Equilibrium of Kirchhoff's rods subject to a distribution of magnetic couples

The equilibrium of magneto-elastic rods, formed of an elastic matrix containing a uniform distribution of paramagnetic particles, that are subject to terminal loads and are immersed in a uniform magnetic field, is studied. The one-dimensional nonlinear equations governing the problem are deduced from a three-dimensional model and are fully consistent with the Kirchhoff's theory in the sense that they hold at the same order of magnitude. Exact solutions of those equations in terms of Weierstrass elliptic functions are presented with reference to magneto-elastic cantilevers that undergo planar deformations under the action of a terminal force and a magnetic field whose directions are either parallel or orthogonal. The exact solutions are applied to the study of a problem of the remotely controlled deformation of a rod and to a bifurcation problem in which the end force and the component of the magnetic field parallel to the undeformed rod axis can be regarded as imperfection parameters; the stability of the different solutions possible under the same load is assessed by comparing the corresponding energies.