regimes based on their ratio.

Findings

Different asymptotic theories depending on the ratio of shell thickness to material oscillations

Complete characterization of models for convex shells in the regime h << ε

Identification of regimes where h << ε and h << ε^2 lead to distinct theories

Abstract

We derive the model of homogenized von K\'arm\'an shell theory, starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the oscillations of the material $\e$ and the thickness of the shell $h$. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case $h\ll\e$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\e^2$. In the case of convex shells we obtain a complete picture in the whole regime $h\ll\e$.