an analytical approach for backward-extrusion forging of regular polygonal hollow components

Int. J. Mech. Sci. Vol. 40, No. 12, pp. 1247—1263, 1998( 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0020—7403/98 $19.00#0.00

PII : S0020–7403(98)00016–2

AN ANALYTICAL APPROACH FOR BACKWARD-EXTRUSIONFORGING OF REGULAR POLYGONAL HOLLOW COMPONENTS

M. M. MOSHKSAR and R. EBRAHIMIDepartment of Materials Science and Engineering, School of Engineering, University of Shiraz, Shiraz, Iran

(Received 13 May 1997; in revised form 16 July 1997)

Abstract—An upper-bound formula was developed to analyze the backward-extrusion forging of regularpolygon cup-shaped components. An admissible spherical velocity field has been suggested for the early stageof the extrusion process. This stage was facilitated by substitution of a nosed punch with cylindrical surfaceinstead of a flat one. The optimum nose contour was determined by the minimization technique. On the otherhand, by this method the semi-prism angle of dead zone was calculated.

For the final stage of the process, an admissible cylindrical velocity field has been applied. In any stage of theprocess, by increasing the number of polygonal sides, the upper-bound results approaches that of circular cupswhich was obtained by other investigators. The comparison of analytical results to that of experimentalinvestigations showed excellent agreement. ( 1998 Elsevier Science Ltd. All rights reserved

Keywords: backward-extrusion forging, regular polygonal hollow components.

NOTATION

di, d

odiameter of inscribed circle of polygonal section of the punch and the die, respectively

h contact length of punchJ* upper bound powerK shear yield stress of material¸ depth of diem constant friction factorN number of sides of polygonal cross-sectionP punch pressurep external pressure applied to emerging annular cross section of the cup

r, h, / general spherical coordinatesRM radius of a cylindrical surface within deformation zone and parallel to S

1and S

2RM

i, RM

ovalues of RM for S

1and S

2, respectively (see Fig. 1)

R, Z, h general cylindrical coordinatesR

i, R

ohalf values of d

iand d

o, respectively

¹i

external tractionºQ punch velocity

ºQr, ºQ h , ºQ ( velocity components in spherical coordinate

ºQR, ºQ h , ºQ Z velocity components in cylindrical coordinate

vf

exit velocity» volume of deformation zone

D*»D magnitude of velocity discontinuity¼Q

epower dissipation due to external pressure

¼Qi, ¼Q

s, ¼Q

fpower dissipation due to strain rate field, velocity discontinuity and friction, respectively

hM angle subtended with the axis by the projection of the radius vector on the plane perpendicular to theside of the cup and passing through the axis (see Fig. 3)

( angle subtended by radius vector with the plane perpendicular to the side of cup and passing throughthe axis.

a semi-prism angle that measured along the normal to a side of polygonal cupb ("n

N)

a015

optimum semi-prism angle for dead zoneaP015

optimum semi-prism angle for nosed punchp0

yield stress of material in compressioneRij

components of strain rate tensor

INTRODUCTION

With progress in cold forming, more and more components of improved quality may be producedeconomically by backward-extrusion forging. The improvements are: better surface finish, better

1247

dimensional control to close tolerances and better mechanical properties, which result from fiberringorientation and from strain hardening.

The upper-bound theorem was first formulated for rigid perfectly plastic materials, and states thatamong all the kinematically admissible strain rate fields, the actual one minimizes the power.Drucker [1] extended this theorem to include the velocity discontinuities. Thus, the expression

J*"2

J3p0 P

V

J12eRijeRijd»#P

S

KD*»iDdS#P

Sm

mKD*»iD dS!P

ST

¹i»idS (1)

is a minimum for the actual velocity field.The first term in Eqn (1) expresses internal power of deformation, due to strain rate field. The

second and third terms represent the power due to shearing over the surface of velocity discontinui-ties and the power due to friction at the interfaces between the tool and the material. The last termreflects the power due to predetermined body traction.

Literature on backward extrusion forging until 1960 is largely limited to summaries of productionexperiences. In a typical analysis, Avitzur, Bishop and Hahn applied the upper-bound approach tobackward-extrusion forging of a thin or thick-walled cup from circular disk [2, 3]. In 1976, Avitzurand his coworkers considered the process of combined backward-forward extrusion [4]. In 1981,Luo considered limitations of the impact extrusion process [5]. The manufacture of bi-layered plainbearings using cold backward extrusion has been described by Mstowski and his coworkers [6]. In1989, Shichun and Miaoquan applied the upper-bound approach to analyze the process of cup—cupaxisymmetric combined extrusion, and they estimated the average punch pressures at differentstages of deformation [7]. Recently, many complex shapes are analyzed by this method [8, 9].

