The 3-d problem of temperature and thermal flux distribution around defects with temperature-dependent material properties

The analytical solution of 3-D heat conduction problem, including the temperature and thermal flux fields, is one of the important problems that have not been com-pletely solved in solid mechanics. Considering the temperature dependence of ma-terial parameters makes the problem more difficult. In this paper, we first reduce the 3-D temperature-dependent heat conduction problem to the solution of 3-D Laplace equation by introducing the intermediate function. Then, the generalized ternary function is proposed, and the general solution of 3-D 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 3-D 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 dis-tribution will re-adjusted so that it overflows at the same rate from all parts of the planar defect boundary.