Die design for axisymmetric extrusion

7i .~ . , c

,

'~7:

ELSEVIER

Jourmd d

Materials Processing Technology Journal of Materials Processing Technology 55 (1995) 331-339

Die design for axisymmetric extrusion N. Venkata Reddy, P.M. Dixit,* G.K. Lal Department of Mechanical Engineering, lndian Institute of Technology, Kanpur-208 016, India Received 23 May 1994

Industrial summary

Determination of total extrusion power and die pressure distribution is very important for die design. In this work, an upper-bound model with strain hardening is proposed, the prediction of the extrusion power of which is as accurate as that determined by the finite-element method (FEM) and is in excellent agreement with published experimental results. The upper-bound model, when combined with the slab method, also predicts the die pressure distribution, which again is in reasonable agreement with FEM results. Further, the computational time taken by the combined upper-bound/slab method is significantly less than that for FEM. The proposed combined upper-bound/slab method is applied to compare eight different die shapes, namely, stream-lined (third and fourth-order polynomial, cosine and modified Blazynski's CRHS), elliptical, hyperbolic, conical and Blazynski's CRHS. Based on the consideration of total extrusion power (under optimal conditions), it is concluded that third- and fourth-order polynomial dies and the cosine die are the best amongst the profiles considered. Parametric study is carried out for the third-order polynomial die to study the effects of reduction ratio, friction factor and strain-hardening on the optimal die length and die pressure distribution.

Notation

# L

length of the die optimal length of the die Lopt m friction factor die pressure P average extrusion pressure Pave r, Z coordinate system %r percentage reduction, (1 - R~/R~) x 100 R(z) die profile R'(z) first derivative of R(z) n"(z) second derivative of R(z) radius at the entry Rt radius at the exit R2 Si;i= 1,2,3 surfaces of velocity discontinuity t time velocity vector ui ram velocity Vo V,, V= velocity in the radial and axial directions, respectively tangential velocity discontinuity on Si IA V, ls, position vector xi homogeneous strain 8H generalized strain strain-rate tensor

* Corresponding author. 0924-0136/96/$15.00 ~) 1996 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 5 ) 0 2 0 2 7 - J

GO ff ij

4,

generalized strain-rate Levy-Mises Coefficient cross-stream coordinate generalized stress generalized stress at zero plastic strain stress tensor deviatoric-stress tensor shear stress on surfaces of velocity discontinuity total power stream function

1. Introduction In the extrusion process, the geometry of the die constitutes an important aspect of die design. The die profile detrmines the extent of redundant work done during the deformation, a profile which minimizes the redundant work, thereby minimizing the extrusion power, being called an optimal profile. The extensive literature exists on optimal die profiles. The slip-line field technique has been used to obtain optimal die shapes in plane strain and wire drawing by Richmond and coworkers [1, 2], in axisymmetric extrusion by Sortais and Kobayashi [3] and in plane-strain drawing and extrusion by Sowerby et al. [4]. There have been several attempts using the upper-bound method

