  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.
