In the regionx>w/2, the velocityVxequals 2V0x/h, so the rate of frictional work is W˙f =4
L/2
w/2
kVxd x =kV0(L2−w2)/h. (8.17) The strain rate in the region between w/2 ≤ x ≤ L/2 is ˙ε=2V0/h, so the rate of homogeneous work is
W˙h=2kε˙(L−w)h=4kV0(L−w). (8.18) Equating the internal work with the rate of external work, 2PLV0, 2P L V0 = W˙c+W˙23+W˙f+W˙hor
P/2k =1+3h/(4L)+L/(4h)−w/L+w2/(4h L). (8.19) The minimum occurs ifw=2hso
P/2k =1+(L/h−h/L)/4. (8.20) 8.8 PLANE-STRAIN DRAWING
Figure 8.13 shows the streamlines and hodograph ˙Wa=W˙h+W˙f+W˙r for plane- strain drawing of a sheet through a die of semi-angleα. All paths are horizontal before crossing AAand after crossing BB. All particles on a vertical line such as CChave the same horizontal component of velocity, Vx, but Vx increases from AA to BB. When a particle crosses AAit suffers a velocity discontinuity, VA∗, which depends on the distanceyfrom the centerline. At the outer surface (y=t/2),VA∗=V0tanα, and at the centerline, VA∗=0. At other points, VA∗=V0ytanα/(t0/2).
The rate of work along this discontinuity is W˙r=2
t/2
0
kV0tanα
(t0/2) ydy=kV0t0tanα/2. (8.21) The rate of sliding along the die isVs=Vx/cosα=V0t0/(tcosα). The rate of frictional work is then
W˙f= t0
tf
mkV0t0
cosαsinα(dt/t)= mkV0t0ε
cosαsinα, (8.22)
whereε=ln(t0/tf).The external work rate is ˙Wa=σdVftf =σdV0t0. Equating external and internal work rates, ˙Wa=W˙h+W˙f+W˙r, and simplifying,
σd/2k =(1+m/sin 2α)ε+(1/2) tanα. (8.23) This can be interpreted asσd =wh +wf+wrwherewh/2k=ε,wf/2k=mε/sin 2α, andwr/2k=(1/2)tanα.
The force balance produces equation8.23 without the (1/2)tanα term if µPin equation7.1is replaced bymk.
8.9 AXISYMMETRIC DRAWING
Consider drawing a rod of diameterD0 to a diameterDfthrough a die of semi-angle α with a constant interface shear stress, mk. For a slab of radius R, the horizontal
8.13.Plane-strain drawing. (a) Flow lines, (b) partial hodograph, and (c) a differential element.
component of velocity isVx=V0(R0/R)2so the sliding velocity at the interface isVx= V0(R0/R)2/cosα. The area of the element in contact with the die is 2πRdR/sinαand the interface stress is, substituting sinαcosα=(1/2) sin 2αand lnR0/Rf =ε/2,
W˙f = R0
Rf
2πmk R20V0
RsinαcosαdR=2πmk R02V0ε/sin 2α. (8.24) The velocity discontinuity, Vr∗=V0(r/R0)tanα, on entering the field depends on the radial distance,r, so the rate of energy dissipation is
W˙r= R0
0
2πr kV0(r/R0) tanαdr =(2/3)πkV0R20tanα. (8.25) The homogeneous work rate is
W˙h =(σd/2k)Rf2Vf =(σd/2k)R20V0. (8.26) Equating the rates of external and internal work,
σd/2k=(σ0/2k+m/sin 2α)ε+(2/3) tanα. (8.27)
PROBLEMS 123 Again this is equal to the slab analysis that produced equation 8.27 without the
redundant work term, (2/3)tanα. Other kinematically admissible fields may be ana- lyzed. Avitzur derived an upper bound for axisymmetric drawing that predicts slightly lower drawing stresses than equation8.27. His velocity field is more complex and the difference between his prediction and equation8.27is small.