332

N. Venkata Reddy et al. /Journal of Materials Processing Technology 55 (1995) 331-339

also. Zimerman and Avitzur [5] obtained the optimal angles for conical dies by assuming generalized plastic boundaries. Chen and Ling I-6] obtained the optimal die lengths of cosine, elliptic and hyperbolic dies, whilst Gunesekara and Hoshino 1-7,81 did the same for streamlined and conical dies. All these solutions are for rigidperfectly plastic materials. The effect of work hardening was considered by Yang and co-workers [9, 10] for predicting the optimal shapes of curved dies, where they assumed a constant value of generalized yield stress for the entire deformation zone which was calculated on the basis of the average effective strain at the exit. Balaji et al. [11] used the finite-element method (FEM) to propose a general methodology for optimal die design in which the die geometry and the plastic boundaries also appear as variables, assuming the material to be rigid-perfectly plastic. More recently Joun and Hwang [12,13] developed an iterative optimization scheme along with the penalty rigid-viscoplastic FE formulation for optimal design in metal forming and applied it for obtaining optimal die profiles in axisymmetric extrusion for various process conditions and strain-rate sensitive materials. All the above studies considered isothermal process conditions. Reddy [14] combined the upper-bound method and the FEM to obtain the optimal die profiles in the case of axisymmetric hot extrusion by considering flow stress dependency on temperature and strain rate. Although the FEM provides a more accurate description of the deformation and stresses than do other methods, it takes a lot of computational time. On the other hand, the upper-bound method takes significantly less time but its accuracy depends on the choice of the kinematically admissible velocity field and on the consideration of the variation in yield stress due to strain hardening. In general, in the upper-bound literature, strain-hardening effects have been neglected. Yang and co-workers [9,10] included strain-hardening in their analysis but only in an average sense. In the present work, an upper-bound model is proposed which takes care of variable yield stress due to strain hardening. In this model, a stream function which produces a kinematically admissible velocity field is assumed. The distribution of generalized strain over the domain is determined by integrating the generalized strain rate along the stream-lines. Then the variable yield stress is calculated using the strain-hardening relationship of the given material. Finally, the total extrusion power and the average extrusion pressure are determined using this variable genrealized stress. It is shown that the proposed upperbound model predicts the extrusion power as accurately as that detrmined by the FEM, the predictions being in excellent agreement with published results. This model is used to obtain the optimal lengths of eight different die profiles by minimizing the extrusion power. The minimization is carried out using the Newton-Raphson tech-

nique. The best profile is determined on the basis of total extrusion power at the optimal length. In cold forming processes, heavy loads as act on the die may lead to an unacceptable level of die deformation or fracture of the die. To estimate the die deformation or predict its fracture, the die pressure distribution along the die-billet interface is required. However, it is not possible to obtain this distribution completely from the upperbound method, but when the upper-bound method is combined with the slab method, the die pressure distribution can then be determined. Normally the slab method gives only an approximate solution, but the proposed combined upper-bound/slab model predicts a die pressure which is in reasonably good agreement with FEM results. Finally, a parametric study is carried out to investigate the effects of reduction ratio, friction factor and strainhardening on the optimal die length and the die pressure distribution.

2. Formulation 2.1. Material behavior

In bulk metal-forming processes such as extrusion, rolling etc., the plastic deformation is very large compared to the elastic deformation. Therefore, the material behavior can be described as that of fluid flow. Under these conditions the measure of deformation is the strain rate ~ij, which is defined as:

The constitutive law of rigid-plastic material relating the deviatoric stress tensor a~'jand the strain rate tensor g~ is expressed as: ~ri~ = 2/~ij.

(2)

For a material yielding according to the von-Mises criterion, the Levy-Mises coefficient/~ is given by = 6./3~,

(3)

where the generalized yield stress 6" and the generalized strain rate ~ are defined by: ?r = x/ _/TZ7-27_,. ~3 a~j ai~,

(4)

The generalized strain ~ is given by: = f l ~dt,

(6)

N. Venkata Reddy et aL /Journal of Materials Processing Technology 55 (1995) 331 339

where the integration is to be carried out along the particle path. In a typical cold-extrusion process where the speeds are low, the strain rates and the temperature rise are quite small. Therefore, in the present work, # is assumed to depend only on ~, i.e.: 5- = 5; (g).

(7)

For most metals, 5; can be modelled by a power law. The specific functional form of 5; for the material under consideration is given in a later section.

2.2. Upper bound theorem Velocity fields that satisfy the constraint of volume constancy and the velocity boundary conditions are called kinematically admissible fields. The upper-bound theorem [15] states that amongst all possible kinematicaily admissible velocity fields, that which minimizes the total power: ~-

rlA r, l~ dSi

~7ij/Cij d O + *

(8)

i

is the actual velocity field. In the above equation O is the plastic deformation zone, z is the shear stress on the surfaces of velocity discontinuity Si: i = 1, 2, 3 and IA V, Is, is the tangential velocity discontinuity on the surfaces Si (Fig. 1). The asterisk indicates admissible values of the respective quantities. In most of the earlier works on optimal die design by the upper-bound method, much emphasis was given to the determination of the complex shapes of the plastic boundaries $1 and $2. The prediction of these shapes is never accurate, since the upper-bound approach does not satisfy the stress balance. The upper-bound solutions obtained by earlier researchers using straight and arbitrarily-shaped plastic boundaries indicate that there is little effect of the shapes of S~ and $2 on the overall solution. In the present work, the deformation zone f2 is assumed to be bounded by straight plastic boundaries at the end sections of the die. This assumption simplifies the

