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A study on determining hardening curve for sheet metal
International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
A study on determining hardening curve for sheet metal Hao-bin Tian, Dachang Kang ∗ School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China Received 22 April 2002; received in revised form 18 November 2002; accepted 7 May 2003
Abstract The hardening curve for sheet metal can be determined from the load–displacement curve of tensile specimen with rectangular cross-section. The previous researches, however, have paid little attention to its use in large deformation. Moreover, it varies with materials, deformation conditions and so on, and there are not enough hardening curves available in manuals. In order to study metal behavior, it is very important to establish a method to create a large strain hardening curve based on the normal mechanical test. In this paper, two new kinds of specimens are proposed, one is a multi-stepped specimen that can be traced to the threestepped specimen, and the other is a tapered specimen which decreases the complexity of the multi-stepped specimen in manufacture. In this study, circle grids are imposed on the specimen surface to calculate true strains at different positions of the specimen. It is found that the load added to the different segments of specimen just before fracture can be determined by the maximum load and breaking load. True strains and corresponding true stresses can be determined after the specimen is pulled to fracture, so the hardening curve can be easily achieved. After a great deal of experiment, the results show that the tapered specimen has almost the same key parameters as the multi-stepped specimen, and the former is more easily used. Meanwhile, the work-hardening exponent (n) and the coefficient of normal anisotropy (r) can be obtained conveniently, and the forming limit line can also be approximately induced. 2003 Elsevier Ltd. All rights reserved. Keywords: Hardening curve; Tapered specimen; Multi-stepped specimen; Circle grid
1. Introduction The hardening curve, which is also called true stress– strain curve or deformation resistance curve, including material response in both the pre- and post-plastic localization regime, becomes essential when large deformation is considered, especially in the metal forming analysis. The tension test is a convenient way to determine the hardening curve of a material. In the forming field, a uniaxial test with a rectangular cross-section specimen is used to obtain global load–displacement data which can be easily normalized to uniaxial material stress–strain data. But this curve is insufficient because it only describes a small portion of the full strain range experienced by the material from zero deformation to final fracture. In a practical sense, this defeats the pur-
Corresponding author. Tel.: +86-451-641-4459; fax: +86-451622-1048. E-mail address: [email protected] (D. Kang). ∗
0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0890-6955(03)00132-9
pose of a tensile specimen as a means to obtain a hardening curve valid over the full range of material deformation. Consequently, a series of methods to determine the hardening curve have been developed. the hydro-forming method to determine hardening curves was presented in 1930, which has the advantage that a full range of hardening curves can be directly determined, while the disadvantage is that a special set-up is needed. the three-stepped specimen method to determine the hardening curve was proposed in 1962 [1], in which the hardening curve is described by the values of n and K calculated by the two-point method, assuming that the hardening curve can be approximately described by the function s = Ken. In the following, a subsequent computational investigation was performed to determine the hardening curve with a tension test [2,3]. Recently, methods to determine the hardening curve have been further researched. For the anisotropic material, how to determine the hardening curve from both a round and rectangular specimen was presented, respectively [4–6], and the method to deter-
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H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
(n) and the coefficient of normal anisotropy (r) can be obtained conveniently, and the forming limit line can also be approximately induced. Fig. 1.
Multi-stepped specimen (unit: mm).
