Equilibrium configurations of mixture thin films undergoing large strains

We describe a mixture thin film as a membrane endowed with multiple out-of-tangent-plane vectors at each point, with vector sequence defined to within a permutation to account for the mixing of the mixture components. Such a description is motivated by a proposal for an atomistic-to-continuum derivation of a representation of multiatomic layer thin films, a view not requir- ing the introduction of phenomenological parameters. Differences between that proposal and the model discussed here are the definition of the values of the layer-descriptor map to within permutations and the explicit introduction of a bending-like term in the energy. The out-of-tangent-plane vectors satisfy a condition forbidding them to fall within the tangent plane after deformation. We consider a generic weakly surface-polyconvex membrane energy and a quadratic bending term involving the out-of-tangent-plane multiple vectors, a term which is also quasiconvex. Under appropriate energy growth assumptions and Dirichlet-type boundary conditions, we prove existence of ground states, i.e., equilibrium configurations described by the solutions to balance equations. An obvious corollary is the existence of equilibrium configurations of single out-of-tangent-plane vector Cosserat surfaces, a natural scheme for plates of simple materials.