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Finite element analysis of material flow in equal channel angular pressing
Scripta mater. 44 (2001) 677– 681 www.elsevier.com/locate/scriptamat
FINITE ELEMENT ANALYSIS OF MATERIAL FLOW IN EQUAL CHANNEL ANGULAR PRESSING Jin-Yoo Suh, Hee-Soo Kim, Jong-Woo Park and Joon-Yeon Chang Alloy Design Research Center, Korea Institute of Science and Technology, P.O.Box 131, Cheongryang, Seoul, 130-650, Korea (Received May 18, 2000) (Accepted in revised form September 19, 2000) Keywords: Equal channel angular pressing; Finite element method; Non-uniform deformation
Introduction Equal-Channel Angular (ECA) pressing is a promising process developed by Segal [1], which can produce large plastic shear deformation. The most unique aspect of this process is that the crosssectional shape of the billet remains unaltered during the process so that there is no geometric restriction on the strain that can be achieved. Recent applications of this process are mainly focused on the production of ultra-fine grained materials by repeated shear deformation [2]. From this point of view, it is important to estimate the magnitude of the strain introduced into the samples, and to understand the deformation behavior of this process. Figure 1 shows a schematic diagram of the ECA pressing process, where ⌽ is an intersection angle of the two equal cross-sectional channels and ⌿ indicates the angle occupied by the curved region at the point of intersection. Segal et al. [1,3] showed that the strain induced is given by ⑀ ⫽ 2/公3 cot ⌽/2 when ⌿ ⫽ 0°. Iwahashi et al. [4] proposed a more general equation to describe the effect of geometric process parameters of the die such as ⌽ and ⌿:
⑀⫽
冑冉 冉 2
3
cot
冊
冉
⌽ ⌿ ⌽ ⌿ ⫹ ⫹ ⌿ cos ec ⫹ 2 2 2 2
冊冊
(1)
According to the equation, the strain is dependent only on the two angular quantities of ⌽ and ⌿. The strain reaches its maximum value of 1.15 at ⌿ ⫽ 0° and monotonously decreases to 0.9 at ⌿ ⫽ 90° when ⌽ is fixed as 90°. Experimental results, however, still show somewhat different deformation patterns including dead zone and non-uniform deformation [5–7]. Figure 2 clearly shows the nonuniform deformation in the lower part of the specimen which passes through the outer corner of the curved die (⌽ ⫽ 90°, ⌿ ⫽ 90°). Since equation 1 basically assumes homogeneous deformation over the whole specimen, it is impossible to explain this non-uniform deformation and resultant effective strain analytically. In order to investigate such a non-uniform deformation in the ECA pressing, the finite element method (FEM) was adopted. There have been a couple of previous investigations addressing the numerical analysis of the deformation behavior in ECA pressing [8,9]. However these did not deal with this type of non-uniform deformation. 1359-6462/01/$–see front matter. © 2001 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(00)00645-X
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Figure 1. Schematic illustration of the ECAP facility showing the angles and ⌿.
Experimental Finite element simulation of the ECA pressing was carried out using the commercial finite element code ABAQUS. Two-dimensional plane strain model was used for simplicity. The die and ram were assumed to be rigid so that there should be no deformation. The specimen was also simplified as being perfectly plastic with a von Mises yield criterion. The friction effect was excluded for comparison with prior theoretical work [3,4]. Dies with the angle ⌿ of 0°, 30°, 60° and 90° at fixed intersection angle of 90° were simulated. The macrostructure of pure Al pressed in a die with ⌿ ⫽ 90° and ⌽ ⫽ 90° was observed in order to compare with the numerical results. The etchant was Barker⬘s reagent consisting of distilled water and fluoboric acid (35%). Results Figure 3 shows the deformation patterns of the specimen calculated at different shape parameters of ⌿ with fixed ⌽ ⫽ 90°. The patterns in the figure present the steady state in which the deformation of the
Figure 2. Non-uniform deformation in the lower part of the material which passes through the curved die (⌽ ⫽ 90°, ⌿ ⫽ 90°).
