The fracture behavior of three cracks originating from a circular hole in a thermoelectric material is studied. The basic theory of thermoelectric materials is given under the condition of the impermeable boundary of the heat flux and electric current. By introducing the conformal mapping function, the infinite plane on the physical plane is mapped into the inner unit circle on the mathematical plane. The formulas of the relevant temperature and stress fields are derived, and the analytical solutions of the complex stress functions are obtained by the complex variable function method. According to Cauchy integral formula, the analytical expressions of the electric current intensity factor and the stress intensity factor for three cracks originating from a circular hole are obtained. The effects of the hole radius and the crack lengths on the electric current intensity factor and stress intensity factors were investigated.

研究了热电材料中源自圆孔的三个裂纹的断裂行为。在热通量和电流的不可渗透边界条件下给出了热电材料的基本理论。通过引入共形映射函数，可以将物理平面上的无限平面映射到数学平面上的内部单位圆中。推导了相关温度和应力场的公式，并通过复变函数法获得了复应力函数的解析解。根据柯西积分公式，得到了三个圆孔裂纹的电流强度因子和应力强度因子的解析表达式。