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Fracture limits of sheet metals under stretch bending
ARTICLE IN PRESS
International Journal of Mechanical Sciences 47 (2005) 1885–1896 www.elsevier.com/locate/ijmecsci
Fracture limits of sheet metals under stretch bending Masatoshi Yoshidaa,, Fusahito Yoshidab, Haruyuki Konishia, Koji Fukumotoa a
Materials, Process and Applied Mechanics Res. Sec., Technical Department, Aluminum & Copper Company KOBE STEEL, LTD., 1-5-5, Takatsukadai Nishi-ku, Kobe HYOGO 651-2271, Japan b Department of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan Received 23 August 2004; received in revised form 30 June 2005; accepted 20 July 2005 Available online 8 September 2005
Abstract Fracture limits in sheet stretch bending were theoretically obtained on the assumption that the fracture occurs when the stretching force reaches its maximum value. From the calculated results, a fracture criterion has been presented where limit wall stretch, Lmax/L0 (Lmax: limit wall length of a sheet, L0: initial wall length), is explicitly given as a function of the non-dimensional bending curvature, t0 =R (t0: sheet thickness, R: bending radius) and the material’s work hardening exponent (n-value). To verify this criterion, three-point stretch bending tests with various punch-radii were performed on three types of aluminum sheets (A5182-O, JIS6061-T4 and JIS6N01-T5). The predicted limit wall stretch, as well as limit forming height, were in good agreement with the experimental results. r 2005 Elsevier Ltd. All rights reserved. Keywords: Sheet metal forming; Stretch bending; Limit wall stretch; Aluminum sheets
1. Introduction In press forming, fracture of sheets mostly occurs at a die (or punch) corner under stretch bending. Therefore, it is of vital importance to have an appropriate forming limit criterion for stretch bending. Corresponding author. Tel.: +81 78 992 5516; fax: +81 78 992 5517.
E-mail address: [email protected] (M. Yoshida). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.07.006
ARTICLE IN PRESS 1886
M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
To evaluate forming limits, several types of stretch bending tests were performed in the past, such as the three-point bending [1–10], L-type stretch bending [11–14] and stretch-drawing over a die-corner [15]. Among these tests, the three-point bending test, as schematically shown in Fig. 1, is the most popular, where a sheet metal is rigidly clamped at its ends and deformed by a V-shaped punch until fracture occurs. To the best of the present authors’ knowledge, important early work was done by Glover [1]. Duncan and his co-workers [2] conducted this type of test, and later, Demeri et al. [3,4] investigated the forming limits on aluminum and steel sheets. Story [7,8] examined the formability for a wide varieties of automotive aluminum alloy sheets, and discussed the effects of sheet thickness and bending radius on the fracture limit. However, these previous works are purely experimental, and so far, a theoretical analysis has not been published. Concerning the fracture criterion in stretch bending, a few papers on theoretical approaches predicting fracture load were published. Hino-Yoshida [13] presented the analytical model for predicting the fracture load on sheet metal laminates based on incremental strain theory of plasticity, where it was assumed that the fracture occurs when the stretching load reaches its maximum value. A similar approach is found in Naka et al. [14], where the effects of temperature and stretching speed on the fracture load under stretch bending were discussed for 5083 aluminum sheet. However, such a forming limit criterion in terms of fracture load would not be so convenient for engineers, since the load acting on the sheet would be difficult to measure. Instead, a limit strain criterion would be more useful in real process design of sheet stamping. Use of FLD (forming limit diagram) is one of the most popular ways of determining the forming limit, however, it would not work well for a case of die-corner fracture, since it is not so easy to accurately measure the localized strain. Furthermore, in numerical simulation of stamping, calculated values of strains at the die-corner are strongly affected by FE mesh size. To overcome this problem, the Aluminum Association [16] made an interesting proposition that, instead of using local strain at the corner, limit wall stretch Lmax/L0 (see Fig. 1, Lmax: limit wall length of a sheet, L0: initial wall length) should be used for a design guideline. In the present paper, a simple theoretical model which can predict the limit wall stretch and forming height in stretch bending is proposed. The limit wall stretch is given as an explicit function of the work hardening exponent (n value) of the material and non-dimensional bending
t0
Blank holder
h
Punch
Rd
L
Rp
Die
Dd L0
Fig. 1. Schematic of stretch bending test.
ARTICLE IN PRESS M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
1887
curvature (t0 =R), which would be very useful for practical process design of sheet metal forming. The predicted results are verified by some stretch bending tests of aluminum alloy sheets.
2. Analytical prediction of fracture criterion 2.1. Analysis of stretch bending Tharrett and Stoughton [9,10] examined the necking limits of sheet metals and reported that necks on steel sheet occurs when the strain on the inner surface reaches a critical value, which agrees with FLD0. However, the rule does not appear to apply for FCC materials. So, in this model, the fracture limit of a sheet is determined on the basis of a maximum stretching force criterion. To obtain an explicit solution, a rigid-plastic analysis of uniform stretch bending under the plane strain condition is performed based on the deformation theory of plasticity. The effect of wall thinning due to cyclic bending–unbending and friction between tools and materials is ignored. The power-law workhardening: seq ¼ Cneq
(1)
is assumed, where seq and eeq denote the equivalent stress and plastic strain, respectively. Considering the normal anisotropy of sheet, characterized by the Lankford value (r value), for the plane-strain condition, we obtain the longitudinal stress s and strain e as 1þr s ¼ sgnðÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi seq , 1 þ 2r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2r jj ¼ eq , 2ð1 þ rÞ where sgnð Þ ¼
(
1
ifð ÞX0
1
ifð Þo0
(2)
(3)
.
