Existence of a weak solution to a nonlinear induction hardening problem with Leblond–Devaux model for a steel workpiece

Abstract

In the present paper, we investigate a mathematical model for an induction hardening process of a steel workpiece. We describe the electromagnetic process by the eddy current equations, which are coupled with the energy balance in the workpiece via the heat equation and the kinetic phase transition. The nonlinearities of the system are caused by a nonlinear relation between the magnetic induction and the magnetic field, together with a nonlinear heat conduction and Joule heating. We propose a time discretization for the variational system, and perform some a priori estimates for discrete solutions. By aid of Rothe’s method and the theory of monotone operators, we show the convergence of the proposed scheme and the existence of a weak solution to the variational problems.