In this study, the upper-bound approach is considered to analyze backward extrusion of regularpolygon cup shape components. A spherical velocity field with cylindrical surfaces of velocitydiscontinuities has been assumed during the early stage of extrusion to account for the redundantwork. A constant frictional stress is assumed on the entire die—workpiece and punch—workpieceinterfaces. The friction factors and process geometry are held constant. The optimal pyramid angle,is determined by the minimization technique. It is shown in this study that for impact extrusion ofpolygonal hollow components the cylindrical velocity field is suitable for the analysis of the finalstage.

ANALYSIS OF REGULAR POLYGONAL CUP SHAPE

¹he early stage of extrusionIn this process the workpiece is a polygonal disk. It is placed into the bottom die with a cavity of

the same dimension of side and a depth larger than the disk’s thickness. The punch advances into themetal, causing the excess material to flow through the gap between the container and the punch.

A representation of the early stage of extrusion is shown in Fig. 1. The punch and the dead zone IVmove downward at a velocity ºQ (a positive number). Zone III moves upward as a rigid body witha velocity v

f, and a pressure p may be applied to the emerging annular cross section. Both the

chamber and zone I, the material which has not yet undergone plastic deformation are stationary.A velocity field must be derived for zone II, where the material is being deformed. For a polygonaldie, punch and workpiece, the point O is the apex of the pyramid formed by extending the die walls,the boundaries of zone II, lateral sides of pyramid and the velocity discontinuity surfaces of S

1and

S2

that both are portions of cylindrical surfaces, whose axes are perpendicular to the axis of die andthrough apex O. The areas of these boundaries are all dependent on the angle a, whose valuechanges with the other process variables so that the required power is minimized.

It would be difficult to write a velocity field for zone II as the process is represented in Fig. 1,because the metal flow from zone II into zone III must be described in addition to the flow ina direction radially downward from point O. However, if v

fis subtracted from all velocities as

indicated in Fig. 1, there is no need to consider a velocity component across the boundary betweenzones II and III. This approach changes the absolute velocities, but not the relative velocities at thesurfaces of velocity discontinuities. When v

fis subtracted, the velocities in Fig. 1 is described by Fig. 2.

»elocity field. For determining the velocity of material in zone II, consider the point P moving inradial direction (see Fig. 3). The rate of flow from miniaturial point O through the differential area

1248 M. M. Moshksar and R. Ebrahimi

Fig. 1. The early stage deformation model, velocity discontinuity and frictional surfaces in the backwardextrusion of regular polygonal component.

Fig. 2. The relative velocities when vf

is subtracted.

around P (on the surface S1) is given by

d»Q "ºQ

1!R2io

cos hMcost

RM 2idhM dt where R

io"

Ri

Ro

. (2)

On the other hand, the rate of flow through a differential area perpendicular to the radius vector OPis given by

d»Q "ºQrr2dhM dt. (3)

Backward-extrusion forging of regular polygonal hollow components 1249

Fig. 3. Geometry of a portion of the deformation zone and the velocity discontinuities.

From Eqns (2) and (3), the radial component of velocity field in zone II is obtained as:

ºQr"

RM 2i

r2

ºQ1!R2

io

cos hMcost

. (4)

From the geometry of the discontinuity surfaces S1

and S2, it may be seen that

cost cos hM "cos h, sin h sin /"sin t, sin hM tan /"tant (5a)

2RMisin a"d

i, 2RM

osin a"d

o. (5b)

By substituting Eqn (5a) into Eqn (4), ºQrbecomes

ºQr"

RM 2i

r2

ºQ1!R2

io

cos h1!sin2 h sin2 /

. (6a)

From the geometry of the forming zone II, and the fact that there appears to be no reason for themetal to rotate in this zone, the circumferential components of velocity field are set at zero, i.e.

