  We consider a class of models motivated by previous numerical studies of
wrinkling in highly stretched, thin rectangular elastomer sheets. The model
used is characterized by a finite-strain hyperelastic membrane energy perturbed
by small bending energy. Without bending energy, the stored-energy density is
not rank-one convex for general spatial deformations but reduces to a
polyconvex function when restricted to the plane, i.e., two-dimensional
hyperelasticity. In addition, it grows unbounded as the local area ratio
approaches zero. The small-bending component of the model is the same as that
in the classical von K\'arm\'an model. Here we prove the existence of energy
minima for a general class of such models.
