Buckling Thin Disks and Ribbons with Non-Euclidean Metrics

TL;DR

This paper investigates how non-Euclidean metrics due to swelling induce buckling in thin disks and ribbons, revealing a critical size beyond which strain cannot be fully relaxed and buckling occurs.

Contribution

It introduces a perturbative approach to analyze the buckling behavior of non-Euclidean thin membranes, linking geometry, topology, and mechanical response.

Findings

A relationship between mean curvature and wrinkle number is established.

A critical radius for buckling in disks is identified.

The approach effectively predicts buckling patterns in swollen thin films.

Abstract

I consider the problem of a thin membrane on which a metric has been prescribed, for example by lithographically controlling the local swelling properties of a polymer thin film. While any amount of swelling can be accommodated locally, geometry prohibits the existence of a global strain-free configuration. To study this geometrical frustration, I introduce a perturbative approach. I compute the optimal shape of an annular, thin ribbon as a function of its width. The topological constraint of closing the ribbon determines a relationship between the mean curvature and number of wrinkles that prevents a complete relaxation of the compression strain induced by swelling and buckles the ribbon out of the plane. These results are then applied to thin, buckled disks, where the expansion works surprisingly well. I identify a critical radius above which the disk in-plane strain cannot be relaxed completely.