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Analysis on earing behavior of textured pure aluminum and A5083 alloy sheets
Journal of Materials Processing Technology 83 (1998) 200 – 208
Analysis on earing behavior of textured pure aluminum and A5083 alloy sheets Jianguo Hu a,*, Takashi Ishikawa a, Keisuke Ikeda b a
Department of Materials Processing Engineering, School of Engineering, Nagoya Uni6ersity, Nagoya 464 -8603, Japan b Department of Materials Processing, Faculty of Engineering, Tohoku Uni6ersity, Sendai 980 -8579, Japan Received 2 May 1997
Abstract Based on the continuum mechanics of textured polycrystals (CMTP), the effects of texture components and their scatter width on the earing behavior of the typical texture orientations appearing in aluminum and its alloy sheets were analyzed quantitatively. A comparison of predicted and experimentally observed earing profiles of pure aluminum and A5083 alloy sheets was conducted. The results have indicated that: (i) for the same texture scatter width, the Goss component {011}100 generates the highest ears, and the surface texture component {111}110 displays the lowest ears in six components of the typical texture in aluminum sheets; (ii) the earing tendency of all of the typical texture components is decreased as the texture scatter width is increased; (iii) the different texture scatter widths may cause the earing profiles to change, regardless of the same texture component; and (iv) the earing profiles predicted for pure aluminum and A5083 alloy sheets are in good agreement with the experimentally measured ears both in shape and in height. The CMTP anisotropic yield function can be employed to predict the earing profiles of textured aluminum sheets. Furthermore, the analytical results have emphasized that the approximate in-plane plastic isotropy of textured aluminum sheets can be realized by controlling an appropriate combination of retained rolling texture and recrystallization texture, in addition to generating {111}110 component. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Aluminum – magnesium–manganese alloy; Anisotropy; Earing; Pure aluminum; Sheet metal; Texture
1. Introduction Aluminum sheets are used widely for the production of cans, the cans being made by the deep drawing of circular blanks, followed by ironing. Undesirable undulations of the rim of cups may form during the deepdrawing operation, this being called earing. The highest parts of the drawn cups are called ears or peaks, and the lowest parts are called troughs. Numerous attempts have been made to predict the earing tendency of drawn cups from the mechanical properties and the plastic anisotropy parameter observed in uniaxial tests at RD, 45° and TD for the initial sheet material. Hill [1] first proposed an explanation for the ear and trough positions based on his * Corresponding author. Current address: Department of Mining and Metallurgical Engineering, McGill University, Frank Dawson Adams Building, 3450 University Street, Montreal, Quebec H3A 2A7, Canada. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00063-6
anisotropic yield function [2] and the assumption of no hold-down pressure (plane stress) or no thickening of the blank (plane strain). Sowerby and Johnson [3] used Hill’s yield function with the plane-strain assumption to calculate earing by means of slip-line field theory. Lee and Kobayashi [4] and Gotoh and Ishise [5] used the finite-element method based on Hill’s yield function and a fourth-degree polynomial yield criterion, respectively. In order to characterize the anisotropy of the material, other phenomenological yield criterions were proposed by Hosford [6], Hill [7], Barlat and Lian [8], Chung and Shah [9], and Hill [10]. None of them can be used well for any textured materials. They cannot explain some rather common behavior encountered in steels as well as in aluminum alloys [11]. Moreover, it is usually difficult to provide a good fit between experimental values and calculated values. It is pointed out that almost all of the previous work depends on the postulated yield functions, which are unable to allow the observed texture orientations to be linked directly
J. Hu et al. / Journal of Materials Processing Technology 83 (1998) 200–208
with the in-plane plastic anisotropy of sheets and consequently lead to somewhat crude and unreasonable results [12]. On the other hand, it is known that earing behavior is an example of the effect of crystallographic texture on the plasticity of sheet metal. In order to take better account of the influence of texture on earing, several models based on the plastic slip of a single crystal or of a polycrystal have been proposed. Tucker’s approach [13], based on the Schmid law for f.c.c. single crystal, is well known and affords relatively reasonable agreement with the results of experiments. Kanetake et al. [14] and Inagaki [15,16] applied Tucker’s work and developed it to include polycrystals by taking advantage of the measured texture of the initial sheets. Van Houtte et al. [17] also made an attempt to predict earing formation from the initial texture data. Even though their calculations are close to the results in earing shape, there is a significant difference in earing height. Panchanadeeswaran et al. [18], Becker et al. [19], Mathur et al. [20] and Kalindindi et al. [21] attempted to use a model based on the Taylor [22]/ Bishop and Hill [23] (TBH) polycrystal theory, combining with finite-element simulation. However, these calculations are very lengthy and are thus necessarily conducted on a super-computer. Even then, some simplifications have to be made: the number of grains at each integration point is generally small and only simple polycrystalline models can be used, consequently leading to over-estimation of the plastic anisotropy of the sheets [11]. Their theoretical earing profiles were unsatisfactory for cold-rolled aluminum sheets with a low reduction. Recently, Lin et al. [24] and Chan [25] made an attempt to predict the earing profiles by adopting the continuum mechanics of textured polycrystals (CMTP) proposed by Lequeu et al. [26,27]. However, there were still some large differences between their predicted and measured results in respect of earing shape because the texture scatter width was not taken into consideration. More recently, Hu et al. [28,29] have employed the CMTP anisotropic yield function to predict the plastic anisotropy parameter of strongly textured pure aluminum and A5083 alloy sheets. Their work has shown that the texture scatter width has an important influence on the in-plane plastic anisotropy of the sheets. In this paper, based on the CMTP anisotropic yield function and an empirical equation between the texture scatter width and the exponent of the CMTP function yield that was formulated by Hu et al. [28], the effect of texture components and their scatter width on the earing behavior of pure aluminum and A5083 alloy sheets is analyzed quantitatively. Comparison of the predicted and observed results is carried out.
