Upper-bound analysis of square-die forward extrusion

Journalof'

Materials

Processing Technology ELSEVIER

Journal of Materials Processing Technology 62 (1996) 242 - 248

Upper-bound analysis of square-die forward extrusion D.K. Kim a, J.R. Cho b, W.B. Bae c'*, Y.H. Kim c aResearch and Development Center, Korea Heavy Industries and Construction Co., Ltd., Changwon, South Korea bSchool of Mechanical System Engineering, Korea Maritime University, Pusan, South Korea ~Engineering Research Center Jot Net Shape & Die Manafactar#zg, Pusan National University, San 30, Jan~on-dong, Kemnjong-gu, Pusan 609-735, South Korea Received 3 July 1995

Industrial summary

The present study is the preliminary research for the derivation of the generalized velocity field in three-dimensional square-die forward extrusion. As a fundamental model, the square-die forward extrusion of circular-shaped bars from regular polygonal billets is chosen and a simple kinematically-admissible velocity field is proposed. From the proposed velocity field, the upper-bound extrusion load, the velocity distribution and the average length of the extruded billets are determined by minimizing the total power consumption with respect to chosen parameters. Experiments have been carried out with hard-solder billets at room temperature. The theoretical predictions of the extrusion load are in good agreement with the experimental results and there is also a reasonable agreement in the average extruded length between theory and experiment. Ke),word~: Square-die forward extrasion; Circular-shaped bars; Polygonal billets

I, Introduction

Square-die forward extrusion has been applied to the manufacture of complicated sections, since this process has many advantages as compared with other conventional methods in terms of product quality, production rate and production cost [1]. During the square-die extrusion of complicated shapes, the metal flow developed is non-uniform over the cross section of the billet. In order to achieve proper die design and process control, systematic analyses are required for predicting the extrusion load and the metal flow effectively. For square-die extrusion, there have been a considerable number of reports related to the finite-element method [2,3], upper-bound method [4-6], slip-line method [7], and other experimental methods [8,9]. Most of the previous studies are concerned with the two-dimensional forward extrusion at the initial stage, but for the proper design of the processes, the non-steady-state analysis of the final stage is more important. However, little attempt is made in analyzing the final-stage square-die forward extrusion of three-dimensional * Corresponding author. 0924-0136/96/$15,00 © 1996 Elsevier Science S.A. All rights reserved SSDI O924~O136(95)O2220.X

shapes, the reason being that the difficulty of analysis of non-steady-state flow and the remeshing problem. Therefore, we need to develop an analytical method for the final-stage square-die forward extrusion of generalized shapes. The present study is the preliminary phase for the derivation of the generalized velocity field in three-dimensional square-die forward extrusion. As a basic model, the square-die forward extrusion of circular-shaped bars from regular polygonal billets is chosen. Experiments are carried out with the hard-solder billets at room temperature for various area reductions, number of axes of symmetry and friction factors; the experimental results are compared with theoretical results. 2. Theoretical development The three-dimensional forward-extrusion process of circular-shaped bars from regular polygonal billets is modeled in Fig. 1. As shown in the figure, the cylindrical coordinates are considered such that the origin lies at the centroid of the bottom surface of the regular-polygonal billet and also such that the 0 = 0 plane lies on the plane of geometrical symmetry. The punch and the container are

D.K. Kim et aL Journal of Materials Processin~ Teclmology 62 (1996) 242 245'

considered as rigid bodies. Element I1| is treated as a pseudo-rigid body [10]. The material undergoes plastic deformation in element ] and element H. The working material is assumed to be isotropic, incompressible and rigid-plastic obeying the yon Mises flow rule.

'

I

1

243

<)%~,

./ , ,

e°~>iq..

2. I. Derivation o f the kinematically u~b~Hssibte celocity field 2,1.1. Element I

The axial velocity component, Uz, which considers velocity boundary conditions, can be expressed as follows: Uz = _ u oT z

(I)

tzi

t

Uo

-- ~

.4 ( R -

Ri)t'¢Ol(O )

i

~.~-

where T is the b o t t o m thickness of the extruded billet. The tangential velocity component, Uo, which considers velocity b o u n d a r y conditions, can be expressed as follows: UO=

:luo )':J O) ~l (% .I r~II c. rl q '. , ,o.1<~-7.: r, I~s-~-'~ r :..w~ --1/

(2)

'

