Optimal Design, Development and Experimental Analysis of a Tension–Torsion Hopkinson Bar for the Understanding of Complex Impact Loading Scenarios

The mechanical system of the proposed TTHB apparatus, shown schematically in Fig. 1, essentially consists of an incident bar, a transmitted bar, with the testing specimen sandwiched in between, a clamp station, and tension and torsion loading devices. The combined tensile and torsional stress pulses are generated by means of the energy-release mechanism. As shown in Fig. 1(a), by tightening the clamp and then applying force and torque at the distal end of the incident bar, an amount of tensile and torsional energies is stored in the section between the clamp and the loading units. A sudden release of the clamp can initiate a combination of a tensile wave and a torsional wave propagating towards the specimen, termed as the incident tensile and shear waves, respectively. Given conservation of momentum, both waves carry half of the corresponding stored energy. Meanwhile another pair of longitudinal and shear waves of equal magnitudes but opposite signs as the incident waves is generated but travels in the other direction, called the release compressive and shear waves, as shown in Fig. 1(b). Due to the large mechanical impedance of the loading device at the bar end, the released waves reflect back without altering their signs, subsequently counteracting the residual energy in the pre-stressed section, as shown in Fig. 1(c). As the released wave returns to the clamp station, the stored energy would have all released, marking the tail of the incident wave. Therefore, the duration of the generated incident tensile wave t_in.l and shear wave t_in.s is the time required for the longitudinal wave and shear wave (at different wave speeds c_l and c_s, respectively, see equation (1)) to travel a length equal to twice the pre-stressed section l_clamp. Note that the wave interactions at the bar-specimen interfaces are not shown in Fig. 1(c) for the sake of clarity to illustrate the generation of the incident waves. Details of the wave propagation and interaction with the specimen, as well as signal interpretation in a typical TTHB experiment, can be found in [41]. The mechanical design of the pre-loads in tension and torsion is elaborated in [42].

The propagation of the longitudinal wave and shear wave in a long bar can be mathematically expressed as wave equations  (1a) and (1b), respectively:where, u and θ are the linear displacement and the angular displacement, respectively; \(\partial t\) and \(\partial X\) denote the derivative with regard to time and space, respectively; E, G, and \(\rho\) the Young’s modulus, shear modulus, and density of the bar material, respectively.

It is worth noting that the shear wave travels at a different speed c_s, typically 1.6–1.7 times lower than the longitudinal wave speed c_l for a vast majority of the materials. As a result, the shear wave would arrive at the specimen later than the longitudinal wave if using a single clamp to generate both waves, as illustrated in Fig. 1(b). Approximate synchronisation of the two types of waves can be fulfilled by positioning the clamp sufficiently close to the specimen. Another consequence of the different wave velocities is that the duration of the longitudinal wave is shorter than that of the torsional wave, resulting in the effective duration of the combined tension–torsion loading limited by the longitudinal wavelength.

The axial particle velocity \(\dot{u}\) and the angular particle velocity \(\dot{\theta }\) of longitudinal and torsional stress waves propagating along a slender rod are given by equations (2a) and (2b).where, \(\sigma\) and \(\tau\) denote the applied stress; r the radius of the bar. Theoretically a higher particle velocity can be achieved by using bars made of high strength lightweight materials, and the maximum strain rate achievable is determined by the yield stress of the material of the bars:where, \({\sigma }_{Y}\) and \({\tau }_{Y}\), are the yield stress of the bar material. It is worth noting that the stress wave analysis utilised for the evaluation of Hopkinson Bar experiments relies on the stresses in the incident and transmitted bars to be elastic. Therefore, the yield stress of the material of the bars should not be reached but dictates the limit of the force or torque that can be stored without permanent deformations. Hence \({\sigma }_{Y}\) and \({\tau }_{Y}\) determine the maximum strain rate achievable for a given material of the bars.

Titanium is a kind of material featuring high values of \({\sigma }_{Y}/\sqrt{\rho E}\) and \({\tau }_{Y}/\sqrt{\rho G}\); however, it is practically difficult to produce a tensile or torsional pulse of an amplitude high enough to benefit from the high yield stress of alloyed titanium. Furthermore, the amount of the stored load is limited by the capacity of the clamp section, and this is generally lower than what the bar material can theoretically withstand [43].

It should also be pointed out that, when the same load is stored in bars of equal dimensions, the longitudinal and rotational particle velocities achievable using aluminium are higher than with alloyed titanium, as these are inversely proportional to the density and the elastic moduli of the material (equations (2a) and (2b)). For the above reasons the aluminium alloy 7075-T6 was preferred and selected as the bar material in the present TTHB design, as it allows for higher strain rates to be achieved [21, 26, 43, 44]. The mechanical properties of aluminium alloy 7075-T6 is provided in Table 1.

