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

J. Valverde & G.H.M. van der Heijden

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.
keywords: rod mechanics, Kirchhoff equations, magnetic buckling,
Hamiltonian-Hopf bifurcation, helical solutions

*International Journal of Non-Linear Mechanics* **47**, 38-53 (2012)