333

mathematical treatment of the problem significantly and provides greater flexibility in the optimization of the die profile $3. Also, the frictional resistance on the surface $3 is assumed to be a constant and equal to m 6 / , / 3 , where m is the friction factor and the value of shear stress on the surfaces $1 and $2 is taken as 6/x/3.

2.3. Veloci~ and strain-rate.fields For steady-state plastic flow through a rigid arbitrarily shaped die, the die profile and the axis of symmetry are stream-lines. Since the material is assumed to be rigid outside the entry and exit sections of the die, the axial velocity profiles at these sections are assumed to be uniform. Further it is assumed that the flow pattern in the deformation zone can be represented in the same functional form as the die profile. Under these assumptions equally spaced stream-surfaces at the die entry remain equally spaced at the die exit also. Hence, the stream surface in the deformation zone is given by:

r = ~R(z),

(9)

where constant ~ represents a stream-line with value ranging from zero at the axis to unity at the die surface and R(z) represents the die profile. Since the stream function has to satisfy the condition of uniform axial velocity Vo at the entry section (radius R1) and the symmetry condition at ~ = 0, it can be assumed as: = VoR~2/2.

(10)

The velocity components, Vr and V~ are obtained in terms of the die profile from the spatial derivatives of the stream function as: V~ =

v,

-

1 o~, -

-

-

=

VoR~

•

r ~r /~(z}-'' 1 0~ YoRf~R'(z) r c?z R(z) z '

I

o~ Vo ------~

0 Fig. 1. Domain and a typical stream surface for axisymmetric extrusion.

(11)

(12)

334

N. Venkata Reddy et aL /Journal of Mawrials Processing Technology' 55 (1995) 331-339

Simpson's rule to find the cumulative generalized strain along each stream-line.

where R'(z) = d R ( z ) / d z .

Since the velocity field is derived from the stream function ~b, it automatically satisfies the incompressibility condition. Further, the axis and the die surface are the stream surfaces, hence the zero-normal-velocity condition on them is satisfied also. Thus, the assumed stream function leads to a kinematically admissible velocity field. At the plastic boundaries $1 and $2 the tangential velocity may be discontinuous when the die surface slope at the entry or exit sections happens to be non-zero. From the velocity field (Eqs (11) and (12)) the non-zero strain-rate components can be calculted as:

2.5. Die profiles and optimization

The shape of the die surface S 3 is chosen to satisfy the reduction requirements with length (L) as the only variable to be optimized. In some cases, it is restricted by constraints of slope or curvature at particular cross-sections. In the present work, eight different die shapes have been considered. Third-order polynomial die [11]

This die profile is represented by:

( •

~rr -~

OV,

VoR~R'(z) --

~r

eoo =

V, =

R3(z)

z2 )

(13a)

(16)

(13b)

and is derived by imposing smooth flow at the end sections.

'

VoR2R'(z).

r

z3

R(z) = R 1 + (R, - R2) 2 ~-3 - 3 ~

R3(z)

' Fourth-order polynomial die [-10]

OVz _

ezz = ~3z

(13c)

VoRfR'(z).

2

RS(z )

This die profile is represented by:

, R(z) = RI +

k,~ = 2 \-~-z + Or J -

R3(z)

~z)

1.5 R ( z ) J ' (13d)

4K(3(cL) 2

L 3 - l!zs +

3K(1 - 2cL) z4 L* '

(17)

where

where R'(z)-

+

6 c L K ( 2 - 3cL) 2 Z L2

d2R(z) dz 2

K=

R1- R2 6(cL) 2 - 6cL + 1

It can be seen that all of the strain-rate components, except ~,z, are uniform over the cross-section.

is obtained by imposing smooth flow at the end sections and by setting the curvature value at z = cL to zero (c = constant).

