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A methodology for evaluating sheet formability combining the tensile test with the M–K model
Materials Science and Engineering A 528 (2010) 480–485
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
A methodology for evaluating sheet formability combining the tensile test with the M–K model Cunsheng Zhang a,∗ , Lionel Leotoing b , Guoqun Zhao a , Dominique Guines b , Eric Ragneau b a b
Shandong University, Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Jinan 250061, Shandong Province, PR China Université Européenne de Bretagne, INSA-LGCGM - EA 3913, 20, avenue des Buttes de Coësmes 35043, Rennes Cédex, France
a r t i c l e
i n f o
Article history: Received 7 June 2010 Accepted 2 September 2010
Keywords: Forming limit curves (FLCs) Marciniak and Kuczynski (M–K) model Marciniak test Digital image correlation (DIC)
a b s t r a c t This paper proposed an approach for evaluating the sheet formability by combining the tensile test with the finite element Marciniak and Kuczynski (M–K) model. Firstly, the tensile test with a notched specimen was carried out to identify an appropriate constitutive law for an AA5086 sheet. A modified Ludwick’s law was used to describe its forming behavior. A technique of digital image correlation associated with a highspeed camera was applied to evaluate specimen’s surface strains during the experiments. The inverse analysis was performed to identify the parameter values in the constitutive law. The initial geometrical imperfection factor in the M–K model was determined. Then by using the commercial finite element software ABAQUS, the M–K model was simulated to evaluate numerically the sheet formability of the alloy. By means of a user-defined FORTRAN subroutine UHARD, the constitutive law was implanted into ABAQUS. Different strain states were obtained by changing displacement ratios and forming limit curves (FLCs) of the sheet were determined too. Finally, an experimental procedure based on the modified Marciniak test was carried out and the FLCs were obtained experimentally. The comparison between the numerical and experimental results showed that the approach developed in this paper could give an appropriate prediction of FLCs. © 2010 Elsevier B.V. All rights reserved.
1. Introduction For sheet metal forming, forming limit curves (FLCs) are an efficient diagnostic tool for evaluating sheet formability and many methods have been developed to determine the FLCs [1–3]. However, the determination of FLCs is a complex task, there is no well-established experimental or numerical procedure for its determination [4]. For experimental predictions of FLCs, two main kinds of forming tests have been developed, the so-called out-of-plane stretching (e.g. the Nakazima test [5], the Hecker test [6]) and the in-plane stretching (e.g. the Marciniak test [7]). During out-of-plane stretching, as illustrated in Fig. 1(a), the blank is deformed under triaxial stress while during in-plane stretching, the stress perpendicular to the sheet surface is small compared to the stresses in the plane and could be neglected, hence the sheet is under near plane stress conditions in the central part (see Fig. 1(b)). Experimental method is a basic way to obtain FLCs of sheet metals. However, there is no precise standard to detect the onset of localized necking and construct more stable and reproductible FLCs, in addition to ISO 12004 which is generally considered as too fuzzy [8]. Moreover, it is a very time consuming procedure to establish
∗ Corresponding author. Tel.: +86 53181696577; fax: +86 53188392811. E-mail address: [email protected] (C. Zhang). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.09.001
FLCs and the scatter in experimental data for a given sheet is usually large [9]. Especially, little research on sheet formability has been reported at high strain rates due to the difficulty in carrying out the experiments. As a result, significant efforts have been made on developing more analytical or numerical models for construction of FLCs. From the viewpoint of numerical research, due to the developments in the methods of modeling and simulation as well as in computational facilities, numerical predictions of FLCs have become more attractive, and FE method has been selected to simulate the necking process. Using the LDH (Limiting Dome Height) test, Narasimhan [10] has predicted the onset of localized necking by the thickness strain gradient across neighboring regions. When the thickness gradient in adjoining regions was 0.92, localized necking was assumed to occur. The predicted and experimental FLCs of a steel sheet are in a good agreement. Based on the Marciniak test, Petek et al. [1] put forward a new method by evaluating the thickness strain as a function of time as well as the first and second time derivative of the thickness strain. They proposed that the maximum of the second temporal derivative of the thickness strain corresponds to the onset of localized necking. Volk [11] identified the onset of localized necking by experimental and numerical methods. With calculated strain rates, the identification was carried out with the two following main effects:
C. Zhang et al. / Materials Science and Engineering A 528 (2010) 480–485
481
Fig. 2. Geometry and dimension of a notched specimen (length in mm).
2.1. Experimental tensile tests Fig. 1. Schematic layouts of out-of-plane and in-plane stretching.
increase of points number with high strain rate (in the localization region) and decrease of the strain rate outside the localization bands. In addition, the Marciniak and Kuczynski model (known as the M–K model) is a widely used analytical one which can help to reduce the experimental effort of formability characterization as well as to predict FLCs of sheet metals [12]. So far, the M–K model has also undergone great improvement. However, for a complex constitutive law, the analytical M–K model does not work well because the inherent system equations cannot be easily resolved. In contrast, with numerical methods, above limits could be overcome by implanting any complex constitutive law into FE code [13]. Therefore, the numerical M–K model is also simulated to construct the FLCs of sheet metals. Banabic et al. [14] determined FLCs by simulated the M–K geometrical model with an inclined groove. For the left-hand side of FLCs, the imperfection orientation was taken according to Hill’s zero extension assumption. A good correlation between predicted and experimental results has been obtained for right-side hand of FLCs, while in left-hand area the predicted FLCs underestimated the experimental ones. Recently, Zhang [13] simulated the M–K model with ABAQUS by means of the implementation of different hardening laws (Swift’s law, Johnson–Cook’s law and Ludwick’s law), it is found that hardening laws influence greatly the determination of FLCs. However, the initial geometrical imperfection factor f0 in the M–K model is an uncertain one. Generally, its value is determined by making the best fit between the numerical and experimental results. Moreover, an appropriate constitutive law is a key to obtaining the practical prediction of FLCs. Hence, the paper begins with the tensile test for an AA5086 sheet. An inverse analysis is applied to identify flow behaviors of this aluminum sheet and the corresponding parameters in the constitutive law are determined. Furthermore, the initial imperfection factor f0 in the M–K model is determined for this given sheet, and the model is simulated to construct the FLCs of this sheet. Finally, an apparatus based on the Marciniak test is developed to experimentally construct the FLCs. A quasi-static experimental procedure is carried out to validate the proposed numerical approach.