mine the hardening curve by a numerical method was also advanced [4,5,7,8]. In addition, the indention method to determine the hardening curve has been developed experimentally and numerically [9,10]. At present, it is still hard to find a curve available in manuals that agrees with that of the material being used, for it varies with materials, temperature and speed of deformation. It also changes with accommodation status and the deformation history of the sheet. While in sheet forming, to acquire a material’s hardening property sufficiently and accurately is important, so tests against a certain material should be carried out to determine the hardening curve. Thus, establishing a method of determining the hardening curve suitable for engineering is essential. As stated above, the uniaxial tensile method is the most popular one to determine the hardening curve, but only a part of the full range of the material deformation can be obtained and this defeats the purpose of a tensile specimen as a means to obtain a hardening curve which is valid over the full range of material deformation. It should be noted that deformation on the specimen is inhomogeneous and the deformation at necking is the largest. Then, large elongation of a specimen can be obtained through short gauge specimen [11]. The practical problem is that it is difficult to measure and record the changes of a specimen’s dimensions. In this regard, two kinds of specimens are proposed, one is a multi-stepped specimen (Fig. 1) and the other is a tapered specimen (Fig. 2). In order to determine true strains, circle grids are drawn on the specimen [12]. In this study, the specimen is placed under tension and pulled until it fails, then the true strains and corresponding true stresses on the specimen can be obtained through the deformation of circle grids and corresponding load (for the circle grids beyond necking, the load is considered the maximum load; while for the circle grids near necking, the load is equal to the load at fracture). Therefore, the hardening curve can be easily determined. After a great deal of experiment, the results show that the tapered specimen has almost the same key parameters as the multi-stepped specimen, and the former can be used more easily. Meanwhile, the work-hardening exponent
Fig. 2.
Tapered specimen (unit: mm).
2. Experimental investigation 2.1. Experiment conditions In the experiments, both the multi-stepped and tapered specimens were subjected to tension loading on a 50 kN material testing machine, and the standard tensile specimen (Fig. 3) was subjected to tension loading on a 100 kN electronic tensile tester. All tests were performed at a rate of 10 mm/min. Materials of LF4, H62 and soft steel sheet of 1 mm thickness are considered in this paper, and the chemical compositions of LF4 and H62 are shown in Tables 1 and 2 respectively. For the tapered specimen, in order to make sure that the stress state is the same as the state of uniaxial stress, its dimension is sufficiently considered to decrease the non-uniform distribution of stress caused by taper to a degree that can be neglected. As for the given dimensions (Fig. 2), half of the taper (a) equals 0.7° (cos4a = 0.9997). According to the mechanical analysis of a tensioned symmetry wedge, the difference between maximum and minimum stress is about 0.03% of mean stress. So the stress distribution of the tapered specimen can be considered as the state of uniaxial stress. Grids of φ2.6 mm are used to LF4 and φ3 mm to H62 and soft steel, which are in a double-row interlaced layout (Fig. 4).
3. Experiment results and discussions Experiments on three kinds of specimens are carried out with materials of LF4, H62 and soft steel. The true stress and true strain can be calculated as follows. The multi-stepped specimen and tapered specimen are pulled until they fail, the maximum load and breaking load can be established. True strains can be obtained through the deformation of grids. After the specimen fractures, circle grids change to an elliptical shape approximately. The major diameter of the ellipse can be measured through a microscope, and then the true strain and true stress at each grid can be calculated by Eqs. (1) and (2), respectively.
Fig. 3.
Standard tensile specimen (unit: mm).
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
1255
Table 1 Chemical composition of LF4 (wt %) Element
Cu
Cr
Fe
Mg
Mn
Si
Zn
Ti
Al
Total impurity
LF4
0.1
0.05–0.25
0.4
4.0–4.9
0.4–1.0
0.4
0.25
0.15
Bal.
0.15
Table 2 Chemical composition of H62 (wt %) Element
Cu
Fe
Bi
P
Pb
Sb
Zn
Total impurity
H62
60.5–63.5
0.15
0.002
0.01
0.08
0.005
Bal.
0.5
Fig. 4.
The grids layout on specimen.
Table 3 Values of su and eu determined by standard tensile specimen Material
su (MPa)
eu
Soft steel LF4 H62
424.7 97.0 526.5
0.23 0.29 0.5
D e ⫽ ln D0 s⫽
FD A0D0
Fig. 5.