ºQ h"ºQ("0. (6b)

»elocity discontinuities on discontinuity surfaces. The velocity discontinuity on the discontinuitysurface S

1has two perpendicular components along PH and PK, respectively (see Fig. 3). Their

respective magnitudes are

ºQ1!R2

io

sin hM andºQ

1!R2io

cos hM tan t. (7)


Therefore, the magnitude of velocity discontinuity on the discontinuity surfaces S1

becomes

D*»1D"

ºQ1!R2

io

(sin2 hM #cos2 hM tan2t)1@2. (8)

From Eqn (5a), D*»1D may be formulated as

D*»1D"

ºQ1!R2

io

sin hM sec/ (1!sin2 hM sin2/)1@2. (9)

Similarly the magnitude of velocity discontinuity D*»2D on the discontinuity surface S

2is given by

D*»2D"

ºQ R2io

1!R2io

sin hM sec/ (1!sin2 hM sin2/)1@2. (10)

Figures 2 and 3 indicate that the velocity discontinuity on the discontinuity surface S3

is given by

D*»3D"ºQ

rDhM /a;r/R

M @#04t .

Therefore

D*»3D"

RM 2i

RM 2ºQ

1!R2io

cos a(1#sin2 a tan2/)1@2

. (11)

As Fig. 2 also shows, the relative velocities at punch—workpiece (S4) and die—workpiece (S

5) interfaces

become

D*»4D"

ºQ R2io

1!R2io

and D*»5D"

ºQ1!R2

io

. (12)

Strain rate field. From the spherical velocity field given by Eqns (6a) and (6b) and the relationbetween strain rates and velocity components in spherical coordinates, the components of the strainrate tensor in the deformation zone II, are calculated as

eR hh"eR(("!

1

2eRrr"

RM 2i

r3ºQ

1!R2io

cos h1!sin2 h sin2 /

,

eRrh"

1

2

RM 2i

r3ºQ

1!R2io

sin h(cos2 h sin2 /!cos2/)

(1!sin2 h sin2/)2,

(13)

eRr("

RM 2i

r3ºQ

1!R2io

sin h cos h sin/ cos/(1!sin2 h sin2 /)2

,

eR h("0.

Internal power of deformation. In zones I, III and IV, no plastic deformation takes place and henceno power is consumed. In zone II, where the deformation occurs, the internal power of deformationis obtained as (see Section A-1 of Appendix A)

¼Qi"¼Q

c#¼Q

p, (14)

where

¼Qc"p

0d2iN

ºQ1!R2

io

lnCdo

diD f

c(a, b) (15a)

and

¼Qp"p

0d2iN

ºQ1!R2

io

lnCdo

diD f

p(a,b). (15b)

The values of functions fc(a,b) and f

p(a,b) are obtained as

fc(a,b)"GC!

67

144!

11

72cos2 aDsin~1(cos a sinb)!

11

144cos a sinbJ1!cos2 a sin2b#

25

36bH

1

sin2 a


and

fp(a,b)"C

cos2b sinb cos a

3J1!cos2 a sin2b!

1

3sin~1(cos a sin b)!

1

6sin 2b#

1

3bD

1

sin2 a.

Power loss due to velocity discontinuity. Assuming Mises yield criterion, the power loss on anydiscontinuity surface is expressed by

¼Qs"P

S

KD*» DdS. (16)

For the discontinuity surfaces S1, S

2and S

3with the velocity discontinuities given by Eqns (9)—(11),

the power loss becomes

¼Qs"P

S1

KD*» D dS1#P

S2

KD*» DdS2#P

S3

KD*» D dS3

(17)

or

¼QS"¼Q

S1#¼Q

S2#¼Q

S3. (18)

The solution of the above integrals yields (see section A.2 of Appendix A)

¼QS1"¼Q

S2"

1

8p0d2iN

ºQ1!R2

io

f1(a, b) (19)

and

¼QS3"p

0d2iN

ºQ1!R2

io

lnCdo

diD f

2(a, b), (20)

where

f1(a,b)"

csc2 a

32J3 GlnCtanAn4#

b2BD (44a#sin 4a!24 sin 2a)#sinb sec2 b (20a!sin 4a!8sin 2a)H

and

f2(a,b)"

cot a csc a

2J3ln[sin a tanb#J1#sin2 a tan2 b].

