Extrusion Through Spherical Dies—An Upper Bound Analysis

An upper bound solution for extrusion through a spherical die has been developed. Equations for the velocity and strain rate fields in the deformation zone are presented. The equations to determine the internal power of deformation, shear power losses along the two surfaces of velocity discontinuity and friction power losses along the die workpiece interface are shown. In order to maintain generality, these power terms have been calculated via numerical integration methods. The shear power losses and the friction power losses for the extrusion through a spherical die are of similar magnitude as for the extrusion through an “equivalent” conical die. The internal power of deformation is greater for the spherical die as compared to the conical die especially at large radius of curvatures for the spherical die. From the model the optimal die curvature can be determined which minimizes the pressure required to extrude through a spherical die. The analysis presented herein can be generalized to any axisymmetric die shape, which produces a cylindrical product from a cylindrical billet. This extension can be accomplished with minimal changes in the model.