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Figure 3. Deformation patterns of the specimen predicted by the numerical analysis. (a) ⌿ ⫽ 0°, (b) ⌿ ⫽ 60°, and (d) ⌿ ⫽ 90°, when ⫽ 90°.
first and the end part of the billet is not considered. The grid distortion directly indicates the magnitude of shear deformation. Figure 3(a) shows uniform shear deformation over the whole material in the case of ⌿ ⫽ 0°, which is in good agreement with the strain predicted by equation 1. In Figure 3(b) simulated in ⌿ ⫽ 30°, sheared lattice can hardly be seen in the lower part, which causes non-uniform shear deformation. As the angle ⌿ increases, the magnitude of shear deformation decreases and the strain difference between upper and lower part of the material becomes more apparent. It is manifest that the bigger the angle ⌿, the more severe the magnitude of the inhomogeneity. The deformation pattern in Figure 3(d) is very close to that of Figure 2. Figure 4 shows the effective plastic strain of deformed material along A-A⬘ sectioned part as defined in Figure 1. The effective plastic strain is uniform in the upper region near the point A. As the angle ⌿ increases, the value of the effective plastic strain in the lower region, around the point A⬘, decreases. The area under the curve, representing the total amount of strain, also decreases as the ⌿ increases. This means that as the ⌿ increases, the pressing load decreases so that the passage of the material through the die becomes easier.
Figure 4. Effect of the curvature angle on the effective plastic strain of the specimen predicted by the numerical analysis.
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Figure 5. Comparison between numerical results and analytical solution (Eq. 1) of the effective plastic strains.
The mean value of the effective plastic strain in the uniform region at each value of ⌿ is compared with the calculated value of equation 1 in Figure 5. The values obtained by the FEM and equation 1 are in good agreement. It is evident that the mean effective strain becomes low as the angle ⌿ increases. Discussion The deformation behavior of ECA pressing was investigated with a fixed intersection angle (⌽) of 90° and ⌿ angles of 0°, 30°, 60° and 90°. In the case of ⌿ ⫽ 0°, the shear deformation takes place uniformly over the whole sample as shown in Figure 3(a). Although non-uniform deformation in the billet has been reported even in the case ⌿ ⫽ 0° [5,6], it is usually due to friction between the surface of the specimen and the die wall. Since this study focused only on the origin of non-uniform deformation with respect to the arc of curvature (⌿), the effect of friction was ignored in the analysis. The deformation behavior and effective plastic strain calculated in this study are in good agreement with the suggestion of Iwahashi [4] when ⌿ ⫽ 0°. But as the ⌿ angle increases, the lower part of the specimen shows a different deformation pattern and the range of the non-uniform deformation grows. For better understanding of the deformation pattern, the contour plot of effective plastic strain is shown in Figure 6, where different curves correspond to different strain levels. In Figure 6(a), at ⌿ ⫽ 0°, the contour lines of plastic strain are concentrated on the plane connecting the two corners of the channel. Thus, the plastic deformation takes place uniformly over the very narrow region along the diagonal line connecting two corners. But as ⌿ increases, as Figure 6(b), the material deforms gradually along the curved, outside corner of the die and the region of non-uniform deformation spreads out from the lower part of the specimen toward the upper part. Especially, the lower region shows a symmet-
Figure 6. Effective plastic strain fields of the specimen predicted by the numerical analysis. (a) ⌿ ⫽ 0°, and (b) ⌿ ⫽ 90°, when ⫽ 90°.
