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Experimental and analytical studies on the prediction of forming limit diagrams
Computational Materials Science 44 (2009) 1252–1257
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Experimental and analytical studies on the prediction of forming limit diagrams S. Ahmadi a,*, A.R. Eivani b, A. Akbarzadeh c a b c
Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, United States Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Department of Materials Science and Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 5 August 2008 Accepted 20 August 2008 Available online 2 October 2008 PACS: 83.50.Uv 81.70.-q 07.05.Tp 02.70.-c
a b s t r a c t Metal forming processes are widely used in industrial productions, automobile bodies, food industries, oil refineries, and liquid and gas transmission systems. Analyzing these processes is very important to reduce wastes and optimize the processes. Study of some main factors such as physical and mechanical properties of material and its formability, die geometry, die material, lubrication and pressing speed has been the topic of many research projects. In this paper, forming limit diagrams (FLDs) for LC and ULC steels and the effect of different parameters like the work-hardening exponent, n, and the plastic strain ratio, r, on these diagrams have been evaluated and simulated using ABAQUS/Standard. In this case, Hill’s quadratic anisotropy function is assumed to be the yield function and the Atkins criterion is used as the failure criterion. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Forming limit diagram FLD Forming limits Finite element method ABAQUS Hill’s quadratic yield function
1. Introduction A forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metals during forming processes such as deep drawing and stretch forming. FLDs are plots of the maximum major principal strains which can be sustained by sheet materials prior to the onset of localized necking. By using the FLD, the strains leaded to material failure through different strain paths can be predicted, so it is considered as an important tool in die design as well as in the optimization and problem corrections in the manufacturing processes. Instead of the experimental implementation of FLDs many researchers have conducted a search for an easier FLD construction. Hecker [1], for instance, proposed an experimental technique which enables one to draw the entire diagram using fewer tests by utilizing a hemispherical punch with different widths of the sheet samples and different types of lubricants. Although, Hecker’s model was a significant improvement to FLD construction, the methodology employed was of great complexity. Tadros and Mellor [2], Gronostajski and Dolny [3] and Raghavan [4], on the other hand, developed techniques to eliminate the tool/material friction effects * Corresponding author. Tel.: +1 801 422 2059. E-mail address: [email protected] (S. Ahmadi). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.08.008
and to keep the blank surface flat. Sing and Rao [5] created a method to determine the FLD through the parameters obtained by a conventional tensile test, which appears to be very effective and less expensive than the previous methods. Recently, several researchers have attempted to predict the FLDs using numerical models. Clift et al. [6] used the ductile fracture criterion to predict the forming limits. Based on these criteria, the occurrence of ductile fracture is estimated using the macroscopic stress and strain that occur during deformation. Takuda et al. [7], Yoshida et al. [8] and Ozturk and Lee [9] also predicted the FLDs by using the ductile fracture criteria. Yoshida et al. simulated the hemispherical punch-stretching using an elasto-plastic three-dimensional finite element model, and Ozturk and Lee simulated the out-of-plane (dome) formability test using ABAQUS/Standard to obtain stress and strain values. In this work, ABAQUS/ Standard software is used to predict the FLD and then this model is compared with the experimental data. 2. Experimental work Two low carbon steel sheets (A and B) and an ultra low carbon steel sheet (C) were utilized for experiments. Tensile tests were carried out using specimens machined in accordance with ASTM standard E8M specifications. The tests were conducted along three
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257 Table 1 Mechanical properties of three sample sheets A–C Sample sheet
r-Value
n-Value
YS (MPa)
UTS (MPa)
A B C
1.68 1.63 1.91
0.246 0.222 0.250
179 168 147
303.9 303.2 287.3
1253
length of the blanks was 200 mm and the width was varied between 25 and 200 mm in the steps of 25 mm. For each blank width, at least four to five specimens were tested to get the maximum number of data points. The circles on the sheet samples became ellipses after deformation, falling into safe, necked, and failed zones. The principal strains in the plane of the sheet are expressed using either the traditional engineering strain or the true strain measures. The engineering strain measure provides a percentage of the length change of the principal directions, a and b, of the ellipse with reference to the initial diameter, d0, of the circle:
a d0 100; d0 b d0 100: eminor ð%Þ ¼ d0
emajor ð%Þ ¼
ð2Þ ð3Þ
Finally, FLDs were drawn by deliminating the safe limiting strains from the unsafe zone containing the necked and fractured ellipses. The accuracy of FLDs lies well within a band of ±2% in the engineering strain values, Fig. 2. 3. Numerical work In the numerical work, the Atkins criterion [12] was used as fracture criterion which is formulated as below in Eqs. (4) and (5):
Z
ef
1 þ ð1=2LÞ ; 1 c rh 1 de1 c¼ ; ; L¼ 3:1YS de2 C¼
ð4Þ
0
Fig. 1. A schematic view of the die and the hydraulic press.