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2. Analytical method
2.1. Deformation model During the deep drawing of a circular sheet, the radial stress is tensile whilst the circumferential stress is compressive, at each point in a flange. Assuming that the flange consists of a series of elements along the circumferential direction, each element is subjected to a compressive stress su and a strain increment dou along the circumferential direction. The strain increment and true-stress tensors are represented by the following Eqs. (1) and (2) when plane-strain deformation is assumed:
Æ1 0 0Ç do= dou Ã0 − 1 0Ã È0 0 0É
(1)
Æ sr s= Ãs ru È0
(2)
s ru 0 Ç su 0 Ã 0 s tÉ
where st is the hold-down compressive stress over the thickness, and sru is the shear stress. The stresses in any direction with respect to the rolling direction (RD) of a blank can be determined by applying the normality rule to the CMTP yield function containing a dispersion of orientations about a single ideal orientation, as shown in Eq. (3): F(S) = a{ S11 − S22 n + S22 − S33 n + S33 − S11 n} + 2b{ S12 n + S23 n + S31 n}= ( 6tc)n
(3)
where the exponent n depends on the scatter width of orientation v, and is given as follows: n=
exp(0.0573v) [1+(0.01v)3]
(4)
The other nomenclatures in Eq. (3), the transformation between an orientation (hkl)[u6w] coordinate frame and [100]–[010]–[001] crystallographic coordinates, and the formulating process of Eq. (4), were all described [28]. If a ratio of the radial strain increment to the circumferential strain increment is introduced by the following equation: Q=
dor dou
(5)
then the radial strain increment, dor, can be expressed as the product of two factors in Eq. (6): dor = Q dou
(6)
The strain increment ratio Q and the stress components normalized by the critical resolved shear stress tc can be
J. Hu et al. / Journal of Materials Processing Technology 83 (1998) 200–208
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found by using the normality rule to the CMTP yield function. Similarly to Lin et al. [24] and Chan [25], it is assumed that the circumferential strain increment is taken to vary inversely with the circumferential stress, as given by: dou su1 = dou1 su
(7)
where dou1 and su1 are the circumferential strain increment and stress in the element that corresponds to the rolling direction. This relationship is based on the observation that there is larger local deformation in the region where the yield stress is lower. Now, consider the parameter: do P= − r dou1
(8)
where substituting it and Eq. (5) into Eq. (6), provides: do do s dor = − r u = − P dou1 = −Q u1 dou1 dou dou1 su
(9)
It should be noted that work hardening must be taken into account when the stress ratio in Eq. (9) is evaluated. This can be done by the power law: su = ko m u
(10)
where su and ou are the flow stress and strain in a uniaxial tensile, and m is the work-hardening index. In the present analysis, isotropic-work hardening is supposed, m= 0.225 and m =0.125 being adopted for annealed aluminum and cold-rolled sheets, respectively. If Eq. (10) is substituted into Eq. (7) and the normality rule is applied, the following equation can be obtained after a relatively complicated deduction.
dou Su1 1−Q1 = dou1 Su 1− Q
m/(m + 1)
(11)
where Su and Su1 are deviator stresses along the circumferential direction in any element and in the element that is in the RD, and Q1 is the ratio defined in Eq. (5) but corresponding to the RD. Therefore, Eq. (12) can be obtained by substituting Eq. (11) into Eq. (9). dor = − Q
su1 s 1 − Q1 dou1 = −Q u1 su su 1 − Q
m/(m + 1)
dou1 (12)
1 H0 = {4(Rb − Rp)+ (4− p)(2rp + t)} 4
(14)
where Rb and Rp are radii of the blank and punch, rp is the radius of the punch profile, and t is the thickness of the blank. The flange elongation can be determined by introducing a constant in Eq. (15): cr =
H( r do¯r
(15)
where do¯r and H( r are the average radial strain increment and the average elongation of the flange, respectively: M p do dori /p= % ri M i=1 i=1 M M
do¯r = %
(16)
where M is the number of elements along the circumferential direction. If the surface area of the cup is approximately the same as that of the blank, then: H( r =
R 2b − R 2p + t−H0 + 0.43rp 2rp
(17)
Thus, the final cup height can be calculated from: H = H0 + cr dor
(18)
A texture component {hkl}u6w in polycrystals generally consists of four equivalent orientations, (hkl)[u6w], (hkl)[u6w], (hkl)[u6w] and (hkl)[u6w]. It is therefore assumed that the final cup height is given by the following equation: H({hkl}u6w)=
1 % H{(hkl)[u6w]} 4 ( 9 hkl)[ 9 u6w]
(19)
The final cup height at any points along the circumferential direction can be obtained for the corresponding orientation using the CMTP yield.
2.3. Deep-drawing tests Circular blanks with a thickness of 1.0 mm and a diameter of 78.0 mm were used during deep-drawing tests with a 40.0 mm diameter flat-bottomed punch and a 42.5 mm diameter die. The punch and die profile radii were 4.0 mm. Commercial Vaseline was employed as a lubricant. The hold-down pressure was constant, 100 kgf (981 N) and 200 kgf (1.96 kN) being used for pure aluminum and A5083 alloy sheets, respectively. In addition, the actual dimensions of all the tools and the initial sheets were adopted in this analysis.
