Thermoelectric fields and associated thermal stresses for an inclined elliptic hole in thermoelectric materials

A theoretical model for the thermoelectric coupling analysis of thermal materials with an inclined elliptic hole is constructed. The extended problem of an elliptic hole under biaxial loading is also discussed. Theoretical and numerical results for the thermoelectric coupling resulting thermal stress are obtained. Results show that, for the biaxial loaded elliptic hole, the solution cannot be the linear superposition of the solutions of two uniaxial loaded cases due to the nonlinear coupling. For inclined elliptic hole, the maximum thermoelectric concentration occurs when the major axis is perpendicular to the loading direction. However, the maximum stress concentration occurs when the major axis is parallel to the loading direction. The solution of a circle hole problem is given as a special case in the framework of elliptic hole problem. The crack problem is also discussed when the ellipse degenerates into a crack. It is found that all field intensity factors exhibit the traditional inverse square-root singularity at the crack tip. The mode I stress intensity factor only depends on the applied electric current. The mode II stress intensity factor only depends on the applied energy flux when the crack line is perpendicular to the loading direction. However, in the biaxial loading case, the mode II stress intensity factor relies on the applied electric current density and energy flux. This is the first paper to conduct a strict closed-form solution for an inclined elliptic hole in thermoelectric materials.