The diameter of the bar also has considerable influence on the achievable strain rate range. Take the torsional wave loading for instance, the shear strain rate of the specimen can be derived, provided dynamic equilibrium, as:where, \({J}_{s}\), \({r}_{s}\) and \({l}_{s}\) are the polar moment of inertia, radius, and gauge length of the specimen, respectively; \(J\) is the polar moment of inertia of the bar; Torque\({T}_{I}\),\({T}_{R}\), and \({T}_{T}\) correspond to the amplitude of the incident, reflected, and transmitted shear waves; \({\tau }_{I}\) and \({\tau }_{s}\) denote the incident shear stress and the shear strength of the specimen material. Considering the limit condition in which the incident bar approaches its yield point at the pre-stressed section, then the incident shear stress \({\tau }_{I}\) equals half of the yield strength \({\tau }_{Y}\) of the bar material,\({\tau }_{I}={\tau }_{Y}/2\), which gives:

This indicates that the shear strain rate is not monotonically changing with the bar radius. The partial derivative of the shear strain rate with respect to bar radius is derived as equation (5). Hence, the optimal radius r that allows for a maximum achievable shear strain rate in the specimen can be obtained by setting equation (5) equal to zero, which gives equation (6).

Here we assume the bar material as Al7075-T6 (Table 1) while the test sample is made of Ti6Al4V to show the variation of the strain rate with respect to the bar radius in Fig. 2(a). Typical tensile strength of Ti6Al4V at high-rate is approximately 1200 MPa [45] and the shear strength is thus estimated, assuming Von Mises plasticity as 693 MPa. Figure 2(a) illustrates the variation of the shear strain rate parametrised by the stress amplitude of the incident wave (as a fraction of the yield stress of the bar material Al7075-T6) and the strength of the sample (as a fraction of the strength of Ti6Al4V). It is clear and intuitive that a higher shear strain rate can be achieved given a higher amplitude of the incident wave and a weaker test material. In practice the stored torque generally corresponds to approximately half of the yield point of bar material or lower, due to the above-mentioned difficulty to reach yielding of the bar during experiments. A bar diameter of 20 mm gives a high shear strain rate for a wide range of applications.

Similarly, the tensile strain rate in the specimen can be calculated as:where, \({A}_{s}\) and \(A\) are the cross section area of the specimen and the bar, respectively. Axial force \({F}_{I}\), \({F}_{R}\), and \({F}_{T}\) correspond to the amplitude of the incident, reflected, and transmitted longitudinal waves; \({\sigma }_{s}\) denotes the tensile strength of the specimen material.

As it is always true that \(\frac{\partial \dot{\varepsilon }}{\partial r}>0\), the tensile strain rate monotonically increases with the radius of the bar. The trend considering different amplitudes of the incident wave and test materials is reported in Fig. 2(b). The bar diameter of 20 mm again allows for achieving a high tensile strain rate in a wide range of applications and hence is selected in the present TTHB system.

The TTHB system is designed on a commercial profile of a length of 6 m (Fig. 3), to provide a proper alignment of the bars with other mechanical components. The bars are supported by a set of spherical bearings through PTFE (polytetrafluoroethylene) sleeves so that the bar system is self-aligning and free to slide and rotate. The TTHB system is functionally composed of the above-mentioned mechanical system and a data acquisition unit.

A linear hydraulic pump is employed as the pre-tension unit in the tension loading station while a harmonic drive actuator as the pre-torsion device in the torsion loading station. The clamp is optimised to enable a favourable initiation of the combined tensile and torsional waves, as will be elaborated in "Critical Mechanical Design" section.

The data acquisition system includes high-speed camera(s) and lighting unit, photon Doppler velocimetry (PDV), and a set of conventional strain gauges. The high-speed camera system is focused on the specimen to record the footage of the deformation process by means of digital image correlation (DIC) analysis to quantify the full-field deformation. Uniaxial strain gauges and shear strain gauges are located on both the incident bar and the transmitted bar to capture the stress waves, as schematically illustrated in Fig. 1(a). The longitudinal and shear strain gauges on the incident bar were located 180 mm and 1063 mm away from the incident end of the specimen respectively, while the distances between the transmitted end of the specimen and the longitudinal and shear strain gauges on the transmitted bar were 345 mm and 575 mm, correspondingly. The position of the strain gauges was chosen to ensure that strain histories were recorded in locations where strains are uniform across the bars cross-section and to avoid the superimposition with stress waves reflected from the far end of the bars prior to the failure of the specimen. The uniaxial strain gauge station consists of four linear strain gauges aligned along the bar axis and equally spaced along the circumference, such that it measures only the axial stress due to the tensile loading but not due to the potential bending of the bars. Whereas the shear strain gauge station is composed of two T-rosette strain gauges that are aligned along the bar axis by their 45-degree bisector and opposite to each other. The shear strain gauges are wired in such a way to measure only the torsional stress induced by torque. The PDV technique is employed to capture the propagating elastic waves of longitudinal and shear nature and complement the measurements of the strain gauges, therefore improving the reliability of the data acquisition system. Each bar is instrumented with one PDV enclosure for the measurements of longitudinal, rotational, and transverse motions. More details of the PDV techniques in the TTHB apparatus can be found in [46].

