Introduction

The rising interest of the forming community for more sustainable and efficient production processes, has driven the development of new high-speed sheet metal forming techniques. Indeed, next to a higher production capacity, increasing the forming speed might have a beneficial influence on the formability of metals. Limits in formability generally originate from the risk of excessive thinning of the sheet, possibly in narrow bands. In high-speed forming operations, the high deformation speed can result in an enhancement of the material ductility due to the combination of several factors, such as a positive strain rate sensitivity of the sheet metal, as well as thermal softening. Indeed, regarding the latter, at high forming velocities, significant temperature rises may occur due to the conversion of plastic work into heat which is not, or only partially, dissipated to the environment during the short duration of the forming process. It is well-known that the formability of metals is highly dependent on the imposed strain state. This observation led to the wide-spread use of forming limit curves (FLCs) representing formability limits in the minor-major strain space for different, assumed linear, strain paths [1]. Several researchers performed experimental and numerical studies showing that the influence of strain state and strain rate is coupled. Balanethiram and Daehn [2] investigated the formability of a commercial interstitial-free iron sheet by electrohydraulic forming experiments with strain rates ranging between 103-104 s-1. The authors found that the plane strain formability increased by nearly three times when the material was tested at high strain rates. Wood [3] carried out explosive forming experiments with strain rates up to 104 s-1 on tensile, tubular and circular samples for a wide range of metals. The experiments showed that increasing the test velocity results in a higher ductility. Based on predictions of the FLCs by means of the Marciniak-Kuczynski model, Verleysen et al. [4] demonstrated a significant improvement of the formability for a TRIP steel at dynamic deformation rates, on the contrary, a deterioration for S235, DC04 and AISI 409 steels was reported. Therefore, to experimentally study the formability of metals at high strain rates, tests imposing different deformation modes, such as uniaxial tensile, shear, compression and biaxial tensile, should be used in a complementary manner. For the design of a specific forming operation, in-depth knowledge of the sheet metal behaviour under the conditions of strain and strain rate met in the actual forming process is of paramount importance. Tensile tests, preferably in different directions if anisotropy is to be expected, are often used to assess the formability. Performed at static and dynamic strain rates, the test results, such as the hardening evolution, the related uniform elongation, and their possible directional dependence, give insight into the influence of strain rate on forming under uniaxial stress conditions. Khan and Baig [5] investigated the formability and anisotropy of AA5182-O sheet performing experiments at different strain rates and in different orientations with respect to the rolling direction (0°, 45° and 90°). However, unambiguous interpretation of the test results was limited to relatively low values of elongation due to the early onset of necking in the samples. Steglich et al. [6] investigated the in-plane and through-thickness compressive behaviour of two magnesium alloy sheets (AZ31 and ZE10) by performing tests on adhesively-bonded, stacked samples. The experimental technique allowed to determine the material flow curve up to relatively large levels of plastic strain, and to assess the material anisotropy by tests along the orthotropic axes of the sheet. High strain levels can also be reached in static and dynamic shear tests as shown for Ti6Al4V sheet in [7]. Abedini et al. [8] employed the same shear sample geometry to investigate the fracture properties in shear loading at different strain rates for a dual phase steel (DP780) and a magnesium alloy (ZEK100). Next to providing large strain data, shear tests allow to reveal the susceptibility of metals to shear band formation. Grolleau et al. [9] used an in-plane torsion setup to determine the plastic deformation of sheet metals up to large strain, e.g. equivalent plastic strains up to 2.15 for DC01, for strain rates ranging from 0.001 to 1 s-1. Compared to tensile tests, also significantly larger strains can be imposed in biaxial tensile tests. Shimamoto et al. [10] performed static, biaxial tests on AA7075-T6 using cruciform specimens. Recently, Quillery et al. [11] developed a dynamic counterpart of biaxial experiments on cruciform samples using an adapted split Hopkinson bar facility. Unfortunately, in both quasi-static and dynamic tests, premature failure at the sample’s gauge corners concealed the full deformation potential of the test material. In bulge tests, on the other hand, circular sheet samples can be deformed under a quasi-equibiaxial stress state up to large levels of plastic strain. Gutscher et al. [12] obtained reliable data for several metals up to strain levels more than double of those obtained in tensile tests. Moreover, the theory proposed by Hill [13], consolidated in standard ISO-16808:2014 [14], allows to derive the constitutive behaviour of isotropic materials from bulge tests using the 3D-displacement and in-plane strain fields around the sample apex. A more accurate calculation of stresses and strains that is additionally valid for anisotropic materials, is presented in [15]. Given the additional ability of bulge tests to cover a highly interesting deformation area for forming and to reveal anisotropic material properties up to large deformation levels, not surprisingly, dynamic versions have been developed. In these tests, the sheet sample is deformed by a high-speed pressure wave. In the electrohydraulic tests performed by Rohatgi et al. [16], a shockwave was generated by high-voltage discharge of capacitors submerged in a liquid medium. High-speed imaging combined with digital image correlation (DIC) was used to obtain the displacement and strain fields of AA5182-O sheets deformed at a strain rate that reached a maximum of about 664 s-1. Kakogiannis et al. [17] used explosives to deform Ti6Al4V sheet and achieved a strain rate close to 1000 s-1 in the centre of the sample. The material anisotropy was revealed using full-field, high-speed stereo DIC. Notwithstanding the possibility of monitoring the full deformation of the specimen during the tests, in both studies no information regarding the evolution of the sample deformation with respect to the loading pressure is reported. Indeed, both in electrohydraulic and explosive bulge experiments, the evolution of the impulse pressure wave is both difficult to control and to measure. To provide a solution here, dynamic bulge tests were developed relying on the principles of split Hopkinson pressure bar (SHPB) testing. The use of SHPB or Kolsky devices is widespread to investigate the mechanical compression response of materials at strain rates ranging between 102 – 5·103 \({s}^{-1}\) [18]. In a SHPB setup, two long bars, i.e., the input and output bar, are used to guide a mechanical compression wave to a specimen and, after deformation, away from the specimen. A major advantage of Hopkinson-based setups is that the force and deformation imposed to the specimen can be determined from straightforward measurements of the elastic waves travelling in the bars. The deformation rate imposed to the sample can be adjusted by changing the amplitude of the loading wave. Grolleau et al. [19] presented a SHPB-based dynamic bulge test. In the setup, the circular sheet sample was attached to a movable hydraulic chamber in contact with the output bar. The input bar served as a piston pressurising the liquid in the chamber at a high speed. Strain gage measurements on the input bar were used to derive the bulge pressure. Since the sample was not visible during the test, the deformation could not be captured using full-field measurement techniques. A strain gage was used to measure the strain evolution at the apex during the experiment. To identify the material properties, the authors proposed an inverse analysis combining (FE) simulations with experimental results. Similarly, Ramezani and Ripin [20] presented a variation of the setup of Grolleau replacing the liquid in the hydraulic chamber by rubber. Major assets of combined SHPB-bulge setups are the fact that, within certain limits, the strain rate imposed to the sample can be adjusted and, that reliable values of the global sample displacement and bulge pressure are obtained from strain measurements on the bars. However, in existing setups the sample surface is hidden during the test, so that the displacement and strain fields on the sample surface cannot be measured. Another drawback is related to the limited force and deformation that can be imposed in SHPB tests which imposes limitations to the sample dimensions. Additionally, small samples are also required to achieve high strain rates. However, using samples with diameter to thickness ratios lower than 100 prevents the application of the above mentioned methodologies used for static bulge tests to calculate the constitutive response of the sheet sample as demonstrated in [21]. Inverse, combined numerical-experimental identification techniques, such as these presented in [22,23,24], offer a solution here. A prerequisite is, however, that sufficient measurement data of high quality is available. In present work, a novel high-speed bulge (HSB) facility is presented. Since, the setup is also SHPB-based, it preserves all positive features of existing setups. Though, by adopting a special configuration of the Hopkinson bars, the setup overcomes the most important shortcoming of existing setups. Indeed, main innovation of the facility is that the sample surface is fully accessible. As such, using high-speed cameras combined with image processing software can be used to calculate the 3D-displacement and in-plane strain fields. Moreover, from strain measurements on the output bar, the time history of the pressure imposed to the sample is obtained. Combining all measurements, unique opportunities to characterise the plastic deformation of sheet metals are obtained. Highly interesting information that would otherwise be hidden, such as the development of anisotropy, the onset of plastic instabilities and fracture in the sample, is revealed. Besides, it allows the assessment of the material formability for quasi-equibiaxial stress states at high-strain rates. Because of the limited sample dimensions, straightforward calculation of the flow curve(s) according to, for example, [14] or [15] is prone to errors. However, the available data can be used in conjunction with full field identification methods to obtain the material’s flow behaviour up to large levels of plastic strain. The potential of the new HSB facility is assessed by tests on AA2024-T3 aluminium sheet. Small samples, i.e., with a diameter to thickness ratio of 75, are used in order to achieve high strain rates. The well-established methodology described in the ISO-16808:2014 standard for bulge experiment, is used to obtain an estimation of the material’s flow curve. FE simulations are performed to assess the obtained flow curve.