2.2. Final cup height 3. Results and discussion The final cup height is composed of two parts, or: H = H0 +Hr
(13)
where H0 is the cup height that is given when the radial strain is zero, and Hr is the elongation of the flange. According to Tucker [7], the former is obtained by:
3.1. Earing profiles of polycrystal sheets with typical texture components Most of the reported experimental investigations [30–36] have shown that the preferred orientations
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existing in recrystallized pure aluminum and its alloy sheets are made up of retained texture that is dominated by rolling texture components {112}111 Copper, {123}634 S and {110}112 Brass, and recrystallization texture components {001}100 Cube and {011}100 Goss. Recent studies [36,37] have found that the surface texture {111}110 appears to a greater extent in the single-roller-driven cold rolling and melt direct-rolling processes. Therefore, the above-mentioned six types of texture components may be regarded as the most typical preferred orientations in aluminum and its alloy sheets. As described in the above analytical method, the earing profiles of the typical texture components were calculated. The effects of the texture components and their scatter widths on the earing profiles are indicated in Fig. 1. From this figure, it can be seen that the troughs of the earing profiles of all of the components have a significant tendency to increase, with the ears tending to decrease a little as the texture scatter width v is increased from 7.5° to 15°. For Cube {001}100, as shown in Fig. 1(a), ears appear at 0° and 90° with respect to the RD, and trough occurs near 45°, regardless of change of the texture scatter width. Moreover, the earing profiles with various scatter widths are completely symmetrical with respect to 45°. For Copper {112}111, as shown in Fig. 1(b), the ears are located near to 40° from the RD when v = 7.5°, and tend to move towards 45° from the RD with increase in v from 7.5° to 15°. The trough at 90° is obviously lower than that at the RD. Its earing profiles are unsymmetrical in the range 0–90°. For S {123}634, as indicated in Fig. 1(c), the earing profiles are somewhat similar to those of Copper. However, its troughs at 0° and 90° have almost the same height. For Brass {011}112 in Fig. 1(d), its profiles are also analogous to those of Copper, but the trough at 90° is significantly higher than that at 0°. For one of the recrystallization textures, Goss {011}100 in Fig. 1(e), somewhat as for Cube {001}100, its ears occur at 0° and 90°, and its trough is located near to 45°. However, the ear at 90° is much higher than that at 0°. Its trough not only increases greatly in height, but also moves towards 30° from the 45° position as the texture scatter width is changed from 7.5° to 15°. For the surface texture component {111}110 in Fig. 1(f), its ears occur at 0° and 60°, and its troughs appear at 30° and 90°. As compared with other components, it is found that {111}110 component gives the lowest ears and the highest troughs. Especially, its ears tend to disappear when v = 15°. Therefore, {111}110 component is the uniquely possible texture to realize in-plane isotropic plasticity in six components of aluminum and its alloy sheets, and it will be most desired for the forming of aluminum sheets.
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3.2. Earing percentage of typical texture components In order to evaluate quantitatively the difference in the earing behavior of the typical texture components in aluminum sheets, the earing percentage (E= ((hmax − hmin)/hmin)× 100%) is calculated. The effect of texture scatter width v on earing percentages is shown in Fig. 2. As seen in this figure, the earing percentages of all the texture components decrease with an increase of the texture scatter width v. The recrystallization texture components, Cube {001}100 and Goss {011}100, display higher earing percentages than three components of retained texture for the same scatter width. As far as the recrystallization texture components are concerned, Goss has a much higher earing percentage than Cube. Brass {011}112 and Copper {112}111 have very close earing percentages. However, the former is slightly lower than the latter when v010°; whilst the former is higher than the latter when vE 12.5°. The surface texture component {111}110 gives a comparatively low earing percentage when v= 15°, its earing percentage becoming near to zero, i.e. the surface texture component will be of very small plastic anisotropy in the rolling plane. In the six types of typical texture components in aluminum and its alloy sheets shown in Fig. 2, Goss exhibits the highest earing percentage, whilst the surface texture component gives the lowest earing percentage.
3.3. Earing profiles of pure aluminum and A5083 alloy sheets From the texture data reported previously for pure aluminum and A5083 alloy sheets [34,35], the earing profiles of both of these aluminum sheets with various textures were calculated, the results being presented in Figs. 3 and 4. From Fig. 3, it can be seen that the earing profiles of pure aluminum sheets have relatively great changes when the materials are treated from cold-rolled to recrystallized. As shown in Fig. 3(a)–(c), because the sheets cold-rolled and incompletely recrystallized contain a little strong rolling or retained texture components S {123}634, Copper {112}111 and Brass {011}112, their ears appear at nearly 45° with respect to the RD, and troughs occur at 0° and 90°. Moreover, the trough at 0° is a little lower than that at 90°. When the annealing temperature is increased to near to the recrystallization temperature, as shown in Fig. 3(d), the occurrence of recrystallization texture and weakening of retained texture components enable the sheets to exhibit different earing profiles that depend on the texture scatter width. Even though the texture components are unchanged, they may have eight ears, four ears or no ears because of the different texture scatter widths. On the whole, the earing profiles in this heat-treated condition have very low ears due to an
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Fig. 1. Effect of texture scatter width on the earing profiles of various texture components.
