9.6 PLANE-STRAIN EXTRUSION

Consider again the frictionless plane stain extrusion treated by upper bounds in Section 8.3 where r=50% and ?=30 . Figure 9.11(a) is the top half of the slip-line field and Figures 9.11(b) and (c) are Mohr s stress circle diagrams along OB and OC. The force balance on the die wall is shown in Figure 9.11(d).

Figure 9.11: (a) Slip-line field for a frictionless extrusion, (b) Mohr s stress circle diagram along OB, (c) Mohr s stress circle diagram along OC, and (d) force balance on die wall.

At the exit the stress ? ₁= ? _x=0 and the stress ? _y is compressive, so line OC is a ? line. On OC, ? _2OC = ?k. Rotating clockwise through ? = ??/6 on an ? line, ? _2OB= ?k+2 k( ??/6). In triangle ABO, ? _2ABO= ?k+2 k( ??/6) so P _ABO= k+2 k( ?/6). Acting against the die wall, P _? =P _ABO +k=2 k(1+ ?/6). . P _? .P _? F _x =F _? sin ?=Pe(1/2) so P _ext/2 k=(0.5) (1+ ?/6)=0.762.

The example above is a special case where the geometry is such that the slip-line field consists of a constant pressure zone and a single centered fan. Figure 9.12 shows such a field. If the entrance thickness