Kirchhoff's analogy for a planar ferromagnetic rod

TL;DR

This paper uses Kirchhoff's kinetic analogy to analyze equilibrium configurations of planar ferromagnetic rods under magnetic fields, revealing bifurcations and localized solutions not seen in purely elastic rods.

Contribution

It extends Kirchhoff's analogy to ferromagnetic rods, identifying new bifurcation phenomena and localized solutions influenced by magnetic fields.

Findings

Identified subcritical and supercritical pitchfork bifurcations under different magnetic fields.

Predicted equilibrium configurations for free and boundary-conditioned ferromagnetic rods.

Discovered localized solutions from homoclinic and heteroclinic orbits absent in elastic rods.

Abstract

Kirchhoff's kinetic analogy relates the equilibrium solutions of an elastic rod or strip to the motion of a spinning top. In this analogy, time is replaced by the arc length parameter in the phase portrait to determine the equilibrium configurations of the rod. Predicted equilibrium solutions from the phase portrait for specific boundary value problems, as well as certain localized solutions, have been experimentally observed. In this study, we employ the kinetic analogy to investigate the equilibrium solutions of planar soft ferromagnetic rods subjected to transverse and longitudinal external magnetic fields. Our analysis reveals a subcritical pitchfork bifurcation in the phase portrait of a ferromagnetic rod subjected to transverse external magnetic field as the axial load is decreased continuously from a large compressive load. Similarly, a supercritical pitchfork bifurcation is observed in the case of longitudinal external magnetic field. We predict equilibrium configurations for a free-standing soft ferromagnetic elastic rod and the same subjected to canonical boundary conditions. Furthermore, we observe novel localized equilibrium solutions arising from homoclinic and heteroclinic orbits, which are absent in the phase portraits of purely elastic rods.