An Existence Theorem for a Class of Wrinkling Models for Highly Stretched Elastic Sheets

TL;DR

This paper proves the existence of energy minima in a class of wrinkling models for highly stretched elastic sheets, combining finite-strain hyperelasticity with small bending energy.

Contribution

It establishes an existence theorem for a broad class of models describing wrinkling in highly stretched elastomer sheets, incorporating non-convex energy densities.

Findings

Existence of energy minima for the models studied.

The model combines hyperelastic membrane energy with classical bending energy.

The energy density is polyconvex in 2D and unbounded as area ratio approaches zero.

Abstract

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.