From Eqs. (1)–(3), we have the stress–strain relationship: rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!nþ1 1þr 2ð1 þ rÞ C jjn . s ¼ sgnðÞ 2 1 þ 2r
(4)
A deformed sheet in stretch bending is schematically illustrated in Fig. 2. We assume that the logarithmic strain is linearly distributed in thickness direction of the sheet as o i ¼ y, (5) t where, t is the thickness of a deformed sheet, y is the distance in the radial direction. This assumption is different from the Kirchhoff-Love hypothesis of linear distribution of the
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y A
Deformed sheet in pure bending(T=0)
ε = εo
A’ t0
B t
Deformed sheet in stretch bending y=p
C
Ri
y=0
B’
ε =ε i
O
C’
ϕ
Tensile force T Punch
Fig. 2. Schematic illustration of deformed sheet in stretch bending.
engineering strain, however, practically, the difference between the Kirchhoff-Love hypothesis and this assumption by using the surface strain eo, ei is small. Let us consider such a deformation process where first a sheet is purely bent (T ¼ 0) until the inner radius reaches the die radius, Ri, (see OABC in Fig. 2) and then it is stretched (T40) over the die (OA0 B0 C0 ). Following large bending theory, from the geometries of OABC and OA0 B0 C0 , and also from Eq. (5), outer surface strain eo and inner surface strain ei under stretch bending are expressed by: ( ) ðRi þ tÞy o i ðt þ pÞ, (6) o ¼ ln ¼ t Ri þ t0 =2 f (
) Ri y o i p, i ¼ ln ¼ t Ri þ t0 =2 f
(7)
where, t0 is the initial thickness of a sheet, p is the y-coordinate at the inner surface of the bent sheet. From Eqs. (5)–(7), we obtain ¼
p¼
lnð1 þ t=Ri Þ y, t ! o 1 t. ln 1 þ t=Ri
(8)
(9)
The volume constancy condition of OABC ¼ OA0 B0 C0 is ffðRi þ t0 Þ2 R2i g yfðRi þ tÞ2 R2i g ¼ . 2 2 From Eqs. (6) and (10), we obtain y 1 2Ri þ t0 o 2Ri þ t0 t0 ¼ e ¼ . f 2 Ri þ t 2Ri þ t t
(10)
(11)
ARTICLE IN PRESS M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
From this, the thickness of the deformed sheet in stretched bending is determined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ t0 eo Ri þ R2i þ ðt0 eo Þ2 . Tensile force T is calculated as n o 1þn Z tþp o lnð1 þ t=Ri Þ1þn 0 o Ct s dy ¼ . T¼ lnð1 þ t=Ri Þ 1 þn p rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!nþ1 1þr 2ð1 þ rÞ C ¼ C. 2 1 þ 2r 0
1889
(12)
(13a)
(13b)
2.2. Fracture limit During stretch bending, flow stress of the sheet, s, is increasing with outer surface strain eo because of its workhardening; on the other hand, the thickness of the sheet t is decreasing. As a result, tensile force T is initially increasing rapidly with eo, but the rate of increasing load is decreasing, and it will reach its maximum value Tmax. The maximum load Tmax and limit outer strain eof are determined by numerically calculating Eq. (13) until qT ¼ 0. q0
(14)
It should be noted that eof is determined uniquely for a given n value and t0 =Ri, since, from Eq. (12), sheet thickness t is a function of eo for a given t0 =Ri. Using the value of Tmax obtained, the limit wall stretch Lmax/L0 is easily calculated. Since the tensile load acting on the wall, Twall, for a wall stretch L/L0 is given by n L L0 0 T wall ¼ st ¼ C ln (15) t0 . L0 L The limit wall stretch Lmax/L0 is calculated by the following equation: n 1þn 1þn Lmax Lmax tf fof jof lnð1 þ tf =Ri Þj g ln ¼ , L0 L0 t0 ð1 þ nÞ lnð1 þ tf =Ri Þ
(16)
where, limit sheet thickness tf is a function of eof for a given t0 =Ri (see Eq. (12)). Here, we emphasize that the limit wall stretch depends only on n value of the material and on the nondimensional bending curvature t0 =Ri (or t0 =R, R ¼ Ri þ t0 =2). As discussed above, the limit wall stretch is numerically determined by Eq. (16). Alternatively, if we could have a simple equation for limit wall stretch as an explicit function of n value and t0 =R, it would be convenient for engineering use. Here we propose the following regression equation: L ¼ en þ ð1 en Þð1 eð0:4=nÞðt0 =RÞ Þ. (17) L0 Fig. 3 shows limit wall stretches Lmax/L0, calculated by both Eq. (16) (solid lines) and (17) (broken lines), as a function of t0 =R for several n values. From this figure, it is found that Lmax/L0
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Limit wall stretch Lmax /L0
1.6
n=0.50 n=0.40
1.5 n=0.30 1.4
n=0.20 n=0.10
1.3
n=0.05
1.2 1.1 1.0 0.0
0.5 1.0 Non-dimensional bending curvature t0 / R
1.5
Fig. 3. Relation between non-dimensional bending curvature t0 =R, n value and limit wall stretch Lmax/L0.