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rically shaped contour lines, which can also be seen in Figure 4. This type of strain distribution indicates that the lower part of the specimen is certainly subjected to different kinds of deformation behavior, similar to bending. From the results, the deformation mechanism of ECA pressing can be inferred as follows. The basic mechanism of shear deformation is the path difference induced by the ECA die. This can be accomplished by hindering the passage of the material. At ⌿ ⫽ 0°, the plunger pushes down the cross-section area of a billet uniformly, and the reaction force of the die acting as a compression to the material is also uniformly distributed. This uniform force makes the specimen push out through another channel. In this process the magnitude of the shear strain is determined by the path difference of the material which advances through the die. However, the curved die allows for material to pass through it more easily. The origin of non-uniform deformation is thought to be a decrease in reaction force of the lower part of the die because of the geometrical effect. Conclusion Numerical analysis with finite element method of ECA pressing was carried out to verify the theoretical deformation model. When ⌿ is zero, homogeneous shear deformation takes place on the whole cross-section of the specimen, which is well predicted by the theoretical model. As ⌿ increases, the shear deformation at the upper part of the specimen is found to be comparatively uniform, and the effective plastic strain is in good agreement with the theoretical value. The shear deformation at the lower part of the specimen, however, becomes smaller and the effective plastic strain is considerably lower than the theoretical result. The non-uniform deformation is governed by the shape of the die, thus the calculation of total effective strain accumulated on the specimen should be carefully considered. References 1. 2. 3. 4. 5. 6. 7. 8. 9.
V. M. Segal, V. I. Reznikov, A. E. Drobyshevskiy, and V. I. Kopylov, Russ. Metall. 1, 99 (1981). M. Nemoto, Z. Horita, M. Furukawa, and T. G. Langdon, Metals Mater. 4, 1181 (1998). V. M. Segal, Mater. Sci. Eng. A. 197, 157 (1995). Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto, and T. G. Langdon, Scripta Mater. 35, 143 (1996). A. Shan, I. G. Moon, H. S. Ko, and J. W. Park, Scripta Mater. 41, 353 (1999). H. S. Ko, J. Y. Chang, S. G. Choi, and I. G. Moon, J. Kor. Inst. Met. Mater. 37, 441 (1999). Y. Wu and I. Baker, Scripta Mater. 37, 437 (1997). P. B. Prangnell, C. Harris, and S. M. Roberts, Scripta Mater. 37, 983 (1997). D. P. Delo and S. L. Semiatin, Met. Mater. Trans. A. 30A, 1391 (1999).
FINITE ELEMENT ANALYSIS OF MATERIAL FLOW IN EQUAL CHANNEL ANGULAR PRESSING Jin-Yoo Suh, Hee-Soo Kim, Jong-Woo Park and Joon-Yeon Chang Alloy Design Research Center, Korea Institute of Science and Technology, P.O.Box 131, Cheongryang, Seoul, 130-650, Korea (Received May 18, 2000) (Accepted in revised form September 19, 2000) Keywords: Equal channel angular pressing; Finite element method; Non-uniform deformation
Introduction Equal-Channel Angular (ECA) pressing is a promising process developed by Segal [1], which can produce large plastic shear deformation. The most unique aspect of this process is that the crosssectional shape of the billet remains unaltered during the process so that there is no geometric restriction on the strain that can be achieved. Recent applications of this process are mainly focused on the production of ultra-fine grained materials by repeated shear deformation [2]. From this point of view, it is important to estimate the magnitude of the strain introduced into the samples, and to understand the deformation behavior of this process. Figure 1 shows a schematic diagram of the ECA pressing process, where ⌽ is an intersection angle of the two equal cross-sectional channels and ⌿ indicates the angle occupied by the curved region at the point of intersection. Segal et al. [1,3] showed that the strain induced is given by ⑀ ⫽ 2/公3 cot ⌽/2 when ⌿ ⫽ 0°. Iwahashi et al. [4] proposed a more general equation to describe the effect of geometric process parameters of the die such as ⌽ and ⌿:
⑀⫽
冑冉 冉 2
3
cot
冊
冉
⌽ ⌿ ⌽ ⌿ ⫹ ⫹ ⌿ cos ec ⫹ 2 2 2 2
冊冊
(1)
According to the equation, the strain is dependent only on the two angular quantities of ⌽ and ⌿. The strain reaches its maximum value of 1.15 at ⌿ ⫽ 0° and monotonously decreases to 0.9 at ⌿ ⫽ 90° when ⌽ is fixed as 90°. Experimental results, however, still show somewhat different deformation patterns including dead zone and non-uniform deformation [5–7]. Figure 2 clearly shows the nonuniform deformation in the lower part of the specimen which passes through the outer corner of the curved die (⌽ ⫽ 90°, ⌿ ⫽ 90°). Since equation 1 basically assumes homogeneous deformation over the whole specimen, it is impossible to explain this non-uniform deformation and resultant effective strain analytically. In order to investigate such a non-uniform deformation in the ECA pressing, the finite element method (FEM) was adopted. There have been a couple of previous investigations addressing the numerical analysis of the deformation behavior in ECA pressing [8,9]. However these did not deal with this type of non-uniform deformation. 1359-6462/01/$–see front matter. © 2001 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(00)00645-X
678
FEM ANALYSIS OF ECAP
Vol. 44, No. 4
Figure 1. Schematic illustration of the ECAP facility showing the angles and ⌿.