For the tensile test in this work, a specimen is specially designed with a notch which may result in fast necking initiation and facilitate the registration of a series of consecutive images of the localized region. The geometry and dimension of the specimens is shown in Fig. 2. The thickness of this sheet is 2.0 mm. To capture the consecutive images during the experiments, a Fastcam ultima APX-RS digital CMOS camera associated with a macro lens is used. The commercial digital imaging program CORRELA2006, developed by LMS at the University of Poitiers, is employed to perform correlation analysis in this work. The DIC program produces the information of the surface strains on the specimen. To find a representative procedure for analyzing the necking progress, the time-sequence of equivalent plastic strain profiles along the longitudinal axis of specimen at different instants of time is displayed in Fig. 3. From the figure, it is clearly observed that at early stage of the forming process, a quasi-homogeneous deformation is distributed along the longitudinal axis of specimen. Then the subsequent development of deformation is concentrated to the central part of the specimen notch, and plastic strain is localized to a smaller and smaller region. When the strain increment in the localized zone exceeds by 7 times that in non-localized zone, a critical moment could be determined corresponding to the onset of localized necking and the principal strains (−0.067, 0.30) in the localized zone are retained as the limit strains to form one point of FLCs of this sheet. A strain path of −0.223 for this tensile test could be calculated by the ratio of minor strain and major strain. Because of its irregular specimen geometry in the present tensile test, the strain path is clearly different from that of a general uniaxial tension, which is about −0.5.
2. Tensile test In this part, the tensile test is carried out on a computercontrolled servo-hydraulic testing machine. A technique of digital image correlation (DIC) associated with a high-speed camera is applied to evaluate surface strains and a complete procedure is built to detect the onset of localized necking during the experiments. Then the use of inverse analysis is performed to identify a constitutive law for this aluminum alloy sheet.
Fig. 3. Strain profiles along the longitudinal axis of specimen.
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test, the initial geometrical imperfection factor f0 in the M–K model is determined, and the constitutive law identified by above inverse analysis is implanted into ABAQUS with its subroutine UHARD. 3.1. Construction of the M–K model with ABAQUS
Fig. 4. Experimental and identified curve of force versus displacement.
2.2. Identification of constitutive model By use of the tensile test with a notched specimen, it is difficult to identify the material’s flow behavior with conventional analytical methods due to the irregular specimen cross-section [15]. Nowadays, the inverse methods are intensively used to adjust the material parameters for more and more complex constitutive laws or irregular specimen geometry. The basic concept of an inverse analysis for parameter identification is to find out a set of unknown material parameters in constitutive equation thanks to a FE simulation of the test. To describe its elasto-plastic behavior for this given sheet, a general constitutive model ¯ = 0 + K ε¯ n
(1)
is proposed to be identified. Here, K and n are material parameters. The detail identification procedure can be found in Diot’s paper [16]. For the uniaxial tensile test under 10 mm/s at 20 ◦ C, the identified results with the inverse analysis are: 0 = 147 MPa, K = 870 MPa, n = 0.335. The comparison between experimental and calculated loads versus displacement is illustrated in Fig. 4. It can be seen that there is a good agreement between the experimental curves and the one identified by inverse analysis. To compare with experiments, the identified data will be used for the following numerical procedure. 3. M–K model In this part, the M–K model is simulated with the commercial finite element code ABAQUS to numerically construct the FLCs of the studied sheet. With experimental results obtained by the tensile
Similarly as the analytical M–K model, an initial defect in the sheet is characterized by two different zone thicknesses in the FE model. Here, it is assumed that the imperfection zone (zone b) is perpendicular to the principal axis-1 [17]. Due to symmetry, only one half of the entire model in the thickness is considered for this FE analysis, as shown in Fig. 5(a). The sheet is meshed by hexahedral elements. To compare deformation states in the two different zones, two different reference elements are required. One of the elements is placed in zone a (Element A), while the other is in zone b (Element B). Essential boundary conditions are imposed by displacement constraints on certain surfaces of the model (see Fig. 5(b)). The elasticity of this material is defined with the Young’s modulus of 70500 MPa and the Poisson’s ratio of 0.33. By means of a user-defined subroutine UHARD, the constitutive model identified by above inverse analysis is implanted into ABAQUS. Because of the relatively smaller thickness in zone b, the equivalent plastic strain in zone b is greater than that in zone a. The maximum values occur at the center of the model, while farther from the center, the strain reduces gradually. Fig. 6(a) clearly shows the evolutions of the equivalent plastic strain of Element A and Element B. As observed from this figure, the strain histories from the two elements are relatively similar until they diverge at approximately t = 8 s. At this stage, the equivalent plastic strain in Element B rises rapidly while that in Element A shows a relative saturation. When the equivalent plastic strain increment in Element B exceeds by 7 times that in Element A (corresponding to t0 in Fig. 6(b)), localized necking is assumed to occur and the final major and minor strains of Element B calculated by linear interpolation are noted as the limit strains for construction of FLCs. 3.2. Identification of the initial imperfection factor f0 Shallow initial grooves are sufficient to cause localization in the M–K model [18]. Generally, the value of the initial imperfection factor f0 is chosen to make the best fit between the numerical and the experimental results, and to some extent, this value denotes the level of sheet formability. Hence, to characterize the practical formability of a given sheet, an appropriate value of f0 should be identified. Different strain states could be covered by imposing different ratios of displacements in the 1 and 2 directions as shown in
Fig. 5. FE M–K model and corresponding boundary conditions in ABAQUS. (a) FE model in ABAQUS. (b) Boundary condition in the M–K model.
