Shape of an elastica under growth restricted by friction

TL;DR

This paper studies the growth and stability of elastic fibers under friction, introducing a new stability criterion, and analyzes the post-critical shapes, including figure-8 formations, with experimental validation.

Contribution

It presents a novel stability definition for growing elastic fibers under friction and develops a semi-analytical method to determine critical lengths and post-critical shapes.

Findings

Identification of a critical length for destabilization

Post-critical shapes include figure-8 configurations

Theoretical predictions agree with physical experiments

Abstract

We investigate the quasi-static growth of elastic fibers in the presence of dry or viscous friction. An unusual form of destabilization beyond a critical length is described. In order to characterize this phenomenon, a new definition of stability against infinitesimal perturbations over finite time intervals is proposed and a semi-analytical method for the determination of the critical length is developed. The post-critical behavior of the system is studied by using an appropriate numerical scheme based on variational methods. We find post-critical shapes for uniformly distributed as well as for concentrated growth and demonstrate convergence to a figure-8 shape for large lengths when self-crossing is allowed. Comparison with simple physical experiments yields reasonable accuracy of the theoretical predictions.