Heat Conduction and Thermal Stress Analyses of a Strip with Semi-Elliptical Notch Under Electric Current

An electric current is applied in a thin strip conductor with an edge crack or a semi-elliptical notch. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. The Joule heat gives rise to the temperature, and thus the heat flux and also free thermal expansion. Using a rational mapping function, problems of the electrical current, the Joule heat, the temperature, the heat flux and the thermal stress are analyzed and their each solution is obtained as a closed form. The heat conduction problem is solved as a temperature boundary value problem. The distributions of the electric current, the Joule heat, the temperature, and the heat flux are shown in figures. After a free thermal expansion is obtained and a support condition is given, the stress components are calculated. The crack problems are analyzed for the problems of the electric current, joule heat, heat flux and stress. The expressions for intensity of the electric current, the heat flux and the Joule heat at the crack tip are derived. The exact expression of the temperature at the crack tip (notch tip) is also given. The notch problems are analyzed, and the approximate expressions of the concentration are given for each problem, of which error is a few percent. The relations between the electric current density and the melting temperature and between the electric current density and Joule heat are investigated for some crack lengths of aluminum strip. The intensity of the Joule heat relates to the creation of the blow hole.