A geometrically exact planar Cosserat shell-model with microstructure: Existence of minimizers for zero cosserat couple modulus

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre-Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus mu(c) = 0, one recovers the infinitesimal-displacement Reissner-Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for mu(c) = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H-1(omega, R-3). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.