directions, with the tensile axis being parallel (0°), diagonal (45°) and perpendicular (90°) to the rolling direction of the sheet. A constant cross-head speed of 0.1 mm min1 was employed in all cases. Three samples were tested for each direction and average values were reported. The standard tensile properties namely, 0.2% yield stress (YS), ultimate tensile stress (UTS), strain-hardening exponent (n) and the plastic strain ratio (r) were determined from the load–elongation data obtained from these tests and shown in Table 1. The normal anisotropy r was calculated using Eq. (1):
r ¼
r 0 þ 2r 45 þ r 90 : 4
ð1Þ
2.1. Measurement of experimental forming limit diagrams Empirical FLDs evaluated using Hecker’s simplified technique [1]. In this method, the experimental procedure mainly involves three stages: grid marking the sheet specimens, punch-stretching the grid-marked samples to failure or onset of localized necking, and measurement of strains. Grid marking on the sheet samples was done using the circle grid analysis (CGA) technique [10]. According to this technique, the grid pattern (circles with 5 mm diameter) was etched on the samples using an electro-chemical etching equipment. Subsequently, punch-stretching experiments were carried out on a double-action hydraulic press. A schematic view of the die and the press is shown in Fig. 1. The sheet samples were subjected to different states of strain, i.e. the tension–tension zone, plane strain and the tension-compression zones by varying the width of the samples [11]. The
f ðrÞ ¼
ð5Þ
where rh is the hydrostatic stress. This ductile fracture criterion requires at least one destructive test for calibration. A tensile test was conducted as a destructive test to calibrate and to attain the C-constant for this criterion. After achieving the value of the C-constant, this fracture criterion was used in all analyses. The analyses were performed in ABAQUS/Standard [13] assuming static conditions and employing shell elements. A simplified modeling approach was used. The die and the punch were modeled as analytical rigid surfaces and the workpiece as a deformable body. The material was modeled as elastic–plastic where the elasticity was taken to be isotropic and the plasticity was assumed as both isotropic and anisotropic conditions. Material properties such as the Young’s modulus (E), the Poisson’s ratio (c), anisotropy coefficients and stress–strain curve statistics were applied to the workpiece in the property module. The S4R element was used in the mesh module, and the small-sliding contact was used in the die/blank and punch/blank interfaces. The friction effect was introduced to the model using the frictionless, lubricated, and dry conditions. This analysis was simulated in three steps. In the first step, the upper die moves down vertically and deforms the sheet metal sample into the drawbead. The hemispherical punch and lower die stay fixed during this step. The punch, bottom die, and upper die remain fixed in the second step, and a 100 kN force is applied to the blank from the upper die. During the third step, while the boundary condition from the second step is still in effect, the punch starts to move up until the desired displacements are achieved. The yield criterion of Hill [14] was adopted and applied as in Eq. (6):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðr22 r33 Þ2 þ Gðr33 r11 Þ2 þ Hðr11 r22 Þ2 þ 2Lr223 þ 2M r231 þ 2Nr212 ;
ð6Þ
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S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
Fig. 2. FLDs, obtained from experimental work for sample sheets A–C.
where rij denotes the stress components and F, G, H, L, M and N are the material constants obtained from tensile tests. These constants
can be expressed in terms of six yield stress ratios R11, R22, R33, R12, R13 and R23, as presented in Eqs. (7)–(10):
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
!
1 1 1 1 ; þ 2 R222 R233 R211 ! 1 1 1 1 G¼ ; þ 2 R233 R211 R222 ! 1 1 1 1 H¼ ; þ 2 R211 R222 R233 F¼
L¼
3 2R223
;
M¼
3 2R231
;
L¼
ð7Þ
sample sheet B has the lowest level of strain among the forming level curves.