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205
Fig. 2. Effect of texture scatter width on the earing percentage of various textures.
appropriate balance existing between recrystallization texture and retained texture. This means that there is another way to realize in-plane isotropy of aluminum sheets, which is to generate a reasonable mixture of retained texture and recrystallization texture, in addition to developing {111}110 component. When the annealing temperature is increased further, as shown in Fig. 3(e) and (f), the earing profiles are characteristic of an ideal Cube texture indicated in Fig. 1(a), because Cube component is mainly preferred and evolved. A trough appears at 45° and ears occur at 0° and 90°. As shown in Fig. 4, the earing profiles for A5083 alloy sheets do not have such a great change as those for pure aluminum sheets because the rolling or retained texture components S {123}634, Copper {112}111, and Brass {011}112 always remain comparatively strong and the recrystallization texture components Cube and Goss are still relatively weak, despite the adopted annealing being above the recrystallization temperature. As a result, the ears appear at 45°, and the troughs occur at 0° and 90°. All of the earing profiles in Fig. 4(a) – (d) keep almost the same shape. However, it is found that the ears become a little lower because of a slight increase of Cube and Goss components. In addition, the comparison regarding the calculated and experimentally observed earing profiles of pure
aluminum and A5083 alloy sheets is presented in Figs. 3 and 4. The predicted earing profiles are in good accordance with the measured profiles. Especially when the texture scatter width is taken into consideration, the predicted and measured profiles are very close in height. As compared with previous results [14,24,25], the present analysis displays a more accurate and reasonable description of the earing profiles of aluminum sheets. However, since the assumption of plane-strain deformation is adopted, changes in thickness cannot be taken into account. Consequently, there is still some slight difference in the predicted and measured heights because the thickness becomes slightly greater or less during the deep drawing of the sheets. The observations of the thickness of the drawn cups have indicated that almost all parts of A5083 alloy cups have a small increase in thickness, whilst the pure aluminum cups have a decrease and an increase in thickness, depending on the circumferential position. Accordingly, there are different degrees of closeness between the predicted and measured profiles for pure aluminum and A5083 alloy. Therefore, the FEM employed, together with reasonable anisotropic yield functions or strain rate potential that can link directly with the crystallographic texture, can be expected to be able to provide even more accurate predictions. In order to achieve a reasonable balance between computational
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Fig. 3. Earing profiles of pure aluminum sheets with various processing treatments.
J. Hu et al. / Journal of Materials Processing Technology 83 (1998) 200–208
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Fig. 4. Earing profiles of A5083 alloy sheets with various processing treatments.
accuracy and time, Bacroix and Gilormini [38] and Van Houtte et al. [39] proposed an adjusted strain rate potential that is capable of being incorporated into FEM simulation. With respect to this new procedure, further work is under way and will be reported later.
4. Conclusions In this paper, based on the CMTP anisotropic yield function and an empirical equation between the texture scatter width and the exponent of the CMTP function,
the effects of texture components and their scatter widths on the earing behavior of the typical texture orientations in aluminum sheets were analyzed quantitatively. Comparisons of the predicted and observed results were carried out. The following conclusions can be drawn from this work. 1. For the same texture scatter width, Goss {011}100 component exhibits the highest ears, whilst the surface texture {111}110 component displays the lowest ears of the six components of the typical texture in aluminum sheets. 2. The earing tendency of all the typical texture
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components decreases as the texture scatter width is increased. 3. The different texture scatter widths may cause the earing profiles to change, regardless of having the same texture component. 4. The earing profiles predicted from pure aluminum and A5083 alloy sheets are in relatively good agreement with the experimentally measured profiles, both in shape and in height.
References [1] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, Oxford, UK, 1950, pp. 328–334. [2] R. Hill, Proc. R. Soc. London A 193 (1948) 281–297. [3] R. Sowerby, W. Johnson, J. Strain Anal. 9 (1974) 102–108. [4] C.H. Lee, S. Kobayashi, J. Eng. Ind. 95 (1973) 863–873. [5] M. Gotoh, F. Ishise, Int. J. Mech. Sci. 20 (1978) 423–435. [6] W.F. Hosford, J. Appl. Mech. 39 (1972) 607–619. [7] R. Hill, Math. Proc. Camb. Philos. Soc. 85 (1979) 281–297. [8] F. Barlat, L. Lian, Int. J. Plast. 5 (1989) 51–66. [9] K. Chung, K. Shah, Int. J. Plast. 8 (1992) 453–476. [10] R. Hill, Int. J. Mech. Sci. 35 (1993) 19–25. [11] M. Arminjon, B. Bacroix, Acta Mech. 88 (1991) 219–243. [12] F. Barlat, S. Panchanadeeswaran, O. Richmond, Metall. Trans. A 22 (1991) 1525 – 1532. [13] G.E. Tucker, Acta Metall. 9 (1961) 275–286. [14] N. Kanetake, Y. Tozawa, T. Otani, Int. J. Sci. 25 (1983) 335 – 345. [15] H. Inagaki, Z. Metallkde. 82 (1991) 361–372. [16] H. Inagaki, Z. Metallkde. 84 (1993) 494–500. [17] P. Van Houtte, G. Gauwenberg, E. Aernoudt, Mater. Sci. Eng. 95 (1987) 115 – 124. [18] S. Panchanadeeswaran, O. Richmond, L.A. Lalli, W.G. Fricke Jr., in: J.S. Kallend, G. Gottsten (Eds.), Proc. 8th Int. Conf. Texture of Materials, TMS, Warrendale, PA, 1988, pp. 1103–1110.
