Helical buckling of a whirling conducting rod in a uniform magnetic field

TL;DR

This paper investigates how a uniform magnetic field influences the buckling and whirling behavior of a conducting elastic rod, deriving explicit critical loads and analyzing secondary instabilities with novel boundary conditions.

Contribution

It introduces exact helical post-buckling solutions for a conducting rod under magnetic influence with new boundary conditions, and provides explicit formulas for critical loads and secondary instability analysis.

Findings

Explicit critical load formulas derived

Secondary instabilities identified via Hopf bifurcations

Helical solutions generated through magnetic and non-magnetic buckling

Abstract

We study the effect of a magnetic field on the behaviour of a conducting elastic rod subject to a novel set of boundary conditions that, in the case of a transversely isotropic rod, give rise to exact helical post-buckling solutions. The equations used are the geometrically exact Kirchhoff equations and both static (buckling) and dynamic (whirling) instability are considered. Critical loads are obtained explicitly and are given by a surprisingly simple formula. By solving the linearised equations about the (quasi-)stationary solutions we also find secondary instabilities described by (Hamiltonian-)Hopf bifurcations, the usual signature of incipient `breathing' modes. The boundary conditions can also be used to generate and study helical solutions through traditional non-magnetic buckling due to compression, twist or whirl.