ð8Þ
4.2. Comparison between theoretical and experimental FLDs
ð9Þ 3 2R212
ð10Þ
:
The yield stress ratios can be rewritten as the function of the plastic strain ratios, Eqs. (11) and (12):
R11 R33
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 90 ðr 0 þ 1Þ ¼ R13 ¼ R23 ¼ 1; R22 ¼ ; r 0 ðr 90 þ 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 90 ðr0 þ 1Þ 3r90 ðr 0 þ 1Þ ¼ ; R12 ¼ ; r0 þ r 90 ð2r 45 þ 1Þðr0 þ r 90 Þ
1255
ð11Þ ð12Þ
where r0, r45 and r90 are the plastic strain ratios in different directions relative to the rolling direction.
4. Results and discussion
The comparison between experimental and theoretical FLDs as calculated and simulated by ABAQUS software is shown in Fig. 4, where Exp, Iso and Aniso stand for experimental and modeling data for isotropic and anisotropic conditions, respectively. On the right-hand side of the FLD, the isotropic curve is the most consistent with the experimental results. However, on the lefthand side, the anisotropic curve is the best fit with the experimental results. This might be due to the effect of the r-value on the FLDs. The plastic strain ratio does not have a large effect on the positive minor strain region, but in the negative region by increasing the r-value, the slope of the curve decreases and eventually the curve approaches the experimental curve. It is also observed that in many regions of the FLDs, the anisotropic simulated curve has a lower value of the major strain than the experimental curve, so the use of anisotropic condition can be more accurate than the isotropic. 4.3. Comparison between the theoretical and the experimental FLD0values
4.1. The effect of n- and r-value on the forming limit curve In general, the strain-hardening exponent and plastic strain ratio of a material are considered to be the most important parameters which influence the FLD. Fig. 3 shows the influences of n- and r-values, respectively, on the experimental FLDs. As shown, sample sheet C has the highest limiting strains. By increasing the n- and rvalues of steel sheets, the forming limit curves are shifted upward. This is because a higher value of the strain-hardening exponent generally delays the onset of instability until the higher strain value, is reached. This delay enhances the limiting strain. The plastic strain ratio, r, also has the great effect on the limiting strain. By increasing the value of this parameter, the formability of the sheet is increased. Therefore, sample sheet C has the highest level and
The importance of the FLD0-value lies in the fact that fractures or cracks appearing in cold-formed parts under press conditions often occur at this strain state. Fig. 5 compares the theoretical and experimental forming limit strains in the plane-strain state, FLD0. The forming-limit strains predicted by the isotropic conditions are in good agreement with the experimental results for all steel sheets; however, the calculated values of FLD0 from the anisotropic model are lower than the experimental results. It is also observed that sample sheet C has the highest value of FLD0, which is due to its physical and mechanical properties, such as its n- and r- values. According to Figs. 4 and 5, the difference between experimental and simulated results using anisotropic conditions on the left and
Fig. 3. FLDs of all sample sheets (using experimental data).
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S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
Fig. 4. Empirical and numerical FLDs. Exp stands for experimental FLDS; Iso or Aniso stands for theoretical FLDs calculated using ABACUS with either fully isotropic or fully anisotropic plasticity model
right sides of the forming limit curves is not significantly large. In fact, using the anisotropic conditions in simulation yields the bet-
ter consistency, while using the isotropic conditions results in values closer to the experimental FLD0.
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
1257
– The value of plastic strain ratio has a large effect on the level of FLC on the left side. By increasing the value of r, the slope of the FLC on the left side is shifted down. – On the right side of the FLD, any variation of r-value has no significant effect on the shape of the diagram. – Higher values of n and r allow for higher values of the final strains and, as a results, the ultimate elongation increases. This also allows the point designating fracture to move to higher values of the major strain. References [1] [2] [3] [4] [5] [6] Fig. 5. Comparison between the calculated and the experimental FLD0-values for the A–C sample sheets.
5. Conclusion Based upon the experimental and simulated results the following conclusions are drawn:
[7] [8] [9] [10] [11] [12] [13] [14]
– By increasing the values of n and r for the sample sheets, formability, deep drawability, and the value of FLD0 are increased.