2.4. Strain hardening

Cosine die

This die profile is represented by: The generalized yield stress # at any point is calculated using the value of generalized strain ~ at that point. The value of ~ in the deformation zone is obtained by integrating the generalized strain rate ~ along the stream-lines according to Eq. (6). The stream-lines are given by Eq. (9). Along any stream-line: dz dr = - dt = -Vz Vr'

dz.

(18)

and leads to smooth flow at the end sections. Elliptical die

This die profile is represented by: (14)

R(z) = x / R 2 - (R~ - R 2) (z/L) 2

therefore Eq. 6 can be written as: =

R1 + R2 R1 -- R2 nz R (z) - - + - cos - 2 2 L

(19)

and introduces velocity discontinuity at the exit section. (15)

Hyperbolic die

This die profile is represented by: Since the material enters the die as a rigid body, the generalized strain ~ is taken as zero on the boundary S~ (Fig. 1). Integration of Eq. (15) is performed using

R(z) = ~ / R ~ + (n~ - R~) (1 -- (zlL) 2)

(20)

and introduces velocity discontinuity at the inlet section.

N. Venkata Reddy et al./Journal of Materials Processing Technology 55 (1995) 331-339

The Newton-Raphson method is used to carry out the minimization process. The average extrusion pressure (P,~) which the ram is required to exert on the inlet surface is determined from:

Conical die This die profile is represented by: R(z) = R1 +

8

2 --

R 1

L

z

(21)

P~*¢ = n R~ Vo"

and introduces discontinuity at the entry as well as at the exit sections.

The slab method is used, along with some of the results of the upper-bound method, to determine the distribution of die pressure. In this method, the equations of motion are written for a slab rather than for a particle by assuming the stresses to be uniform over the cross-section. Neglecting the inertial forces, axial equilibrium of the slab shown in Fig. 2 yields the equation:

R, (22)

Ri

where Ri is the radius at the ith cross-section. Then the concept of CRHS is defined by: ~lti - - ~'H (i - 1 ) ¢ ; H ( i - 1) - -

= q,

R(z)da= - 2dz(azztan~ + (m6/~/3) + ptan~) = 0. (26) whilst equilibrium in the radial direction gives the following equation:

(23)

~'H(i-2)

m~ a,, = ~ tan ~ - p.

An additional equation is needed to eliminate a , , which is provided by the von-Mises yield criterion. In the slab method, the yield criterion is simplified by assuming art, a0o and az, to be the principal stresses. Eqs (13a) and (13b) imply that a , = aoo. Then the yield condition reduces to:

Modified CRHS die The CRHS die surface is modified by imposing smooth flow conditions at the end sections of the die. The optimal length is found by minimizing the extrusion power ¢ by setting: and

(27)

,,/J

where q is an arbitrary constant. Note that q > 1 gives an accelerated rate of deformation, q = 1 provides a uniform rate and q < 1 refers to a decelerated rate. In this work, q is chosen to be unity. This die introduces discontinuity at the entry as well as at the exit sections.

dqS=0 d-L

(25)

2.6. Die pressure distribution

Constancy of ratio of homogeneous strain (CRHS) die 1-16] This die shape is constructed by maintaining a constant ratio of homogeneous strain over each interval. If the die length is divided into intervals of equal length, the homogeneous strain at any section i is given by: on, = 2 In - - ' ,

335

a= - a , = ~.

(28)

Elimination of a,r from Eqs. (27) and (28) leads to: ff

d2~b d~L5 > O.

p = 6 + m~

(24)

~/~

P

tan ~ - azz.

p cos ~ - m (5- sin c~

•"l' .dlR

R(z)

O-zz+dO-zz R

m.

Fdz-

"~"Z

Fig. 2. Equilibrium of a typical slab.

(29)

336

N. Venkata Reddy et al. /Journal of Materials Processing Technology 55 (1995) 331-339

Substitution of p from Eq. (29) into Eq. (26) and integration along the die length gives the following expression for azz: 2 .If ~) + x/3 tan a= = ~-~ __ 6 m(1 + tan 2R(z)

dz

--

P ....

