An analysis of pseudo equal-cross-section lateral extrusion through a curved channel

Journal of Materials Processing Technology 122 (2002) 260±265

An analysis of pseudo equal-cross-section lateral extrusion through a curved channel Chin-Tarn Kwan, Yuan-Chuan Hsu* Department of Mechanical Manufacturing Engineering, National Huwei Institute of Technology, 64 Wun-Hua Road, Huwei, Yunlin, Taiwan, ROC Received 21 February 2001

Abstract A streamline function suitable for pseudo equal-cross-section lateral extrusion through a curved channel is proposed. By using a proposed streamline function, the relative extrusion pressure on the punch, the effective strain on the exit section and the optimum die length for the lateral extrusion are determined. The effects of various process parameters on the solution are also discussed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Pseudo equal-cross-section lateral extrusion; Flow line function; Upper bound method

1. Introduction

2. Theoretical derivation

Equal-cross-section lateral extrusion or equal-channel angular pressing have been used to make ultra®ne grain material [1±3] in recent years. In these processes, the dimensions of the workpiece are not changed after deformation, so the workpiece can be deformed cyclically to very large strain within a die set. Liu et al. [4] performed a strain analysis of equal-cross-section lateral extrusion using a simple pure-shear theorem. Liu et al. [5] also proposed a new changing-channel lateral extrusion and analyzed the process using the ®nite element method. However, no studies have been undertaken for the process of equal-cross-section lateral extrusion through a curved channel. In this study, a streamline function suitable for pseudo equal-cross-section lateral extrusion through a curved channel is proposed. By using the proposed streamline function, the relative extrusion pressure on the punch, the effective strain on the exit section and the optimum die length for the lateral extrusion are determined. The effects of various process parameters on the solution are also discussed.

2.1. Derivation of generalized kinematically admissible velocity field

*

Corresponding author. Fax: ‡886-5-631-0824. E-mail address: [email protected] (Y.-C. Hsu).

The process of equal-cross-section lateral extrusion may be assumed as a plane-strain problem, even if the crosssection is circular, rectangular or any other shape. A schematic diagram of the S-type lateral extrusion of a pseudo equal-cross-section through a curved channel is shown in Fig. 1. The die pro®les are given by the following four-order polynomials:     d0 3e 2e f1 …x† ˆ ‡ Cf L2 ‡ 2 x2 ‡ 2C L x3 ‡ Cf x4 f L L3 2 (1a) and f2 …x† ˆ

  d0 3e 2 2 ‡ Cf L ‡ 2 x L 2



 2e ‡ 2Cf L x3 ‡ Cf x4 L3 (1b)

which are subjected by the boundary conditions of f1 …0† ˆ d0 =2, f2 …0† ˆ d0 =2, f1 …L† ˆ d0 =2 ‡ e, f2 …L† ˆ d0 =2 ‡ e, f10 …0† ˆ f10 …L† ˆ 0, f20 …0† ˆ f20 …L† ˆ 0, where Cf is a geometric parameter related to the relative position of an in¯ection point (Lf/L) given by: Cf ˆ

0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 0 8 0 - 8

3e…1 ‰1

2Lf =L†

6Lf =L ‡ 6…Lf =L†2 ŠL4

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Nomenclature Cf d0 f1(x), f2(x) f10 ; f20 J L Lf Lopt m V V x, V y V 0, Vf DV _ i; W _f W x,y

coefficient in f1(x) and f2(x) related to the position of the inflection point extrusion width continuous die-profile functions df1(x)/dx, df2(x)/dx: the first derivative of dieprofile function f1(x) and f2(x) with respect to x, respectively upper-bound on the lateral extrusion power die length position of an inflection point in a bi-quadratic polynomial die profile die length at which the minimum lateral extrusion pressure exists friction factor at the die±workpiece interface volume of the plastic region velocity components in Cartesian coordinates incoming velocity of the tube and outgoing velocity of the extrusion product, respectively velocity discontinuity power dissipation due to internal deformation and friction, respectively Cartesian coordinates

