Materials Letters 106 (2013) 37–40
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Finite element simulation of accumulative roll-bonding process$ Tadanobu Inoue a,n, Akira Yanagida b, Jun Yanagimoto c a b c
National Institute for Materials Science, Sengen 1-2-1, Tsukuba 305-0047, Japan Tokyo Denki University, Senju-Asahi-Cho 5, Adachi-ku, Tokyo 120-8551, Japan The University of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 11 March 2013 Accepted 25 April 2013 Available online 3 May 2013
The accumulative roll-bonding (ARB) process, which is a severe plastic deformation process, was simulated using finite element analysis, including the influence of friction, stress–strain relations, and roll diameter. The complicated distributions of equivalent strain through the thickness of ARB-processed sheets were quantified. These quantitative strain analyses would be useful for analyzing the evolution of ultrafine-grained structures in the ARB process. & 2013 The Authors. Published by Elsevier B.V. All rights reserved.
Keywords: Finite element analysis Accumulative roll bonding Strain distribution Aluminum
1. Introduction Bulk ultrafine-grained (UFG) materials with grain sizes of tens to hundreds of nanometers, which show improved mechanical properties without the addition of alloying elements, have attracted the attention of researchers in materials science. Since the microstructural evolution of plastically deformed materials is directly related to the magnitude of the plastic strain, understanding of the phenomenon associated with the strain development is very important. In accumulative roll-bonding (ARB), which is a severe plastic deformation (SPD) process for realizing UFG microstructures in metals and alloys, the microstructure and texture in a sheet processed by one ARB cycle without a lubricant dramatically changed depending on the thickness location of the sheet [1–3]. The embedded-pin method is often employed to measure the strain through thickness experimentally [4,5], but magnitude of the strain obtained by this method does not exhibit an exact value [6]. To control the microstructures, it is essential to understand the deformation behavior in the ARB-processed sheets accurately and quantitatively. Although some studies have used finite element analysis (FEA) for other SPD processes, such as equal-channel angular pressing (ECAP) [7], high-pressure torsion [8], and warm caliber rolling [9,10], there have been no reports for the ARB process. This study aims to quantify the equivalent strain
☆ This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. n Corresponding author. Tel.: +81 29 859 2238; fax: +81 29 859 2201. E-mail address:
[email protected] (T. Inoue).
in ARB-processed sheets using FEA, including the influence of friction, stress–strain relations and roll diameter.
2. Modeling of ARB process The two sheets were set in order to analyze three ARB cycles, as shown in Fig. 1a. The sheet is composed of two sheets with dimensions of 1 mm t 12 mm L. The finite element mesh in each sheet with dimensions of 2 mm t 12 mm L shown in Fig. 1b included 4141 nodes and 4000 elements. The mesh size in the thickness direction, tel, is constant, tel ¼0.05 mm. The mesh size in the longitudinal direction, Lel, gradually decreases toward the center from front and back, and the minimum Lel is 0.025 mm at mid-length (center element). In the present study, the condition of a commercial 1100 Al sheet rolled at ambient temperature, as reported by Lee et al. [1], was referred to as the rolling condition: initial thickness, t0 ¼ 2 mm; nominal reduction per pass, r ¼50%; roll diameter, dϕ ¼255 mm; and rolling speed, 170 mm s-1. In addition, the case of a small roll diameter, dϕ ¼ 118 mm, was also examined because the strain depends strongly on the roll diameter as well as the friction [11]. Young's modulus of 70 GPa and Poisson's ratio of 0.35 were used as the elastic modulus. The stress s–strain ε relationships of 1100 Al at 301 K employed in the analysis were described by s¼ 28+105.67ε0.32ε_ 0.017 MPa, but the flow stresses were assumed to remain constant at ε ¼4.0 because hardness does not vary at equivalent strain of over 4.0 on the basis of the ECAP studies [12]. In order to investigate the effect of the s−ε relations, the simulation for the 1100 Al at 473 K was also conducted using s ¼ s0 þ Kεn ε_ m , where s0 ¼20 MPa, K ¼58.40 MPa, n ¼0.24 and m ¼0.0405 with Young's modulus of 60 GPa
0167-577X/$ - see front matter & 2013 The Authors. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matlet.2013.04.093
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Fig. 1. Schematic illustration of three ARB cycles and finite element mesh used.
Fig. 2. Deformation histories of the center part of the FE mesh in the ARB process against roll diameter dϕ and friction coefficient μ.
and Poisson's ratio of 0.35. The Coulomb condition was used as the frictional condition between the rolls and the sheet. Assuming the Coulomb law, the minimum friction coefficient, μ, is 0.13 for dϕ ¼118 mm and 0.09 for dϕ ¼255 mm. Here, as the lubricated conditions, μ¼ 0.14 for dϕ ¼118 mm and μ¼0.1 for dϕ ¼255 mm were adopted; furthermore, as the unlubricated condition, μ¼ 0.25 was adopted for both diameters. In the analysis, the classical metal plasticity models with a von Mises yield surface, (n)PLASTIC, HARDENING¼ISOTROPIC as keyword in ABAQUS, were employed. The equivalent strain, εeq,
imposed by rolling is defined as follows: Z tðsteadyÞ dεeq dt εeq ¼ dt 0
ð1Þ
where dεeq/dt denotes the incremental equivalent strain, and t(steady) is the rolling time [11]. In the ARB simulation shown in Fig. 1a, first, a 2 mm-thick sheet was rolled to a thickness of 1 mm by 50% (1st ARB cycle). Subsequently, the two 1 mm-thick rolled sheets were stacked to be 2 mm in thickness and rolled again to a thickness of 1 mm (2nd ARB cycle).
