Plastic bulging of sheet metals at high strain rates

Int J Adv Manuf Technol (2010) 48:847–858 DOI 10.1007/s00170-009-2335-x ORIGINAL ARTICLE Plastic bulging of sheet metals at high strain rates Maziar Ramezani & Zaidi Mohd Ripin & Roslan Ahmad Received: 17 February 2009 / Accepted: 21 September 2009 / Published online: 17 October 2009 # Springer-Verlag London Limited 2009 Abstract The biaxial bulge test is a material test for sheet metals to evaluate formability and determine the flow stress diagram. Due to the biaxial state of stress induced in this test, the maximum achievable strain before fracture is much larger than in the uniaxial tensile test. A new dynamic bulge testing technique is simulated and analyzed in this study which can be performed on a conventional split Hopkinson pressure bar (SHPB) system to evaluate the strain-rate dependent strength of material at high impact velocities. Polyurethane rubber as pressure carrying medium is used to bulge the OFHC copper sheet. The use of hyperelastic rubber instead of fluid as a pressure medium makes the bulge test simple and easy to perform. The input bar of SHPB is used to apply and measure the bulging pressure. The finite element simulation using ABAQUS/explicit and analytical analysis are compared and show good correlation with each other. The results clearly show that as the strain-rate increases, the strength of the OFHC copper increases. From the study, a robust method to determine the material behavior under dynamically biaxial deformation conditions has been developed. Keywords Bulging . Hyperelastic . Johnson-cook . Split Hopkinson pressure bar (SHPB) . Strain rate 1 Introduction The most widely used test for determining mechanical properties of sheet metal is the standard tensile test which gives the ultimate tensile strength, the yield strength, the total M. Ramezani (*) : Z. M. Ripin : R. Ahmad School of Mechanical Engineering, Universiti Sains Malaysia, 14300 Nibong Tebal, Penang, Malaysia e-mail: [email protected] elongation, and the reduction in area at rupture. The tensile test is simple and inexpensive to conduct, but it has limitations because it only provides the stress–strain behavior of the sheet material under uniaxial deformation conditions. For materials in the form of rod or bar, a number of alternative mechanical tests are used, such as simple compression, torsion, and plane strain compression tests. For sheet material, however, these tests are impractical or inconvenient, and the only widely used alternative to the tensile test is the hydraulic bulge test. In this test, the sheet metal clamped at its edges is stretched against circular die using oil/viscous as a pressure carrying medium. When the medium in the lower chamber is pressurized, the sheet is bulged into the cavity of the upper die (see Fig. 1). The clamping force between the lower and upper die has to be high enough to prevent sliding of the sheet to the die cavity. Thus, the sheet will only be stretched and no draw-in will occur. When the deformation of the material exceeds its formability limit, the sheet will fracture. In this test, the deformation is not affected by friction. Thus, the reproducibility of the test results is good. The basic quasi-static bulge test was developed in the late 1940s to investigate the plasticity [1] and strength of sheet metals [2]. Flow stress of the sheet material can be calculated from bulge test using analytical equations assuming sheet metal as thin membrane. Calculation of the flow stress from the bulge test requires (a) bulging pressure, (b) dome height, (c) radius of curvature at the top of the dome, and (d) thickness at the top of the dome to be measured real time in the experiment. Gutscher et al. [3] used a viscous pressure bulge test for determination of flow stress under quasi-static biaxial state of stress. They performed FEM simulations and experiments in order to study the interrelationship of the geometric and material variables such as dome wall thinning, dome radius, dome 848 Fig. 1 Initial and the pressurized sheet and the geometry of the bulge test height, strain hardening index, material strength coefficient, and anisotropy. They could develop a robust method to determine the flow stress under biaxial deformation conditions using a viscous material as pressure medium. Several years ago, high strain rate metal forming was fairly well developed. These techniques had some advantages over conventional metal forming. These include the ability to use single-sided dies, reduced springback, and improved formability [4]. The strain rate has a significant effect on the material behavior during the deformation process as well as on