decreases considerably as t0 =R increases, and it becomes larger for materials of higher n value. This is natural since strain localization is more likely to occur at a sharp die-corner for a lower workhardening material. All the calculated results by Eq. (17) well agree with those by Eq. (16), hence it is confirmed that we can use the simple Eq. (17) for the determination of the limit wall stretch, instead of performing numerical calculation. The relation between limit outer surface strains eof calculated by Eq. (16) and t0 =R for several n values are shown in Fig. 4. The limit outer surface strain eof dramatically increases with increasing t0 =R. The same tendency was experimentally observed in the past for some sheet metals [9,10,17]. This fact suggests that, for the prediction of sheet fracture, we can use the FLD only for the fracture in almost flat parts of panels, but not for the die-corner fractures. The limit outer surface strain eof becomes larger for materials of higher n values. However, the effect of n value on eof becomes minor for large t0 =R. The relation between limit inner surface strains eif calculated by Eq. (16) and t0 =R for several n values are shown in Fig. 5.The limit inner surface strain eif decreases with increasing t0 =R.
3. Experimental verification For the verification of the above fracture criterion, three-point stretch bending tests were performed. In the tests, both ends of sheet specimens were firmly clamped and stretch-bent by a punch. To examine the effect of the bending radius, four different punches with radii of 0.5, 2.0, 5.0 and 10.0 mm were employed. Test conditions are summarized in Table 1. Three types of aluminum sheets (A5182-O, JIS6061-T4 and JIS6N01-T5) with various thicknesses, 1.0–4.0 mm, with 25.0 mm wide were used. Engineering stress–strain curves of these sheets are shown in Fig. 6.
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1.4
Limit outer surface strain of / %
1.2 1.0 0.8 n=0.50
0.6
n=0.40 n=0.30
0.4
n=0.20 n=0.10
0.2
n=0.05 0.0 0.0
0.5 1.0 Non-dimensional bending curvature t0 /R
1.5
Fig. 4. Relation between non-dimensional bending curvature t0 =R, n value and limit outer surface true strain eof. 0.6 n=0.50 n=0.40
0.5 Limit inner surface strain if / %
n=0.30 n=0.20 0.4
n=0.10 n=0.05
0.3
0.2
0.1
0 0
0.5 1 Non-dimensional bending curvature t0 /R
1.5
Fig. 5. Relation between non-dimensional bending curvature t0 =R, n value and limit inner surface true strain eif.
Mechanical properties of the sheet metals are listed in Table 2. The material constant C and n in the constitutive Eq. (1) were determined from the tensile strength sB (maximum nominal stress in uniaxial tension) and the uniform elongation eB (the engineering strain corresponding to sB) by
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Table 1 Forming conditions AL0 ¼ 118.4 mm (Dd ¼ 102.4 mm, Rd ¼ 8.0 mm) Die
BL0 ¼ 69.8 mm (Dd ¼ 53.8 mm, Rd ¼ 8.0 mm) ARp ¼ 0.5 mm
Tool
BRp ¼ 2.0 mm Punch
V-shape CRp ¼ 5.0 mm DRp ¼ 10.0 mm
Forming speed
4 mm s1
Lubricant
Mineral oil
Blank shape
&25 mm 200 mm
350
Engineering stress / MPa
300 250 200 150 5182-O
100
6061-T4 50 0
6N01-T5 0
10 20 Engineering strain / %
30
Fig. 6. Engineering stress-strain curve.
the following equations: n ¼ lnð1 þ eB Þ
(18)
sB ð1 þ eB Þ (19) nn For example, photographs of fractured 5182-O specimens are shown in Fig. 7. The width reduction of the fractured specimens was not observed obviously, it is considered that the C¼
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Table 2 Mechanical properties of specimens Material
t0 (mm)
sB (MPa)
sY (MPa)
C (MPa)
n Value
r Value
5182-O
1.0 1.4 2.0
266
127
472
0.233
0.605
6061-T4
1.0
238
135
405
0.207
0.530
6N01-T5
2.0 2.4 4.0
265
230
348
0.076
0.470
specimens deformed closer to plane strain condition. The fracture of 5182-O sheets mostly occurred at the punch corner, except for some cases of large punch-radius (the arrows in Fig. 7 indicate the locations of wall fracture). Fig. 8 shows the comparison of experimental data of limit wall stretches Lmax/L0 with the corresponding results of calculation by Eq. (16). Although the experimental conditions were widely scattered, i.e., three different types of metals, sheet thickness of 1.0–4.0 mm, punch radii of 0.5–10.0 mm, and furthermore two different die-spans of 69.8 and 118.4 mm, the calculated results of limit wall stretch for t0 =R are in good agreement with the experimental data. As already mentioned, theoretically the limit wall stretch is determined as a function of n value and nondimensional bending curvature t0 =R. This is confirmed by the experiment. Instead of limit wall stretch Lmax/L0, limit forming height hmax is often used as a measure of forming limit in real stamping operations. For that purpose, here we derive equations for calculating limit forming height hmax (see Eqs. (20)–(22)). They include the limit wall length Lmax which can be determined by either Eq. (16) or (17), as well as some geometrical parameters such as die-radius Rd, punch-radius Rp, die-span L0 and sheet thickness t0. L0 2Ra sin y Lmax ¼ , 2 cos y 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi
2 L0 Lmax hmax Ra ð1 cos yÞ þ Ra sin y ¼ Ra y, 2 2 Ra y þ
Ra ¼ Rp þ Rd þ t0 .