S.S. Hecker, Sheet Met. Ind. 52 (1975) 671–675. A.K. Tadros, P.B. Mellor, Int. J. Mech. Sci. 20 (1978) 121–134. J. Gronostajski, A. Dolny, Memories Sci. Rev. Metall. 4 (1980) 570–578. K.S. Raghavan, Metall. Trans. A 26 (1995) 2075–2084. W.M. Sing, K.P. Rao, J. Mater. Process. Technol. 37 (1993) 37–51. S.E. Clift, P. Hartly, C.E.N. Sturgess, G.W. Rowe, Int. J. Mech. Sci. 32 (1990) 1–17. H. Takuda, K. Mori, N. Hatta, J. Mater. Process. Technol. 95 (1999) 116–121. T. Yoshida, T. Katayama, M. Usuda, J. Mater. Process. Technol. 50 (1995) 226– 237. F. Ozturk, D. Lee, J. Mater. Process. Technol. 147 (2004) 397–404. M.P. Sklad, J. Mater. Process. Technol. 145 (2004) 377–384. George E Dieter, Workability Testing Techniques, American Society for Metals, 1987. A.G. Atkins, Met. Sci. 15 (1981) 81–83. Hibbit, Karlson & Sorensson Inc., ABAQUS, Version 6.3.1, Pawtucket, RI, USA, 2001. R. Hill, Proc. Roy. Soc. Lond. 193A (1948) 197–281.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Experimental and analytical studies on the prediction of forming limit diagrams S. Ahmadi a,*, A.R. Eivani b, A. Akbarzadeh c a b c
Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, United States Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Department of Materials Science and Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 5 August 2008 Accepted 20 August 2008 Available online 2 October 2008 PACS: 83.50.Uv 81.70.-q 07.05.Tp 02.70.-c
a b s t r a c t Metal forming processes are widely used in industrial productions, automobile bodies, food industries, oil refineries, and liquid and gas transmission systems. Analyzing these processes is very important to reduce wastes and optimize the processes. Study of some main factors such as physical and mechanical properties of material and its formability, die geometry, die material, lubrication and pressing speed has been the topic of many research projects. In this paper, forming limit diagrams (FLDs) for LC and ULC steels and the effect of different parameters like the work-hardening exponent, n, and the plastic strain ratio, r, on these diagrams have been evaluated and simulated using ABAQUS/Standard. In this case, Hill’s quadratic anisotropy function is assumed to be the yield function and the Atkins criterion is used as the failure criterion. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Forming limit diagram FLD Forming limits Finite element method ABAQUS Hill’s quadratic yield function
1. Introduction A forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metals during forming processes such as deep drawing and stretch forming. FLDs are plots of the maximum major principal strains which can be sustained by sheet materials prior to the onset of localized necking. By using the FLD, the strains leaded to material failure through different strain paths can be predicted, so it is considered as an important tool in die design as well as in the optimization and problem corrections in the manufacturing processes. Instead of the experimental implementation of FLDs many researchers have conducted a search for an easier FLD construction. Hecker [1], for instance, proposed an experimental technique which enables one to draw the entire diagram using fewer tests by utilizing a hemispherical punch with different widths of the sheet samples and different types of lubricants. Although, Hecker’s model was a significant improvement to FLD construction, the methodology employed was of great complexity. Tadros and Mellor [2], Gronostajski and Dolny [3] and Raghavan [4], on the other hand, developed techniques to eliminate the tool/material friction effects * Corresponding author. Tel.: +1 801 422 2059. E-mail address: [email protected] (S. Ahmadi). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.08.008
and to keep the blank surface flat. Sing and Rao [5] created a method to determine the FLD through the parameters obtained by a conventional tensile test, which appears to be very effective and less expensive than the previous methods. Recently, several researchers have attempted to predict the FLDs using numerical models. Clift et al. [6] used the ductile fracture criterion to predict the forming limits. Based on these criteria, the occurrence of ductile fracture is estimated using the macroscopic stress and strain that occur during deformation. Takuda et al. [7], Yoshida et al. [8] and Ozturk and Lee [9] also predicted the FLDs by using the ductile fracture criteria. Yoshida et al. simulated the hemispherical punch-stretching using an elasto-plastic three-dimensional finite element model, and Ozturk and Lee simulated the out-of-plane (dome) formability test using ABAQUS/Standard to obtain stress and strain values. In this work, ABAQUS/ Standard software is used to predict the FLD and then this model is compared with the experimental data. 2. Experimental work Two low carbon steel sheets (A and B) and an ultra low carbon steel sheet (C) were utilized for experiments. Tensile tests were carried out using specimens machined in accordance with ASTM standard E8M specifications. The tests were conducted along three
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257 Table 1 Mechanical properties of three sample sheets A–C Sample sheet
r-Value
n-Value
YS (MPa)
UTS (MPa)
A B C
1.68 1.63 1.91
0.246 0.222 0.250
179 168 147
303.9 303.2 287.3
1253
length of the blanks was 200 mm and the width was varied between 25 and 200 mm in the steps of 25 mm. For each blank width, at least four to five specimens were tested to get the maximum number of data points. The circles on the sheet samples became ellipses after deformation, falling into safe, necked, and failed zones. The principal strains in the plane of the sheet are expressed using either the traditional engineering strain or the true strain measures. The engineering strain measure provides a percentage of the length change of the principal directions, a and b, of the ellipse with reference to the initial diameter, d0, of the circle:
a d0 100; d0 b d0 100: eminor ð%Þ ¼ d0
emajor ð%Þ ¼
ð2Þ ð3Þ
Finally, FLDs were drawn by deliminating the safe limiting strains from the unsafe zone containing the necked and fractured ellipses. The accuracy of FLDs lies well within a band of ±2% in the engineering strain values, Fig. 2. 3. Numerical work In the numerical work, the Atkins criterion [12] was used as fracture criterion which is formulated as below in Eqs. (4) and (5):
Z
ef
1 þ ð1=2LÞ ; 1 c rh 1 de1 c¼ ; ; L¼ 3:1YS de2 C¼
ð4Þ
0
Fig. 1. A schematic view of the die and the hydraulic press.