(1)
Hardening curves of H62.
mation method which is frequently used in the metal press working theory, the hardening curve can be obtained through suand eu. Curves from three kinds of specimens of the same material have been graphed together, and shown in Figs. 5–7. These figures show that the hardening curve
(2)
Where, D is the major diameter of ellipse and D0 is the initial diameter of grid; A0 is the initial cross-section area of specimen, F is the maximum load for grids beyond necking and the breaking load for grids within necking as well. As stated above, a series of strains and stresses can be gained through grids at different locations on specimen. So the hardening curve can be obtained directly from the test of multi-stepped and tapered specimens. For the standard tensile specimen, tests are carried out by electronic tensile tester in order to get accurate load vs displacement curves. When the load reaches the maximum, true stress (su) and strain (eu) can be obtained from engineering stress and strain, respectively. Values are shown in Table 3. According to the linear approxi-
Fig. 6.
Hardening curves of LF4.
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H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
inhomogeneous, it is possible to find a cross-section on the specimen where the train is within 0.05–0.20. So the r value can be obtained by Eq. (4), and results are shown in Table 4. r⫽
⫺ln(B / B0) ln(D / D0 ⫹ ln(B / B0))
(4)
where, B and B0 represent the current width and initial width of the specimen, respectively. 3.3. The determination of forming limit line
Fig. 7.
Hardening curves of soft steel.
determined by the tapered specimen agrees well with that determined by the multi-stepped specimen. They also show that the hardening curve gained from a standard tensile specimen is tangential to that determined by a tapered specimen. As stated above, specimens proposed in this paper are feasible to determine the hardening curve, and the tapered specimen is preferred. 3.1. The determination of the work-hardening exponent (n) As is well known, the n value is an important parameter reflecting a material’s hardening property. For the tapered specimen, the n value can be obtained through the two-point method that can be described by Eq. (3). The result is shown in Table 4. n⫽
lns2⫺lns1 lne2⫺lne1
(3)
As demonstrated above, the stress state of the tapered specimen that is subjected to uniaxial tension is similar to the state of uniaxial stress. The strain at fracture calculated from grids can be used to determine the point A in the forming limit graph (Fig. 8). SCV (see Fig. 8) can be gained from point A, because the horizontal ordinate of point A is the double length of SCV. Thus, the forming limit line can be determined approximately, as shown in Fig. 8.
4. Conclusions The tapered and multi-stepped specimen with circle grids approach to determine the hardening curve are proposed and verified. The tapered specimen method is preferred in this paper for its simplicity in manufacture and use. A tapered specimen with circle grids is similar to a multi-stepped specimen in view of the variable crosssection areas. In contrast to the three-stepped specimen, the major advantage is that the hardening curve can be directly obtained, which avoids the problem whether the hardening curve agrees with the power function or not. The work-hardening exponent (n) and the coefficient
where, s1 and s2 are different stresses on the hardening curve, and e1 and e2 are the corresponding strains. 3.2. The determination of the coefficient of normal anisotropy (r) The coefficient of normal anisotropy (r) is the ratio of width strain to thickness. In most cases, it is worked out when the strain is within 0.05–0.20. In this study, deformation along the fractured tapered specimen is Table 4 n value and r value determined by tapered specimen
n r
LF4
H62
Soft steel
0.20 0.47
0.21 0.145
0.24 1.72
Fig. 8. Forming limit line obtained by tapered specimens can be determined approximately.
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
of normal anisotropy (r) can be easily obtained from experiments. Point A in the forming limit curves can be easily determined through a tapered specimen with circle grids, and thus the forming limit curve.
