  We consider a class of single-director Cosserat shell models accounting for
both curvature and finite mid-plane strains. We assume a polyconvexity
condition for the stored-energy function that reduces to a physically correct
membrane model in the absence of bending. With appropriate growth conditions,
we establish the existence of energy minimizers. The local orientation of a
minimizing configuration is maintained via the blowup of the stored energy as a
version of the local volume ratio approaches zero. Finally, we specialize our
results to three constrained versions of the theory commonly employed in the
subject.