The design of the clamp plays an essential role in the energy-release mechanism used in this study to generate stress pulses. An ideal design should provide sufficient clamping capacity that prevents the bar from slipping and bending, while on the other hand enable fast and clean release that produces a sharp wave rise and avoids interference from spurious waves due to bar bending. The proposed clamp station is schematically shown in Fig. 4, essentially consisting of a pair of identical clamp jaws that are connected via a notched fracture pin at the top while pressured by a hydraulic cylinder at the bottom. The bar is clamped by increasing the hydraulic pressure not exceeding the fracture load of the pin and then released by further pressure to quickly break the notched pin. A free-moving frame, which is free to slide perpendicular to the bar axis, is designed to allow for equal hydraulic forces applied on both sides of the clamp jaws at the bottom thus preventing transverse movement of the bar in the horizontal plane during clamping. The jaws are attached to a pair of sliding pins at the bottom that guide the jaw to move along a radial slot concentric to the centre of the bar after the breakage of the pin. The jaws are also constrained at the horizontal plane of the bar to prevent the bar from bending in the vertical direction.

The geometry of the jaw is designed to maximise the bar-clamp interaction area that helps increase the gripping capacity without introducing indentation and to prevent vertical deflection of the bar. This part is investigated via finite element (FE) simulations through a non-linear static analysis in ABAQUS/Standard, considering only the initial clamping stage. The FE model of the clamp station without the notched pin is presented in Fig. 5(a). Solid elements C3D8R (ABAQUS/Standard element library, [47]) are used to model the bar, clamp jaws, sliding pins, radial slots, and their interactions. Frictional tangential behaviour is employed in the bar-clamp interaction. The notched fracture pin and the hydraulic cylinder were conceptualised as identical external force demonstrated as the red arrows. Four geometries (Fig. 5(b)) were preliminarily designed by means of a motion analysis to ensure no further interaction between the jaws and the bar during the clamp releasing process. The distribution of von Mises stress and the amplified deflection of the four clamp models are compared in Fig. 5(c). The undeformed model is shaded as a reference to demonstrate the amplified deflection. Particularly the vertical deflection of the bar centroid at the clamp position is extracted in Fig. 5(d). It can be observed that any asymmetry in the load-bearing region of the jaws and in the contact area between the bar and the jaws introduces noticeable vertical deflections (Versions 1.0 and 2.0 in Fig. 5). On the contrary the geometrical symmetry of Versions 3.0 and Version 3.1 results in the absence of any appreciable deflections. Hence Version 3.1 is chosen in the present clamp design as it is exempt from significant deflection of the bar during clamping and undesirable bar-clamp interactions during the clamp releasing process, as confirmed by the experimental data in Fig. 7.

We proceed to assess the influence of the material of the clamp jaws. A more representative FE model is established based on Version 3.1 of the jaw design, as shown in Fig. 6(a). The model of the bar is made of Al7075-T6, 20 mm in diameter and 6 m long, supported by several frictionless bearings. The clamp station is placed at the middle at the bar to reproduce elastic experiments with the incident bar and the transmitted bar connected by an adapter. The notched pin that connects the jaws at the top is modelled since its deformation during the initial clamping stage introduces mechanical asymmetry compared to the direct clamping force at the bottom of the jaws. Clamp jaws made of two materials (aluminium and steel) featuring distinct density, stiffness, and contact friction, are assessed.

The whole procedure of a typical TTHB experiment is simulated in ABAQUS/Explicit, to better reproduce the generation and propagation of stress waves. The simulated responses of the bar at three sections A, B, and C (illustrated in Fig. 6(a)) are shown in Figs. 6(b-d) to evaluate possible slippage and bending of the bar, and the subsequently tensile and shear waves generated upon pin snapping. The simulated results obtained from using the steel clamp are presented in black, while those from the aluminium clamp in red. At the three bar sections, i.e., (A) the pre-stressed section, (B) the clamp section, and (C) the free section (Fig. 6(a)), the node and element at the centroid of cross section are used to measure the axial displacement and direct stress, while those at the circumference to probe the rotation and shear stress, respectively. Note that the axial and shear stress at sections A and C after pin breaking can functionally represent the stress waves recorded on the incident and transmitted bar in an actual TTHB experiment.

