- Email: [email protected]
Sheet Bulge Testing
1.05
Sheet Bulge Testing
T Hallfeldt, Ford Werke GmbH, Köln, Germany W Hotz, Novelis Innovation Centre, Sierre, Switzerland C Leppin, Suisse Technology Partners Ltd., Neuhausen am Rheinfall, Switzerland S Keller, Hydro Aluminium Research & Development, Bonn, Germany H Friebe, GOM mbH, Braunschweig, Germany ET Till, voestalpine Stahl GmbH, Linz, Austria R Mu¨ller, Fraunhofer-Institut für Werkzeugmaschinen und Umformtechnik IWU, Chemnitz, Germany M Vucetic, Leibniz Universität Hannover, Garbsen, Germany H Vegter, Tata Steel Research, Development & Technology, CA Ijmuiden, The Netherlands Ó 2014 Elsevier Ltd. All rights reserved.
1.05.1 1.05.2 1.05.2.1 1.05.2.2 1.05.2.3 1.05.2.4 1.05.3 1.05.3.1 1.05.3.2 1.05.4 1.05.4.1 1.05.4.2 1.05.5 1.05.6 References
1.05.1
Introduction Bulge Test Equipment and Specimens Bulge Test Principle Bulge Test Equipment Test Specimen Strain Rate Effects in Bulge Test Measurement Equipment Pressure Measurement Deformation Measurement Stress–Strain Curve Calculation True Stress–True Strain Calculation A Simple Approach to Determine Uniaxial Stress–Strain Curve from a Bulge Test Summary Acknowledgments
85 85 86 86 87 88 89 89 89 89 89 91 92 93 93
Introduction
Numerical forming simulations require accurate material parameter models, including true stress–true strain curves for the reliable prediction of sheet metal–forming behavior. Uniaxial tension tests are usually accepted as a standardized testing procedure to determine the required material parameters. However, due to the uniaxial loading condition that is present during a tension test, only a limited strain range is considered when determining the true stress–true strain curve. To extend the curves to cover higher true strains in actual forming operations, several empirical functions are available and are used to extrapolate the measured and calculated data, as shown in Figure 1. The tensile test data are valid up to uniform elongation, which typically corresponds to a true strain value of 0.2, depending on the material grade. It is obvious that the extrapolation formulas currently in use lead to a spread of the possible hardening curves beyond the measured tensile test data. The bulge test can be used to extend the validity of the hardening curve. It enables the use of a much larger strain range under biaxial loading conditions. In comparison to extrapolations from uniaxial tensile test data, smaller variations in extrapolated stress values at a strain equal to 1 are found while using different extrapolation methods, as shown in Figures 1 and 2 (1,2). Although the bulge test has been a recognized testing method for several years (1–7), experts are still considering the test conditions and the evaluation of material parameters from the measurement results. Thus, application of the test has not been widespread in the past. It was essential to standardize the test conditions, the measuring procedures, and the test evaluation algorithms (1,2).
1.05.2
Bulge Test Equipment and Specimens
The advantage of the bulge test is that it is a nearly frictionless and biaxial testing procedure. In this test sheet material, behavior will be characterized in terms of formability (in pure stretching). Furthermore, it is possible to determine both a yield locus point as well as the material’s strain-hardening behavior. By knowing the actual thickness in the bulge dome, the curvature of the actual bulge, and the pressure of the fluid, the true stress–true strain curve can be calculated. Tactile measurement systems can be used to determine the curvature, although using optical measurement systems provides more accurate results (1).
Comprehensive Materials Processing, Volume 1
http://dx.doi.org/10.1016/B978-0-08-096532-1.00105-9
85
86
Sheet Bulge Testing
Figure 1
True stress–true strain curves based on tensile test data, with several extrapolations.
Figure 2
True stress–true strain curves based on bulge test data, with several extrapolations.
1.05.2.1
Bulge Test Principle
A circular blank is clamped at its flange in a suitable tool between the die and the blank holder. A bulge is formed by building up pressure with a fluid, usually oil, until tearing of the specimen occurs (Figure 3). The pressure of the fluid is constantly measured during the test, while the specimen’s deformation is recorded by an optical measuring system. By monitoring and measuring the recorded deformations, the local curvature and the true principal strains at the apex of the dome are realized.
1.05.2.2
Bulge Test Equipment
The principles of the test equipment, including the optical measurement system, the location of lock beads, and the pressure measurement system, are shown in Figures 3 and 4. During the test, the optical system measures (without contact) the X, Y, and Z coordinates of the grid points on the bulging surface of the specimen. Based on these coordinates, the principal true strains of ε1 and ε2 for each point of the selected area and the curvature radius r for the apex of the dome are calculated. Note that any movement of the test piece between blank holder and die
Sheet Bulge Testing
Figure 3
87
The principle of the bulge test.
Deformation measurement system
ddie
R1 h t0
p Fluid
Pressure measurement system
Piston
Lock bead
Sealer
d BH Figure 4
Principle drawing of a test rig.
should be prevented. During the test, the bulge pressure is typically acting on parts of the blank holder to reduce the effective blank holder force. This must be considered when defining the necessary blank holder force. Furthermore, the fluid should be in contact with the blank surface without any air remaining to prevent energy storage by compressed air during the test, which leads to higher energy release and greater oil splashing at failure. A lock bead is located between the blank holder and the die. The lock bead geometry should be designed in a manner that prevents wrinkling or premature bulging of the blank while closing the tool, and also prevents the blank from sliding during the test. With the assumption of incompressibility of the material during plastic deformation, the thickness strain ε3 and the actual thickness of the blank can be calculated from the measured principal strains. Furthermore, assuming the stress state of a thin-walled spherical pressure vessel at the apex of the dome, the true stress is calculated from measurements of the fluid pressure, current thickness, and curvature radius. Glass plates in front of the lenses and the illumination unit are recommended to protect the optical measuring system from splashing fluid due to blank failure at the end of the test.
