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A method is proposed to calculate the maximum temperature of the surface of a piecewise-homogeneous half-space heated by a uniformly moving, locally distributed heat flow. Analytical solutions of the corresponding quasistationary heat-conduction problems are obtained for small and large values of the Peclet number. These solutions are used to derive formulas for calculating the maximum temperature in the case of intermediate (moderate) values of the Peclet number.Introduction. Increased interest in solving quasistationary heat-conduction problems is motivated by the formulation of thermal friction problems in [1][2][3][4][5] and studies of other authors. In such formulations, a moving heat flow distributed in the contact area is specified on the working surface of each of the elements of the friction pair. The intensity of this frictional heat flow it is equal to the specific friction power -the product of the relative sliding velocity of the bodies by the shear stress. The latter, in turn, is proportional to the pressure obtained as a result of solution of the corresponding contact problem. In this case, it is customary to use a uniform or elliptic (Hertz) distribution of the contact pressure.This computational scheme is used, in particular, to determine the flash temperature at the sites of frictional contact between the surface protrusions of rubbing bodies [6]. Because the spots of contact have small sizes, the corresponding thermal friction problems are formulated for a semi-infinite body (half-space) whose surface is subjected to a uniformly moving frictional heat flow specified in a bounded region [2,4,7,8]. Analytical solutions of such problems have been obtained for two limiting values of the sliding velocity: stationary and high-velocity. In the latter case, in the heat-conduction equation, the second derivative of the temperature with respect to the independent variable in the sliding direction is ignored [5]. In the case of intermediate (moderate) values of the Peclet number (Pe), solutions are obtained using interpolation methods based on constructing a priori formulas, which in particular cases coincide with the well-known stationary and high-velocity solutions [9, 10].The approach described above is used in the present work to obtain a solution of the three-dimensional quasistationary thermal-conduction problem for a piecewise-homogeneous body consisting of a layer applied onto the surface of a half-space. In solving thermal friction problems, the main objective is to obtain a restriction on the maximum permissible temperature level; therefore, emphasis is placed on constructing engineering formulas for calculating the maximum temperature of compound bodies.1. Formulation of the Problem. We consider a piecewise-homogeneous half-space which consists of a layer of finite thickness d applied onto the surface of a semi-infinite body. A heat flow distributed with intensity q in a square with side a moves at constant velocity V on the free surface of the layer (Fig. 1). We assume that the thermal contact...

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