Background

The novel high-speed bulge test facility combines basic features of both SHPB and conventional hydraulic bulge setups. Therefore, the elementary principles of both test techniques along with the equations used to calculate test results from measurements are briefly recalled in sections "Bulge Test on Sheet Metals" and "Split Hopkinson Pressure Bar".

Bulge Test on Sheet Metals

Fig. 1(a) schematically represents a circular sheet metal sample in a conventional bulge setup. The circular sample of initial thickness \({t}_{0}\) is fully clamped around its perimeter to a circular die of radius \(a\). One side of the sample is in contact with a liquid in a hydraulic chamber. The increment of the pressure \(p\) in the hydraulic chamber leads to the deformation of the sample, which may finally result in fracture. Under certain conditions, the constitutive behaviour of the sheet metal can be derived directly from a bulge test, as elaborated in standard ISO-16808:2014 based on the theoretical framework developed by Hill [13]. When the ratio of the diameter to the thickness of the sample is sufficiently large, bending and shear stresses can be disregarded, and the membrane theory is applicable. In that case, the stresses and strains are constant over the thickness of the sheet. In Fig. 1(b), the two in-plane stress tensor components \({\sigma }_{1}\) and \({\sigma }_{2}\) acting on an infinitesimal membrane element at the sample apex are showed. The principal radii are denoted as \({R}_{1}\) in the meridional plane and \({R}_{2}\) in the perpendicular plane through the surface normal.

