Deformation of an elastic magnetizable square rod due to a uniform electric current inside the rod and an external transverse magnetic field

Mathematics and Mechanics of Solids

Obada

Abou-Dina

et al. 2020

Self Cite

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.

Section: Introductionmentioning

confidence: 99%

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

Mathematics and Mechanics of Solids

Obada

Abou-Dina

et al. 2020

Self Cite

“…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…”

Section: Equations Of Magnetostaticsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Al-Azhar Bulletin of Science

Obada

Abu-Dina

et al. 2020

We use a method recently introduced by some of the authors to obtain a semi-analytical solution to a problem of deformation of a long cylindrical conductor carrying a steady axial current, in the quasi-electrostatic approximation. The method relies on the expansion of the unknown harmonic functions arising in the process of solution in terms of Cartesian polynomial and rational harmonic functions. The normal cross-section of the conductor is taken to be nearly circular and the thermal and magnetic parts are solved independently of each other and of the elastic problem. Numerical results and plots are provided and discussed. A comparison with the case of a circular crosssection allows to assess the influence of imperfection of the cross-sectional shape on the quantities of practical interest. The effect of the dependence of the magnetic permeability on strain is investigated as well.

“…There is little work available in the literature devoted to static thermo-magnetoelasticity of long electrical conducting cylinders carrying a uniform steady current. We cite [1][2][3][4] for circular, elliptic and rectangular boundaries and the calculation of Lorentz force-induced stresses in long straight conductors carrying a uniform current, and [5][6][7][8][9] for elliptic or square boundaries by boundary integrals and under different boundary conditions including a numerical approach.…”

Section: Introductionmentioning

confidence: 99%

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

Mathematics and Mechanics of Solids

Abdusalam

Abou-Dina

et al. 2021

A method proposed earlier, relying on the use of harmonic Cartesian polynomial and rational functions, is extended here to find a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a system of long parallel, non-intersecting, transversely isotropic elastic cylindrical electrical conductors. Results are presented for two conductors of equal circular normal cross-sections carrying currents of equal densities flowing along the same direction, subjected to Robin-type thermal boundary conditions. Quantities of practical interest are represented graphically and discussed. Consideration of a system of electrical conductors is of practical importance in power plants and in various technological instruments, where it is required to assess the interaction between conductors. The obtained formulas for the magnetic vector potential may be of importance for the determination of the coefficients of self- and mutual inductance of long electric conductors, otherwise difficult to calculate by standard methods. Comparing the results with those of a single conductor allows us to assess the interaction between conductors.