An existence theorem for a class of wrinkling models for highly stretched elastic sheets

We consider a class of models motivated by previous numerical studies of wrinkling in highly stretched, thin rectangular elastomer sheets. The model is characterized by a finite-strain hyperelastic membrane energy perturbed by small bending energy. In the absence of the latter, the membrane energy density is not rank-one convex for general spatial deformations but reduces to a polyconvex function when restricted to planar deformations, 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 Karman model. The latter penalizes arbitrarily fine-scale wrinkling, resolving both the amplitude and wavelength of wrinkles. Here, we prove the existence of energy minima for a general class of such models.