[6]
[7]
References [1] R.H. Heyer, D.E. McCabe, J.A. Elias, in: E.W, Earhart (Ed.), Flat Rolled Products, Interscience Publishers, New York, 1962. [2] D.M. Norris, B. Moran, J.K. Scudder, D.F. Quinones, A computer simulation of the tension test, J. Mech. Phy. Solids 26 (1978) 1–19. [3] P. Matic, G.C. Kirby, M.I. Joues, The relation of tensile specimen size and geometry effects to unique constitutive parameters for ductile materials, Proc. R. Soc. Lond. A 147 (1988) 309–333. [4] Z.L. Zhang, J. Ødega˚ rd, O.P. Søvik, C. Thaulow, A study on determining true stress–strain curve for anisotropic materials with rectangular tensile bars, International Journal of Solid and Structures 38 (2001) 4489–4505. [5] Z.L. Zhang, M. Hauge, C. Thaulow, J. Ødega˚ rd, Determining true stress–strain curve for isotropic and anisotropic materials with
[8]
[9]
[10]
[11]
[12]
1257
rectangular tensile bars: method and verifications, Computational Materials Science 20 (2001) 77–85. H.S. Yang, X.P.S. Su, B. Bai, Analysis of diametral strain in uniaxial tensile and compression testing of circle specimens of anisotropic materials, International Journal of Mechanical Sciences 42 (2000) 2395–2415. Z.L. Zhang, M. Hauge, J. Ødega˚ rd, C. Thaulow, Determining material true stress–strain curve from tensile specimens with rectangular cross-section, International Journal of Solid and Structures 36 (1999) 3497–3516. X. Lin, F. Zhai, Z. Zhang, Determination of material flow stress by the method of C-FEM, Journal of Materials Processing Technology 120 (2002) 144–150. A. Nayebi, O. Bartier, G. Mauvoisin, R. Ei. Abdi, New method to determine the mechanical properties of heat treated steels, International Journal of Mechanical Sciences 43 (2001) 2679– 2697. T. Yamamoto, H. Kurishita, H. Matsui, Modeling of the cyclic ball indentation test for small specimens using the finite element method, Journal of Nuclear Materials 271-272 (1999) 440–444. Standard methods and definitions for mechanical testing of steel products, A 370, Annual Book of ASTM Standards, ASTM, Philadelphia, 03.01 (1984) 1–56. G.M. Goodwin, Application of strain analysis to sheet metal forming problems in the press shop, Society of Automotive Engineers, paper no. 680093, PA, 1968.
A study on determining hardening curve for sheet metal Hao-bin Tian, Dachang Kang ∗ School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China Received 22 April 2002; received in revised form 18 November 2002; accepted 7 May 2003
Abstract The hardening curve for sheet metal can be determined from the load–displacement curve of tensile specimen with rectangular cross-section. The previous researches, however, have paid little attention to its use in large deformation. Moreover, it varies with materials, deformation conditions and so on, and there are not enough hardening curves available in manuals. In order to study metal behavior, it is very important to establish a method to create a large strain hardening curve based on the normal mechanical test. In this paper, two new kinds of specimens are proposed, one is a multi-stepped specimen that can be traced to the threestepped specimen, and the other is a tapered specimen which decreases the complexity of the multi-stepped specimen in manufacture. In this study, circle grids are imposed on the specimen surface to calculate true strains at different positions of the specimen. It is found that the load added to the different segments of specimen just before fracture can be determined by the maximum load and breaking load. True strains and corresponding true stresses can be determined after the specimen is pulled to fracture, so the hardening curve can be easily achieved. After a great deal of experiment, the results show that the tapered specimen has almost the same key parameters as the multi-stepped specimen, and the former is more easily used. Meanwhile, the work-hardening exponent (n) and the coefficient of normal anisotropy (r) can be obtained conveniently, and the forming limit line can also be approximately induced. 2003 Elsevier Ltd. All rights reserved. Keywords: Hardening curve; Tapered specimen; Multi-stepped specimen; Circle grid
1. Introduction The hardening curve, which is also called true stress– strain curve or deformation resistance curve, including material response in both the pre- and post-plastic localization regime, becomes essential when large deformation is considered, especially in the metal forming analysis. The tension test is a convenient way to determine the hardening curve of a material. In the forming field, a uniaxial test with a rectangular cross-section specimen is used to obtain global load–displacement data which can be easily normalized to uniaxial material stress–strain data. But this curve is insufficient because it only describes a small portion of the full strain range experienced by the material from zero deformation to final fracture. In a practical sense, this defeats the pur-
Corresponding author. Tel.: +86-451-641-4459; fax: +86-451622-1048. E-mail address: [email protected] (D. Kang). ∗
0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0890-6955(03)00132-9
pose of a tensile specimen as a means to obtain a hardening curve valid over the full range of material deformation. Consequently, a series of methods to determine the hardening curve have been developed. the hydro-forming method to determine hardening curves was presented in 1930, which has the advantage that a full range of hardening curves can be directly determined, while the disadvantage is that a special set-up is needed. the three-stepped specimen method to determine the hardening curve was proposed in 1962 [1], in which the hardening curve is described by the values of n and K calculated by the two-point method, assuming that the hardening curve can be approximately described by the function s = Ken. In the following, a subsequent computational investigation was performed to determine the hardening curve with a tension test [2,3]. Recently, methods to determine the hardening curve have been further researched. For the anisotropic material, how to determine the hardening curve from both a round and rectangular specimen was presented, respectively [4–6], and the method to deter-
1254
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
(n) and the coefficient of normal anisotropy (r) can be obtained conveniently, and the forming limit line can also be approximately induced. Fig. 1.