Power loss due to friction. By assuming a constant friction factor m along the tool material

interfaces S4

and S5

for Mises materials a frictional shear stress q may be defined as q"mp0/J3

where 0)m)1. Therefore, the power loss due to friction effect on these surfaces becomes

¼Qf"P

Sf

qD*» DdS"PS4

mp0

J3D*»

4DdS

4#P

S5

mp0

J3D*»

5D dS

5(21)

or

¼Qf"¼Q

f4#¼Q

f5. (22)

The solution of the above integrals yields (see Section A.3 of Appendix A)

¼Qf4"

mp0

J3

ºQ R2io

1!R2io

MdoN tan b[¸#(R

o!R

i)cot a]N (23)

and

¼Qf5"

mp0

J3

ºQ1!R2

io

(dihN tanb). (24)


Power loss due to external traction. The power loss due to the external pressure p that may beapplied to the emerging annular cross section of the hollow polygonal is given by

¼Qe"P

ST

p DvfDdS

T(25)

where

DvfD"

ºQ R2io

1!R2io

.

Hence, the external traction power is expressed as

¼Qe"

d2o!d2

i4

ºQ R2io

1!R2io

pN tanb (26)

¹he relative extrusion pressure. The upper bound on power is the summation of the termscontributing to the total power, i.e.

J*"¼Qi#¼Q

s#¼Q

f#¼Q

e. (27)

Since the total power requirement is supplied by the punch,

¼Q%95"

d2i4

PºQ N tan b (28)

Now equating these two and using Eqns (14)—(26), the relative extrusion pressure in the early stage ofextrusion forging of the hollow regular polygonal components is given by

P

p0

"

4 cotb1!R2

ioGlnC

do

diD Cfc(a, b)#f

p(a,b)#f

2(a, b)D

#

f1(a,b)

4#

m tanb

J3 Ch

di

#

¸

do

#

do!d

i2d

o

cot aDH#p

p0

. (29)

Optimum prism angle. The total power of deformation consists of seven terms out of which onlyfive (i.e. ¼Q

i, ¼Q

s1, ¼Q

s2, ¼Q

s3and ¼Q

f4) are functions of a. The effect of semi-prism angle on the internal

power of deformation is rather moderate. With low values of a, the area of the discontinuity surfacesS3

and the die—workpiece interface S4

are increased, and thus the values of ¼Qs3

and ¼Qf4

areincreased. Whereas high values of a causes to increase the S

1and S

2surfaces. Thus, there exists an

optimal semi-prism angle, a015

, in which the total power of extrusion becomes minimum. For a fixedgeometry and friction, a method of calculation of the optimum prism angle is to differentiate theexpression of the relative extrusion pressure and set the derivative equal to zero (see Section A.4 ofAppendix A). In the case of polygonal components it leads to

a015

"S 3 tanbGlnCdo

diD#m

do!d

idoH

tanb secb#ln[secb#tan b]. (30)

The expression can be used with a good approximation in the range 0—45°. For nosed punch,optimum prism angle is given by

ap015

"S 6 tanbGlnCdo

diD#m

do!d

idoH

(1#m)Mtanb secb#ln[secb#tanb]N. (31)

¹he final stage of extrusionWhen zone II with the velocity ºQ reaches the bottom of the chamber the representation of metal

flow described above is no longer valid. For this stage as it is shown in Fig. 4 the workpiece is dividedinto two zones, in which the plastic deformation takes place only in zone I.


Fig. 4. The final stage deformation model.

The velocity field in zone I is the same as the velocity field for forging of a solid disk with nobulging. By using volume constancy

2NR¹ tanb ºQR#NR2 tanbºQ "0, hence ºQ

R"!

R

2¹ºQ (32a)

and

2NRZ tanb ºQR#NR2 tanb ºQ

Z"0, hence ºQ

Z"

Z

¹

ºQ . (32b)

Since the problem is axisymmetric, ºQ h"0The velocity of zone II, v

fwhich is moving as a rigid body is determined by volume constancy as

follows:

vf"

ºQ R2io

R2io!1

. (33)

¹he contour of surface S6. The contour of discontinuity surface S6

will be determined through useof the velocity fields for the two zones which it separates. The normal components of the velocitiesmust be equal at any point of discontinuity surface S

6. Thus, S

6is the locus of the points at which

ºQRcos c#ºQ

Zsin c"v

fsin c; therefore

cot c"!

dZ

dR"

vf!ºQ

ZºQR

. (34)

Substituting for ºQZ

and ºQR

and rearranging, the following differential equation is obtained.

