A CLASS OF UPPER-BOUND SOLUTIONS FOR THE EXTRUSION OF SQUARE SHAPES FROM SQUARE BILLETS THROUGH CURVED DIES

An upper-bound analysis is proposed for the extrusion of square sectio ns from square billets through curved dies having prescribed profiles. Kinematically-admissible velocity fields for the purpose are derived using the dual-stream-function technique. Analytical results are prese nted for both frictionless and sticking friction conditions; for the l atter situation the die geometry has been optimised with respect to ap propriate parameters. It is shown that a cosine-shaped die with zero e ntry and exit angles yields the lowest extrusion pressure in the absen ce of friction, whilst the best upper-bound is provided by a straight tapered die under sticking-friction conditions. The internal work of d eformation, however, is still found to be minimum for a straight die f or frictionless extrusion.