General solutions for stationary/moving plane heat source problems in manufacturing and tribology

General solutions (both transient and steady state) for the temperature rise at any point due to stationary/moving plane heat sources of different shapes (elliptical, circular, rectangular, and square) and heat intensity distributions (uniform, parabolic, and normal) are presented using the Jaeger’s classical heat source method (J.C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Proc. Royal Society of NSW 76 (1942) 203–224). Starting from an instantaneous point heat source solution, an elliptical moving heat source with different heat intensity distributions, namely, uniform, parabolic and normal, was used as the basic plane heat source and its solution for the temperature rise at any point was derived. This analysis was then extended to other plane heat sources, such as circular, rectangular, and square heat sources to cover a range of manufacturing processes and tribological problems experienced in engineering practice. In addition, the analysis presented here is valid for both transient and steady state conditions while most analyses to date are strictly for quasi-steady state conditions. The solutions for the stationary heat sources are obtained from the moving heat source solution by simply equating the velocity of sliding to zero. Further, the analysis can be used to determine the temperature distribution not only at the surface but also with respective to the depth which again is a very important consideration in most manufacturing and tribological applications since it effects the subsurface deformation, metallurgical changes, hardness variation, and residual stresses. It can also be used to determine the maximum and average temperatures within the area of the heat source. Thus, the analysis presented here is believed to be comprehensive.