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Finite element analysis of the effect of the inner corner angle in equal channel angular pressing
Materials Science and Engineering A 490 (2008) 438–444
Finite element analysis of the effect of the inner corner angle in equal channel angular pressing Seung Chae Yoon, Hyoung Seop Kim ∗ Department of Nano Material Engineering, Chungnam National University, Yuseong, Daejeon 305-766, Republic of Korea Received 6 January 2008; received in revised form 17 January 2008; accepted 22 January 2008
Abstract The inner corner angle (ICA) is one of the major factors affecting deformation homogeneity in workpieces during equal channel angular pressing (ECAP). In this study, the effect of the ICA on the plastic deformation behavior in ECAP was investigated using the finite element method. A round ICA induces highly inhomogeneous deformation in the head, tail, top and bottom regions of the workpiece due to increasing compressive and decreasing shear deformation components. It was found that a round inner corner with an angle up to 9◦ is acceptable in finite element simulations for reproducing a sharp inner corner. These results can serve as a design guide for processing and dies of ECAP. © 2008 Elsevier B.V. All rights reserved. Keywords: Equal channel angular pressing; Finite element analysis; Severe plastic deformation; Inner corner angle effect; Deformation homogeneity
1. Introduction As nanocrystalline or ultrafine grained metallic materials can show remarkable mechanical properties such as improved strength as well as excellent superplasticity, numerous studies involving these materials have been conducted during the last decade [1–3]. In order to increase the number of possible applications of these materials to industry, great interest in the manufacturing of bulk nanostructured materials (BNMs) has been raised, and many cases for widening the range of their application have been suggested [4,5]. Severe plastic deformation (SPD) processing has come into the spotlight as a process for manufacturing BNMs, and many research groups have been able to make full density BNMs effectively through SPD processing. An important feature of SPD, as a typical top-down approach for manufacturing BNMs, is its resulting sound and clean products that are free from porosities and contamination compared to bottom-up powder metallurgy techniques [6–8]. Equal channel angular pressing (ECAP) [6,8–11] can impose severe plastic strain through repeated processes without reducing the sectional area of the workpiece. ECAP is the most commonly used process in SPD to produce bulk ultrafine grained/nanostructured materials. It has a potential for use with ∗
Corresponding author. Tel.: +82 42 821 6596; fax: +82 42 822 5850. E-mail address: [email protected] (H.S. Kim).
0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.01.066
commercial applications as it involves scale-up and continuous processes. Fig. 1 shows a schematic that outlines the important geometric factors of the ECAP process. In general, geometries in ECAP are described as a channel angle of Φ and a corner angle of Ψ ; in an effort to determine the effect of ICA in this study, the corner angles are expressed as an outer corner angle (OCA) of Ψ o and an inner corner angle (ICA) of Ψ i . As the microstructures and the mechanical properties of the plastic-deformed materials are directly related to the degree of plastic deformation, an understanding of the phenomena associated with strain development is very important in a successful ECAP process design. The theoretical effective strain ε according to the die geometry is given in Eq. (1), as formulated by Segal [11] and Iwahashi et al. [12]: 1 Φ + Ψo Φ + Ψo ε = √ 2 cot + Ψo csc , (1) 2 2 3 Eq. (1) has been successfully used to calculate the average strain developed in ECAP assuming homogeneous and ideal simple shear. The assumption of ideal simple shear is valid if plastic deformation occurs in the immediate vicinity of the plane, i.e., in the shear zone or main deforming zone (MDZ) of region ABC which can be considered the intersection of the two of entry and exit channels. However, this assumption of perfect homogeneous deformation is not valid if ICA Ψ i > 0. In other words, if the inner corner is round, rigid body rotation occurs in MDZ. Although
S.C. Yoon, H.S. Kim / Materials Science and Engineering A 490 (2008) 438–444
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The results can be used for a design guide in the processing of ECAP. 2. Finite element analysis
Fig. 1. Schematic of the ECAP die: Φ denotes channel angle, Ψ i is the inner corner angle (ICA), Ψ o indicates the outer corner angle (OCA), and ABC is the main deformation zone (MDZ).
