The mathematical model of heating process for a friction system made of functionally graded materials (FGMs) was proposed. For this purpose, the boundary-value problem of heat conduction was formulated for two semi-spaces under uniform sliding taking into consideration heating due to friction. Assuming an exponential change in thermal conductivities of the materials, the exact, as well as asymptotic (for small values of time), solutions to this problem were obtained. A numerical analysis was performed for two elements made of ZrO2–Ti-6Al-4V and Al3O2–TiC composites. The influence of the gradient parameters of both materials on the evolution and spatial distributions of the temperature were investigated. The temperatures of the elements made of FGMs were compared with the temperatures found for the homogeneous ceramic materials.

A mathematical model for evaluation of the temperature mode of the disc–pad system during single braking is proposed. The model is based on the thermal problem of friction formulated for two semi-infinite bodies, compressed with pressure increasing over time while reducing the sliding velocity from the initial value to zero at the stop. The exact solution to this problem was obtained by means of Duhamel’s theorem. Validation of the solution was performed by achieving in special cases parameters of known solution to this problem with constant pressure and velocity (under uniform sliding). The results of the numerical calculations are presented for a selected friction pair, made of functionally graded materials with titanium alloy (disc) and aluminum alloy (pad) cores coated with ceramics graded toward friction surfaces. For the established values of the parameters such as the rise time in pressure and the FGM gradients, the ability to quickly obtain spatiotemporal temperature distributions in the disc and pad was presented. The influence of the variability of these parameters on the maximum temperature of the brake system was also investigated.

The model of the frictional heating process during single braking to determine the temperature of the functionally graded friction elements with an account of the thermal sensitivity of materials was proposed. The basis of this model is the exact solution of the one-dimensional thermal problem of friction during braking with constant deceleration. The formulas approximating the experimental data of the temperature dependencies of properties of the functionally graded materials (FGMs) were involved in the model to improve the accuracy of the achieved results. A comparative analysis was performed for data obtained for temperature-dependent FGMs and the corresponding data, calculated without consideration of thermal sensitivity. The results revealed that the assumption of thermal stability of FGMs during braking may cause a significant overestimation of temperature of the friction pair elements.

A mathematical model for determining the temperature distribution in the system consisting of a coating deposited on the surface of substrate was proposed. The foundation material is homogeneous, while the coating is made of a functionally gradient material (FGM) with thermal conductivity increasing exponentially along the thickness. Heating processes of the outer surface of the coating were considered with a constant and linearly decreasing in time intensity of the heat flux. Such thermal loads are common in thermal problems of friction, particularly regarding frictional heating during braking. An exact (in quadrature) solution of the corresponding boundary-value problems of parabolic heat conduction was obtained. Asymptotic solutions to these problems were also found for small and large values of the Fourier number. Calculations were performed for a coating made of two-component FGM ZrO2—Ti-6Al-4V, applied on a cast iron substrate. In order to explain the effect of FGM on temperature, corresponding analysis was carried out for the coating made of a homogeneous (ZrO2) material.

An analytical model was developed to determine the temperature of friction coupling, in which one element was made of a functionally graded material (FGM) and the other was homogeneous. First, for such a system, the boundary–value problem of heat conduction was formulated with consideration of the heat generation due to friction. Then, using the Laplace integral transform, an exact solution to this problem was obtained for uniform sliding, and braking with constant deceleration. A numerical analysis was performed for the selected friction pair consisting of the FGM (zircon dioxide + titanium alloy) and cast iron. It was established that the use of elements made of a FGM consisting of ZrO2 and Ti-6Al-4V can significantly reduce the maximum temperature achieved in the friction system.