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[19] R. Becker, R.E. Smelser, S. Panchanadeeswaran, Modelling Simul. Mater. Sci. Eng. 1 (1993) 203 – 224. [20] K.K. Mathur, P.R. Dawson, U.F. Kocks, Mech. Mater. 10 (1990) 183 – 202. [21] S.R. Kalindindi, C.A. Bronkhorst, L. Anand, J. Mech. Phys. Solids 40 (1992) 265 – 275. [22] G.I. Taylor, J. Inst. Met. 62 (1938) 307 – 324. [23] J.W.F. Bishop, R. Hill, Philos. Mag. 42 (1951) 414 – 427, 1289– 1304. [24] D.W. Lin, D. Dadial, J.J. Jonas, Acta Metall. 33 (1985) 705–714. [25] K.C. Chan, Scr. Metall. 32 (1995) 229 – 234. [26] P. Lequeu, P. Gilormini, J.J. Jonas, Acta Metall. 33 (1985) 705 – 714. [27] P. Lequeu, P. Gilormini, F. Montheillet, B. Bacroxi, J.J. Jonas, Acta Metall. 35 (1987) 439 – 450, 1159 – 1171. [28] J. Hu, K. Ikeda, T. Murakami, Mater. Trans. JIM 36 (1995) 1363 – 1371. [29] J. Hu, K. Ikeda, T. Murakami, J. Mater. Process. Technol. 73 (1998) 49 – 56. [30] K. Lu¨cke, O. Engler, Mater. Sci. Technol. 6 (1990) 1113–1130. [31] X.H. Zheng, M. Ahmad, O. Engler, Mater. Sci. Technol. 10 (1994) 581 – 591. [32] O. Engler, I. Heckemann, T. Rickert, J. Hirsch, K. Lu¨cke, Mater. Sci. Technol. 10 (1994) 771 – 781. [33] O. Engler, J. Hirsch, K. Lu¨cke, Acta Metall. 43 (1995) 121–138. [34] J. Hu, K. Ikeda, T. Murakami, Proc. 87th Conf. Japan Institute of Light Metals, Tokyo, Japan, JILM, 1994, pp. 71 – 72. [35] J. Hu, K. Ikeda, T. Murakami, J. Inst. Light Met. 36 (1996) 126 – 131. [36] J. Hu, K. Ikeda, T. Murakami, J. Jpn. Inst. Met. 60 (1996) 1130 – 1135. [37] Y. Nagai, N. Tsuji, T. Sakai, Y. Saito, J. Jpn. Inst. Met. 60 (1996) 708 – 716. [38] B. Bacroix, P. Gilormini, Modelling Simul. Mater. Sci. Eng. 3 (1995) 1 – 21. [39] P. Van Houtte, A. Van Bael, J. Winters, Textures Microstruct. 25 (1995) 255 – 272.
Analysis on earing behavior of textured pure aluminum and A5083 alloy sheets Jianguo Hu a,*, Takashi Ishikawa a, Keisuke Ikeda b a
Department of Materials Processing Engineering, School of Engineering, Nagoya Uni6ersity, Nagoya 464 -8603, Japan b Department of Materials Processing, Faculty of Engineering, Tohoku Uni6ersity, Sendai 980 -8579, Japan Received 2 May 1997
Abstract Based on the continuum mechanics of textured polycrystals (CMTP), the effects of texture components and their scatter width on the earing behavior of the typical texture orientations appearing in aluminum and its alloy sheets were analyzed quantitatively. A comparison of predicted and experimentally observed earing profiles of pure aluminum and A5083 alloy sheets was conducted. The results have indicated that: (i) for the same texture scatter width, the Goss component {011}100 generates the highest ears, and the surface texture component {111}110 displays the lowest ears in six components of the typical texture in aluminum sheets; (ii) the earing tendency of all of the typical texture components is decreased as the texture scatter width is increased; (iii) the different texture scatter widths may cause the earing profiles to change, regardless of the same texture component; and (iv) the earing profiles predicted for pure aluminum and A5083 alloy sheets are in good agreement with the experimentally measured ears both in shape and in height. The CMTP anisotropic yield function can be employed to predict the earing profiles of textured aluminum sheets. Furthermore, the analytical results have emphasized that the approximate in-plane plastic isotropy of textured aluminum sheets can be realized by controlling an appropriate combination of retained rolling texture and recrystallization texture, in addition to generating {111}110 component. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Aluminum – magnesium–manganese alloy; Anisotropy; Earing; Pure aluminum; Sheet metal; Texture
1. Introduction Aluminum sheets are used widely for the production of cans, the cans being made by the deep drawing of circular blanks, followed by ironing. Undesirable undulations of the rim of cups may form during the deepdrawing operation, this being called earing. The highest parts of the drawn cups are called ears or peaks, and the lowest parts are called troughs. Numerous attempts have been made to predict the earing tendency of drawn cups from the mechanical properties and the plastic anisotropy parameter observed in uniaxial tests at RD, 45° and TD for the initial sheet material. Hill [1] first proposed an explanation for the ear and trough positions based on his * Corresponding author. Current address: Department of Mining and Metallurgical Engineering, McGill University, Frank Dawson Adams Building, 3450 University Street, Montreal, Quebec H3A 2A7, Canada. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00063-6
anisotropic yield function [2] and the assumption of no hold-down pressure (plane stress) or no thickening of the blank (plane strain). Sowerby and Johnson [3] used Hill’s yield function with the plane-strain assumption to calculate earing by means of slip-line field theory. Lee and Kobayashi [4] and Gotoh and Ishise [5] used the finite-element method based on Hill’s yield function and a fourth-degree polynomial yield criterion, respectively. In order to characterize the anisotropy of the material, other phenomenological yield criterions were proposed by Hosford [6], Hill [7], Barlat and Lian [8], Chung and Shah [9], and Hill [10]. None of them can be used well for any textured materials. They cannot explain some rather common behavior encountered in steels as well as in aluminum alloys [11]. Moreover, it is usually difficult to provide a good fit between experimental values and calculated values. It is pointed out that almost all of the previous work depends on the postulated yield functions, which are unable to allow the observed texture orientations to be linked directly
J. Hu et al. / Journal of Materials Processing Technology 83 (1998) 200–208