The horizontal and vertical deflections (Fig. 6(b)) do not show much difference using distinct jaw materials, and both remain at a rather low level due to the optimal geometry design of the jaw. The clamping force was chosen lower than needed to allow potential slipping of the bar during the pre-loading stage, to evaluate the influence of different extents of bar slippage on generating the stress waves. In the case of aluminium clamp, the axial displacement and rotation (Fig. 6(c)) during the first three steps indicate significant longitudinal and rotational slipping of the bar because of the high compliance of the aluminium jaws. As a result, the generated longitudinal stress pulse, if any, oscillates around the zero line (Fig. 6(d)). Even though rotational slipping happened to some extent, a fraction amount of torsional energy was stored, and shear waves are generated but with significant oscillations. In the case of steel jaws, cleaner longitudinal and shear stress waves with a sharp wave rise are generated (Fig. 6(d)). The minor fluctuation can be explained by the slight longitudinal and rotational slipping of the bar during the clamping and energy-storing stage (Fig. 6(c)). It is noted that that after the pin breaks, the shear stress wave generated in the case of aluminium clamp seems to initiate earlier than that using steel clamps due to the larger inertia of the steel clamp. The parametric study of the jaw material indicates that, even though low-density materials would bring less inertial effect and thus shorten the reaction time by few hundreds of microseconds, high-stiffness materials are preferable as they contribute to reducing bar slippage and generating cleaner stress waves.

The distribution of the stresses across the cross section of the bar was assessed via numerical simulation of stress wave generation events using the Version 3.1 of the clamp. The uniformity of the stresses in the bar was evaluated at cross-sections located between 20 and 100 mm away from the clamp. The modelling results revealed the expected linear distribution of the shear stresses along the radius. The distribution of the longitudinal stresses was essentially uniform, with negligible variations. Even at a distance of 20 mm from the clamp the standard deviation of the longitudinal stress along the radius of the bar was only 0.7% of the mean value. It can therefore be concluded that the current clamp design is capable of generating uniform longitudinal and shear stress waves and that one-dimensional stress wave equations still apply in regions close to the clamp.

The parametric studies on the geometry and material of the clamp presented above lead to steel clamps with geometry Version 3.1 being employed in the current design of TTHB. Elastic experiments were conducted to demonstrate the benefits of the clamp design. Representative waves were recorded from strain gauges on the pre-stressed section of the incident bar, namely the uniaxial strain gauge I and the shear strain gauge I illustrated in Fig. 1(a). The longitudinal wave and the shear wave generated using steel clamp jaws of original and optimised designs are compared in Fig. 7. The initial level of the wave indicates the stored tensile or torsional load, while the first drop marks the release of the clamp, followed by the initiated stress wave that lasts for a duration to travel twice the pre-stressed length of the bar, and finally drops to zero. As illustrated in Fig. 1, the amount of the drop, as well as the amplitude of the incident wave, is in theory equal to half of the stored energy. This theoretical value is marked as a dotted line in Figs. 7(b-c) and (e–f) to evaluate the quality of the stress waves. It is noted that higher force and torque were stored in the bar system using the optimised clamp design to demonstrate its increased capacity.

In all cases, the incident waves showed an average amplitude around the theoretical value calculated from the stored loads. However, some undesirable oscillations can be observed in both the longitudinal and shear waves recorded when using the original clamp (Figs. 7(b) and (c)). Whereas the waves generated by the optimised clamp design appeared smooth and exempt from significant spikes or oscillations (Figs. 7(e) and (f)). The small peak at the beginning of the tensile wave in Fig. 7(e) was attributed to the interaction of the stress wave with the bars-specimen connections. It is also evident, presumably due to different release mechanisms of the stored tensile and torsional energies, that the torsional stress wave appears smoother than the tensile wave. The experimental results demonstrate that the optimised design of the clamp helps minimise the adverse deflection of the bar during the clamping and releasing process thus facilitating a better wave generation. It is also clear that the optimised clamp is capable of storing higher tensile and torsional loads, consequently generating stress waves of higher magnitudes.

Simultaneous loading of the longitudinal wave and the shear wave on the specimen is another critical requisite in a combined tension–torsion experiment at high strain rates. When using a single clamp to store and release both axial and torsional energies, the time difference between the arrival of the longitudinal and shear waves largely depends on the location of the clamp with respect to the specimen, because of the distinct wave speeds (equations (1a)-(1b)). In the current TTHB design, to minimise the discrepancy in time between the instants the longitudinal wave and the shear wave approach the specimen, the clamp is positioned in immediate proximity to the sample, 40 mm away from the incident bar-specimen interface.

Synchronisation of the waves from three representative combined tension–torsion experiments on commercially pure titanium, titanium 6Al4V alloy, and commercially pure copper specimens is evaluated in Fig. 8. The stress waves were recorded from strain gauges on the transmitted bar, as illustrated by uniaxial strain gauge O and shear strain gauge O in Fig. 1(a), and then shifted in time to the bar-specimen interface according to their locations. Three levels below 50% of the amplitude of the waves were selected to evaluate the time difference of the wave fronts. Provided the wave speeds of an Al7075 bar, the distance of 40 mm between the clamp and the bar-specimen interface causes the shear wave to be delayed of about 5 microseconds with respect to the longitudinal wave. This theoretical value is validated by the experimental results shown in Fig. 8(a) at level 50% and Fig. 8 (b) at level 10% with a reasonable tolerance of 2 microseconds. Although the difference between the arrivals of the stress waves is minimised by the clamp position, this difference may also be affected by the interaction between the stress waves and the specimen-bar connection as well as by the different release mechanisms for axial and torsional stress waves. Hence in practice, the time difference can be slightly larger, or the shear wave could reach the specimen earlier, within few microseconds. It is demonstrated from Fig. 8 that the current location of the clamp allows the tensile and shear stress waves to reach the specimen within 15 microseconds from each other. The synchronisation of the wave front of the longitudinal and shear waves is therefore satisfactory.