1.05.2.3
Test Specimen
The circular flat test piece is clamped between the blank holder and the die to prevent material flow. The preparation of the blank does not influence the results as long as the surface of the test piece is not damaged by scratches or polishing. For optical full-field
88
Sheet Bulge Testing
Figure 5 Typical measurement images: Left side: full image size; right side: enlarged local area with mathematical subsets (facets) for the calculation of surface point. Reproduced from Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330.
measurement devices, the grid must fulfill two objectives – determination of the specimen’s surface curvature radius as well as the strain calculation of the material deformation. Deterministic grids (such as squares, circles, or dots) should have a strong contrast and must be applied without any notch effect and/or change in microstructure. Some common application techniques include electrochemical etching, photochemical etching, offset printing, and grid transfer. Stochastic (speckled) patterns can be applied by spraying paint on the test piece surface (Figure 5).
1.05.2.4
Strain Rate Effects in Bulge Test
Almost all bulge test equipment contain constant fluid volume rates during the test. This typically leads to an increase in the strain rate at the center of the test specimen and can influence the results, especially when strain-rate–sensitive material grades are tested. Figure 6 shows an example of this condition, where high strain rates lead to increased stress and different slopes of the stress– strain curve, especially in the range between ε ¼ 0.1 and ε ¼ 0.5. As long as the test cannot be conducted with the recommended constant strain rate of 0.05 s1, a constant forming velocity of the punch or fluid should be guaranteed. In order to avoid significant influence on the biaxial stress–strain curve of temperature-sensitive or strain rate–sensitive materials, the bulge test should be conducted in 2–4 min. This time frame guarantees slow and acceptable strain rates, as well as a cost-effective testing duration (8).
700
0.8 Material: mild steel DX54,1.5 mm 0.7
r 1 = 30 mm r 2 = 10 mm
0.6
500
0.5 400
10s (test duration)
0.3 200
0.2
100
0.1
150s (test duration) 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
True strain ε (–) Figure 6
0.4
300
Bulge test curves, calculated at different strain rates.
0.7
0.8
0.9
0.0 1.0
Strain rate dε /dt [1/s]
True stress (MPa)
600
Sheet Bulge Testing
1.05.3
Measurement Equipment
1.05.3.1
Pressure Measurement
89
The pressure of the fluid in the chamber may be measured indirectly or directly. For indirect pressure measurement, a load cell located at the punch will detect the force of the punch during the movement. The pressure can be calculated by dividing the punch force by the punch area. Direct pressure measurement in the bulge chamber is recommended, but consider that the load range of the pressure sensor is aligned with the pressure to be measured in order to reduce the error of measurement.
1.05.3.2
Deformation Measurement
To determine the radius of curvature r and the true principal strains ε1 and ε2, an optical deformation field measurement system with two or more cameras should be used, and the measurement area should be half the size of the die radius or larger. The optical measurement system will provide certain images at a given time interval. In the last image before failure, the area of the dome with the highest deformation is selected and defined as the position at which to determine the true biaxial stress and the true thickness strain ε3. To obtain a stable radius of curvature of the dome, a best-fit sphere can be calculated based on a selected area of points. For this selection, a radius r1 is defined around the apex of the dome in the last image before bursting, and the fit is performed for all forming stages with the same selection of points (Figure 7). For robust values of the true strain and thinning in the apex, the average value of a number of selected points is taken. A second evaluation area is defined by a radius r2 in a similar manner (Figure 7). Based on this procedure, for every forming stage (image) the radius of curvature, the average thickness strains, as well as the corresponding thickness and stress values at the dome apex are calculated. This evaluation can be carried out for different r1 and r2 values. For a good convergence and robust values, the following ranges of r1 and r2 are recommended: r1 ¼ ð0:125 0:025Þddie r2 ¼ ð0:05 0:01Þddie
1.05.4
Stress–Strain Curve Calculation
1.05.4.1
True Stress–True Strain Calculation
For the calculation of the biaxial stress–strain curves, a simple membrane stress state of a thin-walled spherical pressure vessel is assumed at the center of the blank. This implies the following simplifications: 1. An equi-biaxial stress state: s1 ¼ s2 ¼ sB 2. The representation of the curvature by the mean curvature radius: r¼
1 1 ð1=r1 þ 1=r2 Þ 2
As a result, the biaxial true stress can be calculated according to the following equation: sB ¼
rp 2t
Figure 7 Choices of r1 and r2 for calculation of true stress and true strain for each forming stage. Reproduced from Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330.
90
Sheet Bulge Testing
using the fluid pressure p, the curvature radius r, and the current thickness t, with t ¼ t0 expðε3 Þ Assuming a plastic incompressible deformation of the material and neglecting elastic strains, the total thickness strain for the calculation of the actual thickness can be approximated by the total major and minor true strain with the following equation: ε3 z ε1 ε2 Based on the plastic work principle, the biaxial stress–strain curve is a function of the plastic thickness strain: pl sB ε3 Assuming an isotropic linear elastic material behavior and plastic incompressibility, the plastic thickness strain is then given by: pl
ε3 ¼ ε1 ε2 þ 2
1n sB E
For the elasticity modulus E and the Poisson ratio n, literature values are generally sufficient to subtract the elastic contribution; for example, E ¼ 210 GPa and n ¼ 0.3 for steel, and E ¼ 70 GPa and n ¼ 0.33 for aluminum alloys. The ratio of die diameter to thickness should be reasonably high to ensure a near-membrane stress state in the test piece, as well as a negligible bending influence (9). For die diameter to thickness ratios lower than 100, it is recommended to check that the bending strains are relatively small, as compared to the actual thickness strain result ε3, by using the following estimate for the bending strains: t0 εbending z ln 1 expðε3 Þ 2r If these bending strains are significant, an integrated correction of the corresponding error can considerably improve the results (8,9), as demonstrated in the comparison of test results with two different bulge test diameters of the same material, as shown in Figure 8.
