A linear isotropic Cosserat shell model including terms up to $O(h^5)$. Existence and uniqueness

TL;DR

This paper derives a linear elastic Cosserat shell model including effects up to order $O(h^5)$, proves existence and uniqueness of solutions, and connects it to classical shell models.

Contribution

It introduces a higher-order $O(h^5)$ Cosserat shell model and establishes rigorous mathematical existence and uniqueness results.

Findings

Proved existence and uniqueness of solutions for the $O(h^5)$ model.

Established Korn-type inequalities for shell geometries.

Connected the new model to classical Koiter shell theory.

Abstract

In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^3)$ model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted.