The \theta-formulation of the 2D elastica -- Buckling and boundary layer theory

TL;DR

This paper introduces a new ta-formulation for analyzing 2D elastica, simplifying boundary layer analysis and revealing a snapping instability in elastic rings under pressure, validated by simulations and experiments.

Contribution

The paper develops a ta-based formulation that simplifies boundary layer analysis and provides a new elementary approximation for complex elastica shapes.

Findings

Boundary layers occur at inflexion points connecting curvature regions.

A simple composite solution approximates elastica shapes without elliptic functions.

Elastic rings exhibit a snapping instability under pressure, confirmed by simulations and experiments.

Abstract

The equations of a planar elastica under pressure can be rewritten in a useful form by parametrising the variables in terms of the local orientation angle, $\theta$, instead of the arc length. This ``$\theta$-formulation'' lends itself to a particularly easy boundary layer analysis in the limit of weak bending stiffness. Within this parameterization, boundary layers are located at inflexion points, where $\theta$ is extremum, and they connect regions of low and large curvature. A simple composite solution is derived without resorting to elliptic functions and integrals. This approximation can be used as an elementary building block to describe complex shapes. Applying this theory to the study of an elastic ring under uniform pressure and subject to a set of point forces, we discover a snapping instability. This instability is confirmed by numerical simulations. Finally, we carry out experiments and find good agreement of the theory with the experimental shape of the deformed elastica.