Figure 8 True stress–true strain curves from bulge testing, with two different setups for die diameter and corresponding r1 and r2 values. Curves are calculated without bending and elastic strain correction (above) and with bending and elastic strain correction (below).
Sheet Bulge Testing
1.05.4.2
91
A Simple Approach to Determine Uniaxial Stress–Strain Curve from a Bulge Test
From the bulge test, equi-biaxial stress–strain curves are obtained where the average of the major and minor stress from the bulge tests is plotted against the absolute value of the plastic true thickness strain. Usually, the true stress–true strain curve, determined from uniaxial tensile test data in the rolling direction (RD), is used as a reference for the calculation of the stress points of the yield locus. By comparing the curves of the stress–strain data of the equi-biaxial stress state with the uniaxial reference curve, the equibiaxial stress point can be calculated. A convenient way to characterize the yield locus is the calculation of stress ratios that is relative to the uniaxial stress at the RD. The equi-biaxial stress ratio enables us to transform the equi-biaxial stress–strain curve to an equivalent stress–strain curve. This last curve provides work-hardening data at strains higher than the uniform strain of the tensile test. In order to use this approach, some assumptions have been made: the calculation is based on isotropic hardening, the yield locus shape does not change with the strain, and the work hardening is independent from the strain path (loading path). Furthermore, the loading path and strain path of the test scenario are constant, and the strain rate and temperature of the bulge test are close to the strain rate and temperature of the tensile test. The equivalent strain definition is based on the plastic work principle: sf $vεE ¼ s1 $vε1 þ s2 $vε2 þ s3 $vε3 where vεE means the equivalent plastic strain increment; sf is the plastic flow stress; s1, s2, and s3 are the principal stresses in the principal directions 1, 2, 3; and vε1, vε2, and vε3 are the corresponding plastic strain increments. With s2 ¼ s3 ¼ 0 for uniaxial tensile tests, the following equation can be used: sf $vεE ¼ s1 $vε1 For the bulge test, we assume that the two in-plane stresses are equal to s1 ¼ s2 ¼ sB and s3 ¼ 0. By using the plastic incompressibility condition, the following equation can be used: sf $vεE ¼ sB $vε1 þ sB $vε2 ¼ sB $vε3 ¼ sB $jvε3 j Assuming isotropic hardening means that a fixed ratio exists between uniaxial stress and the plastic flow stress fun ¼ s1/sf, and a fixed ratio also exists between the biaxial stress and the plastic flow stress fbi ¼ sB/sf. The uniaxial stress–strain curve in the RD is the reference for the equivalent stress–strain curve, so by definition, fun ¼ s1/sf ¼ 1. A combination of the equations leads to the following definition for the equivalent strain increment: vεE ¼
s1 s3 $vε1 ¼ $jvε3 j5vεE ¼ 1$vε1 ¼ fbi $jvε3 j sf sf
Hence, by the definition of fun ¼ 1 for the uniaxial tensile test in the RD, the equivalent strain is equal to the axial strain. The equivalent strain for the bulge tests is obtained through integration of the equation. The assumption of isotropy (i.e., a constant fbi) leads to the following equation for bulge tests: Z
εE 0
Z vεE ¼
0
jε3 j
fbi $jvε3 j5εE ¼ fbi jε3 j
The biaxial stress–strain curve from bulging can be transformed into an equivalent stress–strain curve for bulging. As the calculation of fbi is the fixed ratio between the equi-biaxial stress and the uniaxial stress in the RD, it is assumed that the yield surface is defined by stress points at constant equivalent strains. Here, the true plastic strain at uniform elongation of the tensile tests in the RD is a possible choice for a reference value for the equivalent strain: εEref ¼ ε1UE This is considered to be the last valid point of the true stress–true strain curve of the tensile test. Accordingly, the stress at the uniform strain of the tensile tests is used as the reference flow stress sf-ref ; namely, the ultimate tensile strength has been transformed to a true stress. Finding the pairs of stresses and strains in the bulge test curve that satisfy the following conditions derived from the Z εE above equation vεE leads to the corresponding reference values of the bulge test: 0
sBref $ε3ref ¼ sf ref $εEref
Since the bulge test curve is given in discrete values, there will not be a pair of stresses and strains that perfectly satisfy the condition above. Therefore, the point m in the bulge test data matches the following condition: sB;m $ε3;m sf ref $εEref and
sB;mþ1 $ε3;mþ1 sf ref $εEref
92
Sheet Bulge Testing
Figure 9 Example of the uniaxial stress strain and the equi-biaxial stress–strain curve of a material, including the biaxial stress calculation of the reference point and the hardening curve, based on scaled bulge test results.
where m is the index of the stress and strain data of the bulge test curve. The requested reference stress of the bulge test can now be computed by simple linear interpolation: sBref ¼ sB;m þ
s s B;mþ1 B;m $ sf ref $εEref sB;m $ε3;m sB;mþ1 $ε3;mþ1 sB;m $ε3;m
The value of the biaxial stress ratio is obtained by fbi ¼ with
Z
εE 0
Z vεE ¼
0
jε3 j
sBref sf ref
fbi $jvε3 j5εE ¼ fbi jε3 j
and, with the definition of the biaxial stress factor, the bulge test curve can be transformed into an equivalent strain–stress curve (Figure 9). When combined with uniaxial stress–strain curves from tensile tests, the transformed curve can be used to generate a hardening curve with data extrapolated beyond strains at a uniform elongation. The described method is a single proposal for handling the stress–strain data from a bulge test under the assumption of isotropic hardening. There are more general methods available, such as those proposed by Kuwabara et al. (10), Barlat et al. (11), Yoon (12), and Sigvant et al. (5), which are based on total plastic work and are not restricted to purely isotropic hardening. With the present method, the assumption of isotropic hardening can be tested by comparing the equivalent stress–strain curve derived from the bulge test and the uniaxial stress–strain curve. When these two curves deviate, then the isotropic hardening assumption is not valid.
