A variational approach to magneto-elastic buckling problems for systems of ferromagnetic or superconducting beams

Abstract

Abstract. Based upon a variational principle derived in a preceding paper, expressions for the magneto-elastic buckling values for ferromagnetic or superconducting systems are given. These relations are evaluated for systems of slender beams. Explicit buckling values are calculated for a single ferromagnetic or superconducting beam of arbitrary cross-section, and for systems of two parallel ferromagnetic or superconducting rods. In the analysis needed for the calculation of the intermediate (i.e., rigid-body) and the perturbed magnetic fields, an intensive use of methods inherent in the theory of complex functions is made. In conclusion our results for a set of two superconducting rods are compared with the results of a mathematically less complicated, but also less rigorous, theory.

A variational approach to magnetoelastic buckling problems for systems of superconducting tori

“…In the integral equations for J and iV the integration over the tangential coordinate 4> is carried out exactly, and then the integral equations are linearized with respect to £. It appears that in both cases J and iV, and therefore also the magnetic field in initial and perturbed state, are in zeroeth order in £ the same as in the case of two parallel rods (see [2]). When the currents in the two tori are equally directed (which is the technically relevant case) only these zeroeth order fields together with the elastic energy of the buckling mode, play a role in the computation of the buckling value.…”

“…In two recent papers, [1], [2], by Van Lieshout, Rongen and Van de Ven, the problem of magnetoelastic buckling was studied on the basis of a variational aproach. In [1] a variational principle, yielding explicit relations for magnetoelastic buckling values, was formulated and in [2] applications to systems of ferromagnetic or superconducting beams were presented.…”

“…In [1] a variational principle, yielding explicit relations for magnetoelastic buckling values, was formulated and in [2] applications to systems of ferromagnetic or superconducting beams were presented. As one of the results of [2], it was proved that a configuration of two equal parallel superconducting rods could become unstable, and the pertinent buckling value for the current was calculated. The mechanical stability of supenXJuJucting structures has been subject of an increasing amount of research; an excellent survey of this field is given by F. Moon in his monograph [3].…”

“…This principle is believed to be more suitable for numerical purposes, because it contains constraints on the fundamental variables which are much weaker than the constraints in [1]. As in [1], [2] and [5] we assume that the electric current is confined to the surface of the superconducting body. In order to arrive at analytical expressions for buckling values, we set up two integral equations, one for the surface current density J and one for a variable ijI, which is related to the perturbation potential ,!,.…”

“…In conclusion, we present some numerical results and we compare these results with those obtained from a mathematically less complicated, but also less rigorous, method. This method, which was also applied in [2], is based upon a generalization of the law of Biot and Savart (cf. [3], (2-6.4» .…”