with the in-plane plastic anisotropy of sheets and consequently lead to somewhat crude and unreasonable results [12]. On the other hand, it is known that earing behavior is an example of the effect of crystallographic texture on the plasticity of sheet metal. In order to take better account of the influence of texture on earing, several models based on the plastic slip of a single crystal or of a polycrystal have been proposed. Tucker’s approach [13], based on the Schmid law for f.c.c. single crystal, is well known and affords relatively reasonable agreement with the results of experiments. Kanetake et al. [14] and Inagaki [15,16] applied Tucker’s work and developed it to include polycrystals by taking advantage of the measured texture of the initial sheets. Van Houtte et al. [17] also made an attempt to predict earing formation from the initial texture data. Even though their calculations are close to the results in earing shape, there is a significant difference in earing height. Panchanadeeswaran et al. [18], Becker et al. [19], Mathur et al. [20] and Kalindindi et al. [21] attempted to use a model based on the Taylor [22]/ Bishop and Hill [23] (TBH) polycrystal theory, combining with finite-element simulation. However, these calculations are very lengthy and are thus necessarily conducted on a super-computer. Even then, some simplifications have to be made: the number of grains at each integration point is generally small and only simple polycrystalline models can be used, consequently leading to over-estimation of the plastic anisotropy of the sheets [11]. Their theoretical earing profiles were unsatisfactory for cold-rolled aluminum sheets with a low reduction. Recently, Lin et al. [24] and Chan [25] made an attempt to predict the earing profiles by adopting the continuum mechanics of textured polycrystals (CMTP) proposed by Lequeu et al. [26,27]. However, there were still some large differences between their predicted and measured results in respect of earing shape because the texture scatter width was not taken into consideration. More recently, Hu et al. [28,29] have employed the CMTP anisotropic yield function to predict the plastic anisotropy parameter of strongly textured pure aluminum and A5083 alloy sheets. Their work has shown that the texture scatter width has an important influence on the in-plane plastic anisotropy of the sheets. In this paper, based on the CMTP anisotropic yield function and an empirical equation between the texture scatter width and the exponent of the CMTP function yield that was formulated by Hu et al. [28], the effect of texture components and their scatter width on the earing behavior of pure aluminum and A5083 alloy sheets is analyzed quantitatively. Comparison of the predicted and observed results is carried out.
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2. Analytical method
2.1. Deformation model During the deep drawing of a circular sheet, the radial stress is tensile whilst the circumferential stress is compressive, at each point in a flange. Assuming that the flange consists of a series of elements along the circumferential direction, each element is subjected to a compressive stress su and a strain increment dou along the circumferential direction. The strain increment and true-stress tensors are represented by the following Eqs. (1) and (2) when plane-strain deformation is assumed:
Æ1 0 0Ç do= dou Ã0 − 1 0Ã È0 0 0É
(1)
Æ sr s= Ãs ru È0
(2)
s ru 0 Ç su 0 Ã 0 s tÉ
where st is the hold-down compressive stress over the thickness, and sru is the shear stress. The stresses in any direction with respect to the rolling direction (RD) of a blank can be determined by applying the normality rule to the CMTP yield function containing a dispersion of orientations about a single ideal orientation, as shown in Eq. (3): F(S) = a{ S11 − S22 n + S22 − S33 n + S33 − S11 n} + 2b{ S12 n + S23 n + S31 n}= ( 6tc)n
(3)
where the exponent n depends on the scatter width of orientation v, and is given as follows: n=
exp(0.0573v) [1+(0.01v)3]
(4)
The other nomenclatures in Eq. (3), the transformation between an orientation (hkl)[u6w] coordinate frame and [100]–[010]–[001] crystallographic coordinates, and the formulating process of Eq. (4), were all described [28]. If a ratio of the radial strain increment to the circumferential strain increment is introduced by the following equation: Q=
dor dou
(5)
then the radial strain increment, dor, can be expressed as the product of two factors in Eq. (6): dor = Q dou
(6)
The strain increment ratio Q and the stress components normalized by the critical resolved shear stress tc can be
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found by using the normality rule to the CMTP yield function. Similarly to Lin et al. [24] and Chan [25], it is assumed that the circumferential strain increment is taken to vary inversely with the circumferential stress, as given by: dou su1 = dou1 su
(7)
where dou1 and su1 are the circumferential strain increment and stress in the element that corresponds to the rolling direction. This relationship is based on the observation that there is larger local deformation in the region where the yield stress is lower. Now, consider the parameter: do P= − r dou1
(8)
where substituting it and Eq. (5) into Eq. (6), provides: do do s dor = − r u = − P dou1 = −Q u1 dou1 dou dou1 su
(9)
It should be noted that work hardening must be taken into account when the stress ratio in Eq. (9) is evaluated. This can be done by the power law: su = ko m u
(10)
where su and ou are the flow stress and strain in a uniaxial tensile, and m is the work-hardening index. In the present analysis, isotropic-work hardening is supposed, m= 0.225 and m =0.125 being adopted for annealed aluminum and cold-rolled sheets, respectively. If Eq. (10) is substituted into Eq. (7) and the normality rule is applied, the following equation can be obtained after a relatively complicated deduction.
