A mathematical model for induction hardening including mechanical e ects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic e ects that lead to the heating of the workpiece as well as thermomechanical e ects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical e ects caused by the phase transitions. We take care of e ects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L 1 data. We prove existence of a weak solution to the complete system using a truncation argument.