Analyses of Thermal Conduction and Stress Induced by Electric Current in an Infinite Thin Plate with an Elliptical Hole

Uniform electric current at infinity is applied to a thin infinite conductor with an elliptical hole disturbing the electric current, which gives rise to Joule heat, temperature increase and heat flux. Joule heat produces uniform and uneven temperature fields which in turn initiate thermal stress. These electrical current, Joule heat, temperature, heat flux and thermal stress analyses are carried out and their closed form solutions are obtained. The heat conduction problem is solved as a temperature boundary value problem. Figures of distribution of Joule heat, temperature, heat flux and stress are shown. A dislocation and a rotation terms for thermal stress analysis appear, which makes problem complex. Solutions of Joule heat, temperature, heat flux and thermal stress are nonlinear for the direction of electric current. For an infinite plate with a circular hole, stress components do not occur on the whole plate. As a special case, a crack problem is analyzed and intensities at the crack tip of each problem are investigated. Relations between melting temperature and electric current density, and between fracture toughness value and electric current density are investigated for some crack lengths for steel.