(20)
(21) (22)
Fig. 9 shows thus calculated limit forming height hmax for a case of L0 ¼ 118.4 mm, together with the experimental data on the three types of aluminum sheets. The calculated limit forming heights agree fairly well with most of the experimental results. In this figure, some discrepancies in hmax between the calculated results and the experimental data are found for some cases of large t0 =R, especially for 6N01-T5. This is not due to the inaccuracy of the present theory, but rather the problem of experiment. In stretch bending of a thick sheet with a sharp punch-radius, the bent radius of the sheet would be unlikely to reach the punch-radius. Such a condition is far from the assumption of the theory that the sheet fully contact with the punch profile.
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M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
Fig. 7. Photographs of fractured 5182-O specimens (L0 ¼ 118.4 mm).
4. Concluding remarks Based on the stretch-bending analysis, the fracture limits in terms of limit wall stretch were examined. It was found from the analysis that the limit wall stretch is determined as a function of n value and non-dimensional bending curvature t0 =R. In the present paper, we propose an equation to determine the limit wall stretch, which can be solved numerically. Alternatively, a
ARTICLE IN PRESS M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896 1.4
Exp. L0 Lmax
Limit wall stretch Lmax /L0
1895
L0 /mm 69.8 118.4
Material 5182-O 6061-T4 6N01-T5
1.3
Cal. 5182-O (n=0.233)
1.2
6061-T4 (n=0.206) 6N01-T5(n=0.076)
1.1
1.0 0.0
0.5
1.0
1.5
2.0
Non-dimensional bending curvature t0 /R
Fig. 8. Comparison of limit wall stretch Lmax/L0 between experiment and calculation. 60 Exp.
Cal. 5182-O
Limit forming height hmax / mm
50
Material
t0
6061-T4
1.0/mm 1.4 5182-O 2.0 6061-T4 1.0 2.0 2.4 6N01-T5 4.0
6N01-T5 40
30
20
10
0 0.0
0.5
1.0
1.5
2.0
Non-dimensional bending curvature t 0 /R
Fig. 9. Comparison of limit forming height hmax between experiment and calculation (L0 ¼ 118.4 mm).
simple equation which gives limit wall stretch as an explicit function of n-value and t0 =R has also been presented. The present theory has been confirmed by performing three-point stretch bending tests on three types of aluminum sheets.
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This fracture criterion for stretch bending is very convenient for real process design of sheet metal stamping. If we combine this explicit form of the criterion with FE simulation of stamping, we can easily predict the die-corner fracture. The great advantage of this criterion, when using in FE simulation, is that we do not need to worry about FE-mesh size for fracture prediction, since the criterion only uses a given die-radius and calculated wall stretch which would be rather insensitive to FE-mesh size.
References [1] Glover G, Duncan JL. Less conventional formability tests. Proceedings of the Australian Institute of Metals Congress 1978:18–23. [2] Uko DK, Sowerby R, Duncan JL. Strain distribution in the bending-under-tension test. Canadian Mining and Metallurgical Bulletin 1977;70(781):127–34. [3] Demeri MY. The stretch-bend forming of sheet metals. Journal of Applied Metalworking 1981;2(1):3–10. [4] Narayanaswamy OS, Demeri MY. Analysis of the angular stretch bend test. Proceedings of the Joint ASM/ASME symposium: novel techniques in metal deformation testing. 1983. p. 99–112. [5] Thomson TR, Brownrigg A. The effect of inclusions on cold formability. Metals Forum 1979;2(2):118–25. [6] Steninger J, Melander A. The relation between bendability, tensile properties and particle structure of low carbon steel. Scandinavian Journal of Metallurgy 1982;11:55–71. [7] Story JM. Geometrical aspects of stretch-bend testing and a correlation with forming performance. Manufacturing Engineering Transactions 1984;12:193–200. [8] Story JM, Jarvis GW, Zonker HR, Murtha SJ. Issues and trends in automotive aluminum sheet forming. SAE technical paper No.930277, 1993. [9] Tharrett MR, Stoughton TB. Stretch-bend forming limits of 1008 AK steel, 70/30 brass, and 6010 aluminum. Dislocations Plasticity & Metal Forming 2003:199–201. [10] Tharrett MR, Stoughton TB. Stretch-bend forming limits of 1008 AK steel. SAE technical paper No.2003-01-1157, 2003. [11] Duncan JL, Bird JE. Die forming approximations for aluminum sheet. Sheet Metal Industries 1978:1015–25. [12] Duncan JL, Shanbel BS, Gerbase-Filho J. A Tensile strip test for evaluating friction in sheet metal forming. SAE technical paper No.780391, 1978. [13] Hino R, Yoshida F. Fracture load in stretch bending for sheet metal laminates. Transactions of Japan Society of Mechanical Engineering 1995;61A(592):2560–5 [in Japanese]. [14] Naka T, Hino R, Yoshida F. Fracture of type 5083 aluminum sheet under warm stretch bending. Key Engineering Materials 2003;233–236:113–8. [15] Damborg FF. Prediction of fracture in bending-under-tension. Proceedings of the 19th Riso International Symposium on Material Science 1998:235–40. [16] The Aluminum Association. Aluminum for automotive bode sheet panels, 1998. [17] Charpentier PL. Influence of punch curvature on the stretching limits of sheet steel. Metallurgical Transactions 1975;6A:1665–9.
