El Sheshtawy, S., Obada, A., Abu-Dina, M., Ghaleb, A. (2020). 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, 31(Issue 2-B), 16-29. doi: 10.21608/absb.2020.40436.1081

Sayed Mohmed El Sheshtawy; A.-S. Obada; Moustafa Abu-Dina; Ahmed Ghaleb. "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, 31, Issue 2-B, 2020, 16-29. doi: 10.21608/absb.2020.40436.1081

El Sheshtawy, S., Obada, A., Abu-Dina, M., Ghaleb, A. (2020). '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, 31(Issue 2-B), pp. 16-29. doi: 10.21608/absb.2020.40436.1081

El Sheshtawy, S., Obada, A., Abu-Dina, M., Ghaleb, A. 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, 2020; 31(Issue 2-B): 16-29. doi: 10.21608/absb.2020.40436.1081

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

^{1}Mathematics Department, Faculty of Science, Al Azhar University

^{2}Department of Mathematics, Faculty of Science, Al Azhar University, Egypt

^{3}Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Abstract

Harmonic Cartesian polynomial and rational functions are used to obtain a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a long, transversely isotropic elastic cylinder of nearly-circular normal cross-section carrying an axial, steady electric current. Numerical results and plots are provided and discussed. Comparison with the case of a circular cross-section allows to assess the in uence of imperfection of the cross-sectional shape on the quantities of practical interest. The effect of dependence of the magnetic permeability on strain is investigated. We use the magnetic vector potential which is parallel to the axis of the cylinder. The mathematical problem reduces to the determination of ve harmonic functions, interrelated by a set of boundary conditions. Two of these harmonic functions combine to yield Airy's stress function in the cross-sectional domain in case the mechanical problem is solved in stresses. A third harmonic function enters in the formulation of the thermal problem for temperature distribution. The two remaining harmonic functions enter in the description of the vector potential distribution inside the conductor and in the surrounding region. The results nd application in calculating the deformation of straight central sections of electric conductors in various instruments, in particular busbars in electric power stations, and the stresses induced in them due to the electric current.