Three-dimensional problem of temperature and thermal flux distribution around defects with temperature-dependent material properties

The analytical solution of three-dimensional heat conduction problem, including the temperature and thermal flux fields, is one of the important problems that have not been completely solved in solid mechanics. Considering the temperature dependence of material parameters makes the problem more difficult. In this paper, we first reduce the three-dimensional temperature-dependent heat conduction problem to the solution of three-dimensional Laplace equation by introducing the intermediate function. Then, the generalized ternary function is proposed, and the general solution of three-dimensional Laplace equation is given. Finally, the analytical solutions of three specific problems are obtained and the corresponding temperature-thermal flux fields are discussed. The results show that the thermal flux field of three-dimensional temperature dependent problem is the same as the classical constant thermal conductivity approach result, while the temperature field is different from the classical result. Thermal flux at a planar defect boundary has r-1/2 singularity, and its intensity is proportional to the fourth root of defect width. On the other hand, when blocked by a planar defect, the thermal flux distribution will re-adjusted so that it overflows at the same rate from all parts of the planar defect boundary.