F ig . 1 Schematic illustration of Equal channel angular pressing

ψ

Plunger

Billet

Die

ϕφ 2=

An UPPER BOUND SOLUTION ANALYSIS OF EQUAL CHANNEL

ANGULAR PRESSING

Wei Wei, Guang Chen Joint Laboratory of Nanostructured Materials and Technology, Nanjing University of

Science and Technology, Nanjing 210094, P.R. China E-mail: Benjamin.wwei@163.com

Keywords: Equal channel angular pressing (ECAP); Upper bound solution; Severe plastic deformation (SPD) Abstract: The purpose of this study is to obtain an upper bound solution of ECAP at 0=ψ and φ =90 deg, which aims at analysis of the relations between ECAP upper bound pressure and the die angles of φ and ψ . The results show that the value of pressing load and the equivalent strain, eε , decreases as the angle ψ increases at φ =90 deg, but the equivalent strain, eε , decreases rapidly and no less than 0.90. The measured maximum load required for ECAP is in good agreement with the values obtained from the upper bound solution.

Introduction

Equal channel angular pressing (ECAP) is an effective route to refining coarse-grain size into submicrometer or nanometer regime by means of large plastic shear deformation, which was first introduced by Segal [1]. The most advantage of ECAP is that the billet undergoes severe plastic deformation (SPD) but remains the same cross-sectional geometry so that it is possible to repeat the pressings for a number of passes in order to achieve a very high total strain. Fig. 1 illustrates a schematic diagram of the ECAP process, where φ is an intersectional angular of the two equal cross-sectional channels and ψ delineating the arc of curvature at the outer point of intersection of the two channels. The strain imposed on a single passage through the die is dependent only on the two angular quantities of φ and ψ . When φ is fixed as 90 deg, the strain reaches its maximum value of 1.155 at ψ =0. Therefore, ECAP is applied to obtain ultra-fine gained or nanostructured ingots as one of the most widely used techniques of SPD [2-4]. In order to optimize the procedure of ECAP for obtaining bulk homogeneous nanostructured ingots, experiments and finite element method (FEM) have recently been applied to analyze material flow [5], the effects of the tooling configuration [6] and the load of ECAP [7]. Previously, plastic flow during ECAP was analyzed by slip line method [1]. An upper bound solution of channel angular deformation has been made by Lee. [8]. However, no study has been made of mechanics analysis of ECAP when considering the outer die angle (ψ ). The purpose of this study is to obtain an upper bound solution of ECAP at ψ =0 and φ =90 deg, which aims at analysis of the relations between upper bound pressing pressure and the die angles of φ and ψ .

Estimation of the ECAP Upper Bound Pressure The flow pattern and velocity hodograph for ECAP without friction are shown in Fig. 2(a) and (b) when 0=ψ . It is supposed that the billet moves progressively inside both channels with the same speed, 10 ==VVt , and the plastic deformation zone is rigid. In Fig.2 (a), The plane 'OO is

Journal of Metastable and Nanocrystalline Materials Vol. 23 (2005) pp 109-112Online available since 2005/Jan/01 at www.scientific.net© (2005) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/JMNM.23.109

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velocity discontinuity plane. In Fig. 2(b), 'OOV is the shear velocity along the plane 'OO . The shear power for the deformation of material of 1 in width, sW& , is given by

)( '''

0'

oots VOOVBOVAOkW ⋅+⋅+⋅=& = ϕctgklV04 The external power, J , is given by 0qlVJ = , where q is the pressing pressure. Therefore the following relation is obtained.

ϕctgklVqlV 00 4≤ ，namely, ϕctgkq 22 ≤ (1)

Fig .2 (a) Flow pat tern Fig.2 (b) Veloci ty hodograph

Fig .3 (a ) Flow pa t te rn F ig .3 (b ) Ve loc i ty hodograph

The flow pattern and velocity hodograph for ECAP without friction are shown in Fig. 3(a) and (b) when φ =90 deg. The plane 'OA , OC and 'OB are all velocity discontinuity planes. The line length of CA' and 'CB is substituted respectively with the arc of CA' and 'CB . In Fig. 3 (a),

24 ψπα += , 42 ψπβ −= , 24 ψπδ −= , αcot'' lBBAA == , αeclOCOBOA cos'' === , αψψ eclOCCBCA cos)2()2('' =⋅== .

In Fig. 3(b), αcos0'' VVV OBOA == , αsin0'' VVV CBCA == ,

αψψ sin)2sin(2))2sin((2 0' VVV CAOC == . The shear power for the deformation of material of 1 in width, sW& , is given by

OCCAOAs VkOCVCAVOAVAAkW ⋅+⋅+⋅+⋅= )(2 ''''

0'&

The following relation can be obtained. )2()2sin()24cot(22 ψψψπ +++≤kq (2)

It is apparent that, equation (2) reduces to equation (1), kq 2 ≤2, when ψ =0.