Fig. 1
figure 1

(a) Schematic representation of a half-section of a deformed sample clamped in a hydraulic bulge setup. (b) Principal stress components \({\upsigma }_{1}\) and \({\upsigma }_{2}\) acting on an infinitesimal membrane element at the sample apex

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Assuming an axisymmetric stress distribution in the sample, equilibrium of the infinitesimal membrane element in Fig. 1(b) in the direction normal to the sample surface results in:

$$\frac{{\sigma_{1} }}{{R_{1} }} + \frac{{\sigma_{2} }}{{R_{2} }} = \frac{p}{t}$$
(1)

where \(t\) is the actual sheet thickness at the location considered. If the material response is isotropic, also the bulge shape is axisymmetric with respect to the Z-axis. In that case, the material at the sample apex is stretched under balanced biaxial tension and the radius of curvature at the sample apex \(R\) is equal in all normal planes. The biaxial stress is then derived from equation (1).

$$\sigma = \frac{pR}{{2t}}$$
(2)

For metals, it is generally assumed that their plastic behaviour is independent of hydrostatic pressure. In that case, the stress state at the sample apex is equivalent to uniaxial compression acting perpendicularly to the sample surface and the corresponding true strain is given as:

$$\varepsilon = \ln \left( {\frac{t}{{t_{0} }}} \right)$$
(3)

For metals that can be described by Mises or J2-plasticity, the von Mises stress and the corresponding equivalent strain are represented by equations (2) and (3), respectively. Different techniques exist to measure the radius of curvature \(R\) and sample thickness \(t\) at the apex. In the ISO-16808:2014 standard, the DIC technique is recommended to calculate the sample’s out-of-plane displacements and in-plane strain components from which \(R\) and \(t\), respectively, can be derived. The two principal in-plane strain components, i.e., \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\), measured by DIC at the visible surface of the sample, are used to calculate the actual thickness at the sample apex under the assumption of volume conservation: \(t={t}_{0}{e}^{-\left({\varepsilon }_{1}+{\varepsilon }_{2}\right)}\). Besides, the curvature needed to derive the stress at the sample apex, is calculated fitting the spatial coordinates of the sample surface with a quadratic polynomial function. This procedure is performed for each measured state. Obviously, the direction-independent stress-strain relation obtained by equations (2) and (3) is only valid for isotropic sheet metals. In [15], a new set of equations valid in case of non-equal biaxial curvatures due to the sample anisotropy was proposed. Based on DIC measurements of the sample deformation, the principal radii \({\mathrm{R}}_{1}\) and \({\mathrm{R}}_{2}\), at the apex are identified. Still relying on the membrane theory, the authors derived equations for the principal stresses, \({\upsigma }_{1}\) and \({\upsigma }_{2}\), at the apex:

$$\sigma_{1} = \frac{{pR_{2} \left( {R_{1} - t} \right)\left( {R_{2} - t} \right)}}{{t\left( {R_{2} - \frac{t}{2}} \right)\left( {R_{1} + R_{2} } \right)}}$$
(4)
$$\sigma_{2} = \frac{{pR_{1} \left( {R_{1} - t} \right)\left( {R_{2} - t} \right)}}{{t\left( {R_{1} - \frac{t}{2}} \right)\left( {R_{1} + R_{2} } \right)}}$$
(5)

Combined with a more accurate calculation of the actual sample thickness at the apex, the new formulation of the biaxial stress at the specimen apex has an accuracy of 1.1% and 1.8% for the two considered materials, i.e., DP600 and AA5518-O. Use of equations (2) and (3), on the other hand, results in an error of 3%. When \({R}_{1},{R}_{2}\gg t\), equations (4) and (5) reduce to the balanced biaxial stress condition:

$$\sigma_{1} = \sigma_{2} = \frac{{pR_{1} R_{2} }}{{t\left( {R_{1} + R_{2} } \right)}}$$
(6)

Important to note is that equation (6) shows that if the ratios of the radii to thickness are sufficiently large, even if the radii are different, the stress state is quasi-equibiaxial. In case of identical radii of curvature, i.e., \({R}_{1}={R}_{2}\), equation (6) is identical to equation (2) proposed by Hill [13] and adopted in the ISO-16808:2014. In case of dynamic bulge experiments based on the SHPB principle, both the force and displacement that can be imposed to the sample are limited. Therefore, to maximise the operation window of the dynamic bulge test in terms of materials, and maximum strains and strain rates that can be imposed, small samples must be used [19]. However, for ratios of die diameter \(D=2a\) (Fig. 1(a)) to initial specimen thickness (\(D/{t}_{0}\)) lower than 100, the assumption of uniform stresses and strains over the sample thickness is no longer valid, and the above mentioned equations merely give a first order approximation of the constitutive response [21].

Split Hopkinson Pressure Bar

A schematic representation of a conventional SHPB facility used to identify the high strain rate compression properties of materials is given in Fig. 2. The basic components are two long bars, i.e., the input and output bar. Between the bars, the small, most often cylindrical, test sample is positioned. To test the sample, a long, cylindrical impactor is accelerated towards the free end the input bar. The impact with the input bar generates a compressive incident wave traveling along the input bar towards the sample. The incident wave deforms the sample at a deformation rate, and is partly reflected back into the input bar and partly transmitted to the output bar. The one-dimensional wave propagation speed in slender bars is a function of the Young’s modulus \(E\) and density \(\rho\): \(C=\sqrt{E/\rho }\). As such, the duration of the incident wave \(T = 2{L}_{imp}/{C}_{imp}\), depends on the material, i.e., \({E}_{imp}\) and \({\rho }_{imp}\), and length \({L}_{imp}\) of the impactor [25]. The subscripts inp and out are further used to refer to parameters of the input and output bar, respectively.