ICA can affect the deformation behavior in ECAP, few readily available studies on this matter exist. Contrary to the assumption of ideal simple shear deformation, actual ECAP deformation characteristics are not perfectly homogeneous. Consequently, the microstructures of materials processed using ECAP are inhomogeneous. The finite element method (FEM) based on continuum plasticity theory has become the most popular and reliable computer-aided-analysis tool in use for metal forming simulations. A number of studies concerning the processing parameters in ECAP have been conducted by using FEM analyses. Most numerical studies in this area concern the effects of the die geometry and the corner angle on deformation patterns [13–19]. Recently, Nagasekhar et al. [19] investigated OCA on flow and strain homogeneity. In addition, many studies of the formation of a corner gap due to the hardening of materials and the resulting non-uniform deformation have been done [17,18]. The die geometry that controls deformation remains an active research area; however there is much less research on ICA. An important point to be noted in FEM simulations of ECAP is that FEM analysts unconsciously control the die geometries to improve the efficiency of their simulations [20–23]. The most frequent arbitrary and the least investigated parameter is ICA. In many cases, a large ICA is used in polymers particularly where convergence is difficult. Cases that are more serious can arise when different ICA values are used in the same paper in which the authors investigated other effects. Indeed, the ICA influences the convergence and calculating speed in ECAP; however, it can be critical to control the ICA without assessing the validity of a round ICA. Therefore, it is meaningful to study the effect of the ICA on ECAP processing especially when it is focused on deformation uniformity. In this paper, the plastic deformation behavior of a workpiece during ECAP is simulated using FEM to investigate the ICA effect and to clarify the validity of using a round ICA.
Isothermal FEM simulations of the ECAP process were carried out using the commercial finite element code DEFORM2D (Version 9.1) [24]. In order to assess only the effect of the ICA, the temperature effect was ignored. According to experimental [25] and theoretical [26] analyses, the isothermal condition can be fulfilled at low pressing speeds. A FEM simulation calculation of ECAP processing was made by assuming plane strain considering the dimensions of the workpiece, which were 10 mm × 10 mm × 60 mm. Fig. 2 shows the initial mesh systems used in the ECAP simulations with various ICAs. The OCA was fixed as sharp (i.e. Ψ o = 0◦ ). Six die geometries with different ICA values were assessed: Ψ i = 0◦ (sharp corner), 9◦ , 18◦ , 27 ◦ , 45◦ and 90◦ , corresponding to fillet radius of curvature values of 0, 1, 2, 3, 5, and 10 mm, respectively, under the channel width of 10 mm. The friction factor between the interfaces of the die channel and the specimen was assumed to be 0.1, corresponding to the typical value in cold metal forming processes [27]. The workpiece material used in the study was fully annealed oxygen-free copper having equi-axed grains that were 50 m in size. As the stress–strain curve of the high strain range, which requires element in the FEM calculations of an ECAP process, cannot be obtained by conventional experimental techniques, the flow curve of pure copper was calculated up to a large strain of 10 using the dislocation cell evolution constitutive model [28]. This model can describe the hardening behavior of dislocation cell-forming crystalline materials at large strains. In all simulations, an automatic remeshing scheme was used to accommodate large strains and to take into account the occurrence of flow localization, which prevents further calculation during the simulation. There are approximately 3000 initial meshes of a four-node plane strain element. This number of elements was found to be sufficient to express local deformation of the strain rate insensitive workpieces through calculations with varying the number of elements in DEFORM2D. A constant ram speed of 1 mm/s was employed. It should be noted that ram displacements at the end of the processing steps all differ, due to the increasing channel width in the intersection of the entry and exit channels as ICA increases: ram displacements in ICA Ψ i = 0◦ , 9◦ , 18◦ , 27◦ , 45◦ , and 90◦ , are 50, 49, 48, 47, 45, and 40 mm, respectively. Beyond these ram displacements, the ram cannot travel further, as entry channel gaps are wider than the ram size of mm. 3. Results and discussion Fig. 3 shows simulated geometries represented by flow nets. It should be noted that a flow net is not a mesh system that varies during processing due to the remeshing of too much distorted elements. However, the deformed geometry of the flow nets maps the corner coordinates of the initial nets. It is clear that the deformed geometries with a sharp ICA and a round
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Fig. 2. Initial mesh for ECAP simulations with various inner corner angles.
ICA are different in MDZ, resulting in differences in the corner regions, the bottom regions and the top regions. For the sharp ICA case (Ψ i = 0◦ ), the deformation shows the typical behavior of strain hardening materials, including corner gap formation
due to strain hardening of the workpiece material, a less sheared bottom zone due to the round corner of the outer corner or the actual workpiece corner, transient front and rear zones, and relatively uniform shear deformation along the thickness direc-
Fig. 3. Deformed geometries represented by flow nets: (a) ICA Ψ i = 0◦ and (b) ICA Ψ i = 90◦ .
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Fig. 4. Predicted load vs. ram displacement curves during ECAP with various inner corner angle conditions.
