A universal velocity field for the extrusion of non-axisymmetric rods with non-uniform velocity distribution in the extrusion direction

A universal velocity ®eld for the extrusion of non-axisymmetric

rods with non-uniform velocity distribution in

the extrusion direction

C.W. Wu, R.Q. Hsu

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan, ROC Received 17 August 1998

Abstract

This study formulates a universal velocity ®eld that is kinematically admissible for application in the extrusion of non-axisymmetric rods. Then upper bound theorem dictates that a better upper bound solution heavily depends on the precise conformity of the velocity ®eld postulated. However, a compromise must frequently be made, since the formulation is in general rather complicated. The kinematically admissible velocity ®eld proposed herein has the following features: (a) it is three-dimensional, (b) it is non-uniformally distributed in the axial direction, and (c) the formulation is straight-forward once the boundary of the deformation zone is speci®ed. In addition, the velocity ®eld is applied to the extrusion of rectangular, hexagonal, and octagonal rods from round billets. Moreover, the extrusion loads are calculated against process variables such as the semi-die angle, the percentage reduction of area, and the friction factor. Furthermore, the velocity ®eld is compared with results from the literature, indicating that the present results render a better upper bound solution for application in extrusion than do previous results. # 2000 Published by Elsevier Science S.A. All rights reserved.

Nomenclature

r, , y cylindrical coordinates

R0, Rf() radius of the billet before extrusion and the product profile function after extru-sion, respectively

Rs0(,y) a function that represents the die surface V0, Vf entrance velocity of the billet and exit velocity of the extruded product, respec-tively

ÿs; ÿf surfaces of shear velocity discontinuities

and friction, respectively

Vr, V, Vy velocity components of the billet in cylindrical coordinates (r,,y), respec-tively

!(,y) angular velocity of the billet

U(y), D(r,,y) uniform and non-uniform velocity com-ponent along the extrusion axis of the billet, respectively

@U(r)/@y first derivative of the uniform velocity component along the extrusion axis of the billet with respect to y

@!(,y)/@y first derivative of the angular velocity of the billet with respect to 

Z(,y) function of (,Rs0,L)

@Z(,y)/@y first derivative of function Z(,y) with respect to y

@Rs0(,y)/@y first derivative of the die surface function with respect to y

@Rs0(,y)/@ first derivative of the die surface function with respect to 

f(y) angle of a surface of geometrical sym-metry

 optimization parameter introduced in the

velocity field

L die length(dimensionless)

J total power consumption in extrusion

_Wi; _Ws; _Wf power dissipation due to internal

defor-mation, internal shear of the billet and friction at the die surface, respectively

Vp volume of the plastic region

y; _" the yield stress and effective strain rate of

the material, respectively

_"ij components of the strain rate tensor

DVÿs; DVÿf the relative slip velocity on ÿsand ÿf

surfaces, respectively

*_{Corrresponding author. Tel.: 886-3-5731934; fax: 886-3-5738061} E-mail address: u8114539@cc.nctu.edu.tw (R.Q. Hsu)

0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 3 6 4 - 7

Pavg average extrusion pressure

a semi-die angle

R.A. reduction of the area of the billet

m friction factor at the die surface

1. Introduction

Non-axisymmetric extrusion, which is used widely to produce rectangular, hexagonal and other non-regular sec-tions, has received increasing interest. Hill [1] demonstrated the feasibility of three-dimensional analysis of the metal-working process. Juneja and Prakash [2] derived an upper bound solution to extrude rod with a polygonal cross-section through straightly converging dies, by utilizing a spherical velocity ®eld with a cylindrical surface of velocity discon-tinuity. In a related investigation, Boer and Webster [3] obtained an upper bound solution to draw square sections from a round billet. Later, Hoshino and Gunasekera [4±7] proposed an upper bound model to extrude polygonal sec-tions from a round billet. Yang and Lee [8] proposed the conformal mapping approach to derive a kinematically admissible velocity ®eld for extrusion through concave and convex shaped dies. Yang et al. [9] also analyzed the three-dimensional extrusion of arbitrarily-shaped sections. Kiuchi et al. [10,11] derived a three-dimensional velocity ®eld for non-symmetric extrusion. However, owing to the complexity of establishing the three-dimensional kinemati-cally admissible velocity ®eld, the above investigations assumed that the velocity component in the extrusion direc-tion is uniform at any cross-secdirec-tion. However, such an assumption does not conform to the actual deformation behavior.