1.05.5
Summary
In order to test sheet metals under biaxial loading conditions, the bulge test was chosen for the following reasons: l
The test conditions are nearly frictionless. The strain range for the determination of stress–strain curves is much larger in comparison to the uniaxial tensile test. l Depending on the material grade used, the maximum strain is three to six times higher. l
Therefore, the bulge test, combined with an online strain measurement system, is an excellent method for reliable testing of a material’s work hardening, as well as the determination and provision of significantly improved yield curves for finite-element (FE)-forming simulations. Moreover, test facilities and test equipment are commercially available. The bulge test has been known for many years, but a first standardization (ISO-standard) will be finalized soon. To improve the accuracy of the test, an optical measurements system is recommended. This optical system will record the evolution of the deformation on the surface during the test, which can then be
Sheet Bulge Testing
93
used to calculate the thickness and curvature of the dome over time. When combined with the measured pressure in the chamber, it is possible to determine a true stress–true strain curve. The usage of the hydraulic bulge test is recommended for use with a die-tothickness ratio of ddie/t0 33. The testing velocity should be limited to reduce deviations for strain rate–sensitive materials. The test velocity should ideally lead to strain rates that are close to the standard tensile test. To calculate the curvature and the thickness of the dome, a simple best-fit sphere approach is suggested. An alternative approach that leads to good results is a biquadratic paraboloid surface approximation (13), which will also be described in the ISO standard, which is currently being prepared (14). Finally, an approach to investigate and calculate true stress–true strain curves is discussed, making the biaxial stress–strain curves comparable to the uniaxial tensile test curves. By using this method, improved hardening curves will be obtained as input for the finite-element method (FEM) simulations.
1.05.6
Acknowledgments
The harmonization and standardization work of the bulge test with optical measurement systems described in this chapter was worked out at the German IDDRG working group ‘bulge test.’ The authors are members of this group and would like to thank all group members for their contributions.
References 1. Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330. 2. Hallfeldt, T.; Keller, S.; Staud, D.; Merklein, M.; Güner, A.; Brosius, A.; Geisler, S. Vereinheitlichung der Versuchsbedigungen fu¨r die Fließkurvenermittlung in der hydraulischen Tiefung. Fortschritte der Kennwerteermittlung fu¨r Forschung und Praxis, 2009, ISBN: 978-3-517-00769-7, pp 129–136. 3. Ranta-Eskola, A. J. Use of the Hydraulic Bulge Test in Biaxial Tensile Testing. Pergamon Press Ltd. Int. J. Mech. Sci. 1979, 21, 457–465. 4. Dziallach, S.; Bleck, W.; Hallfeldt, T. Sheet Metal Testing and Flow Curve Determination under Multiaxial Condition. Adv. Eng. Mater. 2007, 9 (11), 987–994. 5. Sigvant, M.; Mattiasson, K.; Vegter, H.; Thilderkvist, P. A Viscous Pressure Bulge Test for the Determination of a Plastic Hardening Curve and Equibiaxial Material Data. Int. J. Mater. Form. 2009, 2, 235–242. 6. Nasser, A.; Yadav, A.; Pathak, P.; Altan, T. Determination of the Flow Stress of Five AHSS Sheet Materials (DP600, DP780, DP780-CR, DP780-HY and TRIP780) using the Uniaxial Tensile Test and Viscous Pressured Bulge (VBP) Tests. J. Mater. Process. Technol. 2010, 210, 429–436. 7. Koc, M.; Billur, E.; Cora, ÖN. An Experimental Study on the Comparative Assessment of Hydraulic Bulge Test Analysis Methods. Mater. Des. 2011, 32, 272–281. 8. Leppin, C.; Lange, C.; Till, E.; Daniel, D. Evaluation of the Hydraulic Bulge Test for Improved Material Hardening Modeling, Submitted for Publication at 5th Forming Technology Forum, June 5th – 6th 2012, Zurich, Switzerland. 9. Behrens, B. -A.; Bouguecha, A.; Peshekodov, I.; Götze, T.; Huinink, T.; Vucetic, M.; Friebe, H.; Möller, T. Numerical Validation of Analytical Biaxial True Stress – True Strain Curves from Bulge Test. 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Seoul, South Korea, 2011, ISBN: 978-07354-0944-6, pp 107–114. 10. Kuwabara, T.; Van Bael, B.; Iizuka, E. Measurement and Analysis of Yield Locus and Work Hardening Characteristics of Steel Sheets with Different r-Values. Act. Mat. 2002, 20, 3717–3729. 11. Barlat, F.; Aretz, H.; Yoon, J. W.; Karabin, M. E.; Brem, J. C.; Dick, R. E. Linear Transformation-Based Anisotropic Yield Functions. Int. J. Plasticity 2005, 21, 1009–1103. 12. Yoon, J. H.; Cazacu, O.; Yoon, J. W.; Dick, R. E. Earing Predictions for Strongly Textured Aluminium Sheets. Int. J. Mech. Sciences 2010, 52, 1563–1578. 13. Volk, W.; Heinle, I.; Grass, H. Accurate Determination of Plastic Yield Curves and an Approximation Point for the Plastic Yield Locus with the Bulge Test. In Proceedings of the 10th International Conference on Technology of Plasticity; ICTP, 2011; pp 799–804. 14. ISO/CD 16808. Metallic Materials d Sheet and Strip d Determination of Biaxial Stress-Strain Curve by Means of Bulge Test with Optical Measuring Systems, in press.
