A numerical application of the slip line field method to extrusion through conical dies

The industrial importance of cold extrusion is now well known. Unfortunately, the productivity of the process is restricted by the manifestation of a specific defect: the central burst, which is closely linked to ductility and to a depressive stress state in the core of material. An accurate knowledge of these facts is therefore necessary to the extension of the process. In this paper, we develop a slip line field model in order to calculate exactly the hydrostatic pressure on the axis of the system. Its peculiarity is that it can be associated with a mechanical test of material ductility. This method is developed for axisymmetric extrusion through conical dies, with a perfectly plastic material, obeying Tresca's yield criterion with its associated flow rule and with the Haar-Kàrmàn hypothesis. We assume that the pressure has a linear distribution along the die, which makes it possible to construct the slip line field. We then derive the velocity field, and determine by trial and error the pressure gradient along the die so that the velocity field satisfies all the boundary conditions. We apply this method to several extrusion ratios and die-angles; we compute the stress field in the deformed regions, obtaining an upper-bound for the extrusion pressure and a value for the hydrostatic pressure on the axis. Comparisons with experimental results show quite a good agreement between theoretical and experimental values of the extrusion pressure. The method will enable us in a future work to introduce friction along the die, metal work hardening and different profiles of the die.