9.26.Slip-line field for compression with sticking friction. Adapted from W. Johnson and P. B. Hellor, Engineering Plasticity, Van Nostrand Reinhold, 1973.
9.27.Average indentation pressure for the slip-line fields in Figure9.26and slab force analysis (equa- tion7.20).
9.12 PIPE FORMATION
The general variation of extrusion force with displacement is sketched in Figure9.29.
For direct or forward extrusion, after an initial rise the force drops because the friction between work material and chamber walls decreases as the amount of material in the chamber decreases. With indirect extrusion there are no chamber walls. As the ram approaches the die, there may be a drop caused by a change of direction of material flow. See Figure9.30. As material begins to flow along the die wall, a cavity or pipe may be formed. Figure9.31shows such a pipe.
9.28.(a) Slip-line field for interface stress,τ=mk, and (b) corresponding Mohr’s stress circle diagram.
NOTES OF INTEREST
W. L¨uders first noted the networks of orthogonal lines that appear on soft cast steel specimens after bending and etching in nitric acid inDinglers Polytech. J. Stuttgart, 1860. These correspond to slip lines. An example of slip lines revealed by etching
9.29.Schematic of a load-displacement diagram for direct and indirect extrusion.
NOTES OF INTEREST 149
Outside of billet moves to centerline and a pipe may form here.
9.30.Flow pattern at the end of an extrusion. This brings material on the outside of the billet to the centerline and leads to formation of pipe.
is given in Figure9.32. Figure9.33 shows other examples of slip lines on deformed parts.
There is a simple demonstration of the hydrostatic tension that develops at the center of a lightly compressed strip. If a cylinder of modeling clay is rolled back and forth as it is lightly compressed, a hole will start to develop at the center.
The first systematic presentations of the use of slip-line fields for solving practical problems were the books by Hill and by Prager and Hodge listed in the References.
9.31.Billet made by direct extrusion (a) and the pipe at the end (b). Courtesy of W. H. Durrant.
9.32.Network of lines formed by indenting a mild steel. From F. K ¨orber,J. Inst. Metals, 48 (1932), p. 317.
(a) (b)
9.33.(a) Thick wall cylinder deformed under internal pressure and (b) slip lines on the flange of a cup during drawing. From W. Johnson, R. Sowerby, and J. B. Haddow,Plane-Strain Slip Line Fields, American Elsevier, 1970.
REFERENCES
B. Avitzur,Metal Forming: Processes and Analysis, McGraw-Hill, 1968, pp. 250–74.
L. F. Coffin and H. C. Rogers,Trans. ASM, 50 (1967), pp. 672–86.
H. Ford,Advanced Mechanics of Materials, Wiley, 1963.
H. Geringer,Proc 3rd. Int. Congr. Appl. Mech., 29 (1930), pp. 185–90.
A. P. Green,Phil. Mag., 42 (1951), p. 900.
R. Hill,Plasticity, Clarendon Press, 1950.
W. Johnson and P. B. Mellor,Engineering Plasticity, Van Nostrand Reinhold, 1973.
W. Johnson, R. Sowerby, and J. B. Haddow,Plane-Strain Slip Line Fields, American Elsevier, 1970.
W. Prager and P. G. Hodge Jr.,Theory of Perfectly Plastic Solids, Wiley, 1951.
E. G. Thomsen, C. T. Yang, and S. Kobayashi,Mechanics of Plastic Deformation in Metal Processing, Macmillan, 1965.
APPENDIX
Table 9.1 gives thex and y coordinates of the 5◦ slip-line field determined by two centered fans. Figure9.34shows the slip-line field. These values were calculated from tables∗ in which a different coordinate system was use to describe the net. Here the fans have a radius of√
2 and the nodes of the fans are separated by a distance of 2 and the origin is halfway between the nodes.
∗ E. G. Thomsen, C. T. Yang, and S. Kobayashi, Mechanics of Plastic Deformation in Metal Processing(Macmillan, 1965).