Sheet Bulge Testing
T Hallfeldt, Ford Werke GmbH, Köln, Germany W Hotz, Novelis Innovation Centre, Sierre, Switzerland C Leppin, Suisse Technology Partners Ltd., Neuhausen am Rheinfall, Switzerland S Keller, Hydro Aluminium Research & Development, Bonn, Germany H Friebe, GOM mbH, Braunschweig, Germany ET Till, voestalpine Stahl GmbH, Linz, Austria R Mu¨ller, Fraunhofer-Institut für Werkzeugmaschinen und Umformtechnik IWU, Chemnitz, Germany M Vucetic, Leibniz Universität Hannover, Garbsen, Germany H Vegter, Tata Steel Research, Development & Technology, CA Ijmuiden, The Netherlands Ó 2014 Elsevier Ltd. All rights reserved.
1.05.1 1.05.2 1.05.2.1 1.05.2.2 1.05.2.3 1.05.2.4 1.05.3 1.05.3.1 1.05.3.2 1.05.4 1.05.4.1 1.05.4.2 1.05.5 1.05.6 References
1.05.1
Introduction Bulge Test Equipment and Specimens Bulge Test Principle Bulge Test Equipment Test Specimen Strain Rate Effects in Bulge Test Measurement Equipment Pressure Measurement Deformation Measurement Stress–Strain Curve Calculation True Stress–True Strain Calculation A Simple Approach to Determine Uniaxial Stress–Strain Curve from a Bulge Test Summary Acknowledgments
85 85 86 86 87 88 89 89 89 89 89 91 92 93 93
Introduction
Numerical forming simulations require accurate material parameter models, including true stress–true strain curves for the reliable prediction of sheet metal–forming behavior. Uniaxial tension tests are usually accepted as a standardized testing procedure to determine the required material parameters. However, due to the uniaxial loading condition that is present during a tension test, only a limited strain range is considered when determining the true stress–true strain curve. To extend the curves to cover higher true strains in actual forming operations, several empirical functions are available and are used to extrapolate the measured and calculated data, as shown in Figure 1. The tensile test data are valid up to uniform elongation, which typically corresponds to a true strain value of 0.2, depending on the material grade. It is obvious that the extrapolation formulas currently in use lead to a spread of the possible hardening curves beyond the measured tensile test data. The bulge test can be used to extend the validity of the hardening curve. It enables the use of a much larger strain range under biaxial loading conditions. In comparison to extrapolations from uniaxial tensile test data, smaller variations in extrapolated stress values at a strain equal to 1 are found while using different extrapolation methods, as shown in Figures 1 and 2 (1,2). Although the bulge test has been a recognized testing method for several years (1–7), experts are still considering the test conditions and the evaluation of material parameters from the measurement results. Thus, application of the test has not been widespread in the past. It was essential to standardize the test conditions, the measuring procedures, and the test evaluation algorithms (1,2).
1.05.2
Bulge Test Equipment and Specimens
The advantage of the bulge test is that it is a nearly frictionless and biaxial testing procedure. In this test sheet material, behavior will be characterized in terms of formability (in pure stretching). Furthermore, it is possible to determine both a yield locus point as well as the material’s strain-hardening behavior. By knowing the actual thickness in the bulge dome, the curvature of the actual bulge, and the pressure of the fluid, the true stress–true strain curve can be calculated. Tactile measurement systems can be used to determine the curvature, although using optical measurement systems provides more accurate results (1).
Comprehensive Materials Processing, Volume 1
http://dx.doi.org/10.1016/B978-0-08-096532-1.00105-9
85
86
Sheet Bulge Testing
Figure 1
True stress–true strain curves based on tensile test data, with several extrapolations.
Figure 2
True stress–true strain curves based on bulge test data, with several extrapolations.
1.05.2.1
Bulge Test Principle
A circular blank is clamped at its flange in a suitable tool between the die and the blank holder. A bulge is formed by building up pressure with a fluid, usually oil, until tearing of the specimen occurs (Figure 3). The pressure of the fluid is constantly measured during the test, while the specimen’s deformation is recorded by an optical measuring system. By monitoring and measuring the recorded deformations, the local curvature and the true principal strains at the apex of the dome are realized.
1.05.2.2
Bulge Test Equipment
The principles of the test equipment, including the optical measurement system, the location of lock beads, and the pressure measurement system, are shown in Figures 3 and 4. During the test, the optical system measures (without contact) the X, Y, and Z coordinates of the grid points on the bulging surface of the specimen. Based on these coordinates, the principal true strains of ε1 and ε2 for each point of the selected area and the curvature radius r for the apex of the dome are calculated. Note that any movement of the test piece between blank holder and die
Sheet Bulge Testing
Figure 3
87
The principle of the bulge test.
Deformation measurement system
ddie
R1 h t0
p Fluid
Pressure measurement system
Piston
Lock bead
Sealer
d BH Figure 4
Principle drawing of a test rig.
should be prevented. During the test, the bulge pressure is typically acting on parts of the blank holder to reduce the effective blank holder force. This must be considered when defining the necessary blank holder force. Furthermore, the fluid should be in contact with the blank surface without any air remaining to prevent energy storage by compressed air during the test, which leads to higher energy release and greater oil splashing at failure. A lock bead is located between the blank holder and the die. The lock bead geometry should be designed in a manner that prevents wrinkling or premature bulging of the blank while closing the tool, and also prevents the blank from sliding during the test. With the assumption of incompressibility of the material during plastic deformation, the thickness strain ε3 and the actual thickness of the blank can be calculated from the measured principal strains. Furthermore, assuming the stress state of a thin-walled spherical pressure vessel at the apex of the dome, the true stress is calculated from measurements of the fluid pressure, current thickness, and curvature radius. Glass plates in front of the lenses and the illumination unit are recommended to protect the optical measuring system from splashing fluid due to blank failure at the end of the test.