2¹

ºQdR

R"

dZ

vf!(Z/¹)ºQ

. (35)

The following solution results:

CR2Cvf!Z

¹

ºQ D"1 (36)

Assuming that S6

extends from the bottom corner of the punch to the bottom corner of the cavity,one can use the boundary condition that R"R

o/cos/ at Z"0 to evaluate C. When the expressions


for C and vf

are substituted into Eqn (36), the following equation for discontinuity surface S6

willresult:

R

Ro

"

sec/

J1!M1![Ro/R

i]2NZ/¹

(37)

Final stage power. In zone II of the final stage, no plastic deformation takes place and hence nopower is consumed. In zone I, where the deformation occurs, the components of strain rates in thecylindrical coordinate are calculated as

eRRR

"eR hh"!

1

2eRZZ"!

ºQ2¹

. (38)

Therefore, the internal power of deformation is obtained as (see Appendix B)

¼Qi"

1

2p0d2i

N tanbºQ

1!R2io

lnCdo

diD. (39)

The power loss on the velocity discontinuity surface S6

becomes

¼Qs6"

2p0

3J3ºQ N tanbC

1

R3io

!1DGR3

i2¹

#2Ro¹C

1

R2io

!1D~2

H. (40)

The loss of power by friction effect on tool-material interfaces S7—S

10becomes

¼Qf"¼Q

f7#¼Q

f8#¼Q

f9#¼Q

f10, (41)

where

¼Qf7"

mp0NR3i

6J3¹ºQ [secb tan b#ln(sec b#tanb)],

¼Qf8"

mp0NR3o

6J3¹ºQ [sec b tanb#ln(secb#tanb)],

¼Qf9"

mp0

J3

1

1!R2io

2RiN tanbhºQ ,

¼Qf10

"

mp0

J3

R2io

1!R2io

2RoN tanb¸ºQ .

In the final stage, the upper bounds on power is found by summation of Eqns (39)—(41). Equating thisto the external power of Eqn (28), the relative extrusion pressure in this stage is expressed by

P

p0

"

2

1!R2io

lnCdo

diD#

2

3J3R2iC

1

R3io

!1DGR3

i2¹

#2¹Ro C

1

R2io

!1D~2

H#

mb

6J3¹

R3i#R3

oR2

itanb

[secb tanb#ln(secb#tan b)]#2m

J3

1

1!R2ioC

h

Ri

#

¸

RoD#

p

p0

. (42)

RESULTS AND DISCUSSION

In order to test the model developed above, a computer program was used to solve the equationsobtained by the method. Figure 5 summarizes the results obtained by the method for six-sidedregular polygonal hollow component, several wall thickness and friction conditions at the beginningof the process, when the friction surfaces are still small. For R

o/R

i*1.2 (or reduction in area

r)70%) a weak influence of friction is observed. For instance, with m"0.1 (typical value forwell-lubricated condition) the relative extrusion pressure is only 17% less than that of stickingfriction m"1. This indicates that the power dissipation by friction in this instance is relatively smallwhich shows the proper tool design with little tool—material interfaces. For 1(R

o/R

i(3 (or

100%'r'11%) and all friction factors 0.1—1, the minimum relative punch pressure is observed atabout R

o/R

i"1.4 which corresponds to reduction in area of about 50%. For smaller R

o/R

i(larger


Fig. 6. The effect of reduction and friction on punch pressure at the middle of the process.

Fig. 5. The effect of reduction and friction on punch pressure at the beginning of the process.

reductions) the extrusion pressure is increased steeply because of friction effect and for larger Ro/R

i(smaller reductions) it is again increased due to redundant work of non-homogeneous deformationor power dissipation by velocity discontinuity surfaces.

Figure 6 shows again the effect of friction and reduction of area on the punch pressure, but for thesituation that the punch has moved 8 mm into the metal. Now, strong influence of friction isobserved. For instance, for sticking friction condition the relative extrusion pressure increases byabout 75% with respect to m"0.1.

Figure 7 shows for m" 0.14 the difference between considering hollow polygonal componentswith different sides numbers. For R

o/R

i(1.75 (or r'33%), as the number of sides is decreased, the

relative extrusion pressure prediction is increased because of domination of friction effect byincreasing the area of contact. For R

o/R

i'1.75, the nature of the process is reversed, although for

N'6 the shape effect is quite negligible. With increasing number of sides (e.g. N"100), the


Fig. 8. Comparison of the results to other investigators.

Fig. 7. The effect of punch shape on punch pressure.

extrusion force calculated by Eqns (29) and (40) are reduced to that obtained by Avitzur [1, 2] forcircular shape components, as shown in Fig. 8.