In this study, the authors present a novel numerical model based on the upper bound theorem to analyze the extrusion of non-axisymmetric rods with an arbitrary section pro®le (Fig. 1). This model is advantageous in that the three-dimensional kinematically admissible velocity ®eld has a non-uniform velocity component in the extrusion direction. In addition, establishing the velocity ®eld is relatively simple and straight-forward once the die pro®le function is known. The derived velocity ®eld is applied to the extrusion of rectangular, hexagonal, octagonal and round rods to demonstrate the effectiveness of the proposed method. Finally, comparison of the present results with available results from the literature reveals a close corre-spondence.

2. Derivation of the velocity field

When applying the upper bound approach to analyze plastic deformation, a properly constructed admissible velo-city ®eld is deemed essential to ensure the accuracy of the ®nal solution. To be admissible, the velocity components must ful®ll the conditions of incompressibility, and the

boundary conditions required by the geometry, and must be continuous except at particular surfaces where velocity slips are allowed. Thus, the geometrical con®guration of the material during plastic deformation must be known. In the extrusion process, the geometrical con®guration of the billet is con®ned by the die surface. If function Rs0(,y) represents the billet geometry in the deformation zone, then, according to condition of incompressibility: Z _f0 0 Z _R0 0 V0rdrd ˆ Z _fy 0 Z _Rs0;y 0 Vy…r; ; y†rdrd; (1)

where (as shown in Fig. 2) V0denotes the velocity of the billet before entering the die entrance plane A±A0_{, and}

Vy(r,,y) represents the velocity component of billet in the extrusion direction during plastic deformation, i.e. the section bounded by the die surface, die entrance plane and the die exit plane B±B0_{. When the material leaves the die exit}

plane, it moves forward under a uniform speed Vf. If R0is the radius of the billet before extrusion, then Rs0(,y) is a function representing the geometrical configuration of the Fig. 1. Schematic diagram of the extrusion of a hexagonal shaped section rod.

billet during deformation. In addition, f(0) and f(y) are the ranges of integration in the  direction at y ˆ 0 and y ˆ y, respectively. To accommodate a non-uniform component of velocity along the extrusion axis, a convex distribution function D(r,,y) is introduced herein, i.e.:

Vy…r; ; y† ˆ D r; ; y… †U y… † ˆ 1ÿ_Rs4y Lÿy₂ … † 0…; y†L2r

2

 

U y… †; (2) where U(y) is the uniform component of Vy(r,,y) The convexity of Vy(r,,y) is determined by the parameter , which is subjected to optimization when calculating the total energy. L denotes the length of deformation zone. Vy(r,,y) is also subjected to the following boundary conditions: Vy…r; ; 0† ˆ U 0… †; Vy…r; ; L† ˆ U L… †: (3)

Substituting Eq. (2) into Eq. (1) and rearranging yields: U y… † ˆ V0 R_f0 0 Rs20…; 0†d 1ÿ2y Lÿy…_L2 †  _R fy 0 Rs20…; y†d : (4)

The condition of incompressibility can be expressed as: _"rr‡ _"‡ _"yyˆ@Vr…_@rr; ; y†

‡1_r Vr…r; ; y† ‡@V…_@r; ; y†

 

‡@Vy…_@yr; ; y†ˆ 0: (5) To simplify the formulation of the velocity field, the rota-tional velocity component is assumed to be distributed linearly over the radius, i.e.:

V…r; ; y† ˆ r
w…; y†: (6)

At the axis of extrusion, r ˆ 0, the velocity component Vy(r,,y) must be zero. Substituting Eqs. (2) and (6) into Eq. (5) and rearranging leads to:

Vr…r; ; y† ˆ ÿ₂r @U y_@y… †‡@w ; y_@… †   ‡r3 4 Z ; y… † @U y… † @y ‡ @Z ; y… † @y U y… †   : (7) Here, Z ; y… † ˆ 4y Lÿy… † Rs2 0…; y†L2: (8)

On the other hand, at r ˆ Rs0(,y), the material must flow along the die surface. Thus:

Vr…Rs0…; y†; ; y† ˆ V…Rs0…; y†; ; y†_Rs 1 0…; y† @Rs0…; y† @ ‡ Vy…Rs0…; y†; ; y†@Rs0_@y…; y† ˆ w ; y… †@Rs_@0…; y† ‡ Vy…Rs0…; y†; ; y†@Rs0_@y…; y†: (9)

At the angle of symmetry surface f(y), there exist no rotational velocity, thereby leading to !(f(y),y) ˆ 0. Com-bining Eqs. (7) and (9), rearranging leads to:

