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We present a new geometrically nonlinear Cosserat shell model incorporatingeffects up to order $O(h^5)$ in the shell thickness $h$. The method that wefollow is an educated 8-parameter ansatz for the three-dimensional elasticshell deformation with attendant analytical thickness integration, which leadsus to obtain completely two-dimensional sets of equations in variational form.We give an explicit form of the curvature energy using the orthogonalCartan-decomposition of the wryness tensor. Moreover, we consider the matrixrepresentation of all tensors in the derivation of the variational formulation,because this is convenient when the problem of existence is considered, and itis also preferential for numerical simulations. The step by step constructionallows us to give a transparent approximation of the three-dimensional parentalproblem. The resulting 6-parameter isotropic shell model combines membrane,bending and curvature effects at the same time. The Cosserat shell modelnaturally includes a frame of orthogonal directors, the last of which does notnecessarily coincide with the normal of the surface. This rotation-field iscoupled to the shell-deformation and augments the well-known Reissner-Mindlinkinematics (one independent director) with so-called in-plane drill rotations,the inclusion of which is a decisive for subsequent numerical treatment andexistence proofs. As a major novelty, we determine the constitutivecoefficients of the Cosserat shell model in dependence on the geometry of theshell which are otherwise difficult to guess.

The isotropic Cosserat shell model including terms up to $O(h^5)$. Part I: Derivation in matrix notation