A Generalized Velocity Field for Plane Strain Extrusion Through Arbitrarily Curved Dies

In this paper, plane strain extrusion through arbitrarily curved dies is investigated analytically, numerically, and experimentally. Two kinematically admissible velocity fields based on assuming proportional angles, angular velocity field, and proportional distances from the midline in the deformation zone, sine velocity field, are developed for use in upper bound models. The relative average extrusion pressures for the two velocity fields are compared to each other and also with the velocity field of a reference for extrusion through a curved die. The results demonstrate that the angular velocity field is the best. Then, by using the developed analytical model, optimum die lengths which minimize the extrusion loads are determined for a streamlined die and also for a wedge shaped die. The corresponding results for those two die shapes are also determined by using the finite element code and by doing some experiments and are compared with upper bound results. These comparisons show a good agreement.