  Using a geometrically motivated 8-parameter ansatz through the thickness, we
reduce a three-dimensional shell-like geometrically nonlinear Cosserat material
to a fully two-dimensional shell model. Curvature effects are fully taken into
account. For elastic isotropic Cosserat materials, the integration through the
thickness can be performed analytically and a generalized plane stress
condition allows for a closed-form expression of the thickness stretch and the
nonsymmetric shift of the midsurface in bending. We obtain an explicit form of
the elastic strain energy density for Cosserat shells, including terms up to
order $O(h^5)$ in the shell thickness $h$. This energy density is expressed as
a quadratic function of the nonlinear elastic shell strain tensor and the
bending-curvature tensor, with coefficients depending on the initial curvature
of the shell.
