Introduction

This study focuses on the development of a laboratory scale cyclic impact loading test setup, in which well controlled and repeatable loading pulses are applied to the test specimen at a loading frequency of several impacts per second. The motivation for this work is explained in the following. Fatigue failure due to repeated impact loading is a commonplace challenge for example in the mining industry (e.g. in percussive drilling, ore crushing, etc.), in which high efficiency and reliability of operations is required due to economic reasons. In these applications impact loading takes place both in the equipment (cf. the fatigue problem of the piston in a percussive drill [1]) and in the target material (cf. the simulation of rock percussive drilling [2]). Therefore, efficient design of the equipment necessitates accurate and representative data on the cyclic response of the related materials. However, to-the-date, bulk of the fatigue research has been carried out using conventional test devices, which typically apply a smoothly varying (sinusoidal) loading-time-history on the specimen (Fig. 1a). This is in notable contrast to impact loading (Fig. 1b), where the loading on the material is abrupt and very high strain rates can take place locally for example on the contact surface between two impacting components. It is well known that the mechanical properties of most materials are dependent on the imposed loading (strain) rate. Therefore, there is no a priori reason to assume that the use of conventional fatigue data is sufficient in applications involving cyclic impact loading. Dedicated measurements are thus needed.

Based on a review on the open literature, a wide range of experimental approaches has been used to study cyclic impact loading at cycle counts which are impractical to reach by manual operation of the test setup (above ∼ 100 cycles per test). A common approach has been to incorporate an automated reloading mechanism to an impact test setup, such as pendulum impact tester [3,4,5,6,7,8], out-of-plane plate impact by projectile [9, 10], or axial impact setup [11,12,13,14,15,16,17,18,19]. Many of the setups in the last group bear resemblance to the so-called Hopkinson bar technique discussed later in this paper, i.e., axial impact of slender bars, but the details of the setups vary regarding e.g. the striker acceleration/reload mechanism; falling striker [11,12,13] or striker driven by rotating hammer [14,15,16], crank-shaft-slider [17, 18], as well as hydraulic/mechanical spring mechanisms [17, 19] have been used. Examples from the literature include also a variety of other approaches, such as impactor attached to a fast-moving electro-dynamic test machine [20] or pneumatic cylinder [21, 22], impact loading generation via saw-tooth patterned discs rotating against each other [23], thin bending specimen driven against a rigid target by compressive air pulses (so called flapper valve) [