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Fig. 6. Evolutions of the equivalent plastic strain and its increment ratio of two elements selected. (a) Evolution of the equivalent plastic strain of two elements. (b) Evolution of the equivalent plastic strain increment ratio.
4. Experimental validation
Fig. 7. FLCs obtained with the FE M–K model with different imperfection factors.
The necessity to verify the analytical and numerical predictions leads to a further study of the precision and efficiency of definition of FLCs with experiments [1]. During the experiments, one or more high-speed cameras are used to view the sample surface and take consecutive images during experiments. In contrast to the Nakazima test, the investigated region in the Marciniak test remains flat during the experiment, as illustrated in Fig. 1. Because this case is a 2D-application, strains can be measured with only one camera. This is an important advantage of this method and an important reason for us to choose the Marciniak test to determine FLCs in this work. A quasi-static experimental procedure is carried out to experimentally construct FLCs of the AA5086 sheet and compare with above numerical results.
4.1. Experimental preparations Fig. 5(b). Here, the displacement u in direction 1 is fixed at 60 mm; by varying the displacement v in direction 2, strain state changes. To choose an appropriate imperfection factor, the M–K model with different imperfection factor f0 (0.92, 0.95, 0.98) is simulated and the FLCs are obtained as shown in Fig. 7. Regular shapes of forming limit curve are found from this figure. In comparison with f0 = 0.92, there are increases of 45%, 15%, and 38% in major strain for f0 = 0.98 under uniaxial tension, plane strain and equi-biaxial stretching conditions, respectively. Moreover, the level of critical strains for the M–K model with f0 of 0.92 approaches to that of the experimental tensile test (−0.067, 0.30). Therefore, for this sheet formability, the initial imperfection factor with the value of 0.92 is suitable.
In this work, a new experimental apparatus based on the Marciniak test is developed, which includes a reverse experimental setup and an image acquisition system, as shown in Fig. 8(a). In this reverse setup, a bell jar is designed to connect the die with the crosshead of tensile machine. The purpose of this design is to prevent or minimize the vibrations and ensure the rigidity of the experimental setup throughout static and dynamic tests. On the bell jar, there are two small windows, one is for installing the optic mirror and another permits that the reflected light from the specimen surface goes through then focuses on the mirror. During the experiments, the die with the clamped specimen moves downwards and the fixed punch stretches the sheet. With this reverse
Fig. 8. Reverse experimental setup based on the Marciniak test. (a) 3D model of experimental setup. (b) Experimental apparatus.
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Table 1 Specimen dimensions used for the experiments. W (mm)
10
20
30
40
R (mm) Rc (mm) Rm (mm) Re (mm)
45
48
50
52
55
58
60
80
100
50 70 26.5 10
experimental setup, in which the specimen is formed over the punch until fracture appears on the specimen surface, the distance between the mirror and the specimen remains nearly constant throughout the test. This allows the camera to be focused on the specimen surface before the test and take sequential pictures. In order to cover different strain states, ranging from uniaxial through plane strain to equi-biaxial stretching, different specimen geometries are used. In this work, all test samples are shaped by cutting strips of different widths W in a circular flange (Table 1), according to Fig. 9(a). To assure the occurrence of the maximal strains (to trigger localization) on the central part of the blank, the specimens are designed with a reduced central thickness (0.8 mm) compared to the thickness of the sheet (1.5 mm) and the clamping part with a thickness of 2.0 mm. Before the test, the experimental specimen is painted with a random speckle pattern for DIC analysis. The surface which has an applied speckle pattern should be away from the punch contact surface. Finally, the painted specimen is clamped between the blankholder and the die. The bell jar, together with the die, moves down at a crosshead traveling speed of 500 mm/min. Here, a quasi-static procedure at 20 ◦ C is carried out to test the experimental setup and compare with above numerical procedure. The comparison between an undeformed and crack specimen in the experiments is shown in Fig. 10. With the DIC program CORRELA2006, a correlation analysis is carried out on the crack specimen and surface strains are calculated.
Fig. 10. Specimens used in the experiments. (a) Undeformed specimen. (b) Deformed specimen.
4.2. Experimental results As an illustrated example, the specimen of Fig. 10(b) is chosen. In order to carry out a study about the plastic instability associated with the onset and progress of necking in the sheet, two points are chosen. Point B is in (or near) the zone where the rupture occurs while Point A is outside of the zone, as shown in Fig. 11. The evolution of the major strains of Point A and Point B is plotted in Fig. 12. A similar evolution in these two curves is observed up to a moment of about t = 0.9 s, after which they diverge rapidly. Here, the criterion widely used in the M–K model is chosen to pre-
Fig. 9. Specimens specially designed in the experiments. (a) Specimen geometry designed for the experiments. (b) Blank with non-uniform thickness in the experiments.
