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

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.

Elastic buckling analysis of an embedded infinitely long rod under combined axial and torsional loads

“…For example, the critical point at which a rod under combined torsion and axial loading becomes unstable has been obtained from a linear eigenvalue analysis [25][26][27]. The helical buckling of a conducting rod under compression, twist, or whirl generated by magnetic field has also been investigated [28,29]. A general framework for studying the static and dynamic behavior of a rod under torsion and axial loading has also been proposed using the Kirchhoff rod theory [30].…”

“…Later, the effect of boundary conditions on the critical buckling load and buckling modes has been studied extensively for a rod subjected to different loadings and using different methods [27,33]. For example, it was found that under a coat hanger boundary condition, a rod buckles exclusively into a helix and the critical loads can be obtained explicitly unlike the case with welded boundary condition [29]. Unlike the twisting instability evolving from a kink to a knot for a long cylinder rod under pure torsion, wrinkles will occur on a short soft cylinder due to a geometric constraint of zero displacement of the axis of the cylinder and this phenomenon was studied by using the theory of incremental elastic deformation [34].…”

Helical Buckling of Slender Beam Structures Surrounded by an Elastic Medium

“…Under this condition, spiral-shaped helical buckling modes can occur due to the restraining effect of the surrounding medium. Helical buckling phenomena can be widely found in nature (Silverberg [5]), as well as fields of engineering (Sun and Lukasiewicz [6], Thompson et al [7], Valverde and van der Heijden [8]) and nanoscience (Huang and Pattillo [9], Foster and Harland [10], Svensek and Podgornik [11]). …”