Fig. 2
figure 2

Schematic representation of a conventional split Hopkinson pressure bar facility and elastic waves (incident \({\varepsilon }_{i}\), reflected \({\varepsilon }_{r}\) and transmitted \({\varepsilon }_{t}\)) measured from strain gages attached to the two bars

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Strain gages attached to the bar surfaces record the strain histories \({\varepsilon }_{i}\), \({\varepsilon }_{r}\) and \({\varepsilon }_{t}\), corresponding with the incident, reflected and transmitted waves, respectively. The corresponding stress waves \({\sigma }_{i}\), \({\sigma }_{r}\) and \({\sigma }_{t}\) are derived multiplying the strain waves by the Young’s modulus of the bars. From the measured waves, the force, velocity and displacement at the bar/sample interfaces can be calculated using the one-dimensional wave propagation theory [18]. To this purpose, the waves \({\varepsilon }_{i}\) and \({\varepsilon }_{r}\), measured by the strain gage A in the input bar, and \({\varepsilon }_{t}\), measured by the strain gage B in the output bar, are shifted in time from the location where they are measured towards the sample/bar interfaces. Generally, wave dispersion is neglected in the shifting process. However, in case of large bar diameters and/or shifting distances, dispersion might give rise to a non-negligible change of the wave shape and has to be accounted for. The sample force \({\mathrm{F}}_{s}\) can then be calculated from the transmitted wave:

$$F_{s} \left( t \right) = A_{out} \sigma_{t} \left( t \right) = A_{out} E_{out} \varepsilon_{t} \left( t \right)$$
(7)

\({E}_{out}\) and \({A}_{out}\) are the Young’s modulus and the section of the output bar respectively. If the length of the sample is sufficiently small, a quasi-static equilibrium is achieved in the sample. The force equilibrium is achieved when the force at the input bar/sample interface is equal to the force at the output bar/sample interface, and thus the force in every section of the sample is constant. The dynamic equilibrium can be checked verifying that the forces acting at the two bar interfaces are identical:

$$A_{inp} E_{inp} \left[ {\,\left| {\varepsilon_{i} \left( t \right)} \right| - \left| {\varepsilon_{r} \left( t \right)} \right|\,} \right] = A_{out} E_{out} \left| {\varepsilon_{t} \left( t \right)} \right|$$
(8)

\({E}_{\mathrm{inp}}\) and \({A}_{\mathrm{inp}}\) are the Young’s modulus and the section of the input bar respectively.

High-Speed Bulge Test

Working Principle

Fig. 3 gives a schematic representation of the HSB facility, together with the corresponding Lagrangian diagram showing the propagation of the elastic waves in the bars. In traditional SHPB setups, the two elastic bars are aligned, one after the other with the sample sandwiched in between (see Fig. 2). However, in the current setup the input bar is positioned centrically inside the hollow cylindrical output bar. A circular sample clamping system, consisting of a holder and die is connected at the end of the output bar. As a result, the sample surface is visible and can be monitored by high-speed cameras during testing. A piston, inserted into a hydraulic chamber realized inside the hollow output bar, is in contact with the end of the input bar. The impact of long cylindrical impactor at the free end of the input bar generates a compressive wave. The wave has a well-determined length and amplitude, and propagates in the input bar towards the hydraulic piston. The piston is accelerated, thereby pressurizing the liquid in the hydraulic chamber resulting in deformation of the sample. The dynamic load needed imposed to the sample is transferred to the output tube as a tensile wave. At the piston/input bar interface, the compression loading wave is partly reflected as a tensile wave. By changing the amplitude of the loading wave, the deformation rate of the sample can be adjusted. The output bar is used as dynamic load cell, as further elaborated. The input bar is employed to transfer the dynamic compressive wave to the liquid medium. The bars are sufficiently long, so that the reflections of the reflected and transmitted waves, at the free ends of the input and output bar respectively, do not interfere with the sample loading, see the Lagrangian diagram in Fig. 3.

Fig. 3
figure 3

Schematic representation of the high-speed bulge test facility and Lagrangian diagram showing the wave propagation in the bars

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The bulge pressure acting at the sample surface can be derived from the force on the sample given by equation (7):

$$p\ \left( t \right) = \frac{{E_{out} A_{out} }}{{A_{p} }}\varepsilon_{t}\ \left( t \right)$$
(9)

where \({A}_{p}\) is the section of the piston.

Setup

Pictures of the HSB facility installed at Ghent University are shown in Fig. 4.

Fig. 4
figure 4

High-speed bulge setup installed at Ghent University. The left picture shows the side of the projectile, the right picture the side of the sample

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The cylindrical input bar is 6m long and has a diameter of 25mm. The output bar is a 3m long tube with inner and outer diameter of 70mm and 100mm, respectively. Al-7075 in T6 tempered condition is selected for both bars because of its high yield strength of 570MPa and relatively low Young’s modulus of 72000MPa. Because of the high yield strength, high amplitudes of the incident wave are allowed. Moreover, the low Young’s modulus of the selected aluminium grade compere, for instance, to steel, allows to impose high deformation levels to the sample. Indeed, the relative displacement of the bars, and thus the piston displacement, is highly affected by the bars Young’s modulus [18]. An impactor made of Ertalon with a circular cross section of 2922.5 \({mm}^{2}\) is used. The material and size of the impactor are chosen such to obtain an impactor impedance that is slightly lower than the one of the input bar. Because the impedances of the impactor and input bar are close together, good energy transfer is guaranteed; because the impedance of the impactor is slightly lower, only one impact is generated. Moreover, an impactor with slightly round end face is used to obtained the desired rise time without using pulse shaping. With the chosen lengths and material of the bars, the incident wave has a maximum duration of about 1.2ms. Piston, hydraulic chamber and section of the sample subjected to the pressure have the same diameter, i.e., 60mm. However, to clamp the sample, a specimen with diameter of 104mm is required. Indeed, small samples must be used in dynamic bulge experiments in order to achieve high strain rates during the test [19].