Fig. 11. Deformed specimen with rupture.
dict the onset of localized necking. When the equivalent plastic strain increment ratio between Point A and Point B attains 7, the onset of localized necking is assumed to occur and the corresponding major and minor strains (−0.0036, 0.20) of Point B are retained as a point on the FLC. To provide sufficient and reliable data, for each specimen, at least two tests are performed under identical conditions.
Fig. 12. Evolution of major strains for the two Point A and Point B.
C. Zhang et al. / Materials Science and Engineering A 528 (2010) 480–485
Fig. 13. Comparison of numerical and experimental FLCs.
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strain rates by only carrying out the dynamic tensile tests and seeking the appropriate constitutive laws. The same approach will be used to investigate the influence of strain rate on sheet formability. This validation of the approach is in progress by means of a dynamic Marciniak test. • A modified Marciniak test has been developed to experimentally investigate the sheet formability and validate the numerical procedure proposed in this work. To prevent or minimize the vibrations and ensure the rigidity of the experimental setup throughout static and dynamic tests, a specially designed structure with the bell jar is used. With a quasi-static experimental procedure, the comparison with numerical results shows that this experimental setup works well to investigate sheet formability and in turn it indicate that numerical procedure proposed in this work can give a good prediction of FLCs. Acknowledgements
4.3. Comparison of numerical and experimental FLCs Fig. 13 shows the FLCs of the AA5086 sheet obtained by the numerical M–K model and experimental procedure, respectively. Compared with the M–K model, the strain states with the experimental procedure are located in a narrower range, especially at the left-hand side on the FLCs. This phenomenon maybe result from a lack of lubricants during the experiments, which influences the strain path, but not the level of critical limit strains. This first comparative evaluations show a good correlation between limit strains obtained numerically and experimentally. Although some discrepancies due to the variability encountered in forming sheet metal, it should be noted that numerical method could give a reasonable prediction of FLCs. 5. Conclusion This work proposed an approach to construct FLCs of sheet metals combining the tensile test and the FE M–K model. Compared to experimental results obtained by carrying out a modified Marciniak test procedure, the proposed approach could be used to practically predict FLCs of sheet metals. The conclusions are drawn as follows: • With the FE M–K model which may result in fast necking initiation, it is easy to determine the onset of localized necking and numerically construct the FLCs. Especially, the FE M–K model overcomes the limits encountered with the analytical one where the complex constitutive laws cannot be used. • The procedure combining the tensile test with the FE M–K model can be used to predict FLCs of sheet metals. A quasi-static tensile test is conducted to identify an appropriate constitutive law and the initial imperfection factor in the M–K model. By simulating the M–K model, the FLCs of this sheet are determined. More important, it can be used for the determination of FLCs at high
This work was financially supported by Encouragement Foundation for Young Scholars of Shandong Province (BS2010ZZ004). References [1] A. Petek, T. Pepelnjak, K. Kuzman, Journal of Mechanical Engineering 51 (2005) 330–345. [2] H.B. Campos, M.C. Butuc, J.J. Gracio, Journal of Materials Processing Technology 179 (2006) 56–60. [3] P. Eyckens, A. Van Bael, P. Van Houtte, International Journal of Plasticity 25 (2009) 2249–2268. [4] F. Ozturk, D. Lee, Journal of Materials Processing Technology 147 (2004) 397–404. [5] K. Nakazima, T. Kikuma, K. Asuka, Study on the Formability of Steel Sheet, Technical Report, Yawata, 1971. [6] A.K. Ghosh, S.S. Hecker, Metallurgical Transactions 2161 (1974) 5–2164. [7] Z. Marciniak, K. Kuczynski, T. Pokora, International Journal of Mechanical Sciences 15 (1973) 789–805. [8] A. Col, Numerical and Experimental Methods of FLCs Prediction for the Processes of Stamping and Hydroforming of Tubes, Technical Report, 2006. [9] P.D. Wu, S.R. MacEwen, D.J. Lloyd, K.W. Neale, Modelling and Simulation in Materials Science and Engineering 12 (2004) 511–527. [10] K. Narasimhan, in: S. Ghosh, J.M. Castro, J.K. Lee (Eds.), Materials Processing and Design: Modeling, Simulation and Applications. NUMIFORM 2004 Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes, vol. 2, American Institute of Physics, 2004, pp. 850–855. [11] W. Volk, Proceedings of the FLC-Zurich 06, Zurich, Switzerland, 2006. [12] H. Aretz, International Journal of Plasticity 23 (2007) 798–840. [13] C.S. Zhang, Investigations on the Effect of Strain Rate Sensitivity on Formability of Aluminium Alloy Sheets, PhD Thesis, INSA de Rennes, France, 9, 2008. [14] D. Banabic, O. Cazacu, L. Paraianu, P. Jurco, Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Process, vol. 778, MI, USA, 2005, pp. 466–471. [15] B. Davoodi, A. Gavrus, E. Ragneau, Measurement Science and Technology 16 (2005) 2101–2108. [16] C.S. Zhang, L. Leotoing, D. Guines, E. Ragneau, Materials Science and Engineering: A 527 (2010) 967–972. [17] C.S. Zhang, L. Leotoing, D. Guines, E. Ragneau, Proceedings of Numisheet’08, Interlaken, Swiss, 2008, pp. 229–233. [18] W.F. Hosford, Mechanical Behavior of Materials, Cambridge University Press, 2005.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
A methodology for evaluating sheet formability combining the tensile test with the M–K model Cunsheng Zhang a,∗ , Lionel Leotoing b , Guoqun Zhao a , Dominique Guines b , Eric Ragneau b a b
Shandong University, Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Jinan 250061, Shandong Province, PR China Université Européenne de Bretagne, INSA-LGCGM - EA 3913, 20, avenue des Buttes de Coësmes 35043, Rennes Cédex, France
a r t i c l e
i n f o
Article history: Received 7 June 2010 Accepted 2 September 2010
Keywords: Forming limit curves (FLCs) Marciniak and Kuczynski (M–K) model Marciniak test Digital image correlation (DIC)