directions, with the tensile axis being parallel (0°), diagonal (45°) and perpendicular (90°) to the rolling direction of the sheet. A constant cross-head speed of 0.1 mm min1 was employed in all cases. Three samples were tested for each direction and average values were reported. The standard tensile properties namely, 0.2% yield stress (YS), ultimate tensile stress (UTS), strain-hardening exponent (n) and the plastic strain ratio (r) were determined from the load–elongation data obtained from these tests and shown in Table 1. The normal anisotropy r was calculated using Eq. (1):
r ¼
r 0 þ 2r 45 þ r 90 : 4
ð1Þ
2.1. Measurement of experimental forming limit diagrams Empirical FLDs evaluated using Hecker’s simplified technique [1]. In this method, the experimental procedure mainly involves three stages: grid marking the sheet specimens, punch-stretching the grid-marked samples to failure or onset of localized necking, and measurement of strains. Grid marking on the sheet samples was done using the circle grid analysis (CGA) technique [10]. According to this technique, the grid pattern (circles with 5 mm diameter) was etched on the samples using an electro-chemical etching equipment. Subsequently, punch-stretching experiments were carried out on a double-action hydraulic press. A schematic view of the die and the press is shown in Fig. 1. The sheet samples were subjected to different states of strain, i.e. the tension–tension zone, plane strain and the tension-compression zones by varying the width of the samples [11]. The
f ðrÞ ¼
ð5Þ
where rh is the hydrostatic stress. This ductile fracture criterion requires at least one destructive test for calibration. A tensile test was conducted as a destructive test to calibrate and to attain the C-constant for this criterion. After achieving the value of the C-constant, this fracture criterion was used in all analyses. The analyses were performed in ABAQUS/Standard [13] assuming static conditions and employing shell elements. A simplified modeling approach was used. The die and the punch were modeled as analytical rigid surfaces and the workpiece as a deformable body. The material was modeled as elastic–plastic where the elasticity was taken to be isotropic and the plasticity was assumed as both isotropic and anisotropic conditions. Material properties such as the Young’s modulus (E), the Poisson’s ratio (c), anisotropy coefficients and stress–strain curve statistics were applied to the workpiece in the property module. The S4R element was used in the mesh module, and the small-sliding contact was used in the die/blank and punch/blank interfaces. The friction effect was introduced to the model using the frictionless, lubricated, and dry conditions. This analysis was simulated in three steps. In the first step, the upper die moves down vertically and deforms the sheet metal sample into the drawbead. The hemispherical punch and lower die stay fixed during this step. The punch, bottom die, and upper die remain fixed in the second step, and a 100 kN force is applied to the blank from the upper die. During the third step, while the boundary condition from the second step is still in effect, the punch starts to move up until the desired displacements are achieved. The yield criterion of Hill [14] was adopted and applied as in Eq. (6):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðr22 r33 Þ2 þ Gðr33 r11 Þ2 þ Hðr11 r22 Þ2 þ 2Lr223 þ 2M r231 þ 2Nr212 ;
ð6Þ
1254
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
Fig. 2. FLDs, obtained from experimental work for sample sheets A–C.
where rij denotes the stress components and F, G, H, L, M and N are the material constants obtained from tensile tests. These constants
can be expressed in terms of six yield stress ratios R11, R22, R33, R12, R13 and R23, as presented in Eqs. (7)–(10):
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
!