To reduce weight and, thus, inertia effects on the test results, sample clamps, hydraulic chamber and piston are made of high-grade aluminium. In particular, all components of the hydraulic chamber are made of Al-7075 in T6 tempered condition, with the only exception of the stopper ring which is made of S235 steel. The setup allows for testing sheet metals having thicknesses up to 1.8 mm and maximum yield strengths of about 1200 MPa. Water is used as liquid medium because of its high bulk modulus of about 2GPa. Before the start of an experiment, the input bar is put in close contact with the piston, see Fig. 5. Figure 5 schematically presents the hydraulic chamber and sample/clamp assembly, including dimensions.

Fig. 5
figure 5

Schematic representation of the hydraulic chamber and sample/clamp assembly of the high-speed bulge setup installed at Ghent University: 1) input bar, 2) output bar, 3) piston, 4) water, 5) sample, 6) die, 7) stopper ring, 8) sample holder, and 9) external ring threaded to the output bar. Dimensions are given in mm

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Material and Methods

High-speed bulge experiments are performed on 0.8mm thick AA2024-T3 sheet. The incident, reflected and transmitted waves are measured with a frequency of 1MHz using a HBM transient recorder. Prior to testing, a stochastic white/black speckle pattern is painted on the sample surface. Two high-speed cameras, mounted under a certain angle needed for stereo imaging, are used to record the deforming speckle pattern. The cameras used in present study are Photron mini AX200 cameras equipped with two Tamron lenses of 90mm focal length. In order not to damage the cameras, the image of the specimen is projected with the aid of a mirror with high surface quality positioned under an angle of ±45° relative to the sample and cameras. Two Dedocool lamps (see Fig. 4) provide high-intensity lighting. After testing, the recorded images are processed by the commercial DIC software package MatchID. The zero-normalized sum of squared differences (ZNSSD) correlation criteria is used for the processing of the DIC data. This criterion accounts for offset and scaling in intensity variations of the field of view. Table 1 gives an overview of the parameters used for the DIC processing.

Table 1 DIC processing parameters
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A frame rate of 30 000fps and spatial resolution of 512 x 384 pixels are used. The camera and DIC processing settings result in displacement and in-plane strain noise floor levels, i.e., accuracy levels of about 10-3 mm and 10-4, respectively. Maximum out-of-plane displacements of the order of mm’s, and in-plane strains of the order of 10-1, are measured in the experiments and, hence, the noise levels are considered to be acceptable. The spatial coordinates of the data points extracted via DIC are imported in a MATLAB script. As described in the standard procedure ISO-16808:2014, a 2th order polynomial function is used for the determination of the principal curvatures in the rolling and transverse direction i.e., \({k}_{RD}\) and \({k}_{TD}\). The average curvature radius is calculated as \(R=2/\left({k}_{RD} + {k}_{TD}\right)\). The fitting procedure is performed for every image, i.e., time step, and for all the data points located within \(\pm 20 mm\) with respect to the sample apex. The choice of the interpolation window size resulted from a carefully balanced compromise between having a sufficiently high number of data points to guarantee a reliable fitting, on the one hand, and capturing the parameters needed to calculate the stress at the apex, at the other.

Results and Discussion

In the following sections, detailed results are presented of tests performed using three different impactor velocities. Test results based on SHPB-based processing of the strain gage measurements on the bars, and high-speed stereo DIC are reported in sections "SHPB Signals and Sample Force" and "Full-field Displacement and Strain Fields", respectively.

SHPB Signals and Sample Force

In Fig. 6 the strain recordings of gages A and B, see Fig. 3, on the input and the output bar, respectively, are presented for three tests, denoted as Low, Med and High, using different impactor test velocities. The strain gage signals are shifted in time, forward for gage A and backward for gage B, in order to synchronise the incident and transmitted waves at the sample interface. Gage A shows the incident wave immediately followed by the reflected wave. The interaction of the incident wave with the piston generates a peak at the beginning of the reflected wave. Indeed, the acceleration of the piston gives rise to an inertial reaction force in opposite direction with respect to the compressive load imposed by the input bar. Since the reflected wave is propagating away from the sample, the peak does not affect the sample deformation. The duration of the incident wave, and thus the test duration, is approximately of 1.2ms. After an initial rise time of about 80µs, the incident waves reach relatively constant strain amplitudes of 1320, 1500 and 1850µstrain for the Low, Med and High experiments, respectively. The waves recorded by gage B are also presented in Fig. 6, amplified by a factor of 10 for clarity. The transmitted tensile waves increase quasi-monotonically from the onset of loading, up to a maximum value depending on the impactor velocity or sample failure. In the experiment performed at highest velocity, the sample fails during testing. For this test, the sharp drop at the end of the transmitted wave is linked with the sample fracture. For the lowest and medium velocities, on the contrary, the sample did not reach failure and the drop in the transmitted wave corresponds with the end of loading. As such, the peak amplitude for these tests cannot directly be linked with material properties, such as strength, deformation limits or strain rate sensitivity.

Fig. 6
figure 6

Strain signals measured on input bar (Gage A, solid lines) and output tube (Gage B, dotted lines) for different velocities of the impactor (Low, Med and High)

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In Fig. 7, the forces acting at the two bar interfaces plotted together. The good agreement between the forces shows that the force equilibrium condition, previously introduced in section "Split Hopkinson Pressure Bar", and required during Hopkinson experiments, is fulfilled. It is important to note that the hydraulic chamber is placed between the two bars. Thus, when the force equilibrium is verified, i.e., the pressure measured in the chamber at the interface with the input bar is equal to the one measured at the side with the output bar, then it is reasonable to assume that also the pressure in the hydraulic chamber corresponds to the one calculated based on the measurements on the bars.