T. Inoue et al. / Materials Letters 106 (2013) 37–40
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Fig. 3. Distribution of the equivalent strain along the sheet thickness against roll diameter dϕ and friction coefficient μ.
In order to repeat this procedure, a symmetry condition in the y-axis was set before the next rolling, and the sheet was rolled (3rd ARB cycle). Hence, for the 1st and 2nd ARB cycles, a 1/1 model was adopted, and, for the 3rd ARB cycle a 1/2 model was adopted. As the bonding conditions during the simulation, (n)EQUATION keyword in ABAQUS was employed in order to define linear multi-point constrains for the bonded surfaces before the 2nd ARB cycle; furthermore, no slip condition of μ¼1.0 was given on these surfaces.
3. Results and discussion Fig. 2 shows deformed meshes of a center part in each sheet of the FE mesh during the ARB cycles. Under μ¼0.25, the flection is clearly larger in a sheet rolled at dϕ ¼255 mm than in one rolled at dϕ ¼ 118 mm (Fig. 2a and b). This tendency is larger with increasing the number of ARB cycles, N. A similar deformation behavior can be seen in a difference of μ, and the flection under dϕ ¼ 255 mm becomes larger in a sheet rolled at μ¼ 0.25 than in one rolled at μ¼0.1 (Fig. 2b and c). The result means that the evolution of the microstructure and hardness through thickness in the sheet processed by ARB is different depending on the dϕ even if the conditions of reduction and friction are the same. Fig. 3 represents the distributions of the εeq along the sheet thickness. Under the lubricated conditions of a small μ, although the εeq has a slight distribution through the sheet thickness, it is almost constant, and its magnitude corresponds to a value of 0.8 N regardless of the dϕ. On the other hand, under the unlubricated condition, the εeq has a distribution whose magnitude becomes larger than 0.8. In dϕ ¼ 118 mm, the εeq at the surface reaches 1.2 after the 1st cycle. The εeq after the 2nd cycle shows the maximum value of 2.4 at the surface, and it has a peak of 2.0¼1.2 (for the 1st cycle)+0.8 (for the 2nd cycle) at the center. The maximum εeq at the surface represents the value of 3.8 after the 3rd cycle, and the three peaks exist within the 1 mm thickness. The positions with indicated peaks at y¼0.25 and 0.75 correspond to the surface in the 1st cycle. The peak position at the center (y¼0.5) corresponds to the surface in the 2nd cycle; hence, the εeq at the center is represented by 3.2 (¼2.4+0.8). In a larger roll, dϕ ¼ 255 mm, the εeq at the surface is 2.2 after the 1st cycle. After the 2nd cycle, the εeq shows a distribution with the maximum value of 4.6 at the surface, and it has a peak of 3.0 (¼2.2+0.8) at the center. The maximum εeq at the surface showed a value of 7.0 after the 3rd cycle. Fig. 4 shows variations of εeq at the surface against the N, at ambient temperature. For comparison, the results in the Al rolled at 473 K for dϕ ¼118 mm and μ¼ 0.25 are shown in the figure. The result of 0.8 N is also shown as a broken line in the figure. Although, under dϕ ¼255 mm and μ¼0.1, the εeq at the surface is
Fig. 4. Variations of equivalent strain at the surface against the number of ARB cycles at ambient temperature. Here, dϕ and μ denote the roll diameter and friction coefficient used, respectively. The open symbol indicates the results at 473 K. The “ ” is the results of one-pass rolling shown in Ref. [11].
slightly larger than 0.8 N, these values are almost the same. Under μ¼0.25, the εeq increases approximately linearly with respect to N, and the gradients are larger in dϕ ¼255 mm than in dϕ ¼118 mm. This proportionality means that the strain at the surface in the multicycle ARB-processed sheet can be estimated by the conventional one-pass rolling analysis. Furthermore, in Fig. 4, the results at 301 K (solid symbol) are almost the same as those at 473 K (open symbol), under dϕ ¼118 mm and μ¼0.25. Hence, the effect of the s–ε relations of the rolled sheets on the strain is very small in comparison with the effect of friction and dϕ. A similar feature is seen in the IF-steel sheets rolled with the embedded-pin method [13]. The symbol “ ” in the figure denotes the results of one-pass rolling analysis against (dϕ, μ) shown in Fig. 6c of [11]. Using these accurate εeq data, we are able to know the magnitude of the εeq in the multi-cycle ARB-processed sheet against various (dϕ, μ), such as the thin lines shown in Fig. 4. As a result, the strain at the surface, quarter (1/4t), and center (1/2t) in the thickness of the multi-cycle ARB-processed sheet is estimated by the following equations: εeq;surðNÞ ¼ εeq;surðFEAÞ N
for surface
εeq:;1=4tðNÞ ¼ εeq:;1=4tðFEAÞ þ εeq;1=2tðN−1Þ for 1=4t pffiffiffi εeq:;1=2tðNÞ ¼ 2= 3 ln 1=ð1−rÞ þ εeq;surðFEAÞ ðN−1Þ
for 1=2t
where εeq;surðFEAÞ and εeq;1=4tðFEMÞ denote the equivalent strain at the surface and 1/4t, respectively, obtained using FEA for one-pass rolling, and these magnitudes are necessary to accurately analyze
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including not only the friction and roll bite geometry but also the mesh division. 4. Conclusions The exact magnitude and distribution of the equivalent strain in the Al sheet processed during three ARB cycles using FEA were shown in the present study. These quantitative strain analyses would provide useful guidelines for understanding the quantitative correlation between the microstructures and strain in the ARB process. Acknowledgment This study was financially supported in part by the Grant-in-Aid for Scientific Research on Innovative Area, “Bulk Nanostructured Metals”, through MEXT, Japan (No. 2210205), whose support is gratefully appreciated.
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