O'

OO

V0

A

B

Vt V0=

1 Vt

VOO＇

Vt

C

O

O'

O

VOB＇

V0

A

B

A′

B′

Vt

Vt

VOA＇

VOC

VA＇C

V0=

1

VCB＇

110 Science and Technology of Nanomaterials

Evaluation of the Die Angles ofφand Ψ

Segal [9] considered an ECAP die having 0=ψ and φ ϕ2= , and showed that the equal strain on a single passage, eε , is given by

ϕε cot)32(=e (3) The effect of ϕ on the pressing pressure, q , and the equivalent strain, εe, can be plotted in Fig.4 from equations (1) and (3). The results show that the value of pressing pressure, q , and the equivalent strain, εe, decreases as the angle ϕ increases. As ϕ increases from 45 to 49 deg, the value of εe reaches 1.155 and 1.0 respectively. For non-hardening materials, q should be less than the flow stress k2 at ϕ =45 deg. Iwahashi et al. [10] incorporated the angle ψ into the analysis and showed that the equivalent strain is given by

)]22(cos)22cot(2)[31( ψφψψφε +++= ece (4)

Fig. 4 Effect of (a) the inner angle φ on the pressing pressure q (b) equivalent strain εe

Fig. 5 Effect of (a) the outer arc curvature angle ψ on the pressing pressure q (b) equivalent strain εe

The effect of ψ on the pressing pressure, q , and the equivalent strain, εe, can be plotted in Fig.5 from equations (2) and (4). It shows that the value of pressing pressure, q , and the equivalent strain, εe, decreases as ψ increases, but the equivalent strain, εe, decreases rapidly in Fig.5. For ψ =16 deg and ψ =30 deg, the equivalent strain, eε , reaches 1.07 and 1.02 respectively. kq 2 decreases correspondingly from 1.79 to 1.68. At ψ =36 deg, eε and kq 2 comes up to 0.995 and

0. 51

1. 52

2. 53

3. 54

4. 55

5. 56

15 2025 3035 4045 5055 6065 70

q/2k

(a)

00. 5

11. 5

22. 5

33. 5

15 20 25 30 35 40 45 50 55 60 65 70 75

(b)

0. 8

0. 9

1

1. 1

1. 2

0 10 20 30 40 50 60 70 80 90ψ

ε e

(b)

1. 25

1. 5

1. 75

2

2. 25

0 10 20 30 40 50 60 70 80 90ψ

q/2k

(a)

ε e

φ（DEG） φ（DEG）

Journal of Metastable and Nanocrystalline Materials Vol. 23 111

1.64 respectively. The equivalent strain decreases gradually as ψ increases, however, the decrement of the pressing pressure, q , is especially smaller when ψ exceeds 30 deg. This means that the passage of the sample through the die becomes easier as the ψ increases, and the reaction force of the die acting as compression to the sample also decreases. In this process the range of non-uniform shear deformation grows [11].

Fig. 6 Comparison between experimental results and theoretical solution (Eq.2) of the maximum load required for ECAP

The measured maximum load required for ECAP is compared with the calculated value based on the equation (2) in Fig.6. For these load measurements, 12mm×12mm×80mm billets of pure copper (99.98mass%) were used. The values obtained by the upper bond solution and experimental measurements are in good agreement. It is evident that the applied load decreases as as ψ increases, however, this trend becomes smaller when ψ exceeds 30 deg.

Summary and Conclusions

1) Two different relationships have been proposed for estimating the pressing pressure in ECAP. 2) The pressing pressure, q , and the equivalent strain, εe, decreases as the angle 2ϕ (φ =2ϕ )

increases. In this case the value of εe reaches the maximum of 1.155. 3) The results show that the value of pressing pressure, kq 2 , and the equivalent strain, εe,

decreases as the angle ψ increases at φ =90 deg. The measured maximum load required for ECAP is in good agreement with the values obtained from the upper bound solution. The influence of ψ on the shear strain and plastic deformation should been considered carefully during ECAP.

Acknowledgment

This work was supported by the Natural Science Foundation of Jiangsu Province, the People’s Republic of China granted No. BK 2001053.

References

1.V. M. Segal: Mater. Sci. Eng. A271 (1999), p.322 2. R. Z. Valiev, R. K. Islamgaliev, and I. V. Alexandrov: Prog. Mater. Sci. 45(2)(2000), p.103 3. R. Z. Valiev, T. C. Lowe, and A. K. Mukherjee: JOM 52(4) (2000), p.37 4. P. B. Berbon, N. K. Tsenev and R. Z. Valiev, et al.: Metall. Mater. Trans. 29A(9) (1998), p.2237 5. J. R. Bowen, P. B. Prangnell and S. M. Roberts: Mat. Sci. Eng. A287(1) (2000), p.87

6

7.5

9

10.5

12

13.5

15

0 10 20 30 40 50 60 70 80 90

ψ

Load

(ton

)

Theoretical solution

Experimetal results

112 Science and Technology of Nanomaterials

6. S. L. Semiatin, D. P. Delo and E. B. Shell: Acta Mater. 48(8) (2000), p.1841 7. L. Zuyan and W. Zhongjin: J. Mater. Proc. Tech. 94(2/3) (1999), p.193 8.D. N. Lee: Scripta Mater. 43(2) (2000), p.115 9.V. M. Segal: Mater. Sci. Eng. A197 (1995), p.157 10. Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto and T. G. Langdon: Scripta Mater. 35(2) (1996), p.143 11. J. Y. Suh, H. S. Kim, J.W. Park and J.Y. Chang: Scripta Mater. 44(4) (2001), p.677

Journal of Metastable and Nanocrystalline Materials Vol. 23 113

Science and Technology of Nanomaterials 10.4028/www.scientific.net/JMNM.23 An Upper Bound Solution Analysis of Equal Channel Angular Pressing 10.4028/www.scientific.net/JMNM.23.109

DOI References

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doi:10.1046/j.1463-6409.2000.00035.x [10] Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto and T. G. Langdon: Scripta Mater. 35(2) (1996), .143

doi:10.4028/www.scientific.net/MSF.204-206.437 [11] J. Y. Suh, H. S. Kim, J.W. Park and J.Y. Chang: Scripta Mater. 44(4) (2001), p.677

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