When two bodies slide against each other, the resulting frictional heating leads to thermoelastic stresses that can play an important part in the mechanism of wear and surface fatigue failure. The contact pressure and heat input distributions are influenced by the thermal distortion of the bodies and vary in time as a result of wear. To investigate thermally induced failures in such systems, we therefore require a general numerical algorithm for calculating subsurface stresses for any arbitrary distribution of heat input in time and space. Green's Junctions for the surface values of displacement and temperature for the half-plane were given in the literature. In this paper, we derive appropriate Green's Junctions for the stresses and temperatures at a general point below the surface of the half-plane. Particular attention is paid to the presentation of the results in a form convenient for efficient numerical computations.