Analytical Solution To The Problem Of Circular Hole With Multiple Cracks In Thermoelectric Material

Thermoelectric materials are considered to be the most promising technical materials,with a large thermal power,low resistivity and low thermal conductivity,which can convert waste heat into electricity to improve fuel efficiency.The intermal cracking of micro-cracks in brittle thermoelectric materials is often caused by thennal shock.Microcracks can affect thermoelectric properties and mechanical integrity.Thermoelectric materials are usually brittle semiconductor materials.Due to their inherent brittleness and low toughness,when a force,heat or electrie load i5 applied to a thermoelectric material,cracks or microcracks may occur in the material.Therefore,the analysis of defects in thermoelectric materials has become a hot topic gradually.In practical engineering,most of the defects appear in the form of multiple cracks or voids.Thus,the defect analysis of the circular holes with multi-crack in the thermoelectric material has certain prospectiveness.In this paper,the problem of thermoelectric material with circular orifices is studied by the complex function method.The first part of the solution for the collinear crack problem is to analyze the interior of the thermoelectric material under impervious boundary conditions by using the complex variable function method.The temperature field and stress field of the circular orifice problem of the collinear crack in the thermoelectric material are analyzed theoretically,and the goverming equation of the thermoelectric material is directly given.The complex intensity function theory,conformal mapping and analytical continuation method are used to further solve the stress intensity factor.The solution for the second part of the problem of circular holes with three cracks is based on the general form of the temperature field and the stress field inside the thennoelectric material.For the circular aperture problem with three cracks,the outer surface of the circular hole of the z plane is mapped to the inside of the unit circle of the ξplane by using the conformal mapping,and the analytic complex functions of the temperature field and the stress field can be respectively obtained by using the Cauchy integral formula and the boundary condition.In the third part,the problem of circular aperture with Ak periodic cracks in thermoelect:ric material is studied by using the complex variable function method.When the thermoelectric material is only subjected to the current in the y-axis direction at infinity,the temperature field and electric field inside the material are solved.The analytical solution of the stress intensity factor at the crack tip is given.And the variation of the intensity factor with the parameter is analyzed.