Theoretical Analysis for the Redrawing of Cylindrical Cups through Conical Dies without Pressure Sleeves

Abstract

A theory for the redrawing of cylindrical cups through conical dies is presented. In the absence of a pressure sleeve a zone of unconstrained drawing exists prior to the conical zone and the principle of minimum energy is used to establish the extent of the zone and the associated deformation and stresses. A linearization of the von Mises yield criterion for plane stress problems is adopted and use of the flow rule associated with this criterion allows the variation of thickness across an element to be taken into account in the solution of the stress equilibrium equation; the strains can also be determined without resorting to numerical integration. Because the deformation during redrawing takes place in a series of discrete steps, work-hardening can be conveniently included in the analysis. Bending and unbending contribute significantly to the process work and a more detailed examination of the mechanism is justified. The present theory allows the effects of friction, die angle and die intersection radius to be examined but the punch load/redrawing ratio relation has been considered for only one die profile and clearance. The results for the redrawing of 70/30 brass cups with and without interstage annealing are compared with experimental redrawing loads and whilst excellent agreement exists for the near-limiting redrawing ratios, the theory considerably over-estimates the punch loads at the smaller ratios. A possible reason for the difference is discussed.