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An axial ly symmetric quasistatic thermoelasticity problem on the indentation of a flat - ended cylindrical punch with a constant temperature at its base into the functionally - graded half - space which elasticity modulus, Poisson ratio, heat conductivity and expansion coefficients are independently continuously varying in the boundary layer, is considered. Out of the contact area, the surface is perfectly thermally - insulated and stress - free. The earlier solution, obtained through the combined numerical and analytical ap proach (using Hankel integral transform and the modulating fun ction method) to the unmixed pro b lem on the arbitrary thermomechanical effect upon the inhomogeneous in depth thermoelastic half - space, is applied to solve the problem. The original problem is r educed to the system of dual integral equations. The properties of the dual integral equations kernel transforms allow applying a well - grounded bilateral asymptotic technique which is being actively developed at present. The approximate express ions for d et ermining the thermal flux, the half - space surface displacement, and the contact stresses under the heated stamp base, are obtained with the aid of this method. The numerical values of contact stresses for various cases of the thermomechanical properties va riation in the boundary layer of the half - space are provided. The cases either when values of the thermomechanical coating prope r ties are the same as those of the substrate, or when the property value differs twice (upward or dow n ward) on the surface, and linearly decreases (or goes up) in depth to the value in the substrate, are con sidered.