Table9.1.Coordinatesofa5◦NetfortheSlip-LineFieldDeterminedbyTwoCenteredFans φαnm=nm=n+1m=n+2m=n+3m=n+4m=n+5m=n+6m=n+7m=n+8 0◦0y=1.01.08331.15841.22471.28171.32881.36601.39261.4087 x=0.00.09100.18880.29290.40230.51630.63400.75440.8767 5◦11.18261.27411.35721.43121.49511.54841.59071.62141.6399 0.00.10000.20830.32430.94720.57620.71010.84840.9897 10◦21.38311.48451.57701.65971.73201.79251.84071.87601.8975 0.00.11060.23120.36130.49990.64630.79950.95831.1218 15◦31.60501.71771.82141.91461.99632.06532.12062.16112.1861 0.00.12320.25820.40460.56170.72850.90381.08681.2760 20◦41.85191.97812.09462.20012.29292.37182.43512.48202.5108 0.00.13770.28980.45540.63390.82431.02571.23071.4562 25◦52.12832.27012.40182.52152.62722.71762.79052.84462.8781 0.00.15500.32670.51460.71810.93641.16801.41181.6665 30◦62.43902.59912.74842.88463.00563.10933.19343.25603.2948 0.00.17490.36980.58390.81661.06101.33401.61621.9119 35◦72.78972.97133.14133.29683.43563.55493.65193.72453.7696 0.00.19840.42000.66470.93141.21961.52781.85472.1985 40◦83.18743.39403.58793.76623.92574.06324.17554.25954.3121 0.00.22570.47870.75891.06551.39771.75402.13322.5331 45◦93.63943.87554.09764.30234.48594.64474.77474.87324.9335 0.00.25750.54720.86881.22191.60542.01822.45862.9243 50◦104.15614.42594.68084.91625.12815.31175.46265.57605.6472 0.00.29470.62720.99731.40461.84822.32692.83893.3828 55◦114.74705.05655.34965.62115.86596.07866.25376.3856 0.00.3380072051.14721.61792.13182.68733.2831 60◦125.42485.78076.11856.43216.71546.96227.1657 0.00.38860.82961.32231.86702.46313.1091 65◦136.20436.61447.00437.36717.69557.9820 0.00.49820.95731.52692.15842.8505 70◦147.10237.57588.02678.44708.8281 0.00.51721.10551.76582.4986 75◦158.12908.68649.20859.6961 0.00.59810.59811.27942.0455 80◦169.33759.971510.5771 0.00.69251.4827 85◦1710.72611.460 0.00.8031 90◦1812.334 0.0 (Continued) 151
Table9.2.(Continued) φαnn+9n+10n+11n+12n+13n=14n+15n+16n+17n+18 0◦01.41411.40871.39261.36291.32881.28161.22461.15841.08331.000 1.00001.12331.24561.36601.48371.59771.70711.81121.90902.000 5◦11.64631.63991.60681.58921.54491.48791.41891.33791.2455 1.13341.27181.42221.56531.70611.84341.97652.10362.2240 10◦21.90481.89751.87511.83751.78461.71631.63301.5347 1.28911.45821.62821.79791.96582.13052.29092.4452 15◦32.19462.18602.15952.11512.05221.97071.8707 1.47081.66861.86882.06942.26902.46582.6583 20◦42.52072.51072.47972.42722.35272.2555 1.68281.91412.14972.38652.62332.8574 25◦52.88952.87782.84142.77952.6913 1.93042.20142.47772.75803.0372 30◦63.30833.29443.25193.1789 2.21962.53662.86103.1901 35◦73.78533.76923.7191 2.55732.92813.3089 40◦84.33034.3114 2.91583.3859 45◦94.9584 3.4129 152
PROBLEMS 153
9.34.Slip-line field for Table9.1.
PROBLEMS
9.1. Using the slip-line field in Figure9.8for frictionless indentation it was found thatP⊥/2k=2.57. Figure9.35shows an alternate field for the same problem proposed by Hill.
a) Find P⊥/2kfor this field.
b) Construct the hodograph.
c) What percent of the energy is expended along lines of intense shear?
9.35. Slip-line field for plane-strain indentation.
9.2. Figure 9.36 shows a slip-line field with a frictionless punch. Construct the hodograph and findP⊥/2kfor this field.
9.36. Slip-line field for Problem 9.2.
9.3. Plane-strain compression of a hexagonal rod is shown in Figure9.37together with a possible slip-line field.
a) Determine whether this field or penetrating deformation will occur.
b) Find P⊥/2k.
9.37. Possible slip-line field for Problem 9.3.
PROBLEMS 155
9.4. Figure 9.38 is the slip-line field for plane-strain wedge indentation. For the volume of the side mounds to equal the volume displaced, the angleψmust be related toθ by cos(2θ –ψ)=cosψ/(1+sinψ). DetermineF/xin terms of 2kforθ=120◦.
9.38. Slip-line field for plane-strain wedge indentation.
9.5. Figure9.39shows the slip-line field for a 2:1 reduction by indirect or backward frictionless extrusion.
a) DetermineP⊥/2k.
b) Construct a hodograph for the lower half of the field.
c) FindVAD∗ /V0.
d) What percent of the energy is expended by the gradual deformation in the centered fans?