1f !__il

,

where to~(0) = B sin(N0) where N denotes the n u m b e r of axes of symmetry, and A and B are optimizing parameters which consider the a m o u n t of tangential spread. Also, p is an optimizing parameter which considers the order of tangential velocity with respect to the R-direction in element 1. The incompressibility equation in the cylindrical coordinate system is given as follows: ¢~UR

Ua

I dUo

,~--h- + ~ +-~ ~

~.Uz +~ =o

(3)

Substituting Eqs. (I) and (2) into Eq. (3) and integrating the incompressibility condition, the radial velocity component, Ua, is given as follows: &ol(0) U r = -TR(p+ I) dO

i

Fig. I. A general scheme for the deformation model. Uo = -- ~

(6b)

A ( R - Riykol(O)

Uo

Uz = - ~ - z

(6c)

2,1.2. Element 11

The velocity boundary conditions in eletnent 11 are given as I\~llows: U a , = I%(0)

at R = Ri

UoA

URII = 0

(8)

at R = O

x {(R - Ri}~'-)- I -- (Ro(0) _ Ri)t,+ 11 + uo(R z - Ro2 (0)) 2TR

(4)

,/

where the applied radial velocity boundary condition at the container is given as follows: /-JR cos 0 = 0

at R = Ro(O)

,>

/

(I)

(5)

Thus, the velocity field for element ! is given finally in the following form: k ll)

UoA

d~,(0) U r = - T R ( p + i) dO

~

z

× {(R -- Ri)p+ I __ (Ro(O) - Ri)t'+ i} +

uo(R: - Ro 2 (0)) 2TR

o (6a)

Fig. 2. Shear boundary between element I and element II.

D.K. Kim et al./ Journal of Materials Processb~gTechnology62 0996) 242-248

244

5@0 ]--~----T- ~ - - - T - ~ - - - - - ~ - ~ -

--

q

|

270

r

240 ~

2Z

N=6

The7!!---

. . . . i,~

Exp

]

........ I

2~o

"g

~8,o

o

150

%

120

Fig. 3. The extruded circular-shapedbars for various numbers of axes of symmetry.

~.

>"

I /

-t

/

.... i

q

-II

50 .~

]

where Vb(O) is the normal velocity at a point of the shear boundary between element I and element II, as shown in Fig. 2. Vb(O) is given as follows: Vb(0) ffi (UR,)e = R~

(9)

Choosing the velocity field to satisfy the above boundary ,.onditions, the radial velocity component, Ua, is given by:

UR=(~)"VdO)

(lO)

where q is an optimizing parameter which considers the order of the radial velocity component with respect to the R-direction in element II. The tangential velocity component is: (11)

Uo=O and the axial velocity boundary condition:

Uz= - U~

atZ= T

(12)

Substituting Eqs. (10) and {11) into Eq. (3) and integrating the incompressibility condition, the axial velocity component, Uz, is: 240 210

z o J

g ~

!80

0

r(i .tO 50 40 50 60 ;':?, 40 2,, 100 Re,fiucUon

c,f

i

i

Uz----

(£' ('+1 Vt,(O) T

(T-Z)-[1o

Ua =

(£

Vb{O)

(14a)

U. -- 0

(14b)

/R'VI-'

Uz=t-~ )

1 +q

Vb((')(T)(T--Z)-- Uo

Introducing the upper-bound theorem, the following equation should be minimized for the actual velocity distribution:

Z ~d

;50

o _J

12_0

g ,~

90

£s

0 ~.-~.a . 0 5

o

u

c}

i __ 15

~60

r2

u rJ c

-'~0 ,:

Theory Exp.

--~ m . a -

120

× L~J

I

40

t I10

i

R A=45% m=0.1

80

a ,s

u

60 30 I i° n

(14C)

2.2. Upper-bou.dtheorem

m=O I

-4

{13)

Thus the velocity field for element II in the final form is:

~ - - ]

~?A=70%

Arc:c; ~,~

Fig. 5. Comparison between theoretical and expeqmental extrusion loads for various area reductions.

200 bj= 4.t

×

~l

OI

__I 25

Punch Stroke (ram)

Fig. 4. Experimentalload-stroke curve for circular-shaped bars.

I

°4

6

8

Number of axes of Symmetry

Fig. 6. Comparison between theoreticaland experimental extrusion loads for various numbers of axes of symmetry.

D.K. Kim et al. ' Jom',al of Materials ProcessingTechnology 62 (1996) 242 248

: M~

4i: i1.

LL.' : / i :Z

........

3. Expe¢imental

--

t ':7'7 I.

r I, !

I . . . . . .

j

]

L_

;~.

