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A model for resistance welding including phase transitions and Joule heating
Author(s) -
Hömberg Dietmar,
Rocca Elisabetta
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1505
Subject(s) - uniqueness , joule heating , ansatz , quasistatic process , phase transition , mathematics , nonlinear system , phase (matter) , welding , phase field models , thermistor , ohm's law , electric field , field (mathematics) , weak solution , mathematical analysis , thermodynamics , ohm , physics , mechanical engineering , mathematical physics , electrical engineering , pure mathematics , engineering , quantum mechanics
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. Copyright © 2011 John Wiley & Sons, Ltd.