A variational approach to the magneto-elastic buckling problem of an arbitrary number of superconducting beams

Abstract

Based upon a variational principle and the associated theory derived in three preceding papers, an expression for the magneto-elastic buckling value for a system of an arbitrary number of parallel superconducting beams is given. The total current is supposed to be equal both in magnitude and direction for all beams, and the cross-sections are circular. The expression for the buckling value is formulated more explicitly in terms of the so-called buckling amplitudes, the latter following from an algebraic eigenvalue problem. The pertinent matrix is formulated in terms of complex functions, which are replaced by real potentials. The matrix elements are calculated by a numerical method, solving a set of integral equations with regular kernels. Apart from the buckling value(s) the buckling modes are also obtained. Finally, our results are compared with the results of a mathematically less complicated theory, i.e. the method of Blot and Savart.

The refined theory for a magnetoelastic body – II. Plane problems

“…By using Taylor series of the trigonometric differential operators in Eq. (8), it follows from the exact governing Eq. (60) that…”

“…The refined theory for plate bending problems of magnetoelasticity was discussed in our previous work [1]. It was shown that in the magnetoelastic plate problems the approximate governing equations of refined theory have the same form as those of other well-know theoretical models of magnetoelastic plates, such as the Pao-Yeh model [2] from the axiomatic system, the Eringen-Maugin model [3,4] from the rational mechanical theorem, the magnetic couple model [5] and the magnetic force model [6] from the magnetic dipole model, and the variational principle model [7,8]. This property is very helpful to show the validity of the new refined plate theory.…”

The refined theory for a magnetoelastic body–I. plate problems

“…By using Eqs (30), (31) and Taylor series of the trigonometric functions Eq. (8), after tedious manipulation, the result turns out to be…”

“…In order to describe the experimental phenomenon [4,8] and make a good understanding on characteristics for magnetoelastic interaction of ferromagnetic bodies or structures, some theoretical models were proposed by different approaches, such as the Pao-Yeh model [2] from the axiomatic system, the Eringen-Maugin model [5,6] from the rational mechanical theorem, the magnetic couple model [4] and the magnetic force model [7] from the magnetic dipole model, and the variational principle model [9,31]. According to these models, the governing equation for magnetoelastic plate is as follows:…”