Transition between planar and wrinkled regions in uniaxially stretched thin elastic film

TL;DR

This paper analyzes the transition from flat to wrinkled regions in uniaxially stretched thin elastic films by modeling the energy minimization problem and deriving scaling laws as the film thickness approaches zero.

Contribution

It introduces a variational framework for understanding the flat-to-wrinkled transition and simplifies the analysis through scalar constrained problems in the thin film limit.

Findings

Minimal energy characterized by a convex relaxed problem plus a linear thickness term.

Scaling law for energy derived in the vanishing thickness limit.

Simplification of the problem to scalar variational problems for thin films.

Abstract

We study the transition from flat to wrinkled region in uniaxially stretched thin elastic film. We set up a model variational problem, and study energy of its ground state. Using known scaling bounds for the minimal energy, the minimal energy can be written as a minimum of the underlying (convex) relaxed problem plus a term, which grows linearly in the thickness of the film. We show that in the limit of vanishing thickness the prefactor in the scaling law for the original problem can be obtained by minimization of simpler scalar constrained variational problems.