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In this study, we conducted a theoretical analysis of thermal stress around an arbitrarily shaped hole in a thermoelectric material under electric current density and energy flux loading. Based on complex variable methods, conformal mapping, and analytical continuation theorem, the exact solutions of the thermal stress around a hole were obtained for the Seebeck coefficient and electric and heat conductivity. Based on the conversion efficiency equation of thermoelectric materials, higher electrical conductivity and lower heat conductivity should be selected to achieve an optimal design. The theoretical results indicated that higher electrical conductivity could reduce the thermal stress around the hole. However, energy flux and thermal stress concentration might be generated around the adiabatic hole due to the presence of a matrix with lower heat conductivity. Hence, thermoelectric materials with lower thermal conductivity should be selected carefully to avoid premature failure around the hole caused by thermal stress concentration. Finally, we also obtained and discussed the stress intensity factors of a hypocycloid-type crack.

In this paper, the interaction of a generalized screw dislocation with multiple circular inclusions perfectly bonded to an unbounded matrix under remote magnetoelectromechanical loadings is dealt with. The analytical solutions of electric field, magnetic field and displacement field either in the inclusions or the matrix are obtained by use of the complex variable method. The image force acting on the magnetoelectric screw dislocation is calculated by using the generalized Peach-Koehler formula. Finally, the influence of material combinations on the image force is examined for several practical examples. The obtained solutions can be used as Green's functions for the analysis of the corresponding magnetoelectric crack problem.