Multi-stepped specimen (unit: mm).
mine the hardening curve by a numerical method was also advanced [4,5,7,8]. In addition, the indention method to determine the hardening curve has been developed experimentally and numerically [9,10]. At present, it is still hard to find a curve available in manuals that agrees with that of the material being used, for it varies with materials, temperature and speed of deformation. It also changes with accommodation status and the deformation history of the sheet. While in sheet forming, to acquire a material’s hardening property sufficiently and accurately is important, so tests against a certain material should be carried out to determine the hardening curve. Thus, establishing a method of determining the hardening curve suitable for engineering is essential. As stated above, the uniaxial tensile method is the most popular one to determine the hardening curve, but only a part of the full range of the material deformation can be obtained and this defeats the purpose of a tensile specimen as a means to obtain a hardening curve which is valid over the full range of material deformation. It should be noted that deformation on the specimen is inhomogeneous and the deformation at necking is the largest. Then, large elongation of a specimen can be obtained through short gauge specimen [11]. The practical problem is that it is difficult to measure and record the changes of a specimen’s dimensions. In this regard, two kinds of specimens are proposed, one is a multi-stepped specimen (Fig. 1) and the other is a tapered specimen (Fig. 2). In order to determine true strains, circle grids are drawn on the specimen [12]. In this study, the specimen is placed under tension and pulled until it fails, then the true strains and corresponding true stresses on the specimen can be obtained through the deformation of circle grids and corresponding load (for the circle grids beyond necking, the load is considered the maximum load; while for the circle grids near necking, the load is equal to the load at fracture). Therefore, the hardening curve can be easily determined. After a great deal of experiment, the results show that the tapered specimen has almost the same key parameters as the multi-stepped specimen, and the former can be used more easily. Meanwhile, the work-hardening exponent
Fig. 2.
Tapered specimen (unit: mm).
2. Experimental investigation 2.1. Experiment conditions In the experiments, both the multi-stepped and tapered specimens were subjected to tension loading on a 50 kN material testing machine, and the standard tensile specimen (Fig. 3) was subjected to tension loading on a 100 kN electronic tensile tester. All tests were performed at a rate of 10 mm/min. Materials of LF4, H62 and soft steel sheet of 1 mm thickness are considered in this paper, and the chemical compositions of LF4 and H62 are shown in Tables 1 and 2 respectively. For the tapered specimen, in order to make sure that the stress state is the same as the state of uniaxial stress, its dimension is sufficiently considered to decrease the non-uniform distribution of stress caused by taper to a degree that can be neglected. As for the given dimensions (Fig. 2), half of the taper (a) equals 0.7° (cos4a = 0.9997). According to the mechanical analysis of a tensioned symmetry wedge, the difference between maximum and minimum stress is about 0.03% of mean stress. So the stress distribution of the tapered specimen can be considered as the state of uniaxial stress. Grids of φ2.6 mm are used to LF4 and φ3 mm to H62 and soft steel, which are in a double-row interlaced layout (Fig. 4).