Greek letters G1 ; G2 rigid±plastic boundaries defined in Fig. 1 G3 ; G4 die surfaces as the friction boundaries ef final effective strain of the extruded product ef avg average total effective strain at the exit section _ e e; effective strain-rate and effective strain, respectively s effective stress sm average effective stress defined in Eq. (12)

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According to the die-pro®le function, a streamline function for the material ¯owing in the pseudo equal-cross-section channel can be assumed as follows:      3e 2e 3 4 ‡ 2C L x ‡ C x y c ˆ V0 Cf L2 ‡ 2 x2 f f L L3 (2) Next, the horizontal velocity component Vx can be expressed by: Vx ˆ

@c ˆ V0 @y

(3)

and the vertical velocity component Vy by: @c @x   3e 2 ˆ V0 2 Cf L ‡ 2 x L

Vy ˆ

   2e 2 3 3 3 ‡ 2Cf L x ‡ 4Cf x L (4)

The vertical velocity component of Eq. (4) satis®es the following boundary conditions at the entrance and exit: Vy jxˆ0 ˆ Vy jxˆL ˆ 0

(5)

From the derived velocity ®eld of Eqs. (3) and (4), the strainrate components can be computed subsequently as follows: e_ x ˆ

@Vx ˆ0 @x

@Vy ˆ0 @y   1 @Vx @Vy e_ xy ˆ ‡ 2 @y @x   3e ˆ V0 Cf L2 ‡ 2 L

(6a)

e_ y ˆ

(6b)

3

   2e 2 ‡ 2C L x ‡ 6C x (6c) f f L3

Fig. 1. Schematic diagram of S-type pseudo equal-cross-section lateral extrusion.

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2.2. Upper-bound formulation In accordance with the upper-bound theorem [6], the following equation should be minimized for the actual velocity distribution: X X X _i‡ _s‡ _f J ˆ W W W (7) where J is an upper-bound on the total power consumption, _ s the power con_ i the internal power of deformation, W W _ f the frictional power sumption on the shear boundary, and W dissipated over the frictional boundary. _ i , is In Eq. (7), the internal power of deformation, W calculated from Eqs. (6a)±(6c). Z _ W i ˆ sm e_ dv (8) v

where

 1=2 _e ˆ p2 1 …_e2 ‡ e_ 2 † ‡ e_ 2 x y xy 3 2 and sm represents the mean effective stresses for work-hardening material, approximated by the following expression: R ef avg s de sm ˆ 0 (9) ef avg where ef avg is an average total effective strain at the exit section, de®ned as: R ef dA (10) ef avg ˆ RG2 G2 dA

Rt where ef ˆ 0 e_ dt and can be integrated along a constant steamline. Velocity discontinuities do not exist on the entrance and exit boundary, G1 and G2, from the basic nature of the derived velocity ®eld. Therefore, the power of shear defor_ s , vanishes. mation consumed on these boundaries, W The frictional power dissipated over the die surface G3 and the plug surface G4 are identical, as follows: Z msm _ W f ˆ p jDVj ds 3 S Z msm L 2 1=2 ˆ p ‰Vx ‡ Vy2 Šyˆf1 ‰1 ‡ …f10 †2 Š1=2 dx 3 0 (11) for G3 or G4 where m is the friction factor at the die (or plug)±workpiece interface. The upper-bound on the lateral extrusion stress is then determined as: se ˆ

J d 0 V0

(12)

In addition, the relative extrusion stress is obtained by: se J ˆ sm d0 V0 sm

(13)

If the workpiece behaves as a rigid±perfectly plastic material, then sm ˆ s0 automatically.

Fig. 2. Relative lateral extrusion stress se/s0 as a function of relative die length 2L/d0 for m ˆ 0:1, Lf =L ˆ 0:5 and e=d0 ˆ 0:5.

