Explicit solutions of an elliptic hole or a crack problem in thermoelectric materials

The generalized two-dimensional problem of an elliptic hole or a crack in a thermoelectric material subjected to uniform electric current density and energy flux at infinity is investigated based on the complex variable method and the conformal mapping technique. Firstly, the exact solutions of electric potential, temperature and stress fields are presented with the assumption that the surfaces of the elliptic hole are electrically and thermally insulated. It is shown that both the concentration factors of electric current density and stress at hole rim increase as the value of major to minor axis ratio of the elliptic hole increases. Then, explicit solutions are also obtained in closed-form when the elliptic hole degenerates into a crack. It is found that all fields exhibit an inverse square-root singularity at the crack tip, and electric current density and stress intensity factors are defined according to the traditional way. The results show that the mode-I and mode-II stress intensity factors are dependent on the applied electric current density and energy flux, respectively. Furthermore, the mode-I stress intensity factor induced by Joule heating effect has a non-linear relationship with the remote electric loads.