(30)

The procedure for determination of the die pressure distribution is as follows. First, the values of 8 and Pave obtained by the upper-bound method are used to find a= at any section by integrating Eq. (30) numerically. Since # varies over the cross-section, the average value is used. The die pressure p at any section is then obtained from Eq. (29). When there is velocity discontinuity at the inlet and/or exit, there is a power loss. Because of this, there will be a sudden change in the value of a=z at the surfaces of velocity discontinuity. The magnitude of this change is directly proportional to the power loss and inversely proportional to the velocity and the square of the radius (Eq. 25).

1600

"G cl Q) >

.... 1200

0

Present method FEM 0~1 Experimentol [.18] m = 0.12

a.o

,,,~

,9,

j

AISI4t40

80C (/) ~)

0

e3

r- 40C 0 O9

0

L-

laJ

I,

50

~

40

I

50

i

I

60

i,

I

70

I

80

Reduction in oreo (%r) Fig. 3. Comparison of extrusion pressure from the FEM and experimental results. b°

5

Q.

g

Present

3. Results and discussion 3

3.1. Comparison with FEM and experimental results To test the validity of the upper-bound model developed in the previous section, the results obtained are compared with those predicted by the FEM [17] and experimental results of Yang et al. [18] using the fourthorder polynomial die with the value of c as 0.44 and L = 25 mm. Fig. 3 shows the variation of extrusion pressure with reduction ratio for steel and aluminium. It is observed that the upper-bound results are in good agreement with the FEM results of Ref. [17] as well as with the experimental results. The stress-strain relationships for the metals are given by: # = 418.9 + 699.0(g) °'z66 (MPa) for AISI4140 Steel, (31a)

._o 2 e-.

e"

E "o

&

o Z

I

I

0.4

0.8

I

I

1.2

1.6

2.0

Non-dimensionol die length,L/R~ Fig. 4. Comparison of the die pressure distribution with F E M results for a third-order polynomial die.

bo Q.

,=.

5

6 = 118.7 + 229.4(g) °'247 (MPa) for A12024 Aluminium.

(31 b)

The remaining results are obtained by choosing AISI4140 steel as the billet material with an initial radius of 12.5 mm. Figs. 4 and 5 show the die pressure distribution along the die-billet interface for a third-order polynomial die and a conical die respectively, for a typical die length of 25 mm, a reduction in area of 50% and a friction factor of 0.1. It is observed that for the third-order polynomial die the values and the trend obtained by the present method are in very good agreement with the results obtained by the FEM [17]. For the conical die, however, the

g

p= Z

s-

FEM [iT.]

21-

I

0.4

I

0.8

I

1.2

1

1.6

2,0

Non-dimensional die length, L/Rt Fig. 5. Comparison of the die pressure distribution with F E M results for a conical die.

N. Venkata Reddy et al./Journal o]'Material~" Processing Technology 55 (1995) 331 339

agreement is not so good. The present method cannot simulate the sharp changes in the deformation and stresses at the entry and exit sections. Therefore, it cannot predict the pressure peaks in these regions. Thus, the die-pressure values for the conical die are less accurate. The present method takes 6.85 s of user time on a convex-C220 mini super computer to compute the extrusion pressure and the die pressure distribution for a given set of process conditions (i.e. L, % r and m). On the other hand, F E M takes 8 15 iterations to converge and 80s of user time for each iteration. Thus the combined upperbound/slab method takes significantly less time than the FEM, without compromising on the accuracy.

800

'

I

I

I

i

700 600

~

D

T°tal

J

5O0

$ 3

8

400 3OO

J

20O

FP

[00 ,

I

I

1

8

I

i

[6

12

20

Die length, L(mm)