9.39. Slip-line field for the indirect extrusion of Problem 9.5.
9.6. Figure 9.40 shows the slip-line field for the 3:1 frictionless extrusion. Find P⊥/2kandη.
9.40. The slip-line field for the 3:1 frictionless extrusion in Problem 9.6.
9.7. Indentation of a step on a semi-infinite hill is shown in Figure9.41. Show that this field is not possible and draw a correct field.
9.41. An incorrect field for indentation on a semi-infinite hill.
9.8. What is the highest level of hydrostatic tension, expressed asσ2/2k, that can be induced by two opposing flat indenters as shown in Figure9.42?
PROBLEMS 157
9.42. Indentation by two opposing flat indenters.
9.9. Consider the back extrusion in Figure9.43. Assume frictionless conditions.
a) Find P⊥/2k.
b) Construct the hodograph.
9.43. Back extrusion in Problem 9.9.
9.10. Consider punching a long, thin slot as shown in Figure9.44. For punching, shear must occur along AB and CD.
a) Find P⊥/2kas a function oft.
b) If the ratio oft/wis too great, an attempt to punch will result in a plane-strain hardness indentation. What is the largest ratio oft/wfor which a slot can be punched?
c) Consider punching a circular hole of diameter, d. Assume a hardness of P/2k=3. What is the lowest ratio of hole diameter to sheet thickness that can be punched?
9.44. Slot punching.
9.11. A deeply notched tensile specimen much longer than its width and very deep in the direction normal to the drawing is shown in Figure9.45. Calculateσx/2k for the field whereσx=Fx/tn.
9.45. Deeply notched tensile specimen for Problem 9.11.
9.12. Consider a plane-strain tension test on the notched specimen in Figure9.46(a).
The notch angle is 90◦ andwy0. Using the slip-line field in Figure9.46(b), calculateσx/2kwhereσx=Fx/tn.
PROBLEMS 159
9.46. Notched tensile specimen for Problem 9.12.
9.13. If the notches in Problem 9.12 are too shallow (i.e., t0/tn is too small) the specimen may deform by shear between the base of one notch and the opposite side as shown in Figure9.47. What ratio oft0/tnis needed to prevent this?
Fx
tn
to
Fx
9.47. Alternate mode of failure for a notched tensile bar ift0/tn isn’t large enough. See Problem 9.13.
9.14. Figure9.48 shows the appropriate slip-line field for either frictionless plane- strain drawing or extrusion wherer=0.0760 andα=15◦.
a) Find the level ofσ2at point (4, 5) for extrusion.
b) Find the level ofσ2at point (4, 5) for drawing.
c) How might the product depend on whether this is an extrusion or a drawing?
9.48. Slip-line field for Problem 9.14.
9.15. Figure9.49shows two slip-line fields for the compression of a long bar with an octagonal cross section. Which field is appropriate? Justify your answer.
9.49. Two possible slip-line fields for plane-strain compression of an octagonal rod. See Problem 9.15.
PROBLEMS 161
9.16. Consider an extrusion in a frictionless die withα=30◦and such that point (2, 4) in Figure9.18is on the centerline.
a) What is the reduction?
b) CalculatePext/2k.
c) Calculateη.
d) Find the hydrostatic stress,σ2, at the centerline.
9.17. Consider the slip-line field for an extrusion with a constant shear stress along the die wall as shown in Figure9.50.
a) Label theα- andβ-lines.
b) Draw the Mohr’s circle diagram for the state of stress in triangle ACD showingP⊥,τf=mk,α, andβ.
c) Calculate the value ofm.
h0 C
CL
R 45
ht 45 A 30 30
D τf
P
9.50. Slip-line field for extrusion with a constant shear stress die interface. See Problem 9.17.
9.18. At the end of an extrusion in a 90◦ die, a nonsteady-state condition deve- lops. The field in Figure9.51(a) is no longer appropriate. Figure9.51(b) is an
9.51. (a) Slip-line field for a 2:1 extrusion. (b) Upper-bound field for the end of a 2:1 extrusion.
See Problem 9.18.
upper-bound field fort<1. Calculate and plotPext/2kas a function oftfor 0.25
≤t≤5 for the upper-bound field. Include on your plot the value ofPext/2kfor the slip-line field.
9.19. In Problem 9.17, either the slip-line field or the upper bound gives a lower solution. However, as discussed in Section 9.12, a pipe may form at the end of an extrusion. Figure 9.52 gives an upper-bound field that leads to pipe formation. CalculatePext/2kas a function oftfor 0.25≤t≤5 and compare with the solution to Problem 9.17.
9.52. An upper-bound field for pipe formation. See Problem 9.18.