dou Su1 1−Q1 = dou1 Su 1− Q
m/(m + 1)
(11)
where Su and Su1 are deviator stresses along the circumferential direction in any element and in the element that is in the RD, and Q1 is the ratio defined in Eq. (5) but corresponding to the RD. Therefore, Eq. (12) can be obtained by substituting Eq. (11) into Eq. (9). dor = − Q
su1 s 1 − Q1 dou1 = −Q u1 su su 1 − Q
m/(m + 1)
dou1 (12)
1 H0 = {4(Rb − Rp)+ (4− p)(2rp + t)} 4
(14)
where Rb and Rp are radii of the blank and punch, rp is the radius of the punch profile, and t is the thickness of the blank. The flange elongation can be determined by introducing a constant in Eq. (15): cr =
H( r do¯r
(15)
where do¯r and H( r are the average radial strain increment and the average elongation of the flange, respectively: M p do dori /p= % ri M i=1 i=1 M M
do¯r = %
(16)
where M is the number of elements along the circumferential direction. If the surface area of the cup is approximately the same as that of the blank, then: H( r =
R 2b − R 2p + t−H0 + 0.43rp 2rp
(17)
Thus, the final cup height can be calculated from: H = H0 + cr dor
(18)
A texture component {hkl}u6w in polycrystals generally consists of four equivalent orientations, (hkl)[u6w], (hkl)[u6w], (hkl)[u6w] and (hkl)[u6w]. It is therefore assumed that the final cup height is given by the following equation: H({hkl}u6w)=
1 % H{(hkl)[u6w]} 4 ( 9 hkl)[ 9 u6w]
(19)
The final cup height at any points along the circumferential direction can be obtained for the corresponding orientation using the CMTP yield.
2.3. Deep-drawing tests Circular blanks with a thickness of 1.0 mm and a diameter of 78.0 mm were used during deep-drawing tests with a 40.0 mm diameter flat-bottomed punch and a 42.5 mm diameter die. The punch and die profile radii were 4.0 mm. Commercial Vaseline was employed as a lubricant. The hold-down pressure was constant, 100 kgf (981 N) and 200 kgf (1.96 kN) being used for pure aluminum and A5083 alloy sheets, respectively. In addition, the actual dimensions of all the tools and the initial sheets were adopted in this analysis.
2.2. Final cup height 3. Results and discussion The final cup height is composed of two parts, or: H = H0 +Hr
(13)
where H0 is the cup height that is given when the radial strain is zero, and Hr is the elongation of the flange. According to Tucker [7], the former is obtained by:
3.1. Earing profiles of polycrystal sheets with typical texture components Most of the reported experimental investigations [30–36] have shown that the preferred orientations
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existing in recrystallized pure aluminum and its alloy sheets are made up of retained texture that is dominated by rolling texture components {112}111 Copper, {123}634 S and {110}112 Brass, and recrystallization texture components {001}100 Cube and {011}100 Goss. Recent studies [36,37] have found that the surface texture {111}110 appears to a greater extent in the single-roller-driven cold rolling and melt direct-rolling processes. Therefore, the above-mentioned six types of texture components may be regarded as the most typical preferred orientations in aluminum and its alloy sheets. As described in the above analytical method, the earing profiles of the typical texture components were calculated. The effects of the texture components and their scatter widths on the earing profiles are indicated in Fig. 1. From this figure, it can be seen that the troughs of the earing profiles of all of the components have a significant tendency to increase, with the ears tending to decrease a little as the texture scatter width v is increased from 7.5° to 15°. For Cube {001}100, as shown in Fig. 1(a), ears appear at 0° and 90° with respect to the RD, and trough occurs near 45°, regardless of change of the texture scatter width. Moreover, the earing profiles with various scatter widths are completely symmetrical with respect to 45°. For Copper {112}111, as shown in Fig. 1(b), the ears are located near to 40° from the RD when v = 7.5°, and tend to move towards 45° from the RD with increase in v from 7.5° to 15°. The trough at 90° is obviously lower than that at the RD. Its earing profiles are unsymmetrical in the range 0–90°. For S {123}634, as indicated in Fig. 1(c), the earing profiles are somewhat similar to those of Copper. However, its troughs at 0° and 90° have almost the same height. For Brass {011}112 in Fig. 1(d), its profiles are also analogous to those of Copper, but the trough at 90° is significantly higher than that at 0°. For one of the recrystallization textures, Goss {011}100 in Fig. 1(e), somewhat as for Cube {001}100, its ears occur at 0° and 90°, and its trough is located near to 45°. However, the ear at 90° is much higher than that at 0°. Its trough not only increases greatly in height, but also moves towards 30° from the 45° position as the texture scatter width is changed from 7.5° to 15°. For the surface texture component {111}110 in Fig. 1(f), its ears occur at 0° and 60°, and its troughs appear at 30° and 90°. As compared with other components, it is found that {111}110 component gives the lowest ears and the highest troughs. Especially, its ears tend to disappear when v = 15°. Therefore, {111}110 component is the uniquely possible texture to realize in-plane isotropic plasticity in six components of aluminum and its alloy sheets, and it will be most desired for the forming of aluminum sheets.