3.2. Optimal length and choice of the best profiles For a stream-lined die, there is no velocity discontinuity on the surfaces S~ and $2. Hence, the total extrusion power consists of only the internal power of deformation (IPD) and frictional power (FP). Fig. 6 shows the variation of total extrusion power and its two constituents with respect to die length. This result is for a third-order polynomial die and corresponds to a reduction in area of 50% with a friction factor of 0.5. It is observed that the internal power of deformation decreases with die length while the frictional power increases. As a result, the total extrusion power has a minimum value at some length, this length being defined as the optimum length (Lopt). In the case of a conical die, there is power loss due to velocity discontinuity (PVD) on surfaces St and $2. Thus, the total extrusion power has three constituents. Fig. 7 shows the variation of total extrusion power and its three constituents for the conical die. It is observed that the power loss due to velocity discontinuity decreases with die length. Although the optimal die length is less than that for the third-order polynomial die, the total power is greater. The various die profiles considered are compared in Table 1 for 50% reduction in area and m = 0.1 as the friction condition. It is normally believed that streamlined dies require a less amount of power as there are no power losses at the inlet and exit sections, the results of Table 1 in general supporting this assertion. Further, it is shown that the total power consumption is greatest for the conical die. The C R H S die profile is constructed in such a fashion that it minimizes the internal power of deformation (IPD) but there are power losses at the entry as well as at the exit sections. As a result, the total power consumption for the C R H S die is almost as great as that for the conical die. In the modified C R H S die, the discontinuities at the end sections are removed (i.e. it is made stream-lined) whilst retaining the advantage of minimum IPD. N o w there are no power losses at the entry and the exit, but even so the total power consumption of the modified C R H S die is more than that of the other stream-lined dies

n

337

Fig. 6. Variation of total power and its constituents with die length for a third-ordfer polynomial die.

800 Total

700 600 500 v

400 0 n

IPD

500 PVD

200

I I00 0

FP I

I

I

I

4

6

8

10

I

12

14

Die length, L(mm) Fig. 7. Variation of total power and its constituents with die length for a conical die.

Table 1 Comparison of different die profiles Die profile

Lop, (mm)

q5 (W)

3rd order 4th order Cosine Elliptical Hyperbolic Conical CRHS Modi. CRHS

23.730 23.728 24.007 24.774 15.240 11.790 10.620 30.125

404.47 404.67 4(14.88 516.12 480.96 594.88 591.38 436.34

considered. The reason for this is as follows. When the C R H S die is streamlined, the optimal length increases substantially leading to an increase in frictional power. Thus the advantage of smooth flow is achieved at the cost

338

N. l/'enkata Reddy et al. /Journal of Materials Processing Technology 55 (1995) 331 339 70

L.

60

o

50

o c

40

J// •

cO

"~

55

n.~

20

0

reduction ratio and strain hardening by offsetting the increase in internal power of deformation. Similar trends are observed for other die shapes also, and are in conformity with results published earlier I-6,8, 11]. Fig. 9 displays the die pressure distribution for the optimal third-order polynomial die with respect to reduction in area, friction conditions and strain hardening. It is observed that the die pressure increases uniformly with reduction, but when the friction factor is increased the increase is more predominant at the inlet. When the strain hardening is neglected, the die pressure values are under-estimated by as much as 50%.

%

!

I

5

I0

I

15

I

I

20

25

,

:50

Optimum die length, Lopt (mm} Fig. 8. Variation of the optimal die length for a third-order polynomial die.

....

r =50 %,m= 0.1, non- hardening 50 %, 0.1, hardening

bo

--~

.....

50 % , 0.05 harden[nq 60 % ,O.I, hardening

"'

O.

5 u~

P ¢3L "0 O cO

4. Conclusions The proposed model predicts the extrusion power and the die pressure distribution with reasonable accuracy and with significantly less computational time than does the FEM. Further, the predicted trends regarding the optimal length and the die pressure distribution are also in conformity with results published earlier. The method proposed, therefore, can be used successfully for die design in extrusion as well as in drawing. The major advantage of the present model is that it can be used for a large class of die profiles. Amongst the die profiles considered, the third- and fourth-order polynomial dies and the cosine die are the best.

Acknowledgements

E i tO

z

,

I

0.5

I

1.0

Non-dimensional

I

1.5

I

I

2.0

2.5

kO'

die l e n g t h , L / R I

Fig. 9. Variation of die pressure distribution for a third-order polynomial die employing optimal lengths.

of increased frictional power. Therefore, the concept of CRHS does not seem to be useful in minimizing the total power consumption. Based on the consideration of total extrusion power, the third- and fourth-order polynomial dies and the cosine die seem to be the best. This conclusion suggests that no advantage is gained by increasing the order of the polynomial when constructing a streamlined die.