1 1 1 1 ; þ 2 R222 R233 R211 ! 1 1 1 1 G¼ ; þ 2 R233 R211 R222 ! 1 1 1 1 H¼ ; þ 2 R211 R222 R233 F¼
L¼
3 2R223
;
M¼
3 2R231
;
L¼
ð7Þ
sample sheet B has the lowest level of strain among the forming level curves.
ð8Þ
4.2. Comparison between theoretical and experimental FLDs
ð9Þ 3 2R212
ð10Þ
:
The yield stress ratios can be rewritten as the function of the plastic strain ratios, Eqs. (11) and (12):
R11 R33
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 90 ðr 0 þ 1Þ ¼ R13 ¼ R23 ¼ 1; R22 ¼ ; r 0 ðr 90 þ 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 90 ðr0 þ 1Þ 3r90 ðr 0 þ 1Þ ¼ ; R12 ¼ ; r0 þ r 90 ð2r 45 þ 1Þðr0 þ r 90 Þ
1255
ð11Þ ð12Þ
where r0, r45 and r90 are the plastic strain ratios in different directions relative to the rolling direction.
4. Results and discussion
The comparison between experimental and theoretical FLDs as calculated and simulated by ABAQUS software is shown in Fig. 4, where Exp, Iso and Aniso stand for experimental and modeling data for isotropic and anisotropic conditions, respectively. On the right-hand side of the FLD, the isotropic curve is the most consistent with the experimental results. However, on the lefthand side, the anisotropic curve is the best fit with the experimental results. This might be due to the effect of the r-value on the FLDs. The plastic strain ratio does not have a large effect on the positive minor strain region, but in the negative region by increasing the r-value, the slope of the curve decreases and eventually the curve approaches the experimental curve. It is also observed that in many regions of the FLDs, the anisotropic simulated curve has a lower value of the major strain than the experimental curve, so the use of anisotropic condition can be more accurate than the isotropic. 4.3. Comparison between the theoretical and the experimental FLD0values
4.1. The effect of n- and r-value on the forming limit curve In general, the strain-hardening exponent and plastic strain ratio of a material are considered to be the most important parameters which influence the FLD. Fig. 3 shows the influences of n- and r-values, respectively, on the experimental FLDs. As shown, sample sheet C has the highest limiting strains. By increasing the n- and rvalues of steel sheets, the forming limit curves are shifted upward. This is because a higher value of the strain-hardening exponent generally delays the onset of instability until the higher strain value, is reached. This delay enhances the limiting strain. The plastic strain ratio, r, also has the great effect on the limiting strain. By increasing the value of this parameter, the formability of the sheet is increased. Therefore, sample sheet C has the highest level and
The importance of the FLD0-value lies in the fact that fractures or cracks appearing in cold-formed parts under press conditions often occur at this strain state. Fig. 5 compares the theoretical and experimental forming limit strains in the plane-strain state, FLD0. The forming-limit strains predicted by the isotropic conditions are in good agreement with the experimental results for all steel sheets; however, the calculated values of FLD0 from the anisotropic model are lower than the experimental results. It is also observed that sample sheet C has the highest value of FLD0, which is due to its physical and mechanical properties, such as its n- and r- values. According to Figs. 4 and 5, the difference between experimental and simulated results using anisotropic conditions on the left and
Fig. 3. FLDs of all sample sheets (using experimental data).
1256
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
Fig. 4. Empirical and numerical FLDs. Exp stands for experimental FLDS; Iso or Aniso stands for theoretical FLDs calculated using ABACUS with either fully isotropic or fully anisotropic plasticity model
right sides of the forming limit curves is not significantly large. In fact, using the anisotropic conditions in simulation yields the bet-
ter consistency, while using the isotropic conditions results in values closer to the experimental FLD0.
S. Ahmadi et al. / Computational Materials Science 44 (2009) 1252–1257
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– The value of plastic strain ratio has a large effect on the level of FLC on the left side. By increasing the value of r, the slope of the FLC on the left side is shifted down. – On the right side of the FLD, any variation of r-value has no significant effect on the shape of the diagram. – Higher values of n and r allow for higher values of the final strains and, as a results, the ultimate elongation increases. This also allows the point designating fracture to move to higher values of the major strain. References [1] [2] [3] [4] [5] [6] Fig. 5. Comparison between the calculated and the experimental FLD0-values for the A–C sample sheets.
5. Conclusion Based upon the experimental and simulated results the following conclusions are drawn:
[7] [8] [9] [10] [11] [12] [13] [14]
– By increasing the values of n and r for the sample sheets, formability, deep drawability, and the value of FLD0 are increased.
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