the final properties of products [5]. So, the investigation of the influence of the impact rate on the properties and behavior of materials is very demanding. Broomheas and Grieve [6] studied the effect of strain rate on the strain to fracture of sheets using bulge test. They used a drop hammer rig that makes use of a falling weight to impact a punch which applies a pressure loading to the fluid above the sheet material. They could determine the forming limit diagram of low carbon steel for strain rates of up to 70 S−1. Grolleau et al. [7] developed a dynamic bulge testing technique to perform biaxial tests on metals at high Fig. 2 Typical compressive split Hopkinson pressure bar apparatus [9] Int J Adv Manuf Technol (2010) 48:847–858 strain rates. They used split Hopkinson pressure bar apparatus with viscoelastic nylon bars to perform dynamic bulge experiments on aluminum sheets for plastic strain rates of up to 500 S-1. They analyzed the experimental system and measurement accuracy in details and found that bars made of low impedance materials must be used to achieve satisfactory pressure measurement accuracy. The objective of this study is to develop a numerical and theoretical technique for analyzing dynamically loaded sheet materials using the hydraulic bulge test principle where rubber is used as the pressure medium instead of hydraulic fluid. Using rubber simplifies the test rig and overcomes the problem of leakage of high pressure fluid. The new dynamic bulge testing method which is simulated and analyzed in this study can be performed on a conventional split Hopkinson pressure bar (SHPB) apparatus. The commercial finite element code ABAQUS/Explicit has been chosen as numerical test-bed for simulation of dynamic bulging test. Theoretical analysis is based on conventional hydraulic bulge test principle and SHPB relations. In the development of this method, analytical investigations and simulations concerning the dynamic bulge test were carried out. To verify the accuracy of developed method, analytical and finite element results are compared. 2 Dynamic bulge testing system SHPB has become a commonly accepted test method for strain rates in the range of medium and high strain rates (102–104 S−1) [8] and has been used to test various engineering materials. The conventional split Hopkinson bar apparatus consists of two long slender bars that sandwich a short cylindrical specimen between them (see Fig. 2). By Int J Adv Manuf Technol (2010) 48:847–858 849 striking the end of a bar, a compressive stress wave is generated that immediately begins to travel towards the specimen. Upon arrival at the specimen, the wave partially reflects back towards the impact end. The remainder of the wave transmits through the specimen and into the second bar. It is shown that the reflected and transmitted waves are proportional to the specimen’s strain rate and stress, respectively. Specimen strain can be determined by integrating the strain rate. By monitoring the strains in the two bars, specimen stress–strain properties can be calculated [10]. The current SHPB technique which is established by Kolsky [11] is based on one-dimensional wave propagation analysis in pressure bars. The engineering stress, strain rate, and strain, defined on the specimen length, are obtained from: s s ðtÞ ¼ E A "T ðtÞ A0 d"s ðtÞ C0 ¼ 2 "R ðtÞ dðtÞ L0 C0 "s ðtÞ ¼ 2 L0 Zt "R ðtÞdt ð1Þ ð2Þ ð3Þ considered a movable “bulge cell” with polyurethane rubber as pressure medium which can be used to perform dynamic bulge tests in a conventional SHPB system. When bulging with rubber, sealing problems and the possibility of leakage of the high-pressure liquid employed in hydraulic bulging are eliminated. The need for the filling and removal of fluid or the cleaning of the bulged specimen after forming is eliminated. The insertion of the rubber is quick and convenient, and the rubber can be reused [12]. The bulge cell is composed of a thick-walled rubber chamber and a die (Fig. 3). For simulation, the round OFHC copper sheet specimen of thickness 1 mm is clamped between the chamber and the die. The input bar is inserted into the chamber which is filled with polyurethane rubber to transmit the pressure from the input bar to the sheet surface. The outer diameter of the SHPB pressure bars match the inner diameter of the container and die. The pressure bars are made of aluminum. When the striker bar impacts the input bar at a defined velocity, a compressive stress wave is generated propagating towards the input bar/ rubber interface. This stress wave is transmitted through the rubber and ultimately causes the bulging of the sheet specimen, while both the bulge cell and the output