The thermal problem of functionally graded materials (FGM) under linear heat source is studied by a hybrid numerical method. The accuracy of the analytical method and the efficiency of the finite element method are taken into account. The volume fraction of FGM in the thickness direction can be changed by changing the gradient parameters. Based on the weighted residual method, the heat conduction equation under the third boundary condition is established. The temperature distribution of FGM under the action of linear heat source is obtained by Fourier transform. The results show that the closer to the heat source it is, the greater the influence of the heat source is and the influence of the heat source is local. The temperature change trend of the observation points is consistent with the heat source, showing a linear change. The results also show that the higher the value of gradient parameter is, the higher the temperature of location point is. The temperature distribution of observation points is positively correlated with gradient parameter. When the gradient parameter value exceeds a certain value, it has a little effect on the temperature change in the model and the heat conduction in the model tends to be pure metal heat conduction, the optimal gradient parameters combined the thermal insulation property of ceramics and the high strength toughness of metals are obtained.

In this paper, wheel-rail contact model was established and analyzed, some useful results were obtained. The results show that In the early stage of wheel-rail contact, the wheel-rail contact pressure increases gradually and the contact point shape changes from regular polygon to irregular polygon in the process of wheel approaching rail joint. The equivalent stress and elastic deformation trend of the rail are basically the same, both of which decrease rapidly and then slowly increase. The plastic deformation of the rail gradually increases and accumulates, and the stress concentration area occurs at the side of the rail joint. When the wheel starts rolling, the equivalent stress of the wheel gradually decreases at first, and then increase. The peak value of the maximum equivalent stress occurs at the joint where the wheel reaches the rail.

In this paper, a physical model of functionally graded materials (FGM) with a linear change of volume fraction of ceramic and metal in its thickness direction is established. The thermal conductivity of the FGM under a linear heat source is studied by a hybrid numerical method. Based on the weighted residual method, the heat conduction equation under a third boundary condition is studied using a hybrid numerical method which takes into account both the accuracy of the analytical method and the efficiency of the finite element method. A linear heat source is applied to the FGM, the temperature of the heat source changes linearly with time, and the temperature distribution in the space-time domain is obtained by finite element discretization in one direction and Fourier transform in the other. The results show that the closer to the heat source, the greater influence of the heat source it is. The influence of the heat source is local, which is similar to the influence of the force field on structure described by Saint Venant theory. In the heat transfer process, the heat transfer efficiency of each surface is different, that is closely related to the material properties. Using this proposed physical model, it is demonstrated that FGM can avoid the sudden change of temperature and relieve the thermal stress.