3. Experiment results and discussions Experiments on three kinds of specimens are carried out with materials of LF4, H62 and soft steel. The true stress and true strain can be calculated as follows. The multi-stepped specimen and tapered specimen are pulled until they fail, the maximum load and breaking load can be established. True strains can be obtained through the deformation of grids. After the specimen fractures, circle grids change to an elliptical shape approximately. The major diameter of the ellipse can be measured through a microscope, and then the true strain and true stress at each grid can be calculated by Eqs. (1) and (2), respectively.
Fig. 3.
Standard tensile specimen (unit: mm).
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
1255
Table 1 Chemical composition of LF4 (wt %) Element
Cu
Cr
Fe
Mg
Mn
Si
Zn
Ti
Al
Total impurity
LF4
0.1
0.05–0.25
0.4
4.0–4.9
0.4–1.0
0.4
0.25
0.15
Bal.
0.15
Table 2 Chemical composition of H62 (wt %) Element
Cu
Fe
Bi
P
Pb
Sb
Zn
Total impurity
H62
60.5–63.5
0.15
0.002
0.01
0.08
0.005
Bal.
0.5
Fig. 4.
The grids layout on specimen.
Table 3 Values of su and eu determined by standard tensile specimen Material
su (MPa)
eu
Soft steel LF4 H62
424.7 97.0 526.5
0.23 0.29 0.5
D e ⫽ ln D0 s⫽
FD A0D0
Fig. 5.
(1)
Hardening curves of H62.
mation method which is frequently used in the metal press working theory, the hardening curve can be obtained through suand eu. Curves from three kinds of specimens of the same material have been graphed together, and shown in Figs. 5–7. These figures show that the hardening curve
(2)
Where, D is the major diameter of ellipse and D0 is the initial diameter of grid; A0 is the initial cross-section area of specimen, F is the maximum load for grids beyond necking and the breaking load for grids within necking as well. As stated above, a series of strains and stresses can be gained through grids at different locations on specimen. So the hardening curve can be obtained directly from the test of multi-stepped and tapered specimens. For the standard tensile specimen, tests are carried out by electronic tensile tester in order to get accurate load vs displacement curves. When the load reaches the maximum, true stress (su) and strain (eu) can be obtained from engineering stress and strain, respectively. Values are shown in Table 3. According to the linear approxi-
Fig. 6.
Hardening curves of LF4.
1256
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
inhomogeneous, it is possible to find a cross-section on the specimen where the train is within 0.05–0.20. So the r value can be obtained by Eq. (4), and results are shown in Table 4. r⫽
⫺ln(B / B0) ln(D / D0 ⫹ ln(B / B0))
(4)
where, B and B0 represent the current width and initial width of the specimen, respectively. 3.3. The determination of forming limit line
Fig. 7.
Hardening curves of soft steel.
determined by the tapered specimen agrees well with that determined by the multi-stepped specimen. They also show that the hardening curve gained from a standard tensile specimen is tangential to that determined by a tapered specimen. As stated above, specimens proposed in this paper are feasible to determine the hardening curve, and the tapered specimen is preferred. 3.1. The determination of the work-hardening exponent (n) As is well known, the n value is an important parameter reflecting a material’s hardening property. For the tapered specimen, the n value can be obtained through the two-point method that can be described by Eq. (3). The result is shown in Table 4. n⫽
lns2⫺lns1 lne2⫺lne1
(3)
As demonstrated above, the stress state of the tapered specimen that is subjected to uniaxial tension is similar to the state of uniaxial stress. The strain at fracture calculated from grids can be used to determine the point A in the forming limit graph (Fig. 8). SCV (see Fig. 8) can be gained from point A, because the horizontal ordinate of point A is the double length of SCV. Thus, the forming limit line can be determined approximately, as shown in Fig. 8.
