Axisymmetric extrusion through adaptable dies—Part 2: Comparison of velocity fields

Abstract

In Part 1 of this series of papers, six kinematically admissible velocity fields, along with the power terms, were developed for use in upper bound models for arbitrarily shaped dies for axisymmetric extrusion. The three base velocity fields in the deformation zone were derived: (1) assuming proportional angles in the deformation zone, (2) assuming proportional areas in the deformation zone, or (3) assuming proportional distances from the centerline in the deformation zone. In each case the base velocity was modified by an additional term comprised of two functions, each function containing pseudo-independent parameters. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. In Part 2, the results obtained in upper bound models for the six velocity fields for extrusion through a spherical die are compared to one another. The velocity fields are compared based upon: (a) the base velocity field, (b) the number and distribution of pseudo-independent parameters in the flexible functions, and (c) the form of the angular flexible function. A spherical extrusion die shape is used to evaluate and compare the three velocity fields. The results demonstrate that the sine-based velocity field is the best. Furthermore, a natural boundary condition exists which allows the shear surface associated with the streamlined portion of a die to energetically disappear. Part 3 uses the best velocity field to determine an adaptable die shape, which minimizes the extrusion pressure and compares the shape to the arbitrarily curved and streamlined die shape of Yang and Han.