# A homogenization bending shell theory for multiscale materials from $3D$ nonlinear elasticity

## TL;DR

This paper develops homogenized bending shell theories from 3D nonlinear elasticity, accounting for multiple small parameters related to material scales and shell thickness, resulting in different asymptotic models.

## Contribution

It introduces a multiscale homogenization approach for nonlinear shell models considering three small parameters, expanding the theoretical framework for complex materials.

## Key findings

- Derivation of various asymptotic shell theories based on parameter ratios
- Inclusion of multiple homogenization scales in shell modeling
- Extension of nonlinear elasticity to multiscale shell structures

## Abstract

We derive homogenized bending shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains three small parameters: the two homogenization scales $\varepsilon$ and $\varepsilon^2$ of the material properties and the thickness $h$ of the shell. Depending on the asymptotic ratio of these three parameters, we obtain different asymptotic theories.