Quasistationary problem for a cracked body with electrothermoconductivity

In resistance spot welding two workpieces are pressed together by electrodes. Owing to the Joule effect and the high resistivity in the contact area between the workpieces, the welding current leads to an increase in temperature, until finally a weld nugget is formed (see Fig. 1). For a complete description of the process, one has to take into account mechanical, thermal and electrical effects, as well as the free boundary between liquid metal and solid. To the knowledge of the authors, mathematical models up to now have only considered the thermal and electrical effects, neglecting mechanics (see, for example, [15]). A mathematical model for the special case of impulse resistance welding has been developed in [7]. The basic equations to obtain the displacement u = (u1, u2), the temperature θ and the electric potential φ are the quasistatic balance law of momentum, the balance of internal energy as well as the quasistatic balance law of electrical charge. In the framework of isotropic linearized thermoelasticity, we can formulate the balance laws in the undeformed domain.