bar are accelerated [7]. Throughout each simulation, the incident, reflected and transmitted waves, are measured at the center of the pressure bars. 0 qffiffiffi E r is the elastic stress wave speed in pressure where C0 ¼ bars, E and ρ are Young’s modulus and the density of 2 pressure bars, respectively, L0 and A0 (A0 ¼ pd40 ; where d0 is the diameter of the specimen) are the original length and cross-section area of the specimen and A is the crosssection area of pressure bars, εR and εT are the recorded reflected and transmitted strain pulses with the time being shifted from the strain gage locations to the interfaces between pressure bars and specimen according to the elastic wave speed in pressure bars. The dynamic bulge testing setup which is simulated in this study is based on the work of Grolleau et al. [7]. Figure 3 shows a schematic of the dynamic bulge testing setup. We Fig. 3 Schematic of the dynamic bulge testing setup 3 Theoretical investigation 3.1 Membrane theory For a thin spherical shell expanded uniformly by internal pressure, the membrane stress is given very closely by the approximation s¼ pRd 2td ð4Þ where p is the bulge pressure, and td, Rd are the thickness and radius at the top of the dome, respectively (see Fig. 1). 850 Int J Adv Manuf Technol (2010) 48:847–858 The equivalent strain can be calculated using the sheet thickness:   td " ¼ ln ð5Þ t0 and the reflected strain pulse εR(t) at the input bar/rubber interface: The radius at the top of the dome can be calculated by  dc  2 þ h2d  2Rc hd 2 þ Rc Rd ¼ ð6Þ 2hd u in ðtÞ ¼ C0 ½"I ðtÞ þ "R ðtÞ ð10Þ qffiffiffi where C0 ¼ Er is the one dimensional elastic stress wave speed in pressure bars and E, ρ are Young’s modulus and the density of pressure bars, respectively. The velocity of the bulge cell u out ðtÞ is determined from the transmitted wave εT(t) at the output bar/die interface. where Rc is the radius of the fillet of the cavity, dc is the diameter of the cavity, and hd is the dome height. Panknin [13] investigated the hydraulic bulge test experimentally. He measured the radius at the top of the dome of the deformed samples with radii gages. He also calculated the radius at the top of the dome using Eq. 6. The calculated radius agreed well with experimental values for dome heights, normalized by the diameter of the cavity, of up to hd dc ¼ 0:28. For larger dome heights, the radius of the dome determined experimentally was found to be up to 10% smaller than the calculated one. Hill [14] developed analytical methods to describe the deformation in the hydraulic bulge test. For his calculations, he assumed that the shape of the bulge is spherical. With this assumption, the thickness at the top of the dome can be calculated by the following equation: 12 0 B td ¼ t0 @ 1þ 1 C  2 A ð7Þ 2hd dc This equation was improved by Chakrabarty and Alexander [15] who considered the strain hardening coefficient, n, in the equation: 0 12n B td ¼ t0 @ 1þ 1 C  2 A ð8Þ 2hd dc Panknin [13] performed an experimental study of hydraulic bulge with materials with different strain hardening index, n, and found that the strain hardening index has significant influence on the dome height and thickness at the top of the dome. He found that the thickness distribution is more uniform in materials with larger strain hardening. This means that a material with larger strain hardening has, at the same dome height, a larger thickness at the top of the dome [3]. 3.2 Dynamic bulge theory The input bar of SHPB is used to measure the bulging pressure. The rubber pressure in the bulge cell is determined directly from the incident strain pulse εI(t) pðtÞ ¼ E ½"I ðtÞ þ "R ðtÞ ð9Þ The corresponding input bar/rubber interface velocity is: u out ðtÞ ¼ C0 "T ðtÞ ð11Þ The effective bulge velocity is the difference of interface velocities,  ¼ u ðtÞ  u ðtÞ DuðtÞ in out ð12Þ The effective bulge displacement can be determined by integration of Eq. 11 as a function of the measured strain histories: Z  Z ½"I ðt Þ þ "R ðt Þdt þ "T ðt Þdt ð13Þ DuðtÞ ¼ C0 t t 4 Finite element analysis In order to simulate dynamic bulge forming, a finite element model is built in commercial software ABAQUS. An explicit nonlinear approach with negligible temperature effects is assumed for simulations. By taking advantage of axisymmetry, it is possible to simulate the process as two-dimensional axisymmetric model. The die and rubber container are made of mild steel and are modeled as elastic bodies. Polyurethane rubber is used to bulge an OFHC copper blank of diameter 50 mm and thickness of 1 mm. The Coulomb friction model with the coefficient of friction of 0.25 is used to model the interface between rubber and sheet (see