Fig. 7
figure 7

Force equilibrium at the piston/liquid and sample/output bar interfaces for three different velocities of the impactor (Low, Med and High)

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Therefore, the force on the sample can be calculated from the transmitted wave using equation (7) .The apparent shorter duration of the force signal recorded in the input bar is related to the interference between the reflected wave and its reflection at the free end of the input bar. Consequently, gage A does not capture the last part of the reflected wave correctly. It should be noted that a nearly oscillation-free transmitted wave is obtained which guarantees a stable sample loading. This is mainly attributed to the efforts spent during the design to keep impedance mismatches as small as possible by minimizing the mass and dimensions of clamps and hydraulic chamber. To assess the influence of wave dispersion in the output bar, and thus the accuracy of the strain gage measurements, a FE simulation is performed in Abaqus/Explicit [26]. The output tube is modelled as a deformable body meshed with C3D8R, with six elements throughout the thickness with size of 2.5mm. The mesh size along the tube length is 5mm. Values for the density and elastic properties of Al7075 are used, i.e., \(\rho =2700kg/{m}^{3}\), \(E =70 GPa\) and \(v =0.3\), respectively. The measured transmitted wave of the high speed experiment is first fitted to a 4th order polynomial equation, and then imposed as dynamic pressure at one side of the tube. The uniaxial stress at the surface at different locations along the output tube, namely just after the interface (G.1) and at the position corresponding with Gage B (G.2), together with the imposed stress, are plotted in Fig. 8. The FE simulation reveals that the measurement of the imposed uniaxial stress, corresponding with the transmitted wave, becomes less and less accurate moving far from the output bar interface, where the load is applied. Besides, the measurement of the transmitted wave at the location G.2, i.e., location at which the Gage B is placed, leads to an underestimation of about 3% of the imposed stress towards the end of the experiment.

Fig. 8
figure 8

Uniaxial stress at different locations of the output tube length: beginning (G.1), at the corresponding position of the Gage B (G.2)

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Full-Field Displacement and Strain Fields

Out-of-Plane Displacements and Apex Radii

Full-field DIC contour plots of the out-of-plane displacement, for the experiment at the highest velocity, are reported in Fig. 9. The figure shows the sample deflection at different instants of the test, from the onset of deformation (6.9ms) till the frame before fracture (7.8ms). The out-of-plane displacement evolves quasi-symmetrically with respect to the centre of the sample. In Fig. 10 the sample profiles measured via DIC along the central lines aligned with the rolling and transverse direction are presented at several time instants, together with best fitted circles with radius \(R\) derived following the procedure elaborated in section "Material and Methods". At higher deformation, the radius around the apex is smaller than in the surrounding material. The assumption of a spherical sample shape, as in [13], is thus not valid at the later stages of deformation. In Fig. 11 the radii of curvature along the RD and TD, and the average radius of curvature \(R\), calculated from the averaged curvatures as explained in section "Material and Methods", are presented as a function of the apex displacement, together with the difference between the two radii normalized with respect to the \(R\).

Fig. 9
figure 9

Full-field contour plots of the out-of-plane displacement at different instants for the experiment at the highest velocity

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Fig. 10
figure 10

Sample profiles along the central lines aligned with the rolling and transverse directions together with best fitted circles with radius \(R\) (ISO-16808:2014) at different instants for the experiment at the highest velocity

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Fig. 11
figure 11

Fitted radii of curvature along RD and TD, average radius of curvature \(R\), and relative difference with respect to the \(R\), for the experiment at the highest velocity

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The normalized difference at low sample deflections can partly be attributed to measurement inaccuracies. During stable deformation a normalized difference with absolute value below 5% is obtained. Just before sample fracture, more significant differences are observed between the profiles along the rolling and transverse direction.

In-Plane Strain Fields

The contour plots of the strains in the rolling and transverse direction at difference stages during the highest velocity experiment are compiled in Fig. 12. The plots show that no, undesirable, concentration of deformation occurs at the sample clamping. The highest strain values are found at the sample apex. Differences between the two strain components are observed in terms of strain amplitudes and distribution, especially at higher deformations. Indeed, the strain fields, during the last stages of the experiment (7.8-7.9ms), exhibit an elliptical distribution which is more extended in the transverse direction for the rolling strain and in the rolling direction for the transverse strain. The strain in the transverse direction \({\varepsilon }_{TD}\) at the apex is slightly higher than the strain in the rolling direction \({\varepsilon }_{RD}\). The fracture starts at the apex and propagates along the rolling direction.

Fig. 12
figure 12

Full-field contour plots of engineering strain in the rolling (\({\varepsilon }_{RD}\)) and in the transverse (\({\varepsilon }_{TD}\)) directions at different instants during the experiment at the highest velocity

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The differences between the two strain fields clearly indicate the material anisotropy. The evolution of the strain in the rolling and transverse directions, measured at the sample apex, is represented in Fig. 13. The two curves coincide till a strain of approximately 5%, however, with increasing deformation, the two strain components start to deviate. The solid black line represents the coefficient of biaxial anisotropy, defined as \({r}_{b}={\varepsilon }_{TD}/{\varepsilon }_{RD}\), which gives an indication of the evolution of the sheet metal anisotropy. The more pronounced the anisotropy, the more the coefficient deviates from one, indicated by the horizontal line. Starting from \({r}_{b}\) values well below one at the onset of plastic deformation, the coefficient soon reaches a value higher than 1 and then saturates around a value of about 1.2. Towards the end of the deformation, the coefficient slightly decreases. The development of anisotropy during the plastic deformation clearly affects the strain path at the sample apex. Indeed, the deformation path gradually moves away from the pure equibiaxial stretching.