,-o ,

c'

20

q --s

t~]

! i

:zf

# [.'Jr'aL,'Ser

7.,f

CI,0S

# i

:~',rr'ili-rU

,.

Fig. 7. Effect of the number of axes of synlmetry on the extrusiou pressure.

s*= Zw, +Zw, + Z w,

(15>

where J* is the upper-bound on the total power consumption. In Eq. (15), the internal power of deformation, Wi is calculated from the derived strain-rate field. The interhal power of deformation, W, is given by:

W~= f. #~,dV

(16)

where

<~= ~2

245

\(DR~ + D,93_+ ~!z2 + ,~,<,,' + ,!,,.'+

,',,<'.)" s

The power consumption of the shear boundary, W~ is:

w, = Is "lavt" dS

];l;)i i ]

(18)

where m is the friction factor at the die (or punch)-material interface and tA VIf is the velocity discontinuity on the friction boundary. The upper-bound on the extrusion load, L, is then determined as: j* L = -(19)

Uo

where Uo is the velocity of the punch.

The ring-compression test was carried out to determine the friction factors under two different frictional conditions, lubricated and non-lubricated. In the case of lubrication, a mixture of MoS2 powder (10 wt.% ) and grease was used. In this study, the friction factor, m, was used to describe the frictional behavior and was assumed to be constant during deformation. The friction factor, m, was ?ound to be 0.10 in the case of lubrication and 0.28 in the case of non-lubrication. Experiments into square-die forward extrusion were carried out in the above hydraulic press at room temperature. The extrusion load and punch travel were registered simultaneously ,sing a GPIB-PC2A-1 interface board produced by National Instrument Corpora-

(17)

where ~- is the shear yield stress and IA V]~ is the velocity discontinuity of the shear boundary. The frictional power dissipated over the frictional boundary, Wr is:

IV,.= f~ m4AVl, dS

An experimental setup was manufactured for the square-die forward extrusion of circular-shaped bars from regular polygonal billets, the setup being installed in a 100 ton universal testing machine. For the experiments, three regular polygonal-shaped punches and two circular-shaped lower dies were made to obtain various numbers of axes of symmetry and area reductions. As a working material, hard solder was chosen. The billets were cast and then machined to a size slightly smaller than that of the container for smooth insertion of the billet into the container. The square billets were machined to final size of 24.9 mm across flats, and the hexagonal and octagonal billets were machined the same area as those of the square billets. In order to obtain the stress-strain-rate curve, a compression test was carried out in a Greeble 5000 hot-compression testing machine, at room temperature. the stress-strain-rate relationship obtained from the results by curve-fitting being: # = 73.063(~)°°293 MPa (20)

"

] -la li

r

-

[. . . . . . .

rn:., ! E,~,r,,, m , L 2<~ "> rh~ ,i

£q-KL

! I

:

,I,t

'

,

-~' A :+ ! i p

'L' J

%

d ,b

l:

i

,-<

I

L

I'53U

'c

rl

;t

."r,-,l

t '~,~

Fig. 8. Comparison between theoretical and experimentalextrusion loads for various frictionalconditions.

D.K. Kim et el. Journal oI' Materials Processing Tet.lmology 62 (1996) 242-248

246

Theory

~'a_ 150 I

8 ~3 n

12_o

%/ Po'~q

Fb

. . . . . . . . . . . . . . . .

Exp . . . . . .

m&

iiiiiiii=J

i ii

=:::,

iiiiiii N

' ....... I

90 _ Pi

"~-~ 60

"

iiiiiiii"

2

30 k__

P~

0

0.1

0.2

I

i

0.3

04

_

I

I

05

06

. ~

7

Friction Fc~ctor {!-Nil Fig. 9. Effectof the fticlionfactor on the extrusion pressure. lion between transducer and personal computer. The extruded products are shown in Fig. 3. 4. Results and discussion

Since the formulation was not given in closed form, computation were carried out numerically. The total power consumption was calculated for each step of deformation by optimizing it with respect to four optimizing parameters (A, B, p and q) using the polyhedron search method [11]. For the analysis, experiments were carried out to determine the final stage of square-die forward extrusion, the experimental results being shown in Fig. 4. Fig. 4 shows the effect of the punch stroke on the extrusion load for the square-die extrusion of circularshaped bars from regular polygonal billets. The extrusion load increases rapidly at the initial stage, and achieves a peak point, after which it decreases slowly due to the decrease of the shear and friction surfaces, finally increasing again after approximately 80% height reduction. A piping defect under the punch is initiated at about 80% height reduction, being quite enlarged at 92% height reduction, Therefore, it is reasonable to define 80% height reduction as the final stage. 4.1. Extrusion load (pressure)