Fig. 5. Deformed flow net angle with the inner corner angle.
tion due to the balance between the corner gap and the friction effects. On the other hand, for the round ICA case (Ψ i = 90◦ ), the deformed flow net shows a completely filled channel corner due to less movement of the bottom surface, a more sheared bottom zone corner due to less movement of the bottom surface, large transient front and rear zones, and non-uniform shear along the thickness direction, i.e., a top region with less shear and a bottom region with more shear. The top region with less shear is attributed to the round ICA, which induces rigid body motion flow without shear strain due to less constraint against the workpiece flow. It can be found from the final geometries of the workpieces that the deformation histories are different for the head part and the tail part of the workpiece as well as the inner and the outer parts in the steady-state region of the workpiece. Fig. 4 is the calculated load versus the ram displacement curves, showing the deformation steps during ECAP. Three stages of the ECAP load can be distinguished. This trend can be explained using the deformed geometry. Table 1 summarizes the ram displacements of deforming stages I–III under various ICAs. Stage I is the step during which the front part of the workpiece goes through the MDZ (the shaded region ABC in Fig. 1). It is at this point that most of the deformation occurs. The deformed geometry of displacement = 10 mm with ICA = 90◦ corresponds to stage I, as shown in Fig. 3(b). During stage I, the volume of the deformation part of the workpiece increases, as does the internal stress within the workpiece. The load increases as well, as the initially undeformed workpiece passes through the main deformation zone. In stage II, the head part of the workpiece exits in the MDZ after the peak load point. The back part of the workpiece head receives the most severe plastic deforma-
tion during stage I and is released past the peak point until deformation becomes steady; hence, the load decreases during stage II. The deformed geometry of displacement = 9 mm with ICA = 0◦ corresponds to stage II, as shown in Fig. 3(a). It is an interesting observation that a stage II does not exist with a round ICA (Ψ i = 90◦ ). This is a result of the continuous inner corner die shape. Stage III commences when sufficient interactions are established between the workpiece and the channel die, where the load no longer decreases and plateau-like behavior of the load is observed. The deformed geometries of displacements = 30 mm with ICAs = 0◦ and 90◦ correspond to stage III, as shown in Fig. 3(a) and (b). During stage I, the ECAP load is small in a case with a large ICA, due to the low constraint of the round die inner corner. On the other hand, the load increases with the ICA during stages II and III (the steady-state) due to the high heterogeneity in large ICA cases. It should be noted that load overshoot is not related to the overall workpiece but is rather a redundant load due to strain concentration in the back region of the head part of the workpiece. The amount of average shear strain governing the microstructural evolution of plastic-deformed metallic materials is proportional to the slant angle of the initial vertical line of the sample. The slant angle of the flow net with ICA is shown in Fig. 5. As ICA increases, the flow net slant angle decreases monotonously, although OCA is identical in every case. This figure implies that the overall shear strain development is affected by ICA as well as by the channel angle and the OCA, which must be considered. Although the slant angle of the flow net is related to the shear strain that develops in the central region of the workpiece, it does not represent the average shear strain that
Table 1 Ram displacements of deforming stages I–III under various ICAs
Stage I Stage II Stage III
Ψ i = 0◦
Ψ i = 9◦
Ψ i = 18◦
Ψ i = 27◦
Ψ i = 45◦
Ψ i = 90◦
0–4.7 4.7–9.2 9.2–50
0–5.2 5.2–9.7 9.7–49
0–6.6 6.6–13.8 13.8–48
0–8.1 8.1–15.8 15.8–47
0–13.9 13.9–23.3 23.3–45
0–19.7 None 19.7–40
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Fig. 6. Deformed flow net and effective strain distributions. The arrows represent the folding defect regions.
develops all over the sample due to heterogeneity in the deformation. This is described in more detail in the last part of this paper. Deformation homogeneity during ECAP can be analyzed in terms of the effective strain distributions shown in Fig. 6. The deformed flow nets at the end of the ECAP process are shown as well. Transient regions of the head and tail ends receive only small amounts of strain. Effective strains and their distributions at Ψ i = 0◦ and 9◦ are nearly identical: head and tail end regions with low strain deformation, a less deformed bottom region due to strain hardening, and a uniform (εeff = 1.08) deformation zone. As ICA increases, the highly deformed zone starting from the bottom surface point (initially this was the bottom corner point) spreads backward from the sample. Eventually, contrary to expectations, deformation becomes heterogeneous and the strain increases as ICA increases. Interestingly, this surprising result showing increased heterogeneity with ICA is attributed to the outer corner effect. Essentially, as ICA increases, the highly deformed bottom area becomes large and the workpiece fills the die corner gap. As die corner gap decreases, the bottom zone, which is less sheared, decreases and the overall strain increases. It should be noted that deformation homogeneity results from round corner angles: the lesser sheared zone in the bottom region is generated under the round flow of the materials in the bottom region; a round flow can occur in cases involving a round outer corner die and strain hardening of the material [16,17]. Fig. 7 shows the corner gap angle that develops during ECAP due to the strain hardening characteristics as a function of the ICA. The corner gap angle decrease almost linearly with the ICA. With an inner corner angle of 90◦ , a corner gap is not generated at all during ECAP. It is interesting to note that the outer corner of the die was sharp, i.e., OCA Ψ o = 0◦ .
