Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack

Abstract

a b s t r a c tA uniform electric current at infinity was applied to a thin infinite conductor containing an elliptical hole with an edge crack. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. These two states must be considered to analyze the heat conduction problem. The uneven Joule heat gives rise to uneven temperature and thus to heat flux, and to thermal stress.Using a rational mapping function, problems of the electric current, the Joule heat, the temperature, the heat flux, the thermal stress are analyzed, and each of their solutions is obtained as a closed form. The distributions of the electric current, the Joule heat, the temperature, the heat flux and the stress are shown in figures.The heat conduction problem is solved as a temperature boundary value problem. Solving the thermal stress problem, dislocation and rotation terms appear, which complicates this problem. The solutions of the Joule heat, the temperature, the heat flux and the thermal stress are nonlinear in the direction of the electric current. The crack problems are also analyzed, and the singular intensities at the crack tip of each problem are obtained. Mode II (sliding mode) stress intensity factor (SIF) is produced as well as Mode I (opening mode) SIF, for any direction of the electric current. The relations between the electric current density and the melting temperature and between the electric current density and SIF are investigated for some crack lengths in an aluminum plate.

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

“…The mapping function which maps the exterior of the elliptical hole containing an edge crack in the w-plane to the exterior of the unit circle in the f-plane shown in Fig. 2 is given by the following equation (Hasebe and Chen, 1996;Hasebe et al, 2010a):…”

“…The electric current analysis was stated in (Hasebe et al, 2010a). Therefore, only the results are stated in the present paper.…”

“…The intensity was investigated in (Hasebe et al, 2010a). It is noticed from (21) that the magnetic field intensities H x (x, y, z), H y (x, y, z) have the same singularity as that of the electric current at the crack tip (Hasebe, 2009b where r is the distance from the crack tip and h is the angle measured in the counterclockwise direction.…”

“…Electric current gives rise to electromagnetic field, and then causes electromagnetic force, Joule heat, temperature increase, heat flux and thermal stress. In the previous paper, analyses of electric current, Joule heat, temperature, heat flux and thermal stress caused by steady state electric current in a thin infinite plate containing an elliptical hole (Hasebe et al, 2009a) and with an edge crack (Hasebe et al, 2010a) and a strip with an edge notch or crack (Hasebe, 2010b) were reported. The magneto elastic stress caused by the magnetic field induced by electric current in a thin infinite plate with an elliptical hole is analyzed by Hasebe et al (2009a).…”

The Generalized Two Dimensional Thermal-Electro-Elastic Solution for the Cracked-Half-Elliptical-Hole Problem in a Half Plane

“…More recently, Gao et al [6] derived the exact solution for the problem of an elliptical hole or a crack in a thermopiezoelectric solid by using the complex variable method. Hesabe et al [7] used a rational mapping function to analyze the problems of the electric current, the Joule heat, the temperature, the heat flux and the thermal stress induced by an electric current in an infinite thin-plate, containing an elliptical hole with an edge crack. Gao and Wang [8] studied the 2D problem of thermopiezoelectric materials with cracks by means of Parton assumption.…”