International Journal of Mechanical Sciences 47 (2005) 1885–1896 www.elsevier.com/locate/ijmecsci
Fracture limits of sheet metals under stretch bending Masatoshi Yoshidaa,, Fusahito Yoshidab, Haruyuki Konishia, Koji Fukumotoa a
Materials, Process and Applied Mechanics Res. Sec., Technical Department, Aluminum & Copper Company KOBE STEEL, LTD., 1-5-5, Takatsukadai Nishi-ku, Kobe HYOGO 651-2271, Japan b Department of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan Received 23 August 2004; received in revised form 30 June 2005; accepted 20 July 2005 Available online 8 September 2005
Abstract Fracture limits in sheet stretch bending were theoretically obtained on the assumption that the fracture occurs when the stretching force reaches its maximum value. From the calculated results, a fracture criterion has been presented where limit wall stretch, Lmax/L0 (Lmax: limit wall length of a sheet, L0: initial wall length), is explicitly given as a function of the non-dimensional bending curvature, t0 =R (t0: sheet thickness, R: bending radius) and the material’s work hardening exponent (n-value). To verify this criterion, three-point stretch bending tests with various punch-radii were performed on three types of aluminum sheets (A5182-O, JIS6061-T4 and JIS6N01-T5). The predicted limit wall stretch, as well as limit forming height, were in good agreement with the experimental results. r 2005 Elsevier Ltd. All rights reserved. Keywords: Sheet metal forming; Stretch bending; Limit wall stretch; Aluminum sheets
1. Introduction In press forming, fracture of sheets mostly occurs at a die (or punch) corner under stretch bending. Therefore, it is of vital importance to have an appropriate forming limit criterion for stretch bending. Corresponding author. Tel.: +81 78 992 5516; fax: +81 78 992 5517.
E-mail address: [email protected] (M. Yoshida). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.07.006
ARTICLE IN PRESS 1886
M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
To evaluate forming limits, several types of stretch bending tests were performed in the past, such as the three-point bending [1–10], L-type stretch bending [11–14] and stretch-drawing over a die-corner [15]. Among these tests, the three-point bending test, as schematically shown in Fig. 1, is the most popular, where a sheet metal is rigidly clamped at its ends and deformed by a V-shaped punch until fracture occurs. To the best of the present authors’ knowledge, important early work was done by Glover [1]. Duncan and his co-workers [2] conducted this type of test, and later, Demeri et al. [3,4] investigated the forming limits on aluminum and steel sheets. Story [7,8] examined the formability for a wide varieties of automotive aluminum alloy sheets, and discussed the effects of sheet thickness and bending radius on the fracture limit. However, these previous works are purely experimental, and so far, a theoretical analysis has not been published. Concerning the fracture criterion in stretch bending, a few papers on theoretical approaches predicting fracture load were published. Hino-Yoshida [13] presented the analytical model for predicting the fracture load on sheet metal laminates based on incremental strain theory of plasticity, where it was assumed that the fracture occurs when the stretching load reaches its maximum value. A similar approach is found in Naka et al. [14], where the effects of temperature and stretching speed on the fracture load under stretch bending were discussed for 5083 aluminum sheet. However, such a forming limit criterion in terms of fracture load would not be so convenient for engineers, since the load acting on the sheet would be difficult to measure. Instead, a limit strain criterion would be more useful in real process design of sheet stamping. Use of FLD (forming limit diagram) is one of the most popular ways of determining the forming limit, however, it would not work well for a case of die-corner fracture, since it is not so easy to accurately measure the localized strain. Furthermore, in numerical simulation of stamping, calculated values of strains at the die-corner are strongly affected by FE mesh size. To overcome this problem, the Aluminum Association [16] made an interesting proposition that, instead of using local strain at the corner, limit wall stretch Lmax/L0 (see Fig. 1, Lmax: limit wall length of a sheet, L0: initial wall length) should be used for a design guideline. In the present paper, a simple theoretical model which can predict the limit wall stretch and forming height in stretch bending is proposed. The limit wall stretch is given as an explicit function of the work hardening exponent (n value) of the material and non-dimensional bending
t0
Blank holder
h
Punch
Rd
L
Rp
Die
Dd L0
Fig. 1. Schematic of stretch bending test.
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curvature (t0 =R), which would be very useful for practical process design of sheet metal forming. The predicted results are verified by some stretch bending tests of aluminum alloy sheets.
2. Analytical prediction of fracture criterion 2.1. Analysis of stretch bending Tharrett and Stoughton [9,10] examined the necking limits of sheet metals and reported that necks on steel sheet occurs when the strain on the inner surface reaches a critical value, which agrees with FLD0. However, the rule does not appear to apply for FCC materials. So, in this model, the fracture limit of a sheet is determined on the basis of a maximum stretching force criterion. To obtain an explicit solution, a rigid-plastic analysis of uniform stretch bending under the plane strain condition is performed based on the deformation theory of plasticity. The effect of wall thinning due to cyclic bending–unbending and friction between tools and materials is ignored. The power-law workhardening: seq ¼ Cneq
(1)
is assumed, where seq and eeq denote the equivalent stress and plastic strain, respectively. Considering the normal anisotropy of sheet, characterized by the Lankford value (r value), for the plane-strain condition, we obtain the longitudinal stress s and strain e as 1þr s ¼ sgnðÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi seq , 1 þ 2r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2r jj ¼ eq , 2ð1 þ rÞ where sgnð Þ ¼
(
1
ifð ÞX0
1
ifð Þo0
(2)
(3)
.