An important factor is the bottom folding defect, as indicated by the arrows in Fig. 6. Folding defects are typically found in illdesigned metal forging processes. Specifically, surface regions on the back part of the heads penetrate into the workpiece inside. Fig. 8 shows an optical micrograph of a copper workpiece after ECAP, in which clear folding defect are evident. This folding defect can be avoided effectively using a slant perform [29]. The effective strain distributions from the top to bottom along the path of the workpiece normal to the pressing direction in the steady-state region of the workpiece are shown in Fig. 9. As the ICA increases to Ψ i = 45◦ , the effective strain of the top part shows nearly identical values while the effective strain of the bottom part gradually increases and then increases more
Fig. 7. Corner gap angle with the inner corner angle.
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Fig. 8. Folding defect found in the bottom region.
significantly when ICA reaches 90◦ . As seen through this result, the value of the ICA is not an arbitrary parameter for better speed and better convergence in FEM calculations. However, Ψ i = 9◦ is an acceptable value for FEM simulations based on the similarity of the deformed geometry (Fig. 6), the effective strain distribution (Figs. 6 and 9), and the load history (Fig. 4). It may be useful if a proper index for comparing homogeneities under various processing conditions is used in both experimental and theoretical analyses of ECAP. Here, the average effective strain εa and the standard deviation were applied for this purpose. The average effective strain εa of the FEM results can be calculated by averaging the strain distribution with respect to the volume of each finite element. The degree of standard deviation to assess the homogeneity index of ECAP processed materials is expressed as in the following equation: (xi − εa )2 fi σ(x) = , (2) fi Here, xi is the effective strain at the ith element, and fi is the related frequency. As observed in Fig. 10, as the ICA increases, the average effective strain gradually increases. It was also found that the standard deviation also increases as the ICA increases. Through this, it can be confirmed that during ECAP processing,
Fig. 10. Average of the strain and standard deviations (error bar) with the inner corner angle.
the negative effect of the ICA on the overall deformation homogeneity is clear, and due to the increase of the local strain of the bottom part. 4. Conclusions The effects of the ICA on the plastic flow and deformation homogeneity during ECAP were studied using FEM simulations to clarify the ICA effect and to assess the validity of using a round ICA die. With a sharp ICA die, the deformed geometry was predicted to be relatively homogeneous. Deformation heterogeneity develops, however, in dies with a round ICA. The round ICA induces highly inhomogeneous deformation in the head, tail, top and bottom regions of the workpiece due to increasing compressive and decreasing shear deformation components. It was found that a round inner corner with an angle up to 9◦ is acceptable in finite element simulations for reproducing a case with a sharp inner corner. These results can be a design guide for processing and for dies in ECAP processes. Acknowledgement This work was supported by Korea Science & Engineering Foundation through the NRL Program (R0A-2007-000-201040). References [1] [2] [3] [4]
Fig. 9. Effective strain along the path of the workpieces normal to the pressing direction in the steady-state region—0: top and 10: bottom positions.
H. Gleiter, Acta Mater. 48 (2000) 1. H.S. Kim, Y. Estrin, M.B. Bush, Acta Mater. 48 (2000) 493. S.C. Tjong, H. Chen, Mater. Sci. Eng. R 45 (2004) 1. J. Eckert, J. Das, G. He, M. Calin, K.B. Kim, Mater. Sci. Eng. A 449–451 (2007) 24. [5] K.Y. Park, Y.S. Na, S.C. Son, J.H. Lee, V.B. Nguyen, Metal. Mater. Inter. 15 (2007) 433. [6] R.Z. Valiev, R.K. Islamgaliev, I.V. Alexandrov, Prog. Mater. Sci. 45 (2000) 103. [7] N. Tsuji, Y. Ito, Y. Saito, Y. Minamino, Scripta Mater. 47 (2002) 893.
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[8] Z. Horita, M. Furukawa, M. Nemoto, A.J. Barnes, T.G. Langdon, Acta Mater. 48 (2000) 3633. [9] R.Z. Valiev, T.G. Langdon, Prog. Mater. Sci. 51 (2006) 881. [10] D.H. Shin, B.C. Kim, K.-T. Park, W.Y. Choo, Acta Mater. 48 (2000) 3245. [11] V.M. Segal, Mater. Sci. Eng. A 197 (1995) 157. [12] Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto, T.G. Langdon, Scripta Mater. 35 (1996) 143. [13] P.B. Prangnell, C. Harris, S.M. Roberts, Scripta Mater. 37 (1997) 983. [14] A.V. Nagasekhar, Y. Tick-Hon, S. Li, H.P. Seow, Mater. Sci. Eng. A 410–411 (2005) 269. [15] V.M. Segal, Mater. Sci. Eng. A 345 (2003) 36. [16] H.S. Kim, Mater. Sci. Eng. A 315 (2001) 122. [17] H.S. Kim, M.H. Seo, S.I. Hong, Mater. Sci. Eng. A 291 (2000) 86. [18] H.S. Kim, S.I. Hong, M.H. Seo, J. Mater. Res. 16 (2001) 856. [19] A.V. Nagasekhar, W. Wie, Y. Tick-Hon, G. Chen, Adv. Eng. Mater. 9 (2007) 572. [20] A.V. Nagasekhar, Y. Tick-Hon, H.P. Seow, J. Mater. Process. Technol. 192–193 (2007) 449.
