The nonlinear membrane model: Variational derivation under the   constraint "det$\nabla u\not=0$"
  Acerbi, Buttazzo and Percivale gave a variational definition of the nonlinear string energy under the constraint "det$\nabla u>0$" (see [1]). In the same spirit, we obtain the nonlinear membrane energy under the simpler constraint "det$\nabla u\not=0$". 
Omar Anza Hafsa and Jean-Philippe Mandallena