1.05.2.3
Test Specimen
The circular flat test piece is clamped between the blank holder and the die to prevent material flow. The preparation of the blank does not influence the results as long as the surface of the test piece is not damaged by scratches or polishing. For optical full-field
88
Sheet Bulge Testing
Figure 5 Typical measurement images: Left side: full image size; right side: enlarged local area with mathematical subsets (facets) for the calculation of surface point. Reproduced from Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330.
measurement devices, the grid must fulfill two objectives – determination of the specimen’s surface curvature radius as well as the strain calculation of the material deformation. Deterministic grids (such as squares, circles, or dots) should have a strong contrast and must be applied without any notch effect and/or change in microstructure. Some common application techniques include electrochemical etching, photochemical etching, offset printing, and grid transfer. Stochastic (speckled) patterns can be applied by spraying paint on the test piece surface (Figure 5).
1.05.2.4
Strain Rate Effects in Bulge Test
Almost all bulge test equipment contain constant fluid volume rates during the test. This typically leads to an increase in the strain rate at the center of the test specimen and can influence the results, especially when strain-rate–sensitive material grades are tested. Figure 6 shows an example of this condition, where high strain rates lead to increased stress and different slopes of the stress– strain curve, especially in the range between ε ¼ 0.1 and ε ¼ 0.5. As long as the test cannot be conducted with the recommended constant strain rate of 0.05 s1, a constant forming velocity of the punch or fluid should be guaranteed. In order to avoid significant influence on the biaxial stress–strain curve of temperature-sensitive or strain rate–sensitive materials, the bulge test should be conducted in 2–4 min. This time frame guarantees slow and acceptable strain rates, as well as a cost-effective testing duration (8).
700
0.8 Material: mild steel DX54,1.5 mm 0.7
r 1 = 30 mm r 2 = 10 mm
0.6
500
0.5 400
10s (test duration)
0.3 200
0.2
100
0.1
150s (test duration) 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
True strain ε (–) Figure 6
0.4
300
Bulge test curves, calculated at different strain rates.
0.7
0.8
0.9
0.0 1.0
Strain rate dε /dt [1/s]
True stress (MPa)
600
Sheet Bulge Testing
1.05.3
Measurement Equipment
1.05.3.1
Pressure Measurement
89
The pressure of the fluid in the chamber may be measured indirectly or directly. For indirect pressure measurement, a load cell located at the punch will detect the force of the punch during the movement. The pressure can be calculated by dividing the punch force by the punch area. Direct pressure measurement in the bulge chamber is recommended, but consider that the load range of the pressure sensor is aligned with the pressure to be measured in order to reduce the error of measurement.
1.05.3.2
Deformation Measurement
To determine the radius of curvature r and the true principal strains ε1 and ε2, an optical deformation field measurement system with two or more cameras should be used, and the measurement area should be half the size of the die radius or larger. The optical measurement system will provide certain images at a given time interval. In the last image before failure, the area of the dome with the highest deformation is selected and defined as the position at which to determine the true biaxial stress and the true thickness strain ε3. To obtain a stable radius of curvature of the dome, a best-fit sphere can be calculated based on a selected area of points. For this selection, a radius r1 is defined around the apex of the dome in the last image before bursting, and the fit is performed for all forming stages with the same selection of points (Figure 7). For robust values of the true strain and thinning in the apex, the average value of a number of selected points is taken. A second evaluation area is defined by a radius r2 in a similar manner (Figure 7). Based on this procedure, for every forming stage (image) the radius of curvature, the average thickness strains, as well as the corresponding thickness and stress values at the dome apex are calculated. This evaluation can be carried out for different r1 and r2 values. For a good convergence and robust values, the following ranges of r1 and r2 are recommended: r1 ¼ ð0:125 0:025Þddie r2 ¼ ð0:05 0:01Þddie
1.05.4
Stress–Strain Curve Calculation
1.05.4.1
True Stress–True Strain Calculation
For the calculation of the biaxial stress–strain curves, a simple membrane stress state of a thin-walled spherical pressure vessel is assumed at the center of the blank. This implies the following simplifications: 1. An equi-biaxial stress state: s1 ¼ s2 ¼ sB 2. The representation of the curvature by the mean curvature radius: r¼
1 1 ð1=r1 þ 1=r2 Þ 2
As a result, the biaxial true stress can be calculated according to the following equation: sB ¼
rp 2t
Figure 7 Choices of r1 and r2 for calculation of true stress and true strain for each forming stage. Reproduced from Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330.
90
Sheet Bulge Testing
using the fluid pressure p, the curvature radius r, and the current thickness t, with t ¼ t0 expðε3 Þ Assuming a plastic incompressible deformation of the material and neglecting elastic strains, the total thickness strain for the calculation of the actual thickness can be approximated by the total major and minor true strain with the following equation: ε3 z ε1 ε2 Based on the plastic work principle, the biaxial stress–strain curve is a function of the plastic thickness strain: pl sB ε3 Assuming an isotropic linear elastic material behavior and plastic incompressibility, the plastic thickness strain is then given by: pl
ε3 ¼ ε1 ε2 þ 2
1n sB E
For the elasticity modulus E and the Poisson ratio n, literature values are generally sufficient to subtract the elastic contribution; for example, E ¼ 210 GPa and n ¼ 0.3 for steel, and E ¼ 70 GPa and n ¼ 0.33 for aluminum alloys. The ratio of die diameter to thickness should be reasonably high to ensure a near-membrane stress state in the test piece, as well as a negligible bending influence (9). For die diameter to thickness ratios lower than 100, it is recommended to check that the bending strains are relatively small, as compared to the actual thickness strain result ε3, by using the following estimate for the bending strains: t0 εbending z ln 1 expðε3 Þ 2r If these bending strains are significant, an integrated correction of the corresponding error can considerably improve the results (8,9), as demonstrated in the comparison of test results with two different bulge test diameters of the same material, as shown in Figure 8.
