Equilibrium of Kirchhoff's rods subject to a distribution of magnetic couples

TL;DR

This paper develops nonlinear equilibrium equations for magneto-elastic Kirchhoff rods with paramagnetic particles, providing exact solutions and analyzing their deformation and bifurcation behavior under magnetic fields and terminal loads.

Contribution

It introduces exact solutions for magneto-elastic rods using Weierstrass elliptic functions, extending Kirchhoff's theory to magneto-elastic interactions with applications in remote control and bifurcation analysis.

Findings

Exact solutions in terms of elliptic functions for magneto-elastic rods.

Analysis of deformation under parallel and orthogonal magnetic fields.

Bifurcation analysis with magnetic field as a parameter.

Abstract

The equilibrium of magneto-elastic rods, formed of an elastic matrix containing a uniform distribution of paramagnetic particles, that are subject to terminal loads and are immersed in a uniform magnetic field, is studied. The deduced nonlinear equilibrium equations are fully consistent with Kirchhoff's theory in the sense that they hold at the same order of magnitude. Exact solutions of those equations in terms of Weierstrass elliptic functions are presented with reference to magneto-elastic cantilevers that undergo planar deformations under the action of a terminal force and a magnetic field whose directions are either parallel or orthogonal. The exact solutions are applied to the study of a problem of remotely controlled deformation of a rod and to a bifurcation problem in which the end force and the magnetic field act as an imperfection parameter and a bifurcation parameter, respectively.