[21] R.K. Oruganti, P.R. Subramanian, J.S. Marte, M.F. Gigliotti, S. Amancherla, Mater. Sci. Eng. A 406 (2005) 102. [22] S. Xu, G. Zhao, X. Ma, G. Ren, J. Mater. Process. Technol. 184 (2007) 209. [23] B. Aour, F. Za¨ıri, M. Na¨ıt-Abdelaziz, J.M. Gloaguen, O. Rahmani, J.M. Lefebvre, Int. J. Mech. Sci., in press. [24] DEFORM 2D, Scientific Forming Technologies Corporation Columbus, OH, 1997. [25] D. Yamaguchi, Z. Horita, M. Nemoto, T.G. Langdon, Scripta Mater. 45 (2000) 103. [26] H.S. Kim, Mater. Trans. 42 (2001) 536. [27] T. Altan, S.-I. Oh, H.L. Gegel, Metal Forming, Fundamental and Applications, ASM, OH, 1983, p. 85. [28] Y. Estrin, L.S. T´oth, A. Molinari, Y. Brechet, Acta Mater. 46 (1998) 5509. [29] S.C. Yoon, M.H. Seo, H.S. Kim, Scripta Mater. 55 (2006) 159.
Finite element analysis of the effect of the inner corner angle in equal channel angular pressing Seung Chae Yoon, Hyoung Seop Kim ∗ Department of Nano Material Engineering, Chungnam National University, Yuseong, Daejeon 305-766, Republic of Korea Received 6 January 2008; received in revised form 17 January 2008; accepted 22 January 2008
Abstract The inner corner angle (ICA) is one of the major factors affecting deformation homogeneity in workpieces during equal channel angular pressing (ECAP). In this study, the effect of the ICA on the plastic deformation behavior in ECAP was investigated using the finite element method. A round ICA induces highly inhomogeneous deformation in the head, tail, top and bottom regions of the workpiece due to increasing compressive and decreasing shear deformation components. It was found that a round inner corner with an angle up to 9◦ is acceptable in finite element simulations for reproducing a sharp inner corner. These results can serve as a design guide for processing and dies of ECAP. © 2008 Elsevier B.V. All rights reserved. Keywords: Equal channel angular pressing; Finite element analysis; Severe plastic deformation; Inner corner angle effect; Deformation homogeneity
1. Introduction As nanocrystalline or ultrafine grained metallic materials can show remarkable mechanical properties such as improved strength as well as excellent superplasticity, numerous studies involving these materials have been conducted during the last decade [1–3]. In order to increase the number of possible applications of these materials to industry, great interest in the manufacturing of bulk nanostructured materials (BNMs) has been raised, and many cases for widening the range of their application have been suggested [4,5]. Severe plastic deformation (SPD) processing has come into the spotlight as a process for manufacturing BNMs, and many research groups have been able to make full density BNMs effectively through SPD processing. An important feature of SPD, as a typical top-down approach for manufacturing BNMs, is its resulting sound and clean products that are free from porosities and contamination compared to bottom-up powder metallurgy techniques [6–8]. Equal channel angular pressing (ECAP) [6,8–11] can impose severe plastic strain through repeated processes without reducing the sectional area of the workpiece. ECAP is the most commonly used process in SPD to produce bulk ultrafine grained/nanostructured materials. It has a potential for use with ∗
Corresponding author. Tel.: +82 42 821 6596; fax: +82 42 822 5850. E-mail address: [email protected] (H.S. Kim).
