A model for resistance welding including phase transitions and Joule heating

Abstract

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.

A Time Discretization Scheme for a Nonlocal Degenerate Problem Modelling Resistance Spot Welding

“…where g is the derivative ofĝ, [7]. From experimental data we can observe that usually both thermal and electrical conductivities have a discontinuity in the melting point, [17].…”

“…From experimental data we can observe that usually both thermal and electrical conductivities have a discontinuity in the melting point, [17]. Actually we can assume the mixture ansatz k(u, ψ) = (1 − ψ) k 1 (u) + ψ k 2 (u) and τ (u, ψ) = (1 − ψ) τ 1 (u) + ψ τ 2 (u) with possibly different k 1 , k 2 and τ 1 , τ 2 , see [7,18]. Or alternatively we have…”