In an experimental attempt (using cold double backward-extrusion process) the friction factorm was measured to be about 0.14. Figure 9 illustrates the relationship between punch force andpunch stroke for a fixed geometry and friction factor m"0.14. For comparison, the results of anexperimental test with the same condition is also recorded in this figure. In the early stage ofextrusion the punch force is increased with increasing punch penetration, that is depending on thedie wall height (¸), friction factor (m) and punch-material contact length (h). In the final stage (punchstroke'12 mm) the punch force is increased steeply with reducing bottom thickness of theworkpiece and excessive force is required to progress the punch movement.

In the model, it was shown that when plastic zone II reaches the bottom of the chamber, theproposed spherical velocity field is no longer valid and it must be replaced by a two-zone cylindricalvelocity field for the final stage. Figure 9 clearly shows that the force predictions of the model in bothstages agree satisfactorily with the results obtained under experimental conditions.

The effect of dead zone prism angle on the relative extrusion pressure for different punch sides,fixed reduction in area and friction condition is shown in Fig. 10. For the given geometry, the relative


Fig. 9. Comparison of theoretical and experimental results.

Fig. 10. The effect of dead zone prism angle on punch pressure.

extrusion pressure decreases with increasing number of punch sides due to both the frictional andthe discontinuity surface S

3portions of the total relative extrusion pressure. However, for all

punches the dead zone prism angle plays a significant role in the relative extrusion pressure. In fact,the dead zone prism angle is a good guide principle for proper design of nosed punches. For instance,Fig. 10 shows the relative extrusion pressures approach to their minimum values for the dead zoneprism angle between 30 to 40°. It means, for this geometry, the contour of the nosed punch should bemade with the prism angle in the range 30—40°.

CONCLUSION

A kinematically admissible spherical velocity field has been proposed for the early stage of thebackward extrusion forging of regular polygonal hollow components. For the final stage ofextrusion (i.e. when the plastic zone reaches the bottom of the chamber), an admissible cylindricalvelocity field has been applied. From the velocity fields the velocity discontinuities, the upper-boundpower and relative extrusion pressure were found and the configuration of the nosed punches have


been determined by minimizing the total relative extrusion pressure with respect to the dead zoneprism angle. For a fixed geometry and friction condition, the optimum reduction in area for gettingthe minimum extrusion pressure was determined and the effect of friction and punch shape havebeen investigated. For a circular punch the results showed good agreement when compared to thatobtained by other investigators. The theoretical prediction of the extrusion force is in reasonableagreement with the experimental results. Thus, the velocity fields proposed can be used convenientlyfor the prediction of the extrusion load and the configuration of the nosed punch in the backward-extrusion forging of regular polygonal hollow components.

Acknowledgements—Financial support by The Office of Research Council of Shiraz University through grant number74-EN-921-542 is appreciated.

REFERENCES

1. Drucker, D. C. and Providence, R. I., Coulomb friction, plasticity and limit loads, Journal of Applied Mechanics, 1954, 21,71—74.

2. Avitzur, B., Bishop, E. D. and Hahn, W. C. Jr., Impact extrusion upper bound analysis of the early stage. Journal ofEngineering for Industries, ¹ransaction of ASME, 1972, 94, 1079—1086.

3. Hahn, W. C. Jr., Avitzur, B. and Bishop, E. D., Impact extrusion upper bound analysis of the end of the stroke. Journal ofEngineering for Industries, ¹ransaction of ASME, 1973, 95, 849—857.

4. Avitzur, B., Hahn, W. C. Jr. and Mori, M., Analysis of combined backward-forward extrusion, Journal of Engineering forIndustries, ¹ransaction of ASME, 1976, 98, 438—445.

5. Luo, Z. J. and Avitzur, B., Limitations of the impact extrusion process, International Journal of Machines ¹ool DesignResearch, 1982, 22, 41—56

6. Mstowski, J., Montmitonnet, P. and Delamare, F., Geometrical optimization of bi-layered plain bearings manufactured bycold backward co-extrusion. Journal of Mechanics and working ¹echology, 1986, 13, 291—302.

7. Shichun, W. and Miaoquan, L., A study of cup-cup axisymmetric combined extrusion by the upper bound approach, (I.upper-bound solutions for the deformation force). Journal of Mechanics and ¼orking ¹echnology, 1989, 18, 63—84.

8. Yang, D. Y., Kim, Y. U. and Lee, C. M., Analysis of center-shifted backward extrusion of eccentric tubes using roundpunches, Journal of Materials Processes ¹echnology, 1992, 33, 289—298.

