Postbuckling of magnetic rings

We perform for the first time postbuckling of circular magnetic rings consisting of permanently magnetized particles of the same magnetization. Employing variational approach, we first determine the buckling condition in closed form. We then apply this condition to characterize the buckling for two scenarios in which the ring is (i) compressed by dipolar loading due to a central point dipole and (ii) compressed by centrally mechanical loading. Using the concept of effective bending stiffness of a magnetic ring, we next introduce an equivalent loading parameter characterized for both loadings. We find that the critical value of equivalent loading parameter at which the circular magnetic ring buckles for the first scenario is much lower than that for the second scenario. And, in the continuum limit when the number of permanently magnetized particles is very large, the critical value for the former is four times lower than that for the latter. Moreover, the loading scenario decides buckling modes via which the magnetic ring first buckles. For the first scenario, the circular magnetic ring deforms into planar but noncircular shapes via in-plane buckling modes, regardless of ring sizes. For the second scenario, the circular magnetic ring deforms into planar but noncircular or nonplanar shapes via in-plane or out-of-plane buckling modes for, respectively, small ring sizes or sufficiently large ring sizes. Finally, a weakly nonlinear analysis shows that for a magnetic ring subject to dipolar loading, the post-buckled shape is not stable for all ring size and this result is consistent with previous experiments. For a magnetic ring subject to mechanical loading, however, the post-buckled shape is stable for a sufficiently large ring size.