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N Hasebe, C. Bucher, R. Heuer:
"Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack";
International Journal of Solids and Structures, 47 (2010), 138 - 147.

A 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.