Two-dimensional problem of thermoelectric materials with an elliptic hole or a rigid inclusion

The two-dimensional problems of an elliptic hole or a rigid inclusion embedded in a thermoelectric material subjected to uniform electric current density and energy flux at infinity are studied based on the complex variable method of Muskhelishvili and conformal mapping technique. The closed-form solutions of electric potential, temperature and stress components are presented according to electrical insulated and thermal exact boundary conditions on the rim of the hole or inclusion. Numerical results are carried out to illustrate the influence of the value of major to minor axis ratio of the elliptic geometry and heat conductivity of inhomogeneity on thermoelectric and stress fields. It is found that energy flux at surfaces of the hole or rigid inclusion does not vanish due to the Joule heat and Seebeck effect when the electric field is applied. In addition, stress induced by applied electric field has a non-linear relationship with the electric current density. The heat conductivity of the air inside the elliptic hole reduces the concentration factors of energy flux and stress. However, the concentration factors of energy flux and stress at the bonding interface increase with the increasing values of heat conductivity of the flat rigid inclusion.