Fig. 13
figure 13

Time evolution of the engineering strain components at the sample apex and biaxial coefficient of anisotropy for the highest velocity test

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Fig. 14 presents the strain along a radial line starting at the sample apex along the rolling direction at different instances considering a time increment of 0.3ms. The strain is relatively constant in a circular region with a radius of at least 5mm around the apex. The strain increments between the consecutive time steps are almost constant, especially around the apex. However, a significant strain increment, together with a concentration of the strain around the apex, is observed just before the sample fracture (7.5-7.8ms). As is clear from Fig. 10, during deformation, the curvature radius around the apex gradually becomes lower than the radii in the surrounding material. The resulting higher stress combined with the low strain hardening of the aluminium alloy at higher deformation levels, results in a self-amplifying concentration of strains around the apex. In the DIC strain plots just after fracture (Fig. 12), strain values of about 0.29 are observed in a zone around the apex. The equivalent strain obtained for the HSB test at highest speed, calculated using equation (3), is approximately 50%. In case of quasi-static tensile experiments on cylindrical dog bone Al2024 samples along different orientations, the maximum level of equivalent strain reported is about 18% [27]. In [10], the same material was tested at low speed under biaxial loading conditions making use of a cruciform sample. The maximum biaxial strain before fracture is reported to be 1.8%. Indeed, cruciform samples allow to investigate the biaxial response of materials only for low levels of plastic deformation due to premature sample failure at the gauge corners. The significantly higher strains reached in the HSB test compared to other techniques, together with the additional data obtained on the anisotropy, shows that the bulge test can be successfully used to investigate the plastic deformation and formability of sheet metals up to large strain levels.

Fig. 14
figure 14

Evolution of the engineering strain in the rolling direction at different instances for the highest velocity test

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Strain Rate

The strain rate at the sample apex, calculated as time derivative of the through thickness logarithmic strain, given by equation (3), is presented as a function of time in Fig. 15. For the lowest and medium velocity experiments, during plastic deformation, a quasi-constant strain rate ranging between 200-250 \({s}^{-1}\) is obtained. After the stable plateau, the strain rate drops to reach zero at the end of loading. For the highest velocity test, a stable strain rate of about 300-350 \({s}^{-1}\) is imposed to the apex material. After the plateau, the strain rate increases very rapidly until fracture of the sample. The significant rise in strain rate is linked with the concentration of strain around the sample apex, as observed in Fig. 14.

Fig. 15
figure 15

Strain rate evolution with respect to the time for three different velocities of the impactor (Low, Med and High)

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Derivation of the Material Flow Curve

As discussed in section "Bulge Test on Sheet Metals", several approaches can be adopted to determine the material flow curve from a HSB test. The results presented in Fig. 10 (section "SHPB Signals and Sample Force") show the importance of limiting the region around the apex in which the principal radii are calculated to obtain the apex stress using equations (2) or (6). Figure 16 presents the components of the stress along the RD and TD directions calculated using equations (4) and (5) proposed by Min et al. [15] as a function of the apex displacement. Only values of the apex displacement higher than 4mm are considered, which corresponds with onset of plastic deformation at the apex. The discrepancy between the two radii of curvature, see section "Out-of-Plane Displacements and Apex Radii", is not reflected on the two components of the stress, which virtually overlap for the entire test duration.

Fig. 16
figure 16

Evolution stress components along the rolling and transverse directions with respect to the sample deflection at the sample apex for the highest velocity

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The correspondence between the two components of the stress, justifies the assumption of an equibiaxial stress. In this case, the formulation of the biaxial stress proposed by Min et al. [15], i.e., equation (6) , is identical to equation (2) proposed by Hill [13]. Furthermore, as reported by Lafilé et al. [21], Mises plasticity can be used to describe AA2023-T3, so, the thus obtained biaxial stress is equal to the flow or equivalent stress. However, the use of equations (2) and (6) is restricted to samples with a sufficiently large diameter to thickness ratio. Also, the derivation of the thickness or equivalent strain from the in-plane strain values measured at the surface requires large specimens. In the ISO-16808:2014, a \(D/{t}_{0}>100\) is stipulated which is larger than the ratio \(D/{t}_{0}=75\) of the samples used in present study. Bending stresses and strains through the sample thickness might consequently compromise the accuracy of the derived flow curve. Therefore, the error made due to the employment of the membrane theory is further assessed by means of FE simulations. In Fig. 17, the time evolution of the through thickness logarithmic strain, average radius and out-of-plane displacement at the sample apex are represented for the experiment at highest velocity. Fluctuations in the out-of-plane displacement measured by DIC reveal vibrations of the sample during the high-speed bulge test. In the first stage of the test, the vibration of the circular sample is reflected in all curves: the strain and, though to a lesser degree, also the displacement evolve stepwise, the radius decreases accordingly. At the latest stage of deformation (t > 7.5ms), sample ringing disappears.

Fig. 17
figure 17

Time evolution of through thickness strain, radius of curvature R and out-of-plane apex displacement Wc for the experiment at the highest velocity

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Figure 18 presents the biaxial stress, together with the quantities needed to calculate it, i.e., actual thickness, radius of curvature at the apex, and pressure, for the highest velocity experiment. In the first stage of the deformation, clear oscillations are observed in the biaxial stress which are linked with the oscillations of the curvature radius.

Fig. 18
figure 18

Time evolution of equivalent stress (equation (2)) and the quantities necessary for its calculation, i.e., sample thickness t and radius of curvature R at the apex, and bulge pressure, for the experiment at highest velocity.

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The resulting equivalent material curves, for the three experiments, are reported in Fig. 19. For all tests, sample vibrations affect the initial stage of the stress-strain curves. However, it is clear that higher test velocities give rise to higher oscillations. Besides the inevitable oscillations due to interaction of the loading wave with the hydraulic chamber and the specimen clamps, eigenmode vibrations of the specimen give rise to additional oscillations.

