`Journal of Applied MathematicsVolume 2006, Article ID 17936, 24 pageshttp://dx.doi.org/10.1155/JAM/2006/17936`

## A one-dimensional spot welding model

1Department of Mathematics and Statistics, Oakland University, Rochester 48309-4478, MI, USA
2Department of Mechanical Engineering, Oakland University, Rochester 48309-4478, MI, USA

Received 3 July 2006; Revised 3 November 2006; Accepted 22 November 2006

Copyright © 2006 K. T. Andrews et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A one-dimensional model is proposed for the simulations of resistance spot welding, which is a common industrial method used to join metallic plates by electrical heating. The model consists of the Stefan problem, in enthalpy form, coupled with the equation of charge conservation for the electrical potential. The temperature dependence of the density, thermal conductivity, specific heat, and electrical conductivity are taken into account, since the process generally involves a large temperature range, on the order of 1000 K. The model is general enough to allow for the welding of plates of different thicknesses or dissimilar materials and to account for variations in the Joule heating through the material thickness due to the dependence of electrical resistivity on the temperature. A novel feature in the model is the inclusion of the effects of interface resistance between the plates which is also assumed to be temperature dependent. In addition to constructing the model, a finite difference scheme for its numerical approximations is described, and representative computer simulations are depicted. These describe welding processes involving different interface resistances, different thicknesses, different materials, and different voltage forms. The differences in the process due to AC or DC currents are depicted as well.