The magneto-elastic instability of Kirchhoff rings

Understanding the interplay between electromagnetic forces and elasticity is a fundamental problem in the mechanics of deformable structures, particularly when electrical currents interact with magnetic fields to generate distributed body forces. A paradigm for this problem consists of a twisted slender, conducting elastic filament shaped into a closed ring subjected to a uniform magnetic field perpendicular to its plane. The resulting Lorentz force induces stresses that couple non-trivially with bending and torsional deformations, thereby modifying both the stability and post-buckling behaviour of the structure. Here, we analyse a pre-twisted, current-carrying Kirchhoff elastic ring and determine how the electromagnetic loading alters the classical Michell instability. We derive an explicit expression for the critical instability threshold that includes explicitly the magnetic contribution, showing that the Lorentz force can either stabilize or destabilize the circular configuration depending on its direction. A weakly nonlinear analysis further reveals that the magnetic field controls the nature of the bifurcation, enabling transitions between subcritical and supercritical regimes. These analytical predictions are confirmed by numerical solutions of the fully nonlinear equations, demonstrating that magnetic fields provide a direct and effective means of tuning both the onset and the qualitative character of elastic instabilities in conducting rings.