Ramezani et al. [16, 17]). All other interfaces are modeled as friction-free. The interactions between all components are modeled using an automatic surface to surface contact algorithm. All geometric entities are modeled using CAX4R elements. CAX4R is a four-node bilinear axisymmetric quadrilateral, reduced integration, hourglass control element. Since the model is developed by taking advantage of axisymmetry, the component nodes at the symmetry edges are restrained in the appropriate directions. The end of output bar is also constrained in the Y-direction in order to model the momentum trap. The initial velocity is applied to the striker bar to impact the input bar. The axisymmetric finite element model of the bulge cell at the end of the process is shown in Fig. 4. Int J Adv Manuf Technol (2010) 48:847–858 851 Based on the above relations, Johnson and Cook [18] presented the following equation for strength model, where the von-Mises flow stress is given as: h  ih  m i 1  T* ð17Þ s ¼ ½A þ Bð"Þn  1 þ C ln "* where A, B, C, n, m are material constants which are experimentally determined. The expression in the first set of brackets gives the stress as a function of strain for "* ¼ 1 and T* =0. The expressions in the second and third sets of brackets represent the effects of strain rate and temperature. The homologous temperature T* is the ratio of current temperature T to the melting temperature Tm. Fig. 4 Axisymmetric finite element model of the bulge cell at the end of the process The pressure bars mesh comprises only one element row in the radial direction. The pressure bars are made of aluminum and are modeled as linear elastic with the Young’s modulus E=70 GPa and the mass density ρ= 2,700 kg/m3. The Poisson ratios are set to a nonphysical value of v=0. Thus, uniaxial waves in the computational model are not altered when traveling along the bar axis. The length and diameter of the input and output bars are L= 150 mm and d=12 mm. The Johnson–Cook [18] material model is used for the OFHC copper blank. It expresses the equivalent von-Mises flow stress as a function of the equivalent plastic strain, strain rate, and temperature. In quasi-static conditions, metals work harder along the well-known relationship which is known as parabolic hardening rule: s ¼ s 0 þ k"n ð14Þ where σ0 is the yield stress of the metal, n is work hardening exponent, and k is the exponential factor. Dynamic events often involve increases in temperature due to adiabatic heating, and so the thermal softening must be included in the constitutive model. Johnson and Cook [18] described the effect of temperature on the flow stress with following relation: s ¼ sr 1   T  Tr T m  Tr m ð15Þ where Tm is the melting point, Tr is a reference temperature at which reference stress σr is measured, and m is materialdependent constant. The strain rate effect can be simply expressed with the following relationship, which is very often observed at strain rates that are not too high. s / ln " ð16Þ T* ¼ T  Tr Tm  Tr ð18Þ where Tr is the reference temperature at which σ0 is measured. Dimensionless strain rate "* is given as " "* ¼  ð19Þ " 0 where " is the effective plastic strain rate, and "0 is the reference strain rate which can, for convenience, be made equal to 1 ð"0 ¼ 1 s1 Þ. The material constants for OFHC copper as reported by Johnson and Cook [19] are listed in Table 1. Flexible materials have nonlinear stress–strain characteristics for relatively large deformations. Under such conditions, they are generally assumed as nearly incompressible. To model these hyperelastic materials through FEM, a constitutive law based on total strain energy density W has to be adopted [12]. Among several approaches, Mooney– Rivlin theory [20] is used based on the polynomial development of total strain energy. The Mooney–Rivlin material model has previously been used with success to predict the behavior of hyperelastic materials at high strain rates (see, e.g., Shergold et al. [21]). The form of the Mooney–Rivlin strain energy potential is: s ij ¼ W ¼ @W @"ij ð20Þ n X 1 Ckm ðI1  3Þk þ ðI2  3Þm þ k ðI3  1Þ2 2 kþm¼1 ð21Þ where W is the strain energy per unit of reference volume; I1, I2, I3 are the strain invariants; k is the bulk modulus; and Table 1 Material constants for OFHC copper sheet [19] ρ(kg/m3) A(MPa) B(MPa) C n m 8960 90 292 0.025 0.31 1.09 852 Int J Adv Manuf Technol (2010) 48:847–858 I3 =1 for incompressible material behavior. Ckm is the constant of the Mooney–Rivlin material model. Usually two Mooney–Rivlin parameters (C10 and C01) are used to describe hyperelastic rubber deformation. These parameters can be determined by experiments. Sarva et al. [22] studied the large