A three-phase elliptical inhomogeneity in nonlinearly coupled thermoelectric materials

We use complex variable techniques to derive a rigorous closed-form solution to the nonlinearly coupled thermoelectric problem associated with a three-phase elliptical inhomogeneity with two confocal elliptical interfaces when the matrix is subjected to uniform remote electric current density and uniform remote energy flux. Explicit expressions for each of the six analytic functions characterizing the thermoelectric fields in the three phases are obtained. The analysis indicates that the internal temperature exhibits a quadratic distribution within the elliptical inhomogeneity.