  We consider a class of models for nonlinearly elastic surfaces in this work.
We have in mind thin, highly deformable structures modeled directly as
two-dimensional nonlinearly elastic continua, accounting for finite membrane
and bending strains and thickness change. We assume that the stored-energy
density is polyconvex with respect to the second gradient of the deformation,
and we require that it grow unboundedly as the local area ratio approaches
zero. For sufficiently fast growth, we show that the latter is uniformly
bounded away from zero at an energy minimizer. With this in hand, we rigorously
derive the weak form of the Euler-Lagrange equilibrium equations.
