An upper bound solution to extrusion of circular billet to circular shape through cosine dies

Metal forming by extrusion is one of the widely used metal forming processes. Conventionally the extrusion is carried out using the shear faced dies. In these dies the metal is forced to go through abrupt change in cross-section. Hence, they suffer from the practical problems such as formation of dead metal zone, non-uniform flow of metal, more redundant work and designed based on empirical methods. Modifications have been done in the extrusion dies to incorporate gradual reduction in the area of cross-section in order to ensure smooth flow of metal and to dispense with the problems faced by the conventional dies. Such modified dies are known as streamlined extrusion dies. The profile of the streamlined extrusion dies is the crucial parameter to optimize the extrusion process. Many profiles such as third order polynomial equation, fifth order polynomial equation, Bezier curve, etc., have been suggested for the design of streamlined extrusion dies with the view to reduce the extrusion load for the given reduction ratio. In this work the extrusion die is assumed to have the cosine profile to extrude circular billet to circular shape. The plastic deformation work required to extrude circular billet to circular cross-section through cosine profile based is determined using upper bound solution. It has been proved that the die designed based on cosine profile is superior to the conventional shear dies and the straightly converging dies. It is also proposed to validate the results by the experiments.