a b s t r a c t This paper proposed an approach for evaluating the sheet formability by combining the tensile test with the finite element Marciniak and Kuczynski (M–K) model. Firstly, the tensile test with a notched specimen was carried out to identify an appropriate constitutive law for an AA5086 sheet. A modified Ludwick’s law was used to describe its forming behavior. A technique of digital image correlation associated with a highspeed camera was applied to evaluate specimen’s surface strains during the experiments. The inverse analysis was performed to identify the parameter values in the constitutive law. The initial geometrical imperfection factor in the M–K model was determined. Then by using the commercial finite element software ABAQUS, the M–K model was simulated to evaluate numerically the sheet formability of the alloy. By means of a user-defined FORTRAN subroutine UHARD, the constitutive law was implanted into ABAQUS. Different strain states were obtained by changing displacement ratios and forming limit curves (FLCs) of the sheet were determined too. Finally, an experimental procedure based on the modified Marciniak test was carried out and the FLCs were obtained experimentally. The comparison between the numerical and experimental results showed that the approach developed in this paper could give an appropriate prediction of FLCs. © 2010 Elsevier B.V. All rights reserved.
1. Introduction For sheet metal forming, forming limit curves (FLCs) are an efficient diagnostic tool for evaluating sheet formability and many methods have been developed to determine the FLCs [1–3]. However, the determination of FLCs is a complex task, there is no well-established experimental or numerical procedure for its determination [4]. For experimental predictions of FLCs, two main kinds of forming tests have been developed, the so-called out-of-plane stretching (e.g. the Nakazima test [5], the Hecker test [6]) and the in-plane stretching (e.g. the Marciniak test [7]). During out-of-plane stretching, as illustrated in Fig. 1(a), the blank is deformed under triaxial stress while during in-plane stretching, the stress perpendicular to the sheet surface is small compared to the stresses in the plane and could be neglected, hence the sheet is under near plane stress conditions in the central part (see Fig. 1(b)). Experimental method is a basic way to obtain FLCs of sheet metals. However, there is no precise standard to detect the onset of localized necking and construct more stable and reproductible FLCs, in addition to ISO 12004 which is generally considered as too fuzzy [8]. Moreover, it is a very time consuming procedure to establish
∗ Corresponding author. Tel.: +86 53181696577; fax: +86 53188392811. E-mail address: [email protected] (C. Zhang). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.09.001
FLCs and the scatter in experimental data for a given sheet is usually large [9]. Especially, little research on sheet formability has been reported at high strain rates due to the difficulty in carrying out the experiments. As a result, significant efforts have been made on developing more analytical or numerical models for construction of FLCs. From the viewpoint of numerical research, due to the developments in the methods of modeling and simulation as well as in computational facilities, numerical predictions of FLCs have become more attractive, and FE method has been selected to simulate the necking process. Using the LDH (Limiting Dome Height) test, Narasimhan [10] has predicted the onset of localized necking by the thickness strain gradient across neighboring regions. When the thickness gradient in adjoining regions was 0.92, localized necking was assumed to occur. The predicted and experimental FLCs of a steel sheet are in a good agreement. Based on the Marciniak test, Petek et al. [1] put forward a new method by evaluating the thickness strain as a function of time as well as the first and second time derivative of the thickness strain. They proposed that the maximum of the second temporal derivative of the thickness strain corresponds to the onset of localized necking. Volk [11] identified the onset of localized necking by experimental and numerical methods. With calculated strain rates, the identification was carried out with the two following main effects:
C. Zhang et al. / Materials Science and Engineering A 528 (2010) 480–485
481
Fig. 2. Geometry and dimension of a notched specimen (length in mm).
2.1. Experimental tensile tests Fig. 1. Schematic layouts of out-of-plane and in-plane stretching.
increase of points number with high strain rate (in the localization region) and decrease of the strain rate outside the localization bands. In addition, the Marciniak and Kuczynski model (known as the M–K model) is a widely used analytical one which can help to reduce the experimental effort of formability characterization as well as to predict FLCs of sheet metals [12]. So far, the M–K model has also undergone great improvement. However, for a complex constitutive law, the analytical M–K model does not work well because the inherent system equations cannot be easily resolved. In contrast, with numerical methods, above limits could be overcome by implanting any complex constitutive law into FE code [13]. Therefore, the numerical M–K model is also simulated to construct the FLCs of sheet metals. Banabic et al. [14] determined FLCs by simulated the M–K geometrical model with an inclined groove. For the left-hand side of FLCs, the imperfection orientation was taken according to Hill’s zero extension assumption. A good correlation between predicted and experimental results has been obtained for right-side hand of FLCs, while in left-hand area the predicted FLCs underestimated the experimental ones. Recently, Zhang [13] simulated the M–K model with ABAQUS by means of the implementation of different hardening laws (Swift’s law, Johnson–Cook’s law and Ludwick’s law), it is found that hardening laws influence greatly the determination of FLCs. However, the initial geometrical imperfection factor f0 in the M–K model is an uncertain one. Generally, its value is determined by making the best fit between the numerical and experimental results. Moreover, an appropriate constitutive law is a key to obtaining the practical prediction of FLCs. Hence, the paper begins with the tensile test for an AA5086 sheet. An inverse analysis is applied to identify flow behaviors of this aluminum sheet and the corresponding parameters in the constitutive law are determined. Furthermore, the initial imperfection factor f0 in the M–K model is determined for this given sheet, and the model is simulated to construct the FLCs of this sheet. Finally, an apparatus based on the Marciniak test is developed to experimentally construct the FLCs. A quasi-static experimental procedure is carried out to validate the proposed numerical approach.