9. Bae, W. B. and Yang, D. Y., An upper-bound analysis of the backward extrusion of tubes of complicated internal shapesfrom round billets. Journal of Materials Processes ¹echnology, 1993, 36, 157—173.

APPENDIX A

A.1 Internal power of deformationFrom the first term of Eqn (1), the internal power of deformation is

¼Qi"

2

J3p0P

V

J12eRijeRij

d» (A1)

Substitution of the values of eRij

from Eqn (13) into Eqn (A1) yields

¼Qi"2p0RM 2i

ºQ1!R2

ioPV

J1!1112

sin2 hr3(1!sin2/ sin2 h)

d». (A2)

No approximations on the values of h and / have been made in deriving Eqn (A2). Now, neglecting sin4/ sin4 h, we get

¼Qi"¼Q

c#¼Q

p, (A3)

where

¼Qc"2p0RM 2i

ºQ1!R2

ioPV

J1!1112

sin2 hr3

d» (A4)

and

¼Qp"2p0RM 2i

ºQ1!R2

ioPV

sin2 h sin2/J1!1112

sin2 hr3

d» (A5)

where

d»"r2 sin h dhd/ dr (A6)

The limits of variables r, h and / for half-segment O@AEFCOA of deformation zone II are (see Fig. 3): for r fromRM

i/J1!sin2 h sin2/ to RM

o/J1!sin2 h sin2/; for h from 0 to cos~1[cos a/J1#sin2 a tan2/] and for / from 0 to b"n/N.


Fig. A1. Geometry of the velocity discontinuity surfaces.

Substitution of Eqn (A6) into Eqn (A4) and integrating w.r.t. r and simplification of terms in the square root sign, gives

¼Qc"4Np0RM 2i

ºQ1!R2

io

lnCRM

oRM

iD PP sin h(1!11

24sin2 h) dhd/. (A7)

By integrating w.r.t. h and / and substituting Eqns (5a) and (5b), ¼Qccan be simplified to Eqn (15a).

Similarly, substituting Eqn (A6) into (A5) and integrating w.r.t., r the ¼Qp

is obtained as

¼Qp"4Np0RM 2i

ºQ1!R2

io

lnCRM

oRM

iDPP sin3 h sin2/J1!11

12sin2 h dhd/. (A8)

Simplifying the square root term in Eqn (A8) and neglecting sin5 h terms, we get

¼Qp"4Np0RM 2i

ºQ1!R2

io

lnCdo

diDPP sin3 h sin2/dhd/. (A9)

With integration of Eqn (A9) w.r.t. h and /, ¼Qp

can be calculated as Eqn (15b).

A.2. Power loss due to velocity discontinuitiesOn the surface discontinuity S

1, the power loss becomes

¼Qs1"P

s1

KD*»1DdS

1, (A10)

where the magnitude of velocity discontinuity D*»1D is given by Eqn (9). The geometry of this surface is shown in Fig. A1. The

elemental area dS1

is

dS1"RM

idhM dy where y"RM

isin hM tan/ and dy"RM

isin hM sec2/ d/,

hence

dS1"RM 2

isin hM sec2/d/ dhM , (A11)

where the limits for half-segment O@@CF are: for h from 0 to a and for / from 0 to b.


Fig. A2. Geometry of the velocity discontinuity surface S3.

Now substituting Eqns (9) and (A11) into Eqn (A10), the power loss on the discontinuity surface S1

becomes

¼Qs1"

2p0

J3

ºQ1!R2

io

NR2i P

a

0Pb

0

sin2 hM sec3/J1!sin2 hM sin2/ d/dhM (A12)

Simplification of the square root term, by assuming (1!x)1@2"1!(1/2)x, yields

¼Qs1"

2p0

J3

ºQ1!R2

io

NR2i P

a

0Pb

0

(sin2 hM sec3/!12sin4 hM sec3/ sin2/) d/ dhM (A13)

Integration of Eqn (A13) with reference to h and /, yields ¼Qs1

as Eqn (19).The dissipation power on the S

2is calculated to be the same as S

1. Hence,

¼Qs2"¼Q

s1.

Hint: For nosed punch, ¼Qs1"m¼Q

s2.