From Eqs. (1)–(3), we have the stress–strain relationship: rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!nþ1 1þr 2ð1 þ rÞ C jjn . s ¼ sgnðÞ 2 1 þ 2r
(4)
A deformed sheet in stretch bending is schematically illustrated in Fig. 2. We assume that the logarithmic strain is linearly distributed in thickness direction of the sheet as o i ¼ y, (5) t where, t is the thickness of a deformed sheet, y is the distance in the radial direction. This assumption is different from the Kirchhoff-Love hypothesis of linear distribution of the
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y A
Deformed sheet in pure bending(T=0)
ε = εo
A’ t0
B t
Deformed sheet in stretch bending y=p
C
Ri
y=0
B’
ε =ε i
O
C’
ϕ
Tensile force T Punch
Fig. 2. Schematic illustration of deformed sheet in stretch bending.
engineering strain, however, practically, the difference between the Kirchhoff-Love hypothesis and this assumption by using the surface strain eo, ei is small. Let us consider such a deformation process where first a sheet is purely bent (T ¼ 0) until the inner radius reaches the die radius, Ri, (see OABC in Fig. 2) and then it is stretched (T40) over the die (OA0 B0 C0 ). Following large bending theory, from the geometries of OABC and OA0 B0 C0 , and also from Eq. (5), outer surface strain eo and inner surface strain ei under stretch bending are expressed by: ( ) ðRi þ tÞy o i ðt þ pÞ, (6) o ¼ ln ¼ t Ri þ t0 =2 f (
) Ri y o i p, i ¼ ln ¼ t Ri þ t0 =2 f
(7)
where, t0 is the initial thickness of a sheet, p is the y-coordinate at the inner surface of the bent sheet. From Eqs. (5)–(7), we obtain ¼
p¼
lnð1 þ t=Ri Þ y, t ! o 1 t. ln 1 þ t=Ri
(8)
(9)
The volume constancy condition of OABC ¼ OA0 B0 C0 is ffðRi þ t0 Þ2 R2i g yfðRi þ tÞ2 R2i g ¼ . 2 2 From Eqs. (6) and (10), we obtain y 1 2Ri þ t0 o 2Ri þ t0 t0 ¼ e ¼ . f 2 Ri þ t 2Ri þ t t
(10)
(11)
ARTICLE IN PRESS M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896
From this, the thickness of the deformed sheet in stretched bending is determined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ t0 eo Ri þ R2i þ ðt0 eo Þ2 . Tensile force T is calculated as n o 1þn Z tþp o lnð1 þ t=Ri Þ1þn 0 o Ct s dy ¼ . T¼ lnð1 þ t=Ri Þ 1 þn p rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!nþ1 1þr 2ð1 þ rÞ C ¼ C. 2 1 þ 2r 0
1889
(12)
(13a)
(13b)
2.2. Fracture limit During stretch bending, flow stress of the sheet, s, is increasing with outer surface strain eo because of its workhardening; on the other hand, the thickness of the sheet t is decreasing. As a result, tensile force T is initially increasing rapidly with eo, but the rate of increasing load is decreasing, and it will reach its maximum value Tmax. The maximum load Tmax and limit outer strain eof are determined by numerically calculating Eq. (13) until qT ¼ 0. q0
(14)
It should be noted that eof is determined uniquely for a given n value and t0 =Ri, since, from Eq. (12), sheet thickness t is a function of eo for a given t0 =Ri. Using the value of Tmax obtained, the limit wall stretch Lmax/L0 is easily calculated. Since the tensile load acting on the wall, Twall, for a wall stretch L/L0 is given by n L L0 0 T wall ¼ st ¼ C ln (15) t0 . L0 L The limit wall stretch Lmax/L0 is calculated by the following equation: n 1þn 1þn Lmax Lmax tf fof jof lnð1 þ tf =Ri Þj g ln ¼ , L0 L0 t0 ð1 þ nÞ lnð1 þ tf =Ri Þ
(16)
where, limit sheet thickness tf is a function of eof for a given t0 =Ri (see Eq. (12)). Here, we emphasize that the limit wall stretch depends only on n value of the material and on the nondimensional bending curvature t0 =Ri (or t0 =R, R ¼ Ri þ t0 =2). As discussed above, the limit wall stretch is numerically determined by Eq. (16). Alternatively, if we could have a simple equation for limit wall stretch as an explicit function of n value and t0 =R, it would be convenient for engineering use. Here we propose the following regression equation: L ¼ en þ ð1 en Þð1 eð0:4=nÞðt0 =RÞ Þ. (17) L0 Fig. 3 shows limit wall stretches Lmax/L0, calculated by both Eq. (16) (solid lines) and (17) (broken lines), as a function of t0 =R for several n values. From this figure, it is found that Lmax/L0
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Limit wall stretch Lmax /L0
1.6
n=0.50 n=0.40
1.5 n=0.30 1.4
n=0.20 n=0.10
1.3
n=0.05
1.2 1.1 1.0 0.0
0.5 1.0 Non-dimensional bending curvature t0 / R
1.5
Fig. 3. Relation between non-dimensional bending curvature t0 =R, n value and limit wall stretch Lmax/L0.
