  We present a new geometrically nonlinear Cosserat shell model incorporating
effects up to order $O(h^5)$ in the shell thickness $h$. The method that we
follow is an educated 8-parameter ansatz for the three-dimensional elastic
shell deformation with attendant analytical thickness integration, which leads
us to obtain completely two-dimensional sets of equations in variational form.
We give an explicit form of the curvature energy using the orthogonal
Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix
representation of all tensors in the derivation of the variational formulation,
because this is convenient when the problem of existence is considered, and it
is also preferential for numerical simulations. The step by step construction
allows us to give a transparent approximation of the three-dimensional parental
problem. The resulting 6-parameter isotropic shell model combines membrane,
bending and curvature effects at the same time. The Cosserat shell model
naturally includes a frame of orthogonal directors, the last of which does not
necessarily coincide with the normal of the surface. This rotation-field is
coupled to the shell-deformation and augments the well-known Reissner-Mindlin
kinematics (one independent director) with so-called in-plane drill rotations,
the inclusion of which is a decisive for subsequent numerical treatment and
existence proofs. As a major novelty, we determine the constitutive
coefficients of the Cosserat shell model in dependence on the geometry of the
shell which are otherwise difficult to guess.
