A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

TL;DR

This paper develops a nonlinear Cosserat shell model up to order $O(h^5)$, proves the existence of minimizers, and explores its relation to classical models and invariance properties of the bending tensor.

Contribution

It introduces a constrained Cosserat shell model including effects up to order $O(h^5)$ and establishes the existence of minimizers, connecting it to classical shell theories.

Findings

Existence of minimizers for $O(h^5)$ and $O(h^3)$ models.

Analysis of the new bending tensor and its invariance properties.

Relations established between the Cosserat model and classical shell models.

Abstract

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^5)$ and $O(h^3)$ case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya [Int. J. Solids and Struct., 2000] which will be further strengthened.