Semi-analytical solution by Cartesian harmonic polynomials for a problem of deformation of a long, current-carrying elastic cylindrical conductor

Abstract

Harmonic Cartesian polynomial and rational functions are shown to provide a simple way of obtaining a semi-analytical solution to the uncoupled, two-dimensional problem of thermomagnetoelasticity for a long, transversely isotropic, elastic cylinder carrying an axial, steady electric current. The proposed method involves the solution of a difficult inhomogeneous biharmonic equation for the stress function, and may be invariably used for general geometries of the normal cross-section of the cylinder, for various thermal and mechanical boundary conditions, and also to solve the problem of a long non-conducting cylinder in a transverse magnetic field. At all stages of the solution, there is no need to smoothen the cross-sectional contour in the presence of singular points, as for other methods. Results are provided for the elliptic and rectangular normal cross-sections under the Dirichlet thermal condition and prescribed uniform normal boundary displacement. Quantities of practical interest are represented graphically and discussed. The results may be of interest in determining the deformations occurring in current-carrying metallic parts of structures and machines, and in the central regions of busbars in electric power plants.

A problem of thermo-magnetoelasticity for a system of parallel electrical conductors by Cartesian harmonic polynomials and rational functions

“…The present work extends a previous paper [11] to the case of multiple parallel, non-intersecting conductors in which the steady axial currents may be flowing in any one of two possible directions. Its aim is to provide simple means for evaluating the deformation and stresses occurring in these conductors under Joule heat and their own magnetic field, in addition, possibly, to other external factors.…”

Cartesian harmonic polynomials for a problem of deformation of a long, current-carrying elastic cylindrical conductor of nearly-circular normal cross-section

“…and being the harmonic parts of the vector potential inside and outside the domain respectively, and is the expression for the vector potential far away from the cylinder's axis, giving the magnetic vector potential of a straight, infinite current-carrying wire, as may be verified in standard books of Electrodynamics, in the form: (11) The boundary conditions for the magnetic problem illustrate the continuity of the tangential component of the magnetic field and the normal component of the magnetic induction. They are expressed as [8]: (12) (13) and the vanishing behavior at infinity…”

“…Our problem involves five harmonic functions, related by boundary conditions, to be determined: Functions for the temperature, , for the magnetic vector potential in the whole space, and , for the stress function. These functions will be expanded in terms of Cartesian polynomial and rational harmonics in a way explained in [13]. These harmonic functions contain a number of coefficients, to be determined by the method of boundary collocation to satisfy the boundary conditions of the problem.…”

“…We only cite the following references by Ghaleb [4] for the circular boundary, Ayad [5] for the elliptical boundary, Abou-Dina and Ghaleb [6] for a boundary integral formulation of the problem, Deviatkin [7] for cylinders of elliptic or narrow rectangular cross-sections, El Dhaba et al [8,9,10] for the elliptic or square boundaries by boundary integrals, including a numerical approach, El Dhaba [11] and El Dhaba and Ghaleb [12] for a rod with square normal cross-section in an initially uniform transverse magnetic field by boundary integrals. Recently, the authors have investigated the elliptic and the rectangular contours under the Dirichlet thermal condition and uniform normal extension on the boundary [13]. Other work relating to the mechanical-thermal-electromagnetic interaction may be found in the literature in a different context.…”

“…The basic equations and accompanying conditions are presented in brief, more details may be found in [13]. Numerical results are provided and discussed.…”