decreases considerably as t0 =R increases, and it becomes larger for materials of higher n value. This is natural since strain localization is more likely to occur at a sharp die-corner for a lower workhardening material. All the calculated results by Eq. (17) well agree with those by Eq. (16), hence it is confirmed that we can use the simple Eq. (17) for the determination of the limit wall stretch, instead of performing numerical calculation. The relation between limit outer surface strains eof calculated by Eq. (16) and t0 =R for several n values are shown in Fig. 4. The limit outer surface strain eof dramatically increases with increasing t0 =R. The same tendency was experimentally observed in the past for some sheet metals [9,10,17]. This fact suggests that, for the prediction of sheet fracture, we can use the FLD only for the fracture in almost flat parts of panels, but not for the die-corner fractures. The limit outer surface strain eof becomes larger for materials of higher n values. However, the effect of n value on eof becomes minor for large t0 =R. The relation between limit inner surface strains eif calculated by Eq. (16) and t0 =R for several n values are shown in Fig. 5.The limit inner surface strain eif decreases with increasing t0 =R.
3. Experimental verification For the verification of the above fracture criterion, three-point stretch bending tests were performed. In the tests, both ends of sheet specimens were firmly clamped and stretch-bent by a punch. To examine the effect of the bending radius, four different punches with radii of 0.5, 2.0, 5.0 and 10.0 mm were employed. Test conditions are summarized in Table 1. Three types of aluminum sheets (A5182-O, JIS6061-T4 and JIS6N01-T5) with various thicknesses, 1.0–4.0 mm, with 25.0 mm wide were used. Engineering stress–strain curves of these sheets are shown in Fig. 6.
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1.4
Limit outer surface strain of / %
1.2 1.0 0.8 n=0.50
0.6
n=0.40 n=0.30
0.4
n=0.20 n=0.10
0.2
n=0.05 0.0 0.0
0.5 1.0 Non-dimensional bending curvature t0 /R
1.5
Fig. 4. Relation between non-dimensional bending curvature t0 =R, n value and limit outer surface true strain eof. 0.6 n=0.50 n=0.40
0.5 Limit inner surface strain if / %
n=0.30 n=0.20 0.4
n=0.10 n=0.05
0.3
0.2
0.1
0 0
0.5 1 Non-dimensional bending curvature t0 /R
1.5
Fig. 5. Relation between non-dimensional bending curvature t0 =R, n value and limit inner surface true strain eif.
Mechanical properties of the sheet metals are listed in Table 2. The material constant C and n in the constitutive Eq. (1) were determined from the tensile strength sB (maximum nominal stress in uniaxial tension) and the uniform elongation eB (the engineering strain corresponding to sB) by
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Table 1 Forming conditions AL0 ¼ 118.4 mm (Dd ¼ 102.4 mm, Rd ¼ 8.0 mm) Die
BL0 ¼ 69.8 mm (Dd ¼ 53.8 mm, Rd ¼ 8.0 mm) ARp ¼ 0.5 mm
Tool
BRp ¼ 2.0 mm Punch
V-shape CRp ¼ 5.0 mm DRp ¼ 10.0 mm
Forming speed
4 mm s1
Lubricant
Mineral oil
Blank shape
&25 mm 200 mm
350
Engineering stress / MPa
300 250 200 150 5182-O
100
6061-T4 50 0
6N01-T5 0
10 20 Engineering strain / %
30
Fig. 6. Engineering stress-strain curve.
the following equations: n ¼ lnð1 þ eB Þ
(18)
sB ð1 þ eB Þ (19) nn For example, photographs of fractured 5182-O specimens are shown in Fig. 7. The width reduction of the fractured specimens was not observed obviously, it is considered that the C¼
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Table 2 Mechanical properties of specimens Material
t0 (mm)
sB (MPa)
sY (MPa)
C (MPa)
n Value
r Value
5182-O
1.0 1.4 2.0
266
127
472
0.233
0.605
6061-T4
1.0
238
135
405
0.207
0.530
6N01-T5
2.0 2.4 4.0
265
230
348
0.076
0.470
specimens deformed closer to plane strain condition. The fracture of 5182-O sheets mostly occurred at the punch corner, except for some cases of large punch-radius (the arrows in Fig. 7 indicate the locations of wall fracture). Fig. 8 shows the comparison of experimental data of limit wall stretches Lmax/L0 with the corresponding results of calculation by Eq. (16). Although the experimental conditions were widely scattered, i.e., three different types of metals, sheet thickness of 1.0–4.0 mm, punch radii of 0.5–10.0 mm, and furthermore two different die-spans of 69.8 and 118.4 mm, the calculated results of limit wall stretch for t0 =R are in good agreement with the experimental data. As already mentioned, theoretically the limit wall stretch is determined as a function of n value and nondimensional bending curvature t0 =R. This is confirmed by the experiment. Instead of limit wall stretch Lmax/L0, limit forming height hmax is often used as a measure of forming limit in real stamping operations. For that purpose, here we derive equations for calculating limit forming height hmax (see Eqs. (20)–(22)). They include the limit wall length Lmax which can be determined by either Eq. (16) or (17), as well as some geometrical parameters such as die-radius Rd, punch-radius Rp, die-span L0 and sheet thickness t0. L0 2Ra sin y Lmax ¼ , 2 cos y 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi
2 L0 Lmax hmax Ra ð1 cos yÞ þ Ra sin y ¼ Ra y, 2 2 Ra y þ
Ra ¼ Rp þ Rd þ t0 .