REFERENCES
B. Avitzur,Metal Forming: Processes and Analysis, McGraw-Hill, 1968.
C. R. Calladine,Engineering Plasticity, Pergamon Press, 1969.
W. Johnson and P. B. Mellor,Engineering Plasticity, Van Nostrand Reinhold, 1973.
PROBLEMS
8.1. FindPe/2kfor Figure8.2ifθ is 80◦and compare with Figure8.3.
8.2. CalculatePe/2kfor the plane-strain frictionless extrusion illustrated in Figure 8.14. Triangles ABC and CDE are equilateral.
8.14.Upper-bound field for plane-strain drawing for Problem 8.2.
8.3. On which discontinuity in Figure8.14is the largest amount of energy expended?
8.4. Draw the hodograph corresponding to the frictionless indentation illustrated in Figure8.15.
8.15.Upper-bound field for indentation for Problem 8.4.
8.5. For the plane-strain compression illustrated in Figure8.16, calculatePe/2kfor L/Hvalues of 1, 2, 3, and 4. Assume sticking friction.
8.16.Upper-bound field for plane-strain compression for Problem 8.5.
8.6. Reanalyze Problem 8.4 if frictionless conditions prevailed.
8.7. For the indentation shown in Figure8.7,Pe/2k=2.89 if all the angles were 60◦. FindPe/2kif the angles OAB, ABC, and BCD are 90◦ and the other angles are 45◦.
8.8. Figure8.17shows an upper-bound field for a plane-strain extrusion. There are two dead metal zones ADB and FEG.
8.17.Upper-bound field for Problem 8.8.
a) CalculatePe/2kfor the field.
b) Determine the velocity inside triangle ABC.
c) DetermineVAC∗ .
d) Compute the deformation efficiency.
PROBLEMS 125
8.9. a) Use equation8.27 to find the drawing stress,σd, for an axisymmetric rod drawing (Figure 8.18) with reduction of 30%, a semi-die angle of 10◦, and a constant interfacial shear stress of 0.1k. Assume the Tresca crite- rion.
b) Predictσdusing the von Mises criterion.
8.18.Illustration of axisymmetric drawing for Problem 8.9.
8.10. Consider the upper-bound field in Figure8.19for an asymmetric extrusion.
a) Draw the corresponding hodograph.
b) Determine the angleθ.
8.19.Illustration of an asymmetric plane-strain drawing for Problem 8.10.
8.11. For the plane-strain compression illustrated in Figure8.20, calculatePe/2kfor L/hvalues of 1, 2, 3, and 4. Assume sticking friction.
8.20.Upper-bound fields for plane-strain compression in Problem 8.11.
8.12. A proposed upper-bound field for extrusion is shown in Figure 8.21. Draw a hodograph to scale and determine the absolute velocity of particles in the triangle BCD.
8.21.Upper-bound field for the plane strain extrusion of Problem 8.12.
8.13. Consider the plane-strain indentation illustrated in Figure8.22. Assume that the deformation in region AABB is homogeneous. There are discontinuities along AAand BB.
a) Write an expression forVAA∗ andVBB∗ in terms ofV0,z, andt.
b) What is the ratio of the energy expended on these discontinuities to the homogeneous work?
PROBLEMS 127
8.22.Figure for Problem 8.13.
8.14. Figure 8.23 shows two different upper-bound fields for a 2:1 reduction by extrusion. Regions ABC and EFG are dead-metal zones.
a) CalculatePe/2kfor both fields.
b) Determine the deformation efficiency,η, for both cases.
c) What is the absolute velocity of a particle in triangle JGH?
2 2
1 1
45 45
45
45
45 45
45 45
60 60
A B
C
E
D
J
H
F
G
VE
VE
V0
V0
8.23.Two proposed upper-bound fields for a plane-strain extrusion with a 50% reduction for Problem 8.14.