Fig. 19
figure 19

Stress-strain curves obtained from HSB tests for three different velocities of the impactor (Low, Med and High)

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The reliability of the material flow curve obtained for the highest velocity experiment is assessed by FE simulations using Abaqus. The axisymmetric model consists of the sample, die and sample holder reported in Fig. 5. The die and holder are modelled as rigid bodies. The Al2024 sample is modelled as an elasto-plastic body using \(70 GPa\) for the Young’s modulus and \(v =0.3\) for the Poisson’s ratio. J2-plasticity is adopted to describe the plastic response. For the strain hardening, values are used obtained by fitting the experimental data to a quadratic polynomial function. CAX4R elements are used with as many as 9 elements over the thickness in order to accurately capture the through-thickness stress and strain distribution in the sample. A uniform pressure, equal to the one measured by strain gage B, is applied at the bottom surface of the sample, while displacements and rotations are constrained at the outside boundary of the specimen. The differences between the simulated flow curve \({\varepsilon }_{p}-{\sigma }_{VM}\) obtained at the middle plane at the apex, and the flow curve \({\varepsilon }_{p}^{ISO}-{\sigma }_{\mathrm{VM}}^{ISO}\) extracted from the simulation results following the ISO-16808:2014 procedure, are used to assess the experimental errors. Indeed, the latter curve is calculated in an identical way to the tests, i.e., only relying on the simulated in-plane strains and out-of-plane displacements at the outer surface of the sample, and imposed pressure. Figure 20 reports the errors on the equivalent stresses and strains obtained by the ISO standard procedure calculated as: \({E}_{{\varepsilon }_{p}}=100\cdot \left|{\varepsilon }_{p}-{\varepsilon }_{p}^{ISO}\right|/{\varepsilon }_{p}\) and \({E}_{{\sigma }_{VM}}=100\cdot \left|{\sigma }_{VM}-{\sigma }_{\mathrm{VM}}^{ISO}\right|/{\sigma }_{VM}\). Starting from a value close to 70%, the error on the equivalent plastic strain rapidly decreases with increasing strain. For a plastic deformation of about 10%, an error of 9% is observed, while at the maximum strain of the test \({E}_{{\varepsilon }_{p}}\) has a value slightly below 0.5%. On the contrary, the error on the von Mises stress increases during the plastic deformation to reach a maximum of approximately 6.5%. The significant, non-negligible errors highlight the need for more advanced techniques to derive the material flow curve. Inverse, combined numerical-experimental identification techniques, such in [22,23,24], offer a solution here, especially, because the use of high-speed imaging makes the necessary measurement data available.

Fig. 20
figure 20

Error on the equivalent plastic strain calculated by equation (3) (solid line) and error on the von Mises equivalent stress calculated by means of equation (2) (dotted line)

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Summary and Conclusions

A novel high-speed bulge (HSB) facility designed to perform biaxial experiments on sheet metals at an adjustable high strain rate is presented. The setup allows a straightforward, reliable measurement of the bulge pressure. However the main benefit of the facility is that the sample surface is fully accessible for high-speed, full-field optical measurements. The potential of the facility is demonstrated by performing experiments on 0.8mm thick AA2024-T3 aluminium sheets. High-speed camera imaging combined with stereo DIC is used for full-field identification of both the displacement and strain fields. The experiments are carried out for three different impactor velocities resulting in strain rates ranging between 200-350s-1 at the sample apex. The nearly oscillation-free pressure signal indicates a stable sample loading. Furthermore, quasi-static equilibrium is verified. The design of the clamping system prevents strain localization at the sample edges. Instead, the concentration of the deformation at the apex of the sample drives onset of fracture at this location. For the tested material, the fracture is aligned with and propagates along the rolling direction. The ability to capture the displacement and deformation fields of the entire sample surface provides unique opportunities to characterise the material. It allows to reveal information that would otherwise be hidden, such as sample ringing during the early stages of the dynamic loading, the evolution of the material anisotropy, the time evolutions of the sample deflection and curvatures at the apex. Although the anisotropy of the studied aluminium alloy is not that pronounced, the high resolution obtained for the strain fields by the implemented stereo DIC system allows to reveal differences between the strains in the rolling and transverse directions. On the other hand, no traces of anisotropy are revealed in the out-of-plane displacement fields which demonstrates the much higher sensitivity of strain values to anisotropy. As such, the HSB facility opens new possibilities for to identification of sheet metal plasticity till large levels of deformation in quasi-biaxial loading conditions and at high strain rates. In this study, the classical equations proposed by [13], based on the membrane theory and pressure-independent plasticity, are employed following the procedures described in the ISO-16808:2014 to derive the biaxial material flow curve. However, given the limited dimensions of the specimen, i.e., \(D/{t}_{0}=75<100\), the formulas are used outside their validity range. Therefore, their reliability is assessed by finite element (FE) simulations. The assessment shows that non-negligible errors are obtained. These errors combined with the intrinsically lower resolution of high-speed imaging, result in stress-strain curve that only give a first order approximation of the actual constitutive behaviour. Moreover, for materials with a more complex plasticity response, the link between the obtained stress-strain curve and plasticity model, including material parameters, might not be straightforward. Therefore, to take full advantage of the available measurement data, the authors strongly advise the employment of material-independent, advanced identification techniques. The available displacement and strain fields at the sample surface can be used to calibrate and validate complex, strain rate dependent plasticity models. The promising results of the experimental campaign support the use of the novel HSB facility as valid dynamic counterpart of the well-established quasi-static bulge test.