(20)
(21) (22)
Fig. 9 shows thus calculated limit forming height hmax for a case of L0 ¼ 118.4 mm, together with the experimental data on the three types of aluminum sheets. The calculated limit forming heights agree fairly well with most of the experimental results. In this figure, some discrepancies in hmax between the calculated results and the experimental data are found for some cases of large t0 =R, especially for 6N01-T5. This is not due to the inaccuracy of the present theory, but rather the problem of experiment. In stretch bending of a thick sheet with a sharp punch-radius, the bent radius of the sheet would be unlikely to reach the punch-radius. Such a condition is far from the assumption of the theory that the sheet fully contact with the punch profile.
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Fig. 7. Photographs of fractured 5182-O specimens (L0 ¼ 118.4 mm).
4. Concluding remarks Based on the stretch-bending analysis, the fracture limits in terms of limit wall stretch were examined. It was found from the analysis that the limit wall stretch is determined as a function of n value and non-dimensional bending curvature t0 =R. In the present paper, we propose an equation to determine the limit wall stretch, which can be solved numerically. Alternatively, a
ARTICLE IN PRESS M. Yoshida et al. / International Journal of Mechanical Sciences 47 (2005) 1885–1896 1.4
Exp. L0 Lmax
Limit wall stretch Lmax /L0
1895
L0 /mm 69.8 118.4
Material 5182-O 6061-T4 6N01-T5
1.3
Cal. 5182-O (n=0.233)
1.2
6061-T4 (n=0.206) 6N01-T5(n=0.076)
1.1
1.0 0.0
0.5
1.0
1.5
2.0
Non-dimensional bending curvature t0 /R
Fig. 8. Comparison of limit wall stretch Lmax/L0 between experiment and calculation. 60 Exp.
Cal. 5182-O
Limit forming height hmax / mm
50
Material
t0
6061-T4
1.0/mm 1.4 5182-O 2.0 6061-T4 1.0 2.0 2.4 6N01-T5 4.0
6N01-T5 40
30
20
10
0 0.0
0.5
1.0
1.5
2.0
Non-dimensional bending curvature t 0 /R
Fig. 9. Comparison of limit forming height hmax between experiment and calculation (L0 ¼ 118.4 mm).
simple equation which gives limit wall stretch as an explicit function of n-value and t0 =R has also been presented. The present theory has been confirmed by performing three-point stretch bending tests on three types of aluminum sheets.
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This fracture criterion for stretch bending is very convenient for real process design of sheet metal stamping. If we combine this explicit form of the criterion with FE simulation of stamping, we can easily predict the die-corner fracture. The great advantage of this criterion, when using in FE simulation, is that we do not need to worry about FE-mesh size for fracture prediction, since the criterion only uses a given die-radius and calculated wall stretch which would be rather insensitive to FE-mesh size.
References [1] Glover G, Duncan JL. Less conventional formability tests. Proceedings of the Australian Institute of Metals Congress 1978:18–23. [2] Uko DK, Sowerby R, Duncan JL. Strain distribution in the bending-under-tension test. Canadian Mining and Metallurgical Bulletin 1977;70(781):127–34. [3] Demeri MY. The stretch-bend forming of sheet metals. Journal of Applied Metalworking 1981;2(1):3–10. [4] Narayanaswamy OS, Demeri MY. Analysis of the angular stretch bend test. Proceedings of the Joint ASM/ASME symposium: novel techniques in metal deformation testing. 1983. p. 99–112. [5] Thomson TR, Brownrigg A. The effect of inclusions on cold formability. Metals Forum 1979;2(2):118–25. [6] Steninger J, Melander A. The relation between bendability, tensile properties and particle structure of low carbon steel. Scandinavian Journal of Metallurgy 1982;11:55–71. [7] Story JM. Geometrical aspects of stretch-bend testing and a correlation with forming performance. Manufacturing Engineering Transactions 1984;12:193–200. [8] Story JM, Jarvis GW, Zonker HR, Murtha SJ. Issues and trends in automotive aluminum sheet forming. SAE technical paper No.930277, 1993. [9] Tharrett MR, Stoughton TB. Stretch-bend forming limits of 1008 AK steel, 70/30 brass, and 6010 aluminum. Dislocations Plasticity & Metal Forming 2003:199–201. [10] Tharrett MR, Stoughton TB. Stretch-bend forming limits of 1008 AK steel. SAE technical paper No.2003-01-1157, 2003. [11] Duncan JL, Bird JE. Die forming approximations for aluminum sheet. Sheet Metal Industries 1978:1015–25. [12] Duncan JL, Shanbel BS, Gerbase-Filho J. A Tensile strip test for evaluating friction in sheet metal forming. SAE technical paper No.780391, 1978. [13] Hino R, Yoshida F. Fracture load in stretch bending for sheet metal laminates. Transactions of Japan Society of Mechanical Engineering 1995;61A(592):2560–5 [in Japanese]. [14] Naka T, Hino R, Yoshida F. Fracture of type 5083 aluminum sheet under warm stretch bending. Key Engineering Materials 2003;233–236:113–8. [15] Damborg FF. Prediction of fracture in bending-under-tension. Proceedings of the 19th Riso International Symposium on Material Science 1998:235–40. [16] The Aluminum Association. Aluminum for automotive bode sheet panels, 1998. [17] Charpentier PL. Influence of punch curvature on the stretching limits of sheet steel. Metallurgical Transactions 1975;6A:1665–9.