Uncoupled thermomagnetoelastostatics for long cylinders carrying a steady axial electric current by a boundary integral method. A numerical approach

Abstract

The static, uncoupled problem of thermomagnetoelasticity for long cylinders carrying a steady, axial current is investigated in stresses within a numerical approach by a boundary integral method in terms of real harmonic functions.This is the numerical realization of the field equations, boundary conditions and other relations presented in [1]. The method is complemented by the use of boundary collocation method to evaluate some path-independent line integrals needed in the representation of the mechanical displacement field. The material of the cylinder is assumed homogeneous and isotropic, and linear dependence of the magnetic permeability on strain is taken in consideration through two material constants. Formulae are obtained for the boundary values of functions of practical interest like stress and mechanical displacement. Evaluation in the bulk may then be carried out by quadrature on the basis of well-known formulae of the theory of potential. The special case of an elliptic boundary is treated and the results are compared to the analytical solution established in [11]. It is concluded that the proposed numerical scheme performs efficiently in this case, and may thus be used for other forms of the boundary, subject only to smoothness condition.

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

“…The constants μ 1 and μ 2 appearing in these equations express the linear dependence of the magnetic permeability of the cylinder’s material on strain, as in El Dhaba et al [15]; E is the Young modulus.…”

“…The governing equations of static, linear, uncoupled thermomagnetoelasticity are quoted here without proof, as presented by El Dhaba et al [15]. These will be solved under prescribed proper mechanical and thermal boundary conditions.…”

“…The magnetic scalar potential was used for the solution. El Dhaba et al [13][14][15] treated cases with elliptic or square boundaries using boundary integrals and under different boundary conditions, including a numerical approach. The case of a prism of square cross-section in a transverse magnetic field was treated by El Dhaba [16] using boundary integrals.…”

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

“…The governing equations of static, linear uncoupled thermo-magnetoelasticity are quoted here below without proof as presented in [10]. These will be solved under prescribed proper mechanical and thermal boundary conditions.…”

“…The deformation of long, current-carrying wires has been the subject of active research for fifty years or so. 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].…”