The clamp release using the pin fracturing mechanism was investigated during experiments where incident and transmitted bars were connected by an impedance matched adapter. Three types of notched pins made from PAI (polyamide-imide), PEEK (polyetheretherketone), and Al7075, characterised by different fracture strains equal to 15%, 10% and 4%, respectively were employed. Shear stress waves measured on the transmitted bar are shown in Fig. 9 as examples. It is worth noting that, as the incident and the transmitted bars were connected by an impedance matched adapter, the stress pulses displayed in Fig. 9 represents, in fact, the incident pulses generated by the elastic energy storage and release mechanism of the apparatus.

The amplitude of the stress waves varied with the stored energy which was limited by the clamping capacity using different notched pins. It is evident that the metallic notched pin can produce stress waves with higher amplitude, whereas the capacity of the composite notched pins appeared lower but can be improved by modifying the geometry of the clamp and the notched pin if needed. More importantly, a gentler wave rise was produced when using PAI pins while a sharper rise was generated using Al7075, the wave rise produced from PEEK falling in between. As the ductility of the notched pin essentially influences the time for the clamp to detach from the bar, the three pin materials represent different clamp detaching period during the clamp release process. On the other hand, the ramped stress wave generated from the ductile fracture of the pin is beneficial for high-rate testing of brittle materials as it facilitates dynamic equilibrium and prevents pre-mature failure [19, 48]. The choice of more ductile pins is therefore equivalent to the use of pulse shaping techniques when the pulse is produced by direct impact.

Illustrated in Fig. 10 are the SHTB, SHToB, and TTHB methods, and the cylindrical dog-bone specimen, hexagon-ended tubular specimen, and octagon-threaded tubular specimen that suit the different Hopkinson bar methods, respectively. Two metallic materials, commercially pure titanium (CP-Ti) grade 2 and commercially pure copper (Cu, 99.9% purity), are chosen for a series of comparison experiments. The SHTB system employed (Fig. 10(a)) consists of a striker bar, an incident bar, and a transmitted bar, and was designed by Gerlach et al. [49]. The system has been broadly used for dynamic characterisation of metals, polymers, and composites [16, 50, 51] at strain rates ranging from 100 s⁻¹ to 3000 s⁻¹. A long U-shape striker made of Ti6Al4V alloy was specially designed to generate long and clean stress pulses free from significant stress oscillations. The SHToB system (Fig. 10b) was initially designed by Campbell et al. [52, 53] using the energy-release mechanism, where the torsional wave is produced via a rapid release of a clamp that pre-stores the torque load. This torsion bar system has been used to characterise the shear responses of metals and composites [16, 53‐55] at an achievable shear strain rates of approximately 1000 s⁻¹. Both the incident bar and the transmitted bar are made of Ti6Al4V alloy. The hexagonal ends of the tubular specimen mate with the hexagonal slots on the bars to transmit the torsional waves. The TTHB system (Fig. 10(c) and its working principle have been elaborated in "Working Principle" section. The tension–torsion sample is designed based on the thin-walled tube geometry with octagon-threaded ends to transmit both the longitudinal and torsional waves.

The generation of stress pulses of considerable duration leads to the unavoidable superimposition of the incident and reflected stress waves. Hence it is not possible, from the measurements of a single strain gauge, to discern the forward and backward travelling waves. As regards the analysis of experiments conducted on the SHTB and SHToB, the forces and torques at the interface between the bars and the sample and the corresponding particle velocities were obtained by placing an additional strain gauge on the incident bar and by separating the incident and reflected waves using the algorithm described in [56, 57]. An alternative wave separation method, based on Photon Doppler velocimetry was previously presented in [58]. The deformation and failure of the samples during SHTB and SHToB experiments, was captured by means of an ultrafast SI Kirana camera. The high-speed footage was recorded with a resolution of 924 × 748 pixels at a frame rate of 2 × 10⁵ fps and shutter speed of 2 μs.

