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

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