Self-contact in a buckled elastica

TL;DR

This paper investigates the mechanics of a buckled elastica under self-contact, deriving a scale-invariant condition for contact onset and analyzing post-contact configurations across multiple modes.

Contribution

It introduces a novel, boundary-condition independent criterion for self-contact initiation and explores post-contact behaviors in various buckled elastica modes.

Findings

A scale-invariant condition for contact onset is derived.

Multiple post-contact configurations are computed for modes three to ten.

Infinite force is required to transition from point to line contact in certain configurations.

Abstract

We explore the mechanics of a terminally loaded buckled elastica under frictionless self-contact. With the aid of two integrals associated with the elastica, we propose a scale-invariant condition necessary for the onset of contact. The condition is independent of the boundary conditions, does not involve the position vectors of the material points, and delivers the value of the compressive load at which self-contact initiates. Furthermore, we show that one of the two integrals, namely the \emph{Hamiltonian}, persists after contact. We compute post-contact configurations of modes three through ten for a pinned-pinned buckled elastica. At a given value of the compressive load, we report multiple post-contact configurations for modes eight and nine. Finally, we show that an infinite force is required to transition from a point contact to a line contact in symmetric post-contact configurations of odd modes.