Magneto-elastic buckling of an Euler beam

We consider nonlinear elastic deformations of a magneto-elastic beam, using a combined experimental and theoretical approach. In the experiments, a beam had one end clamped with a magnet attached at its free end. When it was placed in an external magnetic field, it was susceptible to Euler beam buckling. However, the classic supercritical bifurcation associated with this buckling became subcritical when an attracting magnet was introduced in close proximity to the beam. To understand these experiments, we develop a model that couples the Euler elastica and dipole magnetic interactions with a uniform external field. The numerical model captures the observed behaviour well and shows that the supercritical magnetic field strength depends almost exclusively on elastic properties of the beam and strength of the permanent magnet, whereas the subcritical behaviour also depends on the separation distance between the attracting pair of magnets. We examine the bifurcation behaviour of the nonlinear system and show that for sufficiently small inter-magnet separation distances, other buckled states coexist with the fundamental mode.