deformation stress–strain behavior of thermoplastic– elastomeric polyurethane. They used SHPB apparatus to perform uniaxial compression tests at different strain rates. The stress–strain diagrams of polyurethane at two different strain rates are illustrated in Fig. 5. As can be seen from Fig. 5, as the strain rate increases, the strength of the material increases. The Mooney–Rivlin constants evaluated by ABAQUS using the compression test data are listed in Table 2. The simulation were performed using two different striker bar velocities, i.e., Vst =15 and 24 m/s. An empirical relationship between the striker bar velocity and the specimen strain rate at SHPB system is Vst " ffi 2L0 ð22Þ where Vst is the striker bar velocity and L0 is the length of specimen. The initial length of the polyurethane in the dynamic bulge test is 30 mm and according to Eq. 22, the approximate strain rate at the polyurethane during the test will be 250 and 400 s−1. 5 Results and discussion Figure 6 shows the strain signals at the middle of pressure bars for simulations performed at two different striker bar velocities of Vst =15 and 24 m/s. The solid curves represent the results of simulation with Vst =24 m/s. The incident Fig. 5 Compressive stress–strain curve of urethane at different strain rates [22] Table 2 Mooney–Rivlin constants for polyurethane rubber at different strain rates Strain rates Mooney–Rivlin constant C10 (MPa) Mooney–Rivlin constant C01 (MPa) Poisson’s ratio 250 400 224.20 77.69 −152.26 −37.66 0.4997 0.4997 compressive strain pulse rises to its plateau strain level of about 33×10−4during a time interval of about 16 μs. The time profile of the incident wave is of rectangular shape, and it remains quite constant during the simulation. The pulse shape of reflected wave is triangular in the tensile range. The maximum tensile stress is about 15×10−4which happens at 16 μs. The reflected strain signal becomes compressive after time duration of 34 μs. The compressive transmitted strain pulse arrives at the die/output bar interface after 38 μs and reaches the maximum amplitude of 16×10−4at the end of the simulation. Furthermore, the strain pulses for the simulation with the striker bar velocity of 15 m/s are shown as dashed curves. The maximum amplitude of the incident strain signal is about 20×10−4 at 19 μs. As can be seen from Fig. 6, the strain levels are considerably different at different impact velocities, but the pulse shapes have only little differences. Figure 7 shows the impact pressure at input bar/rubber interface using different velocities evaluated according to Eq. 9. As can be seen from the solid curve, the pressure– time history shows an initial peak at about 17 μs, and then the pressure level increases monotonically until it reaches its maximum level at 95 μs. The maximum pressures are about 156 and 244 MPa for the simulations carried out at the striker velocities of 15 and 24 m/s. Subsequently, the Int J Adv Manuf Technol (2010) 48:847–858 853 Fig. 6 Representation of the incident, reflected and transmitted strain signals pressure amplitude decreases because of the end of the incident pulse. To compare the results of theoretical and finite element simulation analysis, the pressure–time history at the input bar/rubber interface is monitored directly throughout the simulations. Figure 8 shows the comparison between the pressure curves calculated using Eq. 9 and measured directly during simulation at striker velocity of 15 m/s. The comparison of the curves demonstrates the good correlation between theoretical and simulation results. The comparison at striker velocity of 24 m/s shows similar Fig. 7 Pressure–time history at input bar/rubber interface agreement. In general, the simulation tends to predict slightly lower peak pressure than the theoretical analysis. The maximum error at the peak pressure is 10.2% at striker velocity of 15 m/s and 9.5% at striker velocity of 24 m/s. The velocity–time histories calculated from strain signals using Eqs. 10–12 are shown if Figs. 9 and 10. As can be seen from Fig. 9, the input velocity reaches its maximum of 14.3 m/s at 18 μs after impact and decreases monotonically after that until it reaches to zero. The output bar sets in motion 20 μs after the input velocity reaches its maximum and attains the maximum velocity of 6.5m/s at the time of 854 Int J Adv Manuf Technol (2010) 48:847–858 Fig. 8 Comparison of pressure– time history obtained by finite element simulation and theoretical analysis at striker velocity of 15 m/s 103 μs of the process. According to Fig. 10, at the striker velocity of 24 m/s, the input bar/rubber interface velocity reaches the maximum value of 23 m/s at 16 μs