Deformation of a long, current-carrying elastic cylinder of square cross-section: numerical solution by boundary integrals

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

“…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.…”

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

“…The boundary of the rod is subjected to a given distribution of pressure. The parametric representation for the first quadrant of the square in a set of cylindrical polar coordinates ( r , θ , z ) with origin at the center of the square and z -axis along the axis of the rod is (see [60, 61])…”

“…Following [60] and [61], a smoothing of the square boundary (1), (2) is carried out. This is a necessary requirement for the proposed method to be efficient.…”

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

“…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].…”