0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.01.066
commercial applications as it involves scale-up and continuous processes. Fig. 1 shows a schematic that outlines the important geometric factors of the ECAP process. In general, geometries in ECAP are described as a channel angle of Φ and a corner angle of Ψ ; in an effort to determine the effect of ICA in this study, the corner angles are expressed as an outer corner angle (OCA) of Ψ o and an inner corner angle (ICA) of Ψ i . As the microstructures and the mechanical properties of the plastic-deformed materials are directly related to the degree of plastic deformation, an understanding of the phenomena associated with strain development is very important in a successful ECAP process design. The theoretical effective strain ε according to the die geometry is given in Eq. (1), as formulated by Segal [11] and Iwahashi et al. [12]: 1 Φ + Ψo Φ + Ψo ε = √ 2 cot + Ψo csc , (1) 2 2 3 Eq. (1) has been successfully used to calculate the average strain developed in ECAP assuming homogeneous and ideal simple shear. The assumption of ideal simple shear is valid if plastic deformation occurs in the immediate vicinity of the plane, i.e., in the shear zone or main deforming zone (MDZ) of region ABC which can be considered the intersection of the two of entry and exit channels. However, this assumption of perfect homogeneous deformation is not valid if ICA Ψ i > 0. In other words, if the inner corner is round, rigid body rotation occurs in MDZ. Although
S.C. Yoon, H.S. Kim / Materials Science and Engineering A 490 (2008) 438–444
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The results can be used for a design guide in the processing of ECAP. 2. Finite element analysis
Fig. 1. Schematic of the ECAP die: Φ denotes channel angle, Ψ i is the inner corner angle (ICA), Ψ o indicates the outer corner angle (OCA), and ABC is the main deformation zone (MDZ).
ICA can affect the deformation behavior in ECAP, few readily available studies on this matter exist. Contrary to the assumption of ideal simple shear deformation, actual ECAP deformation characteristics are not perfectly homogeneous. Consequently, the microstructures of materials processed using ECAP are inhomogeneous. The finite element method (FEM) based on continuum plasticity theory has become the most popular and reliable computer-aided-analysis tool in use for metal forming simulations. A number of studies concerning the processing parameters in ECAP have been conducted by using FEM analyses. Most numerical studies in this area concern the effects of the die geometry and the corner angle on deformation patterns [13–19]. Recently, Nagasekhar et al. [19] investigated OCA on flow and strain homogeneity. In addition, many studies of the formation of a corner gap due to the hardening of materials and the resulting non-uniform deformation have been done [17,18]. The die geometry that controls deformation remains an active research area; however there is much less research on ICA. An important point to be noted in FEM simulations of ECAP is that FEM analysts unconsciously control the die geometries to improve the efficiency of their simulations [20–23]. The most frequent arbitrary and the least investigated parameter is ICA. In many cases, a large ICA is used in polymers particularly where convergence is difficult. Cases that are more serious can arise when different ICA values are used in the same paper in which the authors investigated other effects. Indeed, the ICA influences the convergence and calculating speed in ECAP; however, it can be critical to control the ICA without assessing the validity of a round ICA. Therefore, it is meaningful to study the effect of the ICA on ECAP processing especially when it is focused on deformation uniformity. In this paper, the plastic deformation behavior of a workpiece during ECAP is simulated using FEM to investigate the ICA effect and to clarify the validity of using a round ICA.
Isothermal FEM simulations of the ECAP process were carried out using the commercial finite element code DEFORM2D (Version 9.1) [24]. In order to assess only the effect of the ICA, the temperature effect was ignored. According to experimental [25] and theoretical [26] analyses, the isothermal condition can be fulfilled at low pressing speeds. A FEM simulation calculation of ECAP processing was made by assuming plane strain considering the dimensions of the workpiece, which were 10 mm × 10 mm × 60 mm. Fig. 2 shows the initial mesh systems used in the ECAP simulations with various ICAs. The OCA was fixed as sharp (i.e. Ψ o = 0◦ ). Six die geometries with different ICA values were assessed: Ψ i = 0◦ (sharp corner), 9◦ , 18◦ , 27 ◦ , 45◦ and 90◦ , corresponding to fillet radius of curvature values of 0, 1, 2, 3, 5, and 10 mm, respectively, under the channel width of 10 mm. The friction factor between the interfaces of the die channel and the specimen was assumed to be 0.1, corresponding to the typical value in cold metal forming processes [27]. The workpiece material used in the study was fully annealed oxygen-free copper having equi-axed grains that were 50 m in size. As the stress–strain curve of the high strain range, which requires element in the FEM calculations of an ECAP process, cannot be obtained by conventional experimental techniques, the flow curve of pure copper was calculated up to a large strain of 10 using the dislocation cell evolution constitutive model [28]. This model can describe the hardening behavior of dislocation cell-forming crystalline materials at large strains. In all simulations, an automatic remeshing scheme was used to accommodate large strains and to take into account the occurrence of flow localization, which prevents further calculation during the simulation. There are approximately 3000 initial meshes of a four-node plane strain element. This number of elements was found to be sufficient to express local deformation of the strain rate insensitive workpieces through calculations with varying the number of elements in DEFORM2D. A constant ram speed of 1 mm/s was employed. It should be noted that ram displacements at the end of the processing steps all differ, due to the increasing channel width in the intersection of the entry and exit channels as ICA increases: ram displacements in ICA Ψ i = 0◦ , 9◦ , 18◦ , 27◦ , 45◦ , and 90◦ , are 50, 49, 48, 47, 45, and 40 mm, respectively. Beyond these ram displacements, the ram cannot travel further, as entry channel gaps are wider than the ram size of mm. 3. Results and discussion Fig. 3 shows simulated geometries represented by flow nets. It should be noted that a flow net is not a mesh system that varies during processing due to the remeshing of too much distorted elements. However, the deformed geometry of the flow nets maps the corner coordinates of the initial nets. It is clear that the deformed geometries with a sharp ICA and a round
440
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Fig. 2. Initial mesh for ECAP simulations with various inner corner angles.