Experimental Finite element simulation of the ECA pressing was carried out using the commercial finite element code ABAQUS. Two-dimensional plane strain model was used for simplicity. The die and ram were assumed to be rigid so that there should be no deformation. The specimen was also simplified as being perfectly plastic with a von Mises yield criterion. The friction effect was excluded for comparison with prior theoretical work [3,4]. Dies with the angle ⌿ of 0°, 30°, 60° and 90° at fixed intersection angle of 90° were simulated. The macrostructure of pure Al pressed in a die with ⌿ ⫽ 90° and ⌽ ⫽ 90° was observed in order to compare with the numerical results. The etchant was Barker⬘s reagent consisting of distilled water and fluoboric acid (35%). Results Figure 3 shows the deformation patterns of the specimen calculated at different shape parameters of ⌿ with fixed ⌽ ⫽ 90°. The patterns in the figure present the steady state in which the deformation of the
Figure 2. Non-uniform deformation in the lower part of the material which passes through the curved die (⌽ ⫽ 90°, ⌿ ⫽ 90°).
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Figure 3. Deformation patterns of the specimen predicted by the numerical analysis. (a) ⌿ ⫽ 0°, (b) ⌿ ⫽ 60°, and (d) ⌿ ⫽ 90°, when ⫽ 90°.
first and the end part of the billet is not considered. The grid distortion directly indicates the magnitude of shear deformation. Figure 3(a) shows uniform shear deformation over the whole material in the case of ⌿ ⫽ 0°, which is in good agreement with the strain predicted by equation 1. In Figure 3(b) simulated in ⌿ ⫽ 30°, sheared lattice can hardly be seen in the lower part, which causes non-uniform shear deformation. As the angle ⌿ increases, the magnitude of shear deformation decreases and the strain difference between upper and lower part of the material becomes more apparent. It is manifest that the bigger the angle ⌿, the more severe the magnitude of the inhomogeneity. The deformation pattern in Figure 3(d) is very close to that of Figure 2. Figure 4 shows the effective plastic strain of deformed material along A-A⬘ sectioned part as defined in Figure 1. The effective plastic strain is uniform in the upper region near the point A. As the angle ⌿ increases, the value of the effective plastic strain in the lower region, around the point A⬘, decreases. The area under the curve, representing the total amount of strain, also decreases as the ⌿ increases. This means that as the ⌿ increases, the pressing load decreases so that the passage of the material through the die becomes easier.
Figure 4. Effect of the curvature angle on the effective plastic strain of the specimen predicted by the numerical analysis.
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Figure 5. Comparison between numerical results and analytical solution (Eq. 1) of the effective plastic strains.
