Universal Law of Coiling for a Short Elastic Strip Contacting Within a Tube

TL;DR

This paper establishes a universal law of coiling for short elastic strips inside tubes, identifying four deformation types and providing formulas and experiments that apply broadly across physical systems.

Contribution

It introduces a universal coiling law for short elastic strips within tubes, with theoretical formulas and experimental validation for various deformation types.

Findings

Four deformation types identified: two point-contact, three point-contact, continuous-contact, self-contact.

Universal formulas verified across different elastic properties and geometries.

Applicable to diverse physical systems like electronics, materials, and biological structures.

Abstract

We find that there exists a universal law of coiling not only for a long elastic strip contacting within a tube but also for a short one. Here the elastic strip we consider has the ratio of $2 < L/R \le 2\pi$ for its length $L$ to the tube radius $R$. By varying the ratio of $L/R$, we identify four types of deformation for such a short elastic strip, namely, two point-contact, three point-contact, continuous-contact, and self-contact. With theoretical formulas in closed forms and experimental demonstration, these four types are verified for any elastic strips contacting within a tube, irrespective of elastic properties, strip lengths, and tube radius. Our results on coiling can be readily applied to a variety of physical systems, including thin flexible electronic devices, van der Waals materials in scroll shape, and DNA packaging into viral capsids.