Thermoelastic temperature, displacement and stress in heat transfer during laser surface hardening are solved in both Lagrangian formulation and Eulerian formulation. In the Eulerian formulation, the heat flux is fixed in space and the work-piece is moved through a control volume. In the case of uniform velocity and uniform heat flux distribution, the Eulerian formulation leads to a steady-state problem, while the Lagrangian formulation remains transient. In the Eulerian formulation, the reduction to a steady-state problem increases the computational efficiency. Also, in this study, an analytical solution is developed for an uncoupled transient heat conduction equation in which a plane slab is heated by a laser beam. The thermal results of the numerical models are compared with the results of the analytical model. A comparison of the results shows that numerical solutions in the case of uncoupled problem are in good agreement with the analytical solution.