The use of a wave separation algorithm for the analysis of the experiments conducted on the TTHB requires at least four strain gauges, two for tensile and two for shear strain, positioned on the incident bar. It is also worth noting that the algorithm in [56, 57] is not directly applicable, as all strain gauges are located in the portion of the incident bar between the loading units and the clamp and, therefore, the recorded signals include information on both the initial pre-load and the release of the stress waves. For these reasons, the combined dynamic tensile-shear response of materials tested on the TTHB was determined synchronising the stress history recorded on the transmitted bar and the strain history obtained from digital image correlation analysis on the recorded high-speed footage. This was obtained employing a Photron SA-5 Camera operated at a frame rate of 10⁵ fps with a resolution of 256 × 232 pixels and shutter speed of 2 μs.

Force and torque equilibrium conditions were assessed via direct comparison of the load histories recorded by strain gauges on the incident bar and the ones on the transmitted bar. This is possible because the clamp was positioned in close proximity to the sample. Therefore, the signals recorded on the incident bar correspond, after the initial drop in magnitude subsequent to the release of the clamp, to the superposition of the incident and reflected waves. Hence, their histories represent, starting from the release instant, the load applied on the incident side of the specimen.

A grey-scale speckle pattern was applied to the surface of the specimens to determine the full-field displacement using digital image correlation (DIC) analysis. The mean intensity gradient (MIG) [59, 60] was utilised to evaluate the quality of the speckle pattern. The evaluation gave an average value of 60 calculated from images using the Kirana camera and an average value of 24 of images recorded via the Photron camera. Since MIG values greater than 20 suggest a mean bias error of displacement of less than 1% [59], the high-speed images in this study are characterised by good quality. The excellent MIG value calculated from Kirana images can be attributed to the high resolution of the region of interest and appropriate lighting during the experimentation. The commercial DIC software LaVision Davis¹ was used to calculate the time history of the engineering axial and shear strains within the gauge section.

The typical experimental measurements of CP-Ti subjected to tension at a high strain rate using the TTHB method are presented in Fig. 11. The wave signals recorded from strain gauges at the incident bar and the transmitted bar were converted into force and shifted to the specimen interfaces (Fig. 11(a)). The signal recorded from the incident bar indicated the initially stored load up to 30 KN and the subsequent release of the energy upon clamp opening, while the signal from the transmitted bar depicted wave transmitted through the specimen. The small discrepancy between the loads applied on either end of the specimen implies that dynamic equilibrium of the specimen under high-rate tension was satisfied. This deviation from perfect dynamic equilibrium was ascribed to the interaction of the stress waves with the bars-specimen connections. Additionally, the strain gauge was located within the initially preloaded section of the incident bar, which may also affect slightly the assessment of the dynamic equilibrium conditions. Figure 11(a) also demonstrates the ability of the TTHB system to generate clean tensile waves free from spurious oscillations. Constant strain rate was observed during the plastic flow, calculated from the slope of the strain history as 714 s⁻¹ (Fig. 11(b). This further validated the achievement of dynamic equilibrium conditions. The engineering tensile stress–strain curve of CP-Ti (Fig. 11(c)) showed typical plastic characteristics through elastic deformation, slight strain hardening during plastic flow, non-uniform deformation, and finally fracture at the engineering strain of approximately 24%. The DIC analysis showed homogeneous distribution of the axial strain within the gauge section of the specimen during the plastic flow. The strain appeared to localise in the middle of the gauge section approaching the fracture point.

Shown in Fig. 12 are the experimental measurements of CP-Ti subjected to torsion at a high strain rate using the TTHB method. The drop in the initial level of the stored torsional energy indicated the instant at which the clamp was released and the incident wave initiated (Fig. 12(a)). Thereafter the shear wave recorded from the incident bar showed mild oscillations as the specimen started to get engaged with the bars at the beginning of the loading, approximately between 0 µs and 100 µs. After 200 µs the difference between the torque at the two ends of the specimen reduced. Except from the early diversion that may be attributed to the initial slack in the bar-specimen connections, dynamic equilibrium in torsion was reasonably satisfied up to the failure of the specimen. The time histories of stress and strain showed a nearly constant shear strain rate of around 1000 s⁻¹ up to fracture, at which the stress started to drop while the strain gradient displayed discontinuity (Fig. 12(b)). The engineering shear stress–strain curve (Fig. 12(c)) showed strain hardening monotonically up to the fracture strain about 41%. Uniform distribution of the circumferential shear strain was shown in the gauge section during the plastic flow. Note that the post-failure images suggested the macroscopic fracture initiated from the other side of the specimen out of the field of the camera. Hence the calculated fracture strain does not reflect the exact engineering shear strain at the fracture onset but provides a good approximation.

In SHTB and SHToB methods, the engineering stress–strain curves of the dog-bone specimen and the hexagon-ended tubular specimen were determined by synchronising the transmitted stress wave and the average strain within the gauge section of the specimens. The tensile flow curves of CP-Ti obtained from the TTHB were then compared with those from the SHTB, while the shear curves from the TTHB compared with those from the SHToB, as shown in Fig. 13(a) and (b), respectively. Overall, the stress–strain responses measured from the TTHB system in both tension and shear are favourably smoother and free from significant oscillations especially at the early stage of deformation.

