The Upper-Bound Solution as Applied to Three-Dimensional Extrusion and Piercing Problems

Abstract

The upper-bound solution for plane-strain problems is modified and applied to three-dimensional extrusion and piercing. Johnson’s graphical solution is used to solve axisymmetric, eccentric, and two-bar extrusion problems of a lead billet. Kudo’s unit-deforming region approach is employed to solve cylindrical piercing problems of six nonferrous metals. The analytical extrusion pressure is compared with Frisch and Thomsen’s experimental data. A reasonably close agreement is obtained in the first two cases. The calculated piercing pressure is compared with the experimental results of Fukui, et al. A remarkably good agreement is observed for all six cases. Therefore the upper-bound solution after slight modification can be extended to three-dimensional problems in extrusion and piercing. A general conclusion for extending the upper-bound approach to all forming operations cannot be drawn until further study is done along this line.