ICA are different in MDZ, resulting in differences in the corner regions, the bottom regions and the top regions. For the sharp ICA case (Ψ i = 0◦ ), the deformation shows the typical behavior of strain hardening materials, including corner gap formation
due to strain hardening of the workpiece material, a less sheared bottom zone due to the round corner of the outer corner or the actual workpiece corner, transient front and rear zones, and relatively uniform shear deformation along the thickness direc-
Fig. 3. Deformed geometries represented by flow nets: (a) ICA Ψ i = 0◦ and (b) ICA Ψ i = 90◦ .
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Fig. 4. Predicted load vs. ram displacement curves during ECAP with various inner corner angle conditions.
Fig. 5. Deformed flow net angle with the inner corner angle.
tion due to the balance between the corner gap and the friction effects. On the other hand, for the round ICA case (Ψ i = 90◦ ), the deformed flow net shows a completely filled channel corner due to less movement of the bottom surface, a more sheared bottom zone corner due to less movement of the bottom surface, large transient front and rear zones, and non-uniform shear along the thickness direction, i.e., a top region with less shear and a bottom region with more shear. The top region with less shear is attributed to the round ICA, which induces rigid body motion flow without shear strain due to less constraint against the workpiece flow. It can be found from the final geometries of the workpieces that the deformation histories are different for the head part and the tail part of the workpiece as well as the inner and the outer parts in the steady-state region of the workpiece. Fig. 4 is the calculated load versus the ram displacement curves, showing the deformation steps during ECAP. Three stages of the ECAP load can be distinguished. This trend can be explained using the deformed geometry. Table 1 summarizes the ram displacements of deforming stages I–III under various ICAs. Stage I is the step during which the front part of the workpiece goes through the MDZ (the shaded region ABC in Fig. 1). It is at this point that most of the deformation occurs. The deformed geometry of displacement = 10 mm with ICA = 90◦ corresponds to stage I, as shown in Fig. 3(b). During stage I, the volume of the deformation part of the workpiece increases, as does the internal stress within the workpiece. The load increases as well, as the initially undeformed workpiece passes through the main deformation zone. In stage II, the head part of the workpiece exits in the MDZ after the peak load point. The back part of the workpiece head receives the most severe plastic deforma-
tion during stage I and is released past the peak point until deformation becomes steady; hence, the load decreases during stage II. The deformed geometry of displacement = 9 mm with ICA = 0◦ corresponds to stage II, as shown in Fig. 3(a). It is an interesting observation that a stage II does not exist with a round ICA (Ψ i = 90◦ ). This is a result of the continuous inner corner die shape. Stage III commences when sufficient interactions are established between the workpiece and the channel die, where the load no longer decreases and plateau-like behavior of the load is observed. The deformed geometries of displacements = 30 mm with ICAs = 0◦ and 90◦ correspond to stage III, as shown in Fig. 3(a) and (b). During stage I, the ECAP load is small in a case with a large ICA, due to the low constraint of the round die inner corner. On the other hand, the load increases with the ICA during stages II and III (the steady-state) due to the high heterogeneity in large ICA cases. It should be noted that load overshoot is not related to the overall workpiece but is rather a redundant load due to strain concentration in the back region of the head part of the workpiece. The amount of average shear strain governing the microstructural evolution of plastic-deformed metallic materials is proportional to the slant angle of the initial vertical line of the sample. The slant angle of the flow net with ICA is shown in Fig. 5. As ICA increases, the flow net slant angle decreases monotonously, although OCA is identical in every case. This figure implies that the overall shear strain development is affected by ICA as well as by the channel angle and the OCA, which must be considered. Although the slant angle of the flow net is related to the shear strain that develops in the central region of the workpiece, it does not represent the average shear strain that
Table 1 Ram displacements of deforming stages I–III under various ICAs
Stage I Stage II Stage III
Ψ i = 0◦
Ψ i = 9◦
Ψ i = 18◦
Ψ i = 27◦
Ψ i = 45◦
Ψ i = 90◦
0–4.7 4.7–9.2 9.2–50
0–5.2 5.2–9.7 9.7–49
0–6.6 6.6–13.8 13.8–48
0–8.1 8.1–15.8 15.8–47
0–13.9 13.9–23.3 23.3–45
0–19.7 None 19.7–40
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Fig. 6. Deformed flow net and effective strain distributions. The arrows represent the folding defect regions.