It is noted that experiments carried out on the different apparatuses were conducted over a range of different strain rates. However, variations in the flow stress of titanium or copper beyond 10² s-1 become noticeable only for relatively large differences in the strain rate, as reported in [61, 62]. This consideration allows the comparison of dynamic experiments conducted at different strain rates provided that the response of the investigated materials is not sensitive to those differences. The data obtained from SHTB and SHToB experiments (Fig. 13) show that the flow stress of CP-Ti is not sensitive to the variation of strain rates achieved during dynamic experiments. Similar considerations apply to copper (Fig. 14). TTHB experiments were conducted at strain rates in the same order of magnitude. It is therefore concluded that the comparison of the results obtained on the SHTB, SHToB and TTHB apparatuses is appropriate.

It is worth emphasising that the different Hopkinson Bar apparatuses utilised in this study employ different specimen geometries and that the stress state within the sample is affected by the geometry, even when the sample is deformed under the same loading mode (i.e., tension in TTHB and SHTB, and torsion in TTHB and SHToB).

In torsion the tubular specimens used during TTHB and SHToB experiments have slightly different dimensions, but both approximate a stress state of simple shear. Hence, the nominal shear stress–strain curves measured from different methods are consistent, both in terms of strength and strain to failure (Fig. 13(b)). In both cases, strain hardening was clearly observed up to the macroscopic fracture.

In tension, the geometries used during experiments on the TTHB and SHTB apparatuses are substantially different. The stress state induced on a dog-bone specimen approximates the state of uniaxial tension (stress triaxiality \(\eta ={\sigma }_{m}/{\sigma }_{vM}=1/3\), where \({\sigma }_{m}\) and \({\sigma }_{vM}\) are the mean stress and von Mises equivalent stress, respectively) whilst the stress state on a tubular sample loaded in tension is approximately a state of plane strain (stress triaxiality \(\eta =1/\sqrt{3}\)) in circumferential direction and plane stress in radial direction [63].

Figure 13(a) shows that the fracture strains measured on the TTHB-tubular specimen and on the SHTB-dog-bone sample were in the order of 20% and 30%, respectively, within the range of strains to failure reported in previous studies for CP-Ti grade 2 [1, 64].

It has been shown in the literature that stress triaxiality influences noticeably the fracture strain [63, 65‐69]. Several studies have reported the fracture locus of different engineering materials as a function of the stress triaxiality \(\eta\) [63, 65‐69], displaying a considerable reduction in the strain to failure from uniaxial tension to plane strain conditions. The smaller fracture strain measured during tension experiments on thin-walled tubular samples agrees with the trends shown in the above literature. Additionally, dog-bone samples tested in uniaxial tension experience significant strain localisation beyond the onset of necking. Conversely, strain localisation is substantially reduced on thin-walled tubular samples, thus resulting in smaller elongation at break.

With regard to the measured strength, the tensile flow stress obtained on thin-walled tubular specimens exceeded the one measured on dog-bone specimens by 16%. This is because the stress state on a thin-walled tubular specimen loaded in tension approximates a state of plane strain in the circumferential direction and of plane stress in the radial direction [63]. The stress state in a dog-bone specimen loaded in tension is, instead, of uniaxial tensile stress. For Levy-von Mises materials, the flow rule yields the stress state within a tubular specimen under direct tension loading as:where \({\sigma }_{t}\) denotes the axial stress calculated from the axial force over the cross-section area of the tubular specimen. The equivalent stress determined using the von Mises criterion is given by:where \({\sigma }_{vM}\) denotes the von Mises stress, \({J}_{2}=\frac{1}{6}\left[{\left({\sigma }_{1}-{\sigma }_{2}\right)}^{2}+{\left({\sigma }_{2}-{\sigma }_{3}\right)}^{2}+{\left({\sigma }_{3}-{\sigma }_{1}\right)}^{2}\right]\) is the second invariant of the stress deviator, and \({\sigma }_{1},{\sigma }_{2},{\sigma }_{3}\) are the principal stresses. Considering the stress state of a tubular sample loaded in tension we obtain:

Therefore, the tensile stress measured during tensile experiments on tubular specimens is equal to \(\sqrt{4/3}\) times the equivalent stress deforming the gauge area of the sample during plastic deformation:

Differently, the axial stress measured during tensile loading on dog-bone samples is approximately equal to the equivalent stress deforming the sample. This means that, for the same material, assuming the von Mises criterion, the tensile stress measured on tubular specimens is \(\sqrt{4/3}\) times (≈ 15.5%) higher than the uniaxial tensile stress measured on dog-bone specimens. This agrees with the 16% difference between the tensile yield stress measured on the tubular specimens with respect to the yield stress measured on dog-bone specimens.