Figure 8 True stress–true strain curves from bulge testing, with two different setups for die diameter and corresponding r1 and r2 values. Curves are calculated without bending and elastic strain correction (above) and with bending and elastic strain correction (below).
Sheet Bulge Testing
1.05.4.2
91
A Simple Approach to Determine Uniaxial Stress–Strain Curve from a Bulge Test
From the bulge test, equi-biaxial stress–strain curves are obtained where the average of the major and minor stress from the bulge tests is plotted against the absolute value of the plastic true thickness strain. Usually, the true stress–true strain curve, determined from uniaxial tensile test data in the rolling direction (RD), is used as a reference for the calculation of the stress points of the yield locus. By comparing the curves of the stress–strain data of the equi-biaxial stress state with the uniaxial reference curve, the equibiaxial stress point can be calculated. A convenient way to characterize the yield locus is the calculation of stress ratios that is relative to the uniaxial stress at the RD. The equi-biaxial stress ratio enables us to transform the equi-biaxial stress–strain curve to an equivalent stress–strain curve. This last curve provides work-hardening data at strains higher than the uniform strain of the tensile test. In order to use this approach, some assumptions have been made: the calculation is based on isotropic hardening, the yield locus shape does not change with the strain, and the work hardening is independent from the strain path (loading path). Furthermore, the loading path and strain path of the test scenario are constant, and the strain rate and temperature of the bulge test are close to the strain rate and temperature of the tensile test. The equivalent strain definition is based on the plastic work principle: sf $vεE ¼ s1 $vε1 þ s2 $vε2 þ s3 $vε3 where vεE means the equivalent plastic strain increment; sf is the plastic flow stress; s1, s2, and s3 are the principal stresses in the principal directions 1, 2, 3; and vε1, vε2, and vε3 are the corresponding plastic strain increments. With s2 ¼ s3 ¼ 0 for uniaxial tensile tests, the following equation can be used: sf $vεE ¼ s1 $vε1 For the bulge test, we assume that the two in-plane stresses are equal to s1 ¼ s2 ¼ sB and s3 ¼ 0. By using the plastic incompressibility condition, the following equation can be used: sf $vεE ¼ sB $vε1 þ sB $vε2 ¼ sB $vε3 ¼ sB $jvε3 j Assuming isotropic hardening means that a fixed ratio exists between uniaxial stress and the plastic flow stress fun ¼ s1/sf, and a fixed ratio also exists between the biaxial stress and the plastic flow stress fbi ¼ sB/sf. The uniaxial stress–strain curve in the RD is the reference for the equivalent stress–strain curve, so by definition, fun ¼ s1/sf ¼ 1. A combination of the equations leads to the following definition for the equivalent strain increment: vεE ¼
s1 s3 $vε1 ¼ $jvε3 j5vεE ¼ 1$vε1 ¼ fbi $jvε3 j sf sf
Hence, by the definition of fun ¼ 1 for the uniaxial tensile test in the RD, the equivalent strain is equal to the axial strain. The equivalent strain for the bulge tests is obtained through integration of the equation. The assumption of isotropy (i.e., a constant fbi) leads to the following equation for bulge tests: Z
εE 0
Z vεE ¼
0
jε3 j
fbi $jvε3 j5εE ¼ fbi jε3 j
The biaxial stress–strain curve from bulging can be transformed into an equivalent stress–strain curve for bulging. As the calculation of fbi is the fixed ratio between the equi-biaxial stress and the uniaxial stress in the RD, it is assumed that the yield surface is defined by stress points at constant equivalent strains. Here, the true plastic strain at uniform elongation of the tensile tests in the RD is a possible choice for a reference value for the equivalent strain: εEref ¼ ε1UE This is considered to be the last valid point of the true stress–true strain curve of the tensile test. Accordingly, the stress at the uniform strain of the tensile tests is used as the reference flow stress sf-ref ; namely, the ultimate tensile strength has been transformed to a true stress. Finding the pairs of stresses and strains in the bulge test curve that satisfy the following conditions derived from the Z εE above equation vεE leads to the corresponding reference values of the bulge test: 0
sBref $ε3ref ¼ sf ref $εEref
Since the bulge test curve is given in discrete values, there will not be a pair of stresses and strains that perfectly satisfy the condition above. Therefore, the point m in the bulge test data matches the following condition: sB;m $ε3;m sf ref $εEref and
sB;mþ1 $ε3;mþ1 sf ref $εEref
92
Sheet Bulge Testing
Figure 9 Example of the uniaxial stress strain and the equi-biaxial stress–strain curve of a material, including the biaxial stress calculation of the reference point and the hardening curve, based on scaled bulge test results.
where m is the index of the stress and strain data of the bulge test curve. The requested reference stress of the bulge test can now be computed by simple linear interpolation: sBref ¼ sB;m þ
s s B;mþ1 B;m $ sf ref $εEref sB;m $ε3;m sB;mþ1 $ε3;mþ1 sB;m $ε3;m
The value of the biaxial stress ratio is obtained by fbi ¼ with
Z
εE 0
Z vεE ¼
0
jε3 j
sBref sf ref
fbi $jvε3 j5εE ¼ fbi jε3 j
and, with the definition of the biaxial stress factor, the bulge test curve can be transformed into an equivalent strain–stress curve (Figure 9). When combined with uniaxial stress–strain curves from tensile tests, the transformed curve can be used to generate a hardening curve with data extrapolated beyond strains at a uniform elongation. The described method is a single proposal for handling the stress–strain data from a bulge test under the assumption of isotropic hardening. There are more general methods available, such as those proposed by Kuwabara et al. (10), Barlat et al. (11), Yoon (12), and Sigvant et al. (5), which are based on total plastic work and are not restricted to purely isotropic hardening. With the present method, the assumption of isotropic hardening can be tested by comparing the equivalent stress–strain curve derived from the bulge test and the uniaxial stress–strain curve. When these two curves deviate, then the isotropic hardening assumption is not valid.