after impact. The die/output bar interface attains the maximum velocity of 8.5 m/s at the end of the process. As can be seen from Figs. 9 and 10, the effective piston velocity (dashed line) follows closely the evolution of the input velocity. Figure 11 shows the bulge dome height, hd (see Fig. 1) as a function of time which is measured directly during the simulations. As illustrated in Fig 11, the bulge height Fig. 9 History of the interfaces velocities at striker velocity of 15 m/s increases continuously until it reaches its maximum at the end of simulation. The maximum bulge height is 6.3 mm for striker velocity of 15 m/s and 8 mm for striker velocity of 24 m/s. Figure 12 shows the thickness changes at the top of the dome during the bulge forming process. The thickness is calculated using Eq. 8 and Fig. 11. As can be seen in Fig. 12, the thickness of the top of the dome decreases during the process from the initial value of 1 mm until it reaches its minimum thickness of 7.6 mm with Vst = 15 m/s and 6.5 mm with Vst =24 m/s. Int J Adv Manuf Technol (2010) 48:847–858 Fig. 10 History of the interfaces velocities at striker velocity of 24 m/s Fig. 11 Bulge dome height history measured during simulation Fig. 12 Thinning history at the top of the dome 855 856 Fig. 13 Strain history of sheet metal under dynamic bulging process Fig. 14 Pressure–strain curves of sheet metal under dynamic bulging process Fig. 15 Comparison of pressure–strain curves of biaxial bulge test and uniaxial tensile test at forming speed of 24 m/s Int J Adv Manuf Technol (2010) 48:847–858 Int J Adv Manuf Technol (2010) 48:847–858 The strain history of copper sheet during dynamic bulging process calculated using Eq. 5 is shown in Fig. 13. According to Fig. 13, the sheet deformation starts at about 40 μs after the input bar impacts the rubber and reaches the maximum strain of 0.27 at striker velocity of 15 m/s and 0.43 at striker velocity of 24 m/s. Combining Figs. 7 and 13, we arrive at Fig. 14 which shows the pressure–strain curve of OFHC copper at two different impact velocities during dynamic bulging process. As depicted in Fig. 14, as the strain rate (impact velocity) increases, the strength of the material increases. This is comparable with the results of Follansbee et al. [23] which dynamic loading of copper at different strain rates showed similar results. To compare the results of biaxial bulge test with uniaxial tensile test, finite element model of I-shape flat specimen is build in ABAQUS/Explicit according to ASTM E 8 M standard. I-shape model is then stretched in the forming speed of 24 m/s. The result of finite element simulation is shown in Fig. 15 and is compared with that of biaxial bulge test. As can be seen from the figure, the pressure–strain curves for OFHC copper determined from biaxial dynamic bulge test and uniaxial dynamic tensile test are quite similar. The figure shows that biaxial test attains a higher strain level as compared to the tensile test before material fracture. Therefore, due to the biaxial state of stress induced in this test, the pressure– strain curve can be determined up to larger strains than in the tensile test. This is important for process simulations using FE analysis. 6 Conclusions In this study, a dynamic bulge testing technique is simulated and analyzed to perform biaxial tests on metals at high strain rates. The main conclusions of this research are summarized below: & & & The incident compressive strain signal is rectangular shape, and its amplitude remains quite constant throughout the simulation. The pulse shape of reflected wave is triangular in the tensile range, and it becomes compressive after time duration of 34 μs. The compressive transmitted strain pulse arrives at the die/output bar interface after 38 μs. The amplitudes of strain signals are considerably different at different impact velocities, but the pulse shapes have only little differences. The pressure–time history at input bar/rubber interface shows an initial peak at about 17 μs, and then the pressure level increases monotonically until it reaches 857 & & & the maximum pressures of 156 and 244 MPa for the simulations performed at the striker velocities of 15 and 24 m/s. The pressure–time curves obtained by analytical analysis and finite element simulation show good agreement. The maximum error at the peak pressure is 10.2% at striker velocity of 15 m/s and 9.5% at striker velocity of 24 m/s. The bulge height increases continuously until it reaches its maximum at the end of simulation. At this point, the minimum sheet thickness is 0.76 mm with Vst =15 m/s and 0.65 mm with Vst =24 m/s. 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