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3.2. Earing percentage of typical texture components In order to evaluate quantitatively the difference in the earing behavior of the typical texture components in aluminum sheets, the earing percentage (E= ((hmax − hmin)/hmin)× 100%) is calculated. The effect of texture scatter width v on earing percentages is shown in Fig. 2. As seen in this figure, the earing percentages of all the texture components decrease with an increase of the texture scatter width v. The recrystallization texture components, Cube {001}100 and Goss {011}100, display higher earing percentages than three components of retained texture for the same scatter width. As far as the recrystallization texture components are concerned, Goss has a much higher earing percentage than Cube. Brass {011}112 and Copper {112}111 have very close earing percentages. However, the former is slightly lower than the latter when v010°; whilst the former is higher than the latter when vE 12.5°. The surface texture component {111}110 gives a comparatively low earing percentage when v= 15°, its earing percentage becoming near to zero, i.e. the surface texture component will be of very small plastic anisotropy in the rolling plane. In the six types of typical texture components in aluminum and its alloy sheets shown in Fig. 2, Goss exhibits the highest earing percentage, whilst the surface texture component gives the lowest earing percentage.
3.3. Earing profiles of pure aluminum and A5083 alloy sheets From the texture data reported previously for pure aluminum and A5083 alloy sheets [34,35], the earing profiles of both of these aluminum sheets with various textures were calculated, the results being presented in Figs. 3 and 4. From Fig. 3, it can be seen that the earing profiles of pure aluminum sheets have relatively great changes when the materials are treated from cold-rolled to recrystallized. As shown in Fig. 3(a)–(c), because the sheets cold-rolled and incompletely recrystallized contain a little strong rolling or retained texture components S {123}634, Copper {112}111 and Brass {011}112, their ears appear at nearly 45° with respect to the RD, and troughs occur at 0° and 90°. Moreover, the trough at 0° is a little lower than that at 90°. When the annealing temperature is increased to near to the recrystallization temperature, as shown in Fig. 3(d), the occurrence of recrystallization texture and weakening of retained texture components enable the sheets to exhibit different earing profiles that depend on the texture scatter width. Even though the texture components are unchanged, they may have eight ears, four ears or no ears because of the different texture scatter widths. On the whole, the earing profiles in this heat-treated condition have very low ears due to an
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Fig. 1. Effect of texture scatter width on the earing profiles of various texture components.
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Fig. 2. Effect of texture scatter width on the earing percentage of various textures.
appropriate balance existing between recrystallization texture and retained texture. This means that there is another way to realize in-plane isotropy of aluminum sheets, which is to generate a reasonable mixture of retained texture and recrystallization texture, in addition to developing {111}110 component. When the annealing temperature is increased further, as shown in Fig. 3(e) and (f), the earing profiles are characteristic of an ideal Cube texture indicated in Fig. 1(a), because Cube component is mainly preferred and evolved. A trough appears at 45° and ears occur at 0° and 90°. As shown in Fig. 4, the earing profiles for A5083 alloy sheets do not have such a great change as those for pure aluminum sheets because the rolling or retained texture components S {123}634, Copper {112}111, and Brass {011}112 always remain comparatively strong and the recrystallization texture components Cube and Goss are still relatively weak, despite the adopted annealing being above the recrystallization temperature. As a result, the ears appear at 45°, and the troughs occur at 0° and 90°. All of the earing profiles in Fig. 4(a) – (d) keep almost the same shape. However, it is found that the ears become a little lower because of a slight increase of Cube and Goss components. In addition, the comparison regarding the calculated and experimentally observed earing profiles of pure
aluminum and A5083 alloy sheets is presented in Figs. 3 and 4. The predicted earing profiles are in good accordance with the measured profiles. Especially when the texture scatter width is taken into consideration, the predicted and measured profiles are very close in height. As compared with previous results [14,24,25], the present analysis displays a more accurate and reasonable description of the earing profiles of aluminum sheets. However, since the assumption of plane-strain deformation is adopted, changes in thickness cannot be taken into account. Consequently, there is still some slight difference in the predicted and measured heights because the thickness becomes slightly greater or less during the deep drawing of the sheets. The observations of the thickness of the drawn cups have indicated that almost all parts of A5083 alloy cups have a small increase in thickness, whilst the pure aluminum cups have a decrease and an increase in thickness, depending on the circumferential position. Accordingly, there are different degrees of closeness between the predicted and measured profiles for pure aluminum and A5083 alloy. Therefore, the FEM employed, together with reasonable anisotropic yield functions or strain rate potential that can link directly with the crystallographic texture, can be expected to be able to provide even more accurate predictions. In order to achieve a reasonable balance between computational
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Fig. 3. Earing profiles of pure aluminum sheets with various processing treatments.
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Fig. 4. Earing profiles of A5083 alloy sheets with various processing treatments.
accuracy and time, Bacroix and Gilormini [38] and Van Houtte et al. [39] proposed an adjusted strain rate potential that is capable of being incorporated into FEM simulation. With respect to this new procedure, further work is under way and will be reported later.
4. Conclusions In this paper, based on the CMTP anisotropic yield function and an empirical equation between the texture scatter width and the exponent of the CMTP function,
the effects of texture components and their scatter widths on the earing behavior of the typical texture orientations in aluminum sheets were analyzed quantitatively. Comparisons of the predicted and observed results were carried out. The following conclusions can be drawn from this work. 1. For the same texture scatter width, Goss {011}100 component exhibits the highest ears, whilst the surface texture {111}110 component displays the lowest ears of the six components of the typical texture in aluminum sheets. 2. The earing tendency of all the typical texture
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components decreases as the texture scatter width is increased. 3. The different texture scatter widths may cause the earing profiles to change, regardless of having the same texture component. 4. The earing profiles predicted from pure aluminum and A5083 alloy sheets are in relatively good agreement with the experimentally measured profiles, both in shape and in height.
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