We conclude that the discrepancy in the tensile stress–strain curves measured using different apparatuses is ascribed to the different stress states associated with different specimen geometries. Tensile loads generate on thin-walled tubular specimens a multiaxial stress state that is different from the uniaxial tensile stress of the dog-bone specimen. Both stress states however contribute to the definition of the failure stress locus of the investigated material and to the description of its behaviour.

We proceed to conduct the same comparison campaign on Cu specimens. The stress–strain curves measured from the TTHB, SHTB, and SHToB method are compared in Fig. 14. In tension (Fig. 14(a)), the tubular specimen fractured at strain of less than 10% which is considerably smaller than that of the dog-bone specimen, of about 40%. A difference of about 13% measured from the yield stress of the tubular specimens over that of the dog-bone specimens was shown, while the ultimate stress was consistent between the two, with a small difference of 1.9%. Similar to CP-Ti (Fig. 13(b)), the shear stress–strain curves of Cu measured from TTHB and SHToB methods (Fig. 14(b)) are comparable regarding the flow stress and the engineering strain to fracture. The consistency can be again attributed to the comparable tubular geometries of the specimens used in both methods.

It is noted that the fracture strain measured from SHToB experiments varied over a wide range of strains, from 100 to 200%. Given the ductile nature of Cu, the macroscopic fracture can be initiated anywhere on the circumference of the tube, while the rest of the circumference keeps deforming under torsion. It is thus suggested to employ multiple cameras surrounding the circumference of the tubular specimen if the testing space allows, to increase the chance to record the deformation up to fracture onset. This technique has been introduced in quasi-static torsion experiments [41, 55, 70] but remains challenging at high-rate loading due to the limited space in split Hopkinson bar testing and the availability of multiple high-speed cameras.

The measured experimental results using the SHTB, SHToB, and TTHB methods are summarised in the \(\left({\sigma }_{1},{\sigma }_{2}\right)\) plane to depict the ultimate stress loci of CP-Ti and copper under investigation, as shown in Fig. 15. The von Mises criterion and its modified versions are evaluated on the basis of the experimental measurements.

The equation of the original von Mises model is given as equation (9) in "Experimental Results and Comparison" section. There is no predicted effect of hydrostatic stress \({\sigma }_{\mathrm{h}}={I}_{1}/3\) (where \({I}_{1}={\sigma }_{1}+{\sigma }_{2}+{\sigma }_{3}\) the first stress invariant) in this failure criterion. The pressure dependence was afterwards included in the model by introducing a polynomial term in \({I}_{1}\) [71]:where, \({k}^{2}\) is a critical value related to the strength of the material, and \({\alpha }^{i}\) are constants reflecting the effects of hydrostatic stress. As the pressure dependency introduces a difference in terms of the strength in compression \({\sigma }_{\mathrm{c}}\) and tension \({\sigma }_{\mathrm{t}}\), a coefficient \(m\) that describes the asymmetry level is introduced as:

When n = 1 and 2 respectively, equation (10) yields [2, 72, 73]:

Equation 12 is the Drucker-Prager criterion [74], the conically modified von Mises model, which considers the linear effect of \({I}_{1}\); while equation (13) reflects the quadratic effect of \({I}_{1}\), and it is known as the parabolically modified von Mises model.

The von Mises criterion (equation (9)) and its conically and parabolically modified models (equations (12)–(13)) are fitted to the experimental measurements in the \(\left({\sigma }_{1},{\sigma }_{2}\right)\) plane via least square optimisation. Models with different asymmetry levels m are presented. The model that yields the minimum cost is highlighted in red. Figure 15(a) indicates that CP-Ti grade 2 features a significant level of asymmetry m = 1.42. The original von Mises model fails to describe the ultimate stress locus of CP-Ti grade 2, as it overestimates the uniaxial tension measured from SHTB and the plane strain tension from TTHB, but underpredicts the pure shear from SHToB and TTHB. The strong tension–compression asymmetry has been reported for predominantly hexagonal close-packed titanium alloys [1, 75]. As it concerns copper (Fig. 15(b), the original von Mises model well identifies the measured uniaxial tensile strength and pure shear strength but fails to capture the plane strain tension. A level of the tension–compression asymmetry of 1.10 was approximated for copper at the ultimate stress point.

The depicted stress loci show that the TTHB system and the thin-walled tube specimen, by generating monotonic high-rate torsional loading, can reproduce the pure shear stress state as the SHToB method; a plane strain tension stress state, complementary to the uniaxial tensile stress state produced from the dog-bone specimen, can be obtained by generating monotonic high rate tensile loading. Most importantly, as elaborated in sections "Split Tension–Torsion Hopkinson Bar" and "Critical Mechanical Design", the TTHB system can produce simultaneous tensile and torsional loading. Any arbitrary combination of the tension and shear can be reproduced and thus populate the stress locus presented in Fig. 15. In this sense, a wide range of stress states at high rates of strain can be directly measured from a single testing apparatus with identical specimens. This ensures the compatibility of the experimental data while avoiding systematic errors induced by using different testing systems and distinct specimen geometries.