WIAS Preprint No. 1496, (2010)

A model for resistance welding including phase transitions and Joule heating


  • Hömberg, Dietmar
  • Rocca, Elisabetta

2010 Mathematics Subject Classification

  • 80A22 35K55 35M10 35B65


  • Phase transitions, thermistor, welding, weak solutions, well-posedness results, regularity results


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 modelling 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 quasistatic electric charge conservation law. We prove existence of weak solutions in the 3D case, while the regularity result and the uniqueness of solution is stated only in the 2D case. Indeed, uniqueness for the three dimensional system is still an open problem.

Appeared in

  • Math. Methods Appl. Sci., 34 (2011) pp. 2077--2088.

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