3.3. Parametric study The variation of optimal length with reduction ratio for two friction and two hardening conditions is shown in Fig. 8 for a third-order polynomial die. The optimal die length decreases with friction factor in order to offset the tendency for increase in the frictional power with m. For a similar reason, the optimal length increases with

The authors thank Dr. T. Sundararajan, Department of Mechanical Engineering, Indian Institute of Technology, Madras, for his helpful suggestion.

References [1] M.L. Devenpeck and O. Richmond, Strip-drawing experiments with a sigmoidal die profile, ASME J. Eng. Ind., 87 (1965) 425. [2] O. Richmond and H.L. Morrison, Streamlined wire drawing dies of minimum length, J. Mechs. and Phys. Solids, 15 (1967) 195. [3] H.C. Sortais and S. Kobayashi, An optimum die profile for axisymmetric extrusion, Int. J. Mach. Tool Des. Res., 8 (1968) 61. [4] R. Sowerby, W. Johnson and S.K. Samanta, Plane strain drawing and extrusion of a rigid perfectly plastic material through concave dies, Int. J. Mech. Sci., 10 (1968) 231. [5] Z. Zimerman and B. Avitzur, Metal flow through conical converging dies - a lower upper-bound approach using generalized boundaries of the plastic zone, ASME J. Eng. Ind., 92 (1970) 119. [6] C.T. Chen and F.E. Ling, Upper bound solutions to axisymmetric extrusion problems, lnt. J. Mech. Sci., 10 (1968) 863. [7] J.S. Gunesekera and S. Hoshino, Analysis of extrusion or drawing of polygonal sections through straightly converging dies, ASME J. Eng. Ind., 104 (1982) 38. [8] J.S. Gunesekera and S. Hoshino, Analysis of extrusion of polygonal sections through streamlined dies, ASME, J. Eng. Ind., 107 (1985) 229.

N. Venkata Reddy et al. /Journal of Materials Processing Technology 55 (1995) 331-339 [9] D.Y. Yang, C.H. Han and B.C. Lee, The use of generalized deformation boundaries for the analysis of axisymmetric extrusion through curved dies, Int. J. Mech. Sci., 27 (1985) 653. [10] D.Y. Yang and CH. Hart, A new formulation of generalized velocity field for axisymmetric forward extrusion through arbitrarily curved dies, ASME J. Eng. Ind., 109 (1987) 161. [11] P.A. Balaji, T. Sundararajan and G.K. Lal, Viscoplastic deformation analysis and extrusion die design by FEM, ASME J. Appl. Mech., 58 (1991) 644. [12] M,S. Joun and S.M. Hwang, Optimal process design in steadystate metal forming by finite element method-l. Theoretical considerations, Int. d. Mach. Tool Manuf, 33 (1993) 51. [13] MS. Joun and SM. Hwang, Optimal process design in steadystate metal forming by finite element method-lI. Application to die profile design in extrusion, Int. J. Mach. Tool Manuf, 33 (1993) 63.

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[-14] N.V. Reddy, Upper bound and finite element analysis of hot extrusion, M. Tech Thesis, Indian Institute of Technology Kanpur, March 1992. [15] B. Avitzur and S.H. Talbert, Upper-bound solutions and the balance-of-power approach, in: T.Z. Blazynski (Ed.I, Plasticity and Modern Metal-Forming Technology, Elsevier Applied Science, London, 1989, p. 17. [16] T.Z. Blazynski, Mode of deformation and material response, in: T.Z. Blazynski (Ed.), Plasticity and Modern Metal-Forming Technology, Elsevier Applied Science, London, 1989, p. 115. [17] N.V. Reddy, P.M. Dixit and G.K. Lal, Central bursting and optimal die profile in axisymmetric extrusion, ASME J. Eng. Ind., in press. [18] D.Y. Yang, C.M. Lee and J.R. Cho, Analysis of axisymmetric extrusion of rods by the method of weighted residuals using body-fitted coordinate transformation, Int. J. Mech., Sci., 32 (1990) 101.