For the tensile test in this work, a specimen is specially designed with a notch which may result in fast necking initiation and facilitate the registration of a series of consecutive images of the localized region. The geometry and dimension of the specimens is shown in Fig. 2. The thickness of this sheet is 2.0 mm. To capture the consecutive images during the experiments, a Fastcam ultima APX-RS digital CMOS camera associated with a macro lens is used. The commercial digital imaging program CORRELA2006, developed by LMS at the University of Poitiers, is employed to perform correlation analysis in this work. The DIC program produces the information of the surface strains on the specimen. To find a representative procedure for analyzing the necking progress, the time-sequence of equivalent plastic strain profiles along the longitudinal axis of specimen at different instants of time is displayed in Fig. 3. From the figure, it is clearly observed that at early stage of the forming process, a quasi-homogeneous deformation is distributed along the longitudinal axis of specimen. Then the subsequent development of deformation is concentrated to the central part of the specimen notch, and plastic strain is localized to a smaller and smaller region. When the strain increment in the localized zone exceeds by 7 times that in non-localized zone, a critical moment could be determined corresponding to the onset of localized necking and the principal strains (−0.067, 0.30) in the localized zone are retained as the limit strains to form one point of FLCs of this sheet. A strain path of −0.223 for this tensile test could be calculated by the ratio of minor strain and major strain. Because of its irregular specimen geometry in the present tensile test, the strain path is clearly different from that of a general uniaxial tension, which is about −0.5.
2. Tensile test In this part, the tensile test is carried out on a computercontrolled servo-hydraulic testing machine. A technique of digital image correlation (DIC) associated with a high-speed camera is applied to evaluate surface strains and a complete procedure is built to detect the onset of localized necking during the experiments. Then the use of inverse analysis is performed to identify a constitutive law for this aluminum alloy sheet.
Fig. 3. Strain profiles along the longitudinal axis of specimen.
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test, the initial geometrical imperfection factor f0 in the M–K model is determined, and the constitutive law identified by above inverse analysis is implanted into ABAQUS with its subroutine UHARD. 3.1. Construction of the M–K model with ABAQUS
Fig. 4. Experimental and identified curve of force versus displacement.
2.2. Identification of constitutive model By use of the tensile test with a notched specimen, it is difficult to identify the material’s flow behavior with conventional analytical methods due to the irregular specimen cross-section [15]. Nowadays, the inverse methods are intensively used to adjust the material parameters for more and more complex constitutive laws or irregular specimen geometry. The basic concept of an inverse analysis for parameter identification is to find out a set of unknown material parameters in constitutive equation thanks to a FE simulation of the test. To describe its elasto-plastic behavior for this given sheet, a general constitutive model ¯ = 0 + K ε¯ n
(1)
is proposed to be identified. Here, K and n are material parameters. The detail identification procedure can be found in Diot’s paper [16]. For the uniaxial tensile test under 10 mm/s at 20 ◦ C, the identified results with the inverse analysis are: 0 = 147 MPa, K = 870 MPa, n = 0.335. The comparison between experimental and calculated loads versus displacement is illustrated in Fig. 4. It can be seen that there is a good agreement between the experimental curves and the one identified by inverse analysis. To compare with experiments, the identified data will be used for the following numerical procedure. 3. M–K model In this part, the M–K model is simulated with the commercial finite element code ABAQUS to numerically construct the FLCs of the studied sheet. With experimental results obtained by the tensile
Similarly as the analytical M–K model, an initial defect in the sheet is characterized by two different zone thicknesses in the FE model. Here, it is assumed that the imperfection zone (zone b) is perpendicular to the principal axis-1 [17]. Due to symmetry, only one half of the entire model in the thickness is considered for this FE analysis, as shown in Fig. 5(a). The sheet is meshed by hexahedral elements. To compare deformation states in the two different zones, two different reference elements are required. One of the elements is placed in zone a (Element A), while the other is in zone b (Element B). Essential boundary conditions are imposed by displacement constraints on certain surfaces of the model (see Fig. 5(b)). The elasticity of this material is defined with the Young’s modulus of 70500 MPa and the Poisson’s ratio of 0.33. By means of a user-defined subroutine UHARD, the constitutive model identified by above inverse analysis is implanted into ABAQUS. Because of the relatively smaller thickness in zone b, the equivalent plastic strain in zone b is greater than that in zone a. The maximum values occur at the center of the model, while farther from the center, the strain reduces gradually. Fig. 6(a) clearly shows the evolutions of the equivalent plastic strain of Element A and Element B. As observed from this figure, the strain histories from the two elements are relatively similar until they diverge at approximately t = 8 s. At this stage, the equivalent plastic strain in Element B rises rapidly while that in Element A shows a relative saturation. When the equivalent plastic strain increment in Element B exceeds by 7 times that in Element A (corresponding to t0 in Fig. 6(b)), localized necking is assumed to occur and the final major and minor strains of Element B calculated by linear interpolation are noted as the limit strains for construction of FLCs. 3.2. Identification of the initial imperfection factor f0 Shallow initial grooves are sufficient to cause localization in the M–K model [18]. Generally, the value of the initial imperfection factor f0 is chosen to make the best fit between the numerical and the experimental results, and to some extent, this value denotes the level of sheet formability. Hence, to characterize the practical formability of a given sheet, an appropriate value of f0 should be identified. Different strain states could be covered by imposing different ratios of displacements in the 1 and 2 directions as shown in
Fig. 5. FE M–K model and corresponding boundary conditions in ABAQUS. (a) FE model in ABAQUS. (b) Boundary condition in the M–K model.