! ; y… † ˆ ÿ 2 Rs2 0…; y† Z _fy 0  Vy…Rs0…; y†; ; y†Rs0…; y† @Rs0…; y† @y ‡ Rs2 0…; y† 2 @U y… † @y ÿ Rs4 0…; y† 4  @Z ; y…_@y †U y… † ‡ Z ; y… †@U y_@y… †   d: (10)

The velocity components becomes:

Vy…r; ; y† ˆ D r; ; y… †U y… † ˆ 1ÿ_Rs4y Lÿy₂ … † 0…; y†L2r 2   U y… †; (11) V…r; ; y† ˆ r
!…; y†; (12) Vr…r; ; y† ˆ ÿr₂ @U y_@y… †‡@! ; y…_@ †   ‡r₄3 Z ; y… †@U y_@y… †‡@Z ; y…_@y †U y… †   : (13) Here, U y… † ˆ V0 R_f0 0 Rs20…; 0†d 1ÿ2y Lÿy… † L2  _R fy 0 Rs20…; y†d ; (14) ! ; y… † ˆ ÿ_Rs22 0…; y† Z _fy 0  Vy…Rs0…; y†; ; y†Rs0…; y† @Rs0_@y…; y†‡Rs20…; y† 2 @U y… † @y ÿ Rs4 0…; y† 4  @Z ; y…_@y †U y… † ‡ Z ; y… †@U y_@y… †   d; (15) Z ; y… † ˆ 4y Lÿy… † Rs2 0…; y†L2: (16)

One of the characteristics of this velocity field state is that when function Rs0(,y) is known, all of the velocity com-ponents can be readily formulated. According to the velocity field formulated above, the strain rate can be calculated as follows: _"rrˆ@Vr…_@rr; ; y†; _" ˆ1_r@V…_@r; ; y†‡Vr…r; ; y_r †; _"yyˆ@Vy…_@yr; ; y†; (17) _"rˆ1₂ @V…_@rr; ; y†ÿV…r; ; y_r †‡1_r@Vr…_@r; ; y†   ; _"yˆ1₂ @V…_@yr; ; y†‡1_r@Vy…_@r; ; y†   ; _"yrˆ1₂ @Vr…_@yr; ; y†‡@Vy…_@rr; ; y†   :

3. Upper bound solution

The upper bound theorem speci®es that the power, J, consumed in the plastic deformation zone should be mini-mized with respect to  for the actual velocity ®eld:

J ˆ _Wi‡ _Ws‡ _Wf: (18)

The internal deformation power, _Wi, is:

_Wiˆ Z Vp y
_"dV ˆ 2 3 p Z _L 0 Z _fy 0 Z _Rs0;y 0 y
1 2_"ij_"ij  1=2 rdrddy: (19) The shear power _Ws is attributed to the velocity

disconti-nuity (slip) on the rigid-plastic zone interface, plane A±A0

and B±B0_{; i.e.:} _Wsˆ Z ÿs 1_ 3 p
y
DVÿsds ˆ 1 3 p Z _f0 0 Z _Rs0;0 0 y
V 2 …r; ; 0†‡Vr2…r; ; 0† h i1=2 rdrd ‡ 1 3 p Z _fL 0 Z _Rs0;L 0 y
V 2 …r; ; L†‡Vr2…r; ; L† h i₁₌₂ rdrd: (20) In this study, the friction factor m is used to calculate the friction energy loss on the die surface and m is deemed constant during the extrusion process. Thus, the friction power dissipated on die surface, _Wf, becomes:

_Wfˆ Z ÿf m_ 3 p
y
DVÿfds ˆ m_ 3 p Z _L 0 Z _fy 0 y
DVÿf
1 ‡  @Rs0…; y† @y  ₂ ‡ _Rs 1 0…; y† @Rs0…; y† @  ₂₁₌₂ Rs0…; y†ddy; (21) where: DVÿf ˆ Vr2…r; ; y† ‡ V2…r; ; y† ‡ Vy2…r; ; y† n o₁₌₂ : (22)

4. Results and discussion

To demonstrate the effectiveness of the proposed velocity ®eld, several non-axisymmetric extrusions are selected as the objects of study, i.e. from round rod to square, hexagonal and octagonal sections. The dies used herein are equal-angle divided and linearly connected converging dies as shown in Fig. 3. In this manner, Rs0(,y) can be expressed as: Rs0…; y† ˆ R0ÿ R… 0ÿRf…††
_Ly

h i

n o

; (23)

where Rf() is the exit profile function of the billet. Owing to the entrance and exit cross-sections not being of the same shape, the semi-die angle in this study is defined as half of the die surface inclination in all of the calculations that follow. The results are shown below.

