An Eshelby inclusion in a nonlinearly coupled thermoelectric bi-material and tri-material

We first derive an analytic solution to Eshelby's problem concerning a circular inclusion within one of two perfectly bonded nonlinearly coupled thermoelectric half-planes. The inclusion undergoes a prescribed uniform electric current-free thermoelectric potential gradient and a prescribed uniform energy flux-free temperature gradient. Closed-form expressions for the six analytic functions characterizing the thermoelectric fields of electric current density and energy flux in all three phases of the composite are obtained primarily with the aid of analytic continuation. A general method is then proposed for the analytic solution of Eshelby's problem of an inclusion of arbitrary shape in a nonlinearly coupled thermoelectric bi-material. Finally, we extend our ideas to the case of a tri-material by developing an analytic solution to the thermoelectric problem associated with a circular Eshelby inclusion in a nonlinearly coupled thermoelectric tri-material composed of two semi-infinite thermoelectric media bonded together through an intermediate thermoelectric layer of finite thickness.