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Fig. 6. Evolutions of the equivalent plastic strain and its increment ratio of two elements selected. (a) Evolution of the equivalent plastic strain of two elements. (b) Evolution of the equivalent plastic strain increment ratio.
4. Experimental validation
Fig. 7. FLCs obtained with the FE M–K model with different imperfection factors.
The necessity to verify the analytical and numerical predictions leads to a further study of the precision and efficiency of definition of FLCs with experiments [1]. During the experiments, one or more high-speed cameras are used to view the sample surface and take consecutive images during experiments. In contrast to the Nakazima test, the investigated region in the Marciniak test remains flat during the experiment, as illustrated in Fig. 1. Because this case is a 2D-application, strains can be measured with only one camera. This is an important advantage of this method and an important reason for us to choose the Marciniak test to determine FLCs in this work. A quasi-static experimental procedure is carried out to experimentally construct FLCs of the AA5086 sheet and compare with above numerical results.
4.1. Experimental preparations Fig. 5(b). Here, the displacement u in direction 1 is fixed at 60 mm; by varying the displacement v in direction 2, strain state changes. To choose an appropriate imperfection factor, the M–K model with different imperfection factor f0 (0.92, 0.95, 0.98) is simulated and the FLCs are obtained as shown in Fig. 7. Regular shapes of forming limit curve are found from this figure. In comparison with f0 = 0.92, there are increases of 45%, 15%, and 38% in major strain for f0 = 0.98 under uniaxial tension, plane strain and equi-biaxial stretching conditions, respectively. Moreover, the level of critical strains for the M–K model with f0 of 0.92 approaches to that of the experimental tensile test (−0.067, 0.30). Therefore, for this sheet formability, the initial imperfection factor with the value of 0.92 is suitable.
In this work, a new experimental apparatus based on the Marciniak test is developed, which includes a reverse experimental setup and an image acquisition system, as shown in Fig. 8(a). In this reverse setup, a bell jar is designed to connect the die with the crosshead of tensile machine. The purpose of this design is to prevent or minimize the vibrations and ensure the rigidity of the experimental setup throughout static and dynamic tests. On the bell jar, there are two small windows, one is for installing the optic mirror and another permits that the reflected light from the specimen surface goes through then focuses on the mirror. During the experiments, the die with the clamped specimen moves downwards and the fixed punch stretches the sheet. With this reverse
Fig. 8. Reverse experimental setup based on the Marciniak test. (a) 3D model of experimental setup. (b) Experimental apparatus.
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Table 1 Specimen dimensions used for the experiments. W (mm)
10
20
30
40
R (mm) Rc (mm) Rm (mm) Re (mm)
45
48
50
52
55
58
60
80
100
50 70 26.5 10
experimental setup, in which the specimen is formed over the punch until fracture appears on the specimen surface, the distance between the mirror and the specimen remains nearly constant throughout the test. This allows the camera to be focused on the specimen surface before the test and take sequential pictures. In order to cover different strain states, ranging from uniaxial through plane strain to equi-biaxial stretching, different specimen geometries are used. In this work, all test samples are shaped by cutting strips of different widths W in a circular flange (Table 1), according to Fig. 9(a). To assure the occurrence of the maximal strains (to trigger localization) on the central part of the blank, the specimens are designed with a reduced central thickness (0.8 mm) compared to the thickness of the sheet (1.5 mm) and the clamping part with a thickness of 2.0 mm. Before the test, the experimental specimen is painted with a random speckle pattern for DIC analysis. The surface which has an applied speckle pattern should be away from the punch contact surface. Finally, the painted specimen is clamped between the blankholder and the die. The bell jar, together with the die, moves down at a crosshead traveling speed of 500 mm/min. Here, a quasi-static procedure at 20 ◦ C is carried out to test the experimental setup and compare with above numerical procedure. The comparison between an undeformed and crack specimen in the experiments is shown in Fig. 10. With the DIC program CORRELA2006, a correlation analysis is carried out on the crack specimen and surface strains are calculated.
Fig. 10. Specimens used in the experiments. (a) Undeformed specimen. (b) Deformed specimen.
4.2. Experimental results As an illustrated example, the specimen of Fig. 10(b) is chosen. In order to carry out a study about the plastic instability associated with the onset and progress of necking in the sheet, two points are chosen. Point B is in (or near) the zone where the rupture occurs while Point A is outside of the zone, as shown in Fig. 11. The evolution of the major strains of Point A and Point B is plotted in Fig. 12. A similar evolution in these two curves is observed up to a moment of about t = 0.9 s, after which they diverge rapidly. Here, the criterion widely used in the M–K model is chosen to pre-
Fig. 9. Specimens specially designed in the experiments. (a) Specimen geometry designed for the experiments. (b) Blank with non-uniform thickness in the experiments.