Fig. 5 shows the effect of the area reduction on the extrusion load, the latter increasing with increasing area reduction for a fixed number of axes of symmetry and a given friction factor. Theoretical extrusion loads computed from the present method were compared with the experimental values, there being found to be in good agreement. Fig. 6 shows the effect of the number of axes of symmetry on the extrusion load. The extrusion load

Fig. I0, Effectof the area reductionon the configurationof the free surface and the velocitydistributionof the extruded billeton the R- Z plane. decreases slightly with increasing number of axes of symmetry for a constant area reduction and a given friction factor, the reason being that the increase in the number of axes of symmetry makes the material flow more simple, such simple flow requiring lesser deformation energy, as shown in Fig. 7. The theoretical extrusion loads computed from the present method were compared with the experimental values, and again being found in good agreement

Fig, 8 shows the effect of the friction factor on the extrusion load; the latter increases with increasing friction factor for a given number of axes of symmetry, the reason being that the increase in the friction factor makes the friction energy greater, as shown in Fig. 9. The theoretical extrusion loads computed from the present method were compared with the experimental values, and again being found in good agreement. 4.2. Velocity distribution o f the extruded billet

The effect of area reduction on the veloc;ty distribution of the extruded billet for a fixed number of axes of symmetry and a given friction factor is shown in Figs. 10 and 11. In the R - Z plane, the axial extrusion velocity increases with increasing area reduction and the magnitude of the inner point velocity is greater than that of outer point velocity, this being consistent with the experimental results. In the R - O plane, the tangential velocity increases with increasing area reduction: from R,A=70~ N=4

/

/i (

m=O

I

m ={).28

Fig. 1I, Effectof the area reductionon the velocitydistributionof the extruded billet on the R-O plane.

D.K. Khn el al. Jour~mlof Materials Processing Techmdogy 62 (1996) 242-248 Exp

Theory N=4

m=O l

............

N:6

247

N=[I !

n

Ol

'

R.A=70 .

!::iii: iiiii!~

!i~i! ,

. . . . .

\

L=-:-=--_

I ~-Tt

• .....

~o,~ i i ! ! ! ! ' i!!?!i

i

( 1~ i::: t

iiiill

i ,

i F--

~

/~

:

,

;

0 =30'

,

0=22S"

,

~o.

o

o.

N=4

N=6

N=B

Fig. 13. Effect of the number of axes of symmetry on the velocity distribution of the extruded billet on the R-O plane. Fig. 11 it is known that the tangential flow is rotating from the 0 = []/N ° plane to the 0 = 0 ° plane. The effect o f the number o f axes o f symmetry on the velocity distribution o f the extruded billet for a fixed area reduction and a given friction Factor is shown in Figs. 12 and 13. In the R - Z plane, the axial extrusion velocity increases slightly wilh increasing number o f axes o f symmetry. In tile R - O plane, tile amount o f rotation o f the tangential velodty decreases slightly and the material flow becomes simple with increasing number o f axes o f symmetry, which is the reason for the decrease o f the extrusion load decreases with increase in the number o f axes o f symmetry.

Theory -

-

Fig. 16 shows the effect o f the area reduction on the average length o f the extruded billet, the average length o f the extruded billet increasing with increasing area reduction for a fixed number o f axes of symmetry and a given friction factor. Fig. 17 shows the effect o f number o f axes o f symmetry on the average length o f the extruded billet, the average length o f the extruded billet increasing slightly with increasing number o f axes o f symmetry. The theoretical predictions of the average length o f the extruded billets are in general agreement with the experimental results.

5. Conclusions As the preliminary study for the derivation o f tile generalized velocity field in three-dimensional square-

L '7,C t f "-

0=45"

-~ 120 7

'!iiii"

I _

160

I!=6 in=O

l,,~ ............... !

~'F

1

II--~ll

-i

Exp ......

0=0"

nl=Q.| R.A=45% N=4

The effect of the friction factor on the velocity distribution o f the extruded billet for a fixed area reduction and a given number o f axes o f symmetry is shown in Figs. 14 and 15. The friction factor does not affect the velocity distribution o f the extruded billet at all: the velocity field is not a function of the friction factor. 4.3. At:er~lge length o f the extruded billets

0 =45.

.

•

R ¢1=70~

Fig, 15. Effect of the friction factor on the velocitydistribution of the extruded bifiet on the R (J plane.

