Upper Bound Solutions for Extrusion with Two-Stage Conical Die and Curved Die

Abstract

The upper bound solutions for the axi-symmetric extrusions with two-stage die and curved die were studied. For the two-stage extrusion the radial flow was used as the admissible velocity field. The optimum shape of the surface of velocity discontinuity was given as a sphere having its center at a certain position on the extrusion axis. The extrusion pressure and the distribution of strain in the extruded billet were calculated based on the optimum velocity field. Discussions were made on the effects of die angle and extrusion ratio of each stage on the calculated values. The upper bound solution for curved die (sigmoidal die) was obtained adopting a flow function, where the shape of die section was expressed with Fourier series. The extrusion pressure and the distribution of strain in the extruded billet was calculated. The hydrostatic extrusion of copper was made and the extrusion pressure and the distribution of Vickers hardness in the extruded billet were measured. The experimental results showed good agreement with the calculated values.