develops all over the sample due to heterogeneity in the deformation. This is described in more detail in the last part of this paper. Deformation homogeneity during ECAP can be analyzed in terms of the effective strain distributions shown in Fig. 6. The deformed flow nets at the end of the ECAP process are shown as well. Transient regions of the head and tail ends receive only small amounts of strain. Effective strains and their distributions at Ψ i = 0◦ and 9◦ are nearly identical: head and tail end regions with low strain deformation, a less deformed bottom region due to strain hardening, and a uniform (εeff = 1.08) deformation zone. As ICA increases, the highly deformed zone starting from the bottom surface point (initially this was the bottom corner point) spreads backward from the sample. Eventually, contrary to expectations, deformation becomes heterogeneous and the strain increases as ICA increases. Interestingly, this surprising result showing increased heterogeneity with ICA is attributed to the outer corner effect. Essentially, as ICA increases, the highly deformed bottom area becomes large and the workpiece fills the die corner gap. As die corner gap decreases, the bottom zone, which is less sheared, decreases and the overall strain increases. It should be noted that deformation homogeneity results from round corner angles: the lesser sheared zone in the bottom region is generated under the round flow of the materials in the bottom region; a round flow can occur in cases involving a round outer corner die and strain hardening of the material [16,17]. Fig. 7 shows the corner gap angle that develops during ECAP due to the strain hardening characteristics as a function of the ICA. The corner gap angle decrease almost linearly with the ICA. With an inner corner angle of 90◦ , a corner gap is not generated at all during ECAP. It is interesting to note that the outer corner of the die was sharp, i.e., OCA Ψ o = 0◦ .
An important factor is the bottom folding defect, as indicated by the arrows in Fig. 6. Folding defects are typically found in illdesigned metal forging processes. Specifically, surface regions on the back part of the heads penetrate into the workpiece inside. Fig. 8 shows an optical micrograph of a copper workpiece after ECAP, in which clear folding defect are evident. This folding defect can be avoided effectively using a slant perform [29]. The effective strain distributions from the top to bottom along the path of the workpiece normal to the pressing direction in the steady-state region of the workpiece are shown in Fig. 9. As the ICA increases to Ψ i = 45◦ , the effective strain of the top part shows nearly identical values while the effective strain of the bottom part gradually increases and then increases more
Fig. 7. Corner gap angle with the inner corner angle.
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Fig. 8. Folding defect found in the bottom region.
significantly when ICA reaches 90◦ . As seen through this result, the value of the ICA is not an arbitrary parameter for better speed and better convergence in FEM calculations. However, Ψ i = 9◦ is an acceptable value for FEM simulations based on the similarity of the deformed geometry (Fig. 6), the effective strain distribution (Figs. 6 and 9), and the load history (Fig. 4). It may be useful if a proper index for comparing homogeneities under various processing conditions is used in both experimental and theoretical analyses of ECAP. Here, the average effective strain εa and the standard deviation were applied for this purpose. The average effective strain εa of the FEM results can be calculated by averaging the strain distribution with respect to the volume of each finite element. The degree of standard deviation to assess the homogeneity index of ECAP processed materials is expressed as in the following equation: (xi − εa )2 fi σ(x) = , (2) fi Here, xi is the effective strain at the ith element, and fi is the related frequency. As observed in Fig. 10, as the ICA increases, the average effective strain gradually increases. It was also found that the standard deviation also increases as the ICA increases. Through this, it can be confirmed that during ECAP processing,
Fig. 10. Average of the strain and standard deviations (error bar) with the inner corner angle.
the negative effect of the ICA on the overall deformation homogeneity is clear, and due to the increase of the local strain of the bottom part. 4. Conclusions The effects of the ICA on the plastic flow and deformation homogeneity during ECAP were studied using FEM simulations to clarify the ICA effect and to assess the validity of using a round ICA die. With a sharp ICA die, the deformed geometry was predicted to be relatively homogeneous. Deformation heterogeneity develops, however, in dies with a round ICA. The round ICA induces highly inhomogeneous deformation in the head, tail, top and bottom regions of the workpiece due to increasing compressive and decreasing shear deformation components. It was found that a round inner corner with an angle up to 9◦ is acceptable in finite element simulations for reproducing a case with a sharp inner corner. These results can be a design guide for processing and for dies in ECAP processes. Acknowledgement This work was supported by Korea Science & Engineering Foundation through the NRL Program (R0A-2007-000-201040). References [1] [2] [3] [4]
Fig. 9. Effective strain along the path of the workpieces normal to the pressing direction in the steady-state region—0: top and 10: bottom positions.
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