1.05.5
Summary
In order to test sheet metals under biaxial loading conditions, the bulge test was chosen for the following reasons: l
The test conditions are nearly frictionless. The strain range for the determination of stress–strain curves is much larger in comparison to the uniaxial tensile test. l Depending on the material grade used, the maximum strain is three to six times higher. l
Therefore, the bulge test, combined with an online strain measurement system, is an excellent method for reliable testing of a material’s work hardening, as well as the determination and provision of significantly improved yield curves for finite-element (FE)-forming simulations. Moreover, test facilities and test equipment are commercially available. The bulge test has been known for many years, but a first standardization (ISO-standard) will be finalized soon. To improve the accuracy of the test, an optical measurements system is recommended. This optical system will record the evolution of the deformation on the surface during the test, which can then be
Sheet Bulge Testing
93
used to calculate the thickness and curvature of the dome over time. When combined with the measured pressure in the chamber, it is possible to determine a true stress–true strain curve. The usage of the hydraulic bulge test is recommended for use with a die-tothickness ratio of ddie/t0 33. The testing velocity should be limited to reduce deviations for strain rate–sensitive materials. The test velocity should ideally lead to strain rates that are close to the standard tensile test. To calculate the curvature and the thickness of the dome, a simple best-fit sphere approach is suggested. An alternative approach that leads to good results is a biquadratic paraboloid surface approximation (13), which will also be described in the ISO standard, which is currently being prepared (14). Finally, an approach to investigate and calculate true stress–true strain curves is discussed, making the biaxial stress–strain curves comparable to the uniaxial tensile test curves. By using this method, improved hardening curves will be obtained as input for the finite-element method (FEM) simulations.
1.05.6
Acknowledgments
The harmonization and standardization work of the bulge test with optical measurement systems described in this chapter was worked out at the German IDDRG working group ‘bulge test.’ The authors are members of this group and would like to thank all group members for their contributions.
References 1. Keller, S.; Hotz, W.; Friebe, F. Yield Curve Determination Using the Bulge Test Combined with Optical Measurements; IDDRG, 2009; pp 319–330. 2. Hallfeldt, T.; Keller, S.; Staud, D.; Merklein, M.; Güner, A.; Brosius, A.; Geisler, S. Vereinheitlichung der Versuchsbedigungen fu¨r die Fließkurvenermittlung in der hydraulischen Tiefung. Fortschritte der Kennwerteermittlung fu¨r Forschung und Praxis, 2009, ISBN: 978-3-517-00769-7, pp 129–136. 3. Ranta-Eskola, A. J. Use of the Hydraulic Bulge Test in Biaxial Tensile Testing. Pergamon Press Ltd. Int. J. Mech. Sci. 1979, 21, 457–465. 4. Dziallach, S.; Bleck, W.; Hallfeldt, T. Sheet Metal Testing and Flow Curve Determination under Multiaxial Condition. Adv. Eng. Mater. 2007, 9 (11), 987–994. 5. Sigvant, M.; Mattiasson, K.; Vegter, H.; Thilderkvist, P. A Viscous Pressure Bulge Test for the Determination of a Plastic Hardening Curve and Equibiaxial Material Data. Int. J. Mater. Form. 2009, 2, 235–242. 6. Nasser, A.; Yadav, A.; Pathak, P.; Altan, T. Determination of the Flow Stress of Five AHSS Sheet Materials (DP600, DP780, DP780-CR, DP780-HY and TRIP780) using the Uniaxial Tensile Test and Viscous Pressured Bulge (VBP) Tests. J. Mater. Process. Technol. 2010, 210, 429–436. 7. Koc, M.; Billur, E.; Cora, ÖN. An Experimental Study on the Comparative Assessment of Hydraulic Bulge Test Analysis Methods. Mater. Des. 2011, 32, 272–281. 8. Leppin, C.; Lange, C.; Till, E.; Daniel, D. Evaluation of the Hydraulic Bulge Test for Improved Material Hardening Modeling, Submitted for Publication at 5th Forming Technology Forum, June 5th – 6th 2012, Zurich, Switzerland. 9. Behrens, B. -A.; Bouguecha, A.; Peshekodov, I.; Götze, T.; Huinink, T.; Vucetic, M.; Friebe, H.; Möller, T. Numerical Validation of Analytical Biaxial True Stress – True Strain Curves from Bulge Test. 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Seoul, South Korea, 2011, ISBN: 978-07354-0944-6, pp 107–114. 10. Kuwabara, T.; Van Bael, B.; Iizuka, E. Measurement and Analysis of Yield Locus and Work Hardening Characteristics of Steel Sheets with Different r-Values. Act. Mat. 2002, 20, 3717–3729. 11. Barlat, F.; Aretz, H.; Yoon, J. W.; Karabin, M. E.; Brem, J. C.; Dick, R. E. Linear Transformation-Based Anisotropic Yield Functions. Int. J. Plasticity 2005, 21, 1009–1103. 12. Yoon, J. H.; Cazacu, O.; Yoon, J. W.; Dick, R. E. Earing Predictions for Strongly Textured Aluminium Sheets. Int. J. Mech. Sciences 2010, 52, 1563–1578. 13. Volk, W.; Heinle, I.; Grass, H. Accurate Determination of Plastic Yield Curves and an Approximation Point for the Plastic Yield Locus with the Bulge Test. In Proceedings of the 10th International Conference on Technology of Plasticity; ICTP, 2011; pp 799–804. 14. ISO/CD 16808. Metallic Materials d Sheet and Strip d Determination of Biaxial Stress-Strain Curve by Means of Bulge Test with Optical Measuring Systems, in press.