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Fig. 3. Relationship of relative extrusion stress se/s0 with die length 2L/d0 and with eccentric distance e/d0 for m ˆ 0:1 and Lf =L ˆ 0:5.

3. Results and discussion

3.1. Effect of die length

In order to study the effect of various parameters for pseudo cross-section lateral extrusion through a curved channel, the derived velocity ®eld is applied to analyze the process. In this study, the workpiece material is assumed to be rigid±perfectly plastic.

For given workpiece dimensions, friction factor, and the position of the in¯ection point of the die pro®les, the variation of the lateral extrusion stress with the die length is shown in Fig. 2. This ®gure reveals that the internal deformation and the friction dominate the optimum die

Fig. 4. Variation of effective strain on the exit cross-section with die length and with eccentric distance for m ˆ 0:1 and Lf =L ˆ 0:5.

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Fig. 5. Relative extrusion stress se/s0 as a function of friction factor m (Lf =L ˆ 0:5, and e=d0 ˆ 0:5).

length, i.e. the length that makes the extrusion pressure minimum. Fig. 3 shows the variation of the extrusion stress with die length (2L/d0) and with eccentric distance (e/d0) for given workpiece dimensions, friction factor and the position of the in¯ection point of the die pro®les. The ®gure shows that the optimum die length increases with increasing eccentric

distance, and that the relative extrusion pressure also increases with increasing eccentric distance. Fig. 4 shows the variation of effective strain at the exit cross-section with die length and with eccentric distance for a given friction factor, workpiece dimensions and the position of in¯ection point of the die pro®les. This ®gure shows that the effective strain decreases with increasing die length,

Fig. 6. Effect of friction factor on optimal die length 2Lopt/d0 and extrusion stress se/s0 (Lf =L ˆ 0:5 and e=d0 ˆ 0:5).

C.-T. Kwan, Y.-C. Hsu / Journal of Materials Processing Technology 122 (2002) 260±265

whereas the effective strain increases with increasing eccentric distance. 3.2. Effect of friction Fig. 5 shows the relative extrusion pressure (se/s0) calculated using Eq. (13), and FEM results (using DEFORM 2D [7]) for lateral extrusion through a curved channel. The relative extrusion pressure increases linearly with friction factor. As expected, the relative extrusion pressures obtained using Eq. (13) are greater than those obtained by FEM. For given lateral extrusion dimensions, the optimum die length and the related extrusion pressure are shown in Fig. 6. The ®gure shows that the optimum die length decreases with increasing friction factor. 4. Conclusions A streamline function suitable for pseudo equal-crosssection lateral extrusion through a curved channel has been proposed. By using the proposed streamline function, the

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relative extrusion pressure on the punch, the effective strain on the exit section and the optimum die length for the lateral extrusion were determined. The results of the analysis indicate pressure, the effective strain and the optimum die length all increase with increase in the eccentric distance, whereas the effective strain decreases with increase in the die length. References [1] R.Z. Valiev, A.V. Korzmkov, R.R. Mulyukov, Mater. Sci. Eng. A 168 (1993) 141. [2] V.M. Segal, Mater. Sci. Eng. A 197 (1995) 157. [3] R.Z. Valiev, E.V. Kozlov, Yu.F. Ivanov, J. Lian, A.A. Nazarov, B. Baudelet, Acta Metall. Mater. 42 (1994) 2467. [4] Z.Y. Liu, G.X. Liang, E.D. Wang, Z.R. Wang, Mater. Sci. Eng. A 242 (1998) 137. [5] Z.Y. Liu, G. Liu, Z.R. Wang, J. Mater. Process. Technol. 102 (2000) 30. [6] B. Avitzur, Metal Forming, Processes and Analysis, Huntington, New York, 1979. [7] DEFORM User's Guide, Scientific Forming Technologies Corporation, Columbus, OH, 1999.