Fig. 4 indicates that for extrusion at small semi-die angles, the pressures required are larger than those for larger semi-die angles. Increasing the semi-semi-die angle implies a gradual decrease of extrusion pressure, which then reaches a mini-mum pressure. Beyond this optimal semi-die angle, the extrusion pressure increases with the semi-die angle. Such an increase is due to that the two dominant components of

Fig. 3. Extrusion die for a hexagonal cross-section.

the power consumed in the extrusion process, i.e., the internal deformation power and the friction power con¯ict with each other. With a smaller semi-die angle, the length of contact between the billet and the die is longer, causing signi®cantly high friction losses. On the other hand, for a larger semi-die angle, the die length is shorter; thus, the internal deformation becomes a predominant factor. Accord-ing to Fig. 4, the extrusion pressure increases with an increase of the reduction of area at the same semi-die angle. For the working conditions indicated in the ®gure, the extrusion pressures apparently reach a minimum at a semi-die angle of between 108 and 208.

Fig. 5 illustrates the effects of friction factor m on the extrusion pressure. Herein, the extrusion pressure increases with an increasing friction factor. If the friction factor m is zero, no power losses occur on the die surface. The total extrusion pressure consists mainly of the internal deforma-tion component. Fig. 6 displays the parameter  against friction factor m. Eq. (2) clearly indicates that when  is large, the Vy(r,,y) at die surface is small. This ®nding suggests that the distribution of Vy(r,,y) over the radius has a more prominent convexity. For a larger friction factor

m, the die surfaces are more sticky; the velocity component along the extrusion axis tends to be distorted more severely also. Therefore,  increases with an increase of friction factor m.

Fig. 7 presents the in¯uence of the product shape by plotting the extrusion pressure against the number of section sides. According to this ®gure, the extrusion pressure is larger when the number of sides of the exit cross-section is smaller. Although the difference between two product shapes in terms of the extrusion pressure is appreciable, it decreases with an increase of the number of section sides. When the number of section sides increases, the product cross-section approaches circular and the extrusion pressure is closer to that of axisymmetric extrusion.

An important features in this study is that the velocity component in the extrusion axis is not uniform. Fig. 8 summarizes the effects of  on the extrusion pressure. This ®gure reveals that for the given set of process variables, the extrusion pressure calculated with a uniform velocity along the extrusion axis exceeds that of the non-uniform velocity ®eld. On the other hand, the larger the semi-die angle implies a more severe deformation. Thus, the effect of  Fig. 5. Effects of the friction factor m on the extrusion pressure.

Fig. 6. Effects of the friction factor m on .

Fig. 7. Effects of the product shape on the extrusion pressure.

is more prominent, as indicated from the difference in the extrusion pressure.

Fig. 9 compares the results of this study with the upper bound solution of Gunasekera and Hoshino [7]. According to this ®gure, extrusion pressures are shown against the relative die length (L/R0). These comparisons reveal that the proposed method offers a superior upper bound solution. 5. Conclusions

This study present a novel model based on the upper bound theorem and, then, applies it to the extrusion of non-axisymmetric-shaped sections. The kinematically admissi-ble velocity ®eld formulated has a convex velocity distribu-tion along the extrusion axis. The model proposed herein is applied to the extrusion of rectangular, hexagonal, and octagonal sections. The extrusion pressures are plotted against various variables such as semi-die angle, friction factor and reduction ratio. Based on the results in this study, the following can be concluded.

1. The extrusion pressure is the lowest for semi-die angle of between 108 and 208.

2. A higher friction factor renders more severe velocity distortions in the extrusion direction for a given die geometrical configuration.

3. Shape complexity significantly influences the extrusion pressure. For the same round billet, the extrusion pressure decreases with an increase of the number of sides of the section of the product.

4. The proposed three-dimensional velocity field renders a better upper bound solution than the existing mode [7]. It can also be applied to extrude sections with arbitrary cross-sectional shapes.

References

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[11] M. Kiuchi, M. Ishikawa, Upper bound analysis of extrusion and/or drawing of L-, T-, and H- sections Ð study on non-symmetric extrusion and drawing II, J. JSTP 24(270) (1983) 722±729. Fig. 9. Comparison of the present result with those of Hoshino [7] for the