The power loss on the discontinuity surface S3

is expressed as

¼Qs3"P

s3

KD*»3DdS

3, (A14)

where (see Fig. A2) dS3"dRM dy and y"RM tant"RM sin a tan/; hence dy"RM sin a sec2/d/. Therefore

dS3"RM sin a sec2/d/ dRM . (A15)

The limits of variables for the surface of half-segment of ACFE are: for RM from RMito RM

oand for / from 0 to b. By substituting

Eqns (11) and (A15) into Eqn (A14) we get

¼Qs3"

2p0

J3

ºQ1!R2

io

RM 2iN P

b

0PR1 0

RM i

cos a sin a sec2 /

RM J1#sin2 a tan2/dRM d/. (A16)

Integrating Eqn (A16) w.r.t. RM and /, the value of ¼QS3

is obtained as Eqn (20).

A.3. Power loss due to frictionIn Eqn (21) the elementary area on the interfaces S

4and S

5are

dS4"R

otan b dZ and dS

5"R

itanbdZ (A17)

where for the interface S4, Z varies from (R

i!R

o) cot a to ¸, but for the interface S

5from 0 to h, respectively. Substituting

Eqns (12) and (A17) in Eqn (21), ¼Qf4

and ¼Qf5

are calculated as Eqns (23) and (24).

A.4. Optimum prism angleIn order to minimize the relative extrusion pressure in the early stage of extrusion,

LLaA

P

p0B"0. (A18)


Fig. B1. Geometry of the velocity discontinuity surface S6.

Hence

lnCdo

diDM f @

c(a, b)#f @

p(a,b )#f @

2(a,b )N#1

4f @1(a,b)!

m tanb

J3

do!d

i2d

osin2 a

"0. (A19)

The terms f @c(a, b ) and f @

p(a, b) are negligible compared to other terms. For differentiation of f @

1(a, b ) and f @

2(a, b ) the following

assumptions are made:

sina+a, a cot a+1!a2

3

and

ln[sin a tan b#J1#sin2 a tan2b]+ln[1#sin a tanb#12sin2 a tan2b]+sin a tanb.

thus

f @2(a, b)+!

tanb

2J3 sin2 a+!

tanb

2J3a2

and

f @1(a, b)+

2

3J3 Gtanb secb#lnCtanAn4#

b2BDH"

2

3J3Mtanb secb#ln[secb#tan b]N.

Substituting in Eqn (A19), it yields

C!tanb

2J3a2D lnCdo

diD#

1

6J3Mtan b secb#ln[secb#tanb]N!

m tan b

2J3a2

do!d

ido

"0.

Then, the optimum prism angle of dead zone is calculated as Eqn (30).

APPENDIX B

The internal power of deformation for the final stage is obtained by substituting the values of eRij

from Eqn (36) into Eqn(A1). It yields

¼Qi"p0

ºQ¹ P

V

d», (B1)

where d»"R2cos2/tanb dZ and R is evaluated from Eqn (37). Thus,

¼Qi"!

p0ºQ N tanb¹ P

0

T

R2odZ

1![1!1/R2io][Z/¹]

. (B2)

The solution of the above integral yields Eqn (39).


For the calculation of velocity discontinuity power, the magnitude of velocity discontinuity on the discontinuity surfaceS6

(see Fig. B1) is obtained as

D*»6D"v

fcos c#ºQ

Rsin c!º

Zcos c. (B3)

From the geometry of this surface, the elemental area is calculated as

dS6"!2NR cos/ tanb

dZ

cos cor dS

6"2NR cos2/

dR

sin cd/. (B4)

Substitution of Eqns (B3) and (B4) into Eqn (A10) yields

¼Qs6"

2p0

J3

RM 2ioºQ

1!R2io

N tan b P0

T

R cos/dZ!

p0NºQ

J3¹ Pb

0PR0@#04(

Ri@#04(

R2 cos/dRd/#

2p0ºQ

J3¹N tan b P

0

T

ZR cos/dZ. (B5)

Using Eqn (37), the solution of Eqn (B5) will result in Eqn (40).For frictional power, the elementary area on frictional interfaces S

7to S

10are obtained as

dS7"dS

8"R d/dR, dS

9"R

itanb dZ and dS

10"R

otan bdZ. (B6)

The corresponding relative velocities are, respectively,

D*»7D"D*»

8D"!

RºQ2¹

, D*»9D"

ºQ1!R2

io

and D*»10

D"R2

ioºQ

1!R2io

. (B7)

Now, as before, assuming a constant frictional stress along the tool—material interfaces and substituting the above equationsin Eqn (21), the frictional power ¼Q

ffor the final stage is obtained as Eqn (41).


an analytical approach for backward-extrusion forging of regular polygonal hollow components

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