Fig. 11. Deformed specimen with rupture.
dict the onset of localized necking. When the equivalent plastic strain increment ratio between Point A and Point B attains 7, the onset of localized necking is assumed to occur and the corresponding major and minor strains (−0.0036, 0.20) of Point B are retained as a point on the FLC. To provide sufficient and reliable data, for each specimen, at least two tests are performed under identical conditions.
Fig. 12. Evolution of major strains for the two Point A and Point B.
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Fig. 13. Comparison of numerical and experimental FLCs.
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strain rates by only carrying out the dynamic tensile tests and seeking the appropriate constitutive laws. The same approach will be used to investigate the influence of strain rate on sheet formability. This validation of the approach is in progress by means of a dynamic Marciniak test. • A modified Marciniak test has been developed to experimentally investigate the sheet formability and validate the numerical procedure proposed in this work. To prevent or minimize the vibrations and ensure the rigidity of the experimental setup throughout static and dynamic tests, a specially designed structure with the bell jar is used. With a quasi-static experimental procedure, the comparison with numerical results shows that this experimental setup works well to investigate sheet formability and in turn it indicate that numerical procedure proposed in this work can give a good prediction of FLCs. Acknowledgements
4.3. Comparison of numerical and experimental FLCs Fig. 13 shows the FLCs of the AA5086 sheet obtained by the numerical M–K model and experimental procedure, respectively. Compared with the M–K model, the strain states with the experimental procedure are located in a narrower range, especially at the left-hand side on the FLCs. This phenomenon maybe result from a lack of lubricants during the experiments, which influences the strain path, but not the level of critical limit strains. This first comparative evaluations show a good correlation between limit strains obtained numerically and experimentally. Although some discrepancies due to the variability encountered in forming sheet metal, it should be noted that numerical method could give a reasonable prediction of FLCs. 5. Conclusion This work proposed an approach to construct FLCs of sheet metals combining the tensile test and the FE M–K model. Compared to experimental results obtained by carrying out a modified Marciniak test procedure, the proposed approach could be used to practically predict FLCs of sheet metals. The conclusions are drawn as follows: • With the FE M–K model which may result in fast necking initiation, it is easy to determine the onset of localized necking and numerically construct the FLCs. Especially, the FE M–K model overcomes the limits encountered with the analytical one where the complex constitutive laws cannot be used. • The procedure combining the tensile test with the FE M–K model can be used to predict FLCs of sheet metals. A quasi-static tensile test is conducted to identify an appropriate constitutive law and the initial imperfection factor in the M–K model. By simulating the M–K model, the FLCs of this sheet are determined. More important, it can be used for the determination of FLCs at high
This work was financially supported by Encouragement Foundation for Young Scholars of Shandong Province (BS2010ZZ004). References [1] A. Petek, T. Pepelnjak, K. Kuzman, Journal of Mechanical Engineering 51 (2005) 330–345. [2] H.B. Campos, M.C. Butuc, J.J. Gracio, Journal of Materials Processing Technology 179 (2006) 56–60. [3] P. Eyckens, A. Van Bael, P. Van Houtte, International Journal of Plasticity 25 (2009) 2249–2268. [4] F. Ozturk, D. Lee, Journal of Materials Processing Technology 147 (2004) 397–404. [5] K. Nakazima, T. Kikuma, K. Asuka, Study on the Formability of Steel Sheet, Technical Report, Yawata, 1971. [6] A.K. Ghosh, S.S. Hecker, Metallurgical Transactions 2161 (1974) 5–2164. [7] Z. Marciniak, K. Kuczynski, T. Pokora, International Journal of Mechanical Sciences 15 (1973) 789–805. [8] A. Col, Numerical and Experimental Methods of FLCs Prediction for the Processes of Stamping and Hydroforming of Tubes, Technical Report, 2006. [9] P.D. Wu, S.R. MacEwen, D.J. Lloyd, K.W. Neale, Modelling and Simulation in Materials Science and Engineering 12 (2004) 511–527. [10] K. Narasimhan, in: S. Ghosh, J.M. Castro, J.K. Lee (Eds.), Materials Processing and Design: Modeling, Simulation and Applications. NUMIFORM 2004 Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes, vol. 2, American Institute of Physics, 2004, pp. 850–855. [11] W. Volk, Proceedings of the FLC-Zurich 06, Zurich, Switzerland, 2006. [12] H. Aretz, International Journal of Plasticity 23 (2007) 798–840. [13] C.S. Zhang, Investigations on the Effect of Strain Rate Sensitivity on Formability of Aluminium Alloy Sheets, PhD Thesis, INSA de Rennes, France, 9, 2008. [14] D. Banabic, O. Cazacu, L. Paraianu, P. Jurco, Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Process, vol. 778, MI, USA, 2005, pp. 466–471. [15] B. Davoodi, A. Gavrus, E. Ragneau, Measurement Science and Technology 16 (2005) 2101–2108. [16] C.S. Zhang, L. Leotoing, D. Guines, E. Ragneau, Materials Science and Engineering: A 527 (2010) 967–972. [17] C.S. Zhang, L. Leotoing, D. Guines, E. Ragneau, Proceedings of Numisheet’08, Interlaken, Swiss, 2008, pp. 229–233. [18] W.F. Hosford, Mechanical Behavior of Materials, Cambridge University Press, 2005.