4. Conclusions The tapered and multi-stepped specimen with circle grids approach to determine the hardening curve are proposed and verified. The tapered specimen method is preferred in this paper for its simplicity in manufacture and use. A tapered specimen with circle grids is similar to a multi-stepped specimen in view of the variable crosssection areas. In contrast to the three-stepped specimen, the major advantage is that the hardening curve can be directly obtained, which avoids the problem whether the hardening curve agrees with the power function or not. The work-hardening exponent (n) and the coefficient
where, s1 and s2 are different stresses on the hardening curve, and e1 and e2 are the corresponding strains. 3.2. The determination of the coefficient of normal anisotropy (r) The coefficient of normal anisotropy (r) is the ratio of width strain to thickness. In most cases, it is worked out when the strain is within 0.05–0.20. In this study, deformation along the fractured tapered specimen is Table 4 n value and r value determined by tapered specimen
n r
LF4
H62
Soft steel
0.20 0.47
0.21 0.145
0.24 1.72
Fig. 8. Forming limit line obtained by tapered specimens can be determined approximately.
H.-b. Tian, D. Kang / International Journal of Machine Tools & Manufacture 43 (2003) 1253–1257
of normal anisotropy (r) can be easily obtained from experiments. Point A in the forming limit curves can be easily determined through a tapered specimen with circle grids, and thus the forming limit curve.
[6]
[7]
References [1] R.H. Heyer, D.E. McCabe, J.A. Elias, in: E.W, Earhart (Ed.), Flat Rolled Products, Interscience Publishers, New York, 1962. [2] D.M. Norris, B. Moran, J.K. Scudder, D.F. Quinones, A computer simulation of the tension test, J. Mech. Phy. Solids 26 (1978) 1–19. [3] P. Matic, G.C. Kirby, M.I. Joues, The relation of tensile specimen size and geometry effects to unique constitutive parameters for ductile materials, Proc. R. Soc. Lond. A 147 (1988) 309–333. [4] Z.L. Zhang, J. Ødega˚ rd, O.P. Søvik, C. Thaulow, A study on determining true stress–strain curve for anisotropic materials with rectangular tensile bars, International Journal of Solid and Structures 38 (2001) 4489–4505. [5] Z.L. Zhang, M. Hauge, C. Thaulow, J. Ødega˚ rd, Determining true stress–strain curve for isotropic and anisotropic materials with
[8]
[9]
[10]
[11]
[12]
1257
rectangular tensile bars: method and verifications, Computational Materials Science 20 (2001) 77–85. H.S. Yang, X.P.S. Su, B. Bai, Analysis of diametral strain in uniaxial tensile and compression testing of circle specimens of anisotropic materials, International Journal of Mechanical Sciences 42 (2000) 2395–2415. Z.L. Zhang, M. Hauge, J. Ødega˚ rd, C. Thaulow, Determining material true stress–strain curve from tensile specimens with rectangular cross-section, International Journal of Solid and Structures 36 (1999) 3497–3516. X. Lin, F. Zhai, Z. Zhang, Determination of material flow stress by the method of C-FEM, Journal of Materials Processing Technology 120 (2002) 144–150. A. Nayebi, O. Bartier, G. Mauvoisin, R. Ei. Abdi, New method to determine the mechanical properties of heat treated steels, International Journal of Mechanical Sciences 43 (2001) 2679– 2697. T. Yamamoto, H. Kurishita, H. Matsui, Modeling of the cyclic ball indentation test for small specimens using the finite element method, Journal of Nuclear Materials 271-272 (1999) 440–444. Standard methods and definitions for mechanical testing of steel products, A 370, Annual Book of ASTM Standards, ASTM, Philadelphia, 03.01 (1984) 1–56. G.M. Goodwin, Application of strain analysis to sheet metal forming problems in the press shop, Society of Automotive Engineers, paper no. 680093, PA, 1968.