The mean value of the effective plastic strain in the uniform region at each value of ⌿ is compared with the calculated value of equation 1 in Figure 5. The values obtained by the FEM and equation 1 are in good agreement. It is evident that the mean effective strain becomes low as the angle ⌿ increases. Discussion The deformation behavior of ECA pressing was investigated with a fixed intersection angle (⌽) of 90° and ⌿ angles of 0°, 30°, 60° and 90°. In the case of ⌿ ⫽ 0°, the shear deformation takes place uniformly over the whole sample as shown in Figure 3(a). Although non-uniform deformation in the billet has been reported even in the case ⌿ ⫽ 0° [5,6], it is usually due to friction between the surface of the specimen and the die wall. Since this study focused only on the origin of non-uniform deformation with respect to the arc of curvature (⌿), the effect of friction was ignored in the analysis. The deformation behavior and effective plastic strain calculated in this study are in good agreement with the suggestion of Iwahashi [4] when ⌿ ⫽ 0°. But as the ⌿ angle increases, the lower part of the specimen shows a different deformation pattern and the range of the non-uniform deformation grows. For better understanding of the deformation pattern, the contour plot of effective plastic strain is shown in Figure 6, where different curves correspond to different strain levels. In Figure 6(a), at ⌿ ⫽ 0°, the contour lines of plastic strain are concentrated on the plane connecting the two corners of the channel. Thus, the plastic deformation takes place uniformly over the very narrow region along the diagonal line connecting two corners. But as ⌿ increases, as Figure 6(b), the material deforms gradually along the curved, outside corner of the die and the region of non-uniform deformation spreads out from the lower part of the specimen toward the upper part. Especially, the lower region shows a symmet-
Figure 6. Effective plastic strain fields of the specimen predicted by the numerical analysis. (a) ⌿ ⫽ 0°, and (b) ⌿ ⫽ 90°, when ⫽ 90°.
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rically shaped contour lines, which can also be seen in Figure 4. This type of strain distribution indicates that the lower part of the specimen is certainly subjected to different kinds of deformation behavior, similar to bending. From the results, the deformation mechanism of ECA pressing can be inferred as follows. The basic mechanism of shear deformation is the path difference induced by the ECA die. This can be accomplished by hindering the passage of the material. At ⌿ ⫽ 0°, the plunger pushes down the cross-section area of a billet uniformly, and the reaction force of the die acting as a compression to the material is also uniformly distributed. This uniform force makes the specimen push out through another channel. In this process the magnitude of the shear strain is determined by the path difference of the material which advances through the die. However, the curved die allows for material to pass through it more easily. The origin of non-uniform deformation is thought to be a decrease in reaction force of the lower part of the die because of the geometrical effect. Conclusion Numerical analysis with finite element method of ECA pressing was carried out to verify the theoretical deformation model. When ⌿ is zero, homogeneous shear deformation takes place on the whole cross-section of the specimen, which is well predicted by the theoretical model. As ⌿ increases, the shear deformation at the upper part of the specimen is found to be comparatively uniform, and the effective plastic strain is in good agreement with the theoretical value. The shear deformation at the lower part of the specimen, however, becomes smaller and the effective plastic strain is considerably lower than the theoretical result. The non-uniform deformation is governed by the shape of the die, thus the calculation of total effective strain accumulated on the specimen should be carefully considered. References 1. 2. 3. 4. 5. 6. 7. 8. 9.
V. M. Segal, V. I. Reznikov, A. E. Drobyshevskiy, and V. I. Kopylov, Russ. Metall. 1, 99 (1981). M. Nemoto, Z. Horita, M. Furukawa, and T. G. Langdon, Metals Mater. 4, 1181 (1998). V. M. Segal, Mater. Sci. Eng. A. 197, 157 (1995). Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto, and T. G. Langdon, Scripta Mater. 35, 143 (1996). A. Shan, I. G. Moon, H. S. Ko, and J. W. Park, Scripta Mater. 41, 353 (1999). H. S. Ko, J. Y. Chang, S. G. Choi, and I. G. Moon, J. Kor. Inst. Met. Mater. 37, 441 (1999). Y. Wu and I. Baker, Scripta Mater. 37, 437 (1997). P. B. Prangnell, C. Harris, and S. M. Roberts, Scripta Mater. 37, 983 (1997). D. P. Delo and S. L. Semiatin, Met. Mater. Trans. A. 30A, 1391 (1999).