~

R A=70% m=Ol

//

%

R A=4E';=

q

~- ~

Fig. 12. Effect of the number of axes of symmetry on the configuration of the fi'ee surface and the velocity distribution of the extruded billet on the R-Z plane.

/1

,

....

~3

i

i

' 3;

i

i •

t

III 1

i

1 ........ >

i 1. . . . i _ _ _ _ f - -

0

1C! .20

1

}7@duCtiUrl

Fig. 14, Effect of friction factor on the configuration of the free surface and the velocity distribution of the extruded billet on the R - Z plane.

__ I

5'_~

_±___

~n ,L:t

Areo

A ___~

i:,,]

---J

r,~ ~¢,

i~i

Fig. 16. Comparison between the theoretical and experimental average extruded lengths for various area reductions.

D.K. Kim et al,. Journal c~' Materials Processing Technology 62 (1996) 242-248

248

N Pave. I

R,O,Z Ri Ro(O) S

C ,< ,a.

4 C:

. . . . . . .

--

Sr

u~ c

T Uo u ~ , u,,, u~

t

rhJ ":t .p

~f

21.62

2,f

-_.1"i1]-~2~

Fig. 17. Comparison between the theoretical and experimental average extruded lengths for various numbers of axes of symmetry.

I~xvI

die forward extrusion, the square-die forward extrusion of circular-shaped bars from regular polygonal billets has been chosen and a kinematically admissible velocity field has been proposed. From the proposed velocity field, the extrusion load, the velocity distribution and the average length of the extruded billets have been determined by minimizing the total power with respect to optimizing parameters. In order to verify the validity of the theory, experiments have been carried out with hard-solder billets at room temperature. For various area reductions, number of axes of symmetry and friction factors, there is good agreement in the load between theory and experiment. The theoretical prediction of the average extruded length is also in reasonable agreement with the experimental measure. ments, except for large area reductions. The proposed velocity field can be used conveniently for the prediction of the extrusion load in the squaredie forward extrusion of circular-shaped bars from regular polygonal billets. However, for the prediction of the average length for large area reductions, the proposed velocity field should be improved.

~'b(0)

6. List of symbols

A,B,p,q J* L fll

optimizing parameters relating to the velocity field upper-bound on the extrusion power upper-bound on the extrusion load friction factor at the die (or punch)material interface

V

w,, w,, a'r

number of axes of symmetry average extrusion pressure of the punch cylindrical coordinates lower die radius and outer radius of the extended billet punch profile function and container profile function cross-sectional area of the shear boundary plane frictional surface area of the diematerial and punch-material interfaces bottom thickness of the extruded billet velocity of the punch velocity components in cylindrical coordinates magnitude of velocity discontinuity volume of an element in the plastic region normal velocity of the boundary between deforming elements power consumptions due to internal deformation, shear and friction

Greek letters 8 If

~)(0)

effective strain rate effective stress for the working material shear yield stress function to satisfy the velocity boundary conditions on the axes of symmetry in the cross section

References [I] K. Lange, Haadbot~k of Metal Forming, McGraw-Hill, New York, 1985. [2] K. lwata, K. Osakada and S. Fujino, J. Eng. b,d., 94 (1972) 697-703. [3] Y,S. Kang and D.Y. Yang, J. Korean See. TechaoL Plast., 3 (1994) 156-166. [4] C.T. Chen and F.F. Ling, bit. J. Mech. Sci., 10 (1968) 863-879. [5] H. Takuda, N. Hatta and H. Lippman, J. dpn. Soc. TeehaoL PlastMty, 31 (1990) 202-207. [6] M. Kiuchi, M. Hoshino and S. lijima, Simulation of unsteady flow in non-axisymmetric reward extrusion-II, Res. Rep. h~st. b:dustrial ScL, Universityof Tokyo, 40 (1988) 184-187. [7] N.S. Das and W. Johnson, hit. J. Mech. St'L, 30 (1988) 61-69. [8] H. Keife,J. Mater. Prec. Technol., 37 (1993) 189-202, [9] M. Tokizawa, N. Takatsuji, K. Murotani and K. Matsuki, J. Jpn. Soc. TedmoL Plasticity, 23 (1982) 437- 443. [I0] W.B. Bae and D.Y. Yang, J. Mater. Prec. Technol., 36 (1993) 175-185. [I 1] J.L. Kuester, J.H. Mize, Optimi:ation Techniques with Fortran, McGraw-Hill, New York, 1973.