An e$cient "nite element procedure has been developed to calculate the temperatures and stresses arising due to a moving source of heat. The procedure is applied to calculate the thermal stresses produced in hardened steels during grinding. The thermal load during grinding is modeled as a uniformly or triangularly distributed, 2D heat source moving across the surface of a half-space, which is insulated or subjected to convective cooling. The grinding of elastic and elastic}plastic workpiece materials has been simulated. The calculated transient stresses and temperatures in an elastic solid are found to be in good agreement with prior analytical and numerical results. In an elastic}plastic workpiece material, for which no analytical solution is available for the residual stress distributions, the "nite element calculations show that the near surface residual stress is predominantly tensile and that the magnitude of this stress increases with increasing heat #ux values. Based on an analysis of the e!ects of workpiece velocity, heat #ux magnitude and convective cooling, on the residual stress distributions in an elastic}plastic solid, it is seen that the calculated thermal stress distributions are consistent with experimentally measured residual stresses on ground surfaces. Furthermore, the results explain often cited observations pertaining to thermally induced grinding stresses in metals.