A model for resistance welding including phase transitions and Joule heating

In this paper, we introduce a new model for solid–liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase-field model and the consideration of phase-dependent physical parameters through a mixture ansatz. The PDE system resulting from our modeling approach couples a strongly nonlinear heat equation, a non-smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with a quasistatic electric charge conservation law. We prove the existence of weak solutions in the three-dimensional (3D) case, whereas the regularity result and the uniqueness of solution is stated only in the two-dimensional case. Indeed, uniqueness for the 3D system is still an open problem.