We are concerned with the derivation of the $\Gamma$-limit to a three-dimensional geometrically exact
Cosserat model as the relative thickness $h>0$ of a flat domain tends to zero. The Cosserat bulk model involves
already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear.
It is shown that the $\Gamma$-limit based on a natural scaling assumption
consists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with $h$,
augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with $h$.
No specific bending term appears in the dimensional homogenization process. The formulation
exhibits an internal length scale $L_c$ which survives the homogenization process.
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A major technical difficulty, which we encounter in applying the $\Gamma$-convergence arguments,
is to establish equi-coercivity of the sequence of
functionals as the relative thickness $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption.
While the three-dimensional problem is well-posed for the Cosserat couple modulus $\mu_c\ge 0$, equi-coercivity forces us
to assume a strictly positive Cosserat couple modulus $\mu_c>0$. The $\Gamma$-limit model determines the
midsurface deformation $m\in H^{1,2}(\omega,\R^3)$. For the case of zero Cosserat couple modulus $\mu_c=0$
we obtain an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$, without equi-coercivity which is then strenghtened to a $\Gamma$-convergence result for zero Cosserat couple modulus. The classical linear
Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for $\mu_c=0$
apart from a stabilizing shear energy term.
