On a mathematical model for case hardening of steel

The applications of steel in industry are very diverse and widespread. The basic principle involved in heat treatment is the process of heating and cooling. The industrial process of case hardening aims to harden just the workpiece case, letting the inner part softer. The macrosopical model presented here takes into account the diffusion of carbon in the workpiece at austenitic phase, the slow diffusion at high temperature and the rapid cooling, which produces the formation of the martensitic microstructure. During this process, phase transformations in steel take place, influenced by the non homogeneous carbon distribution. The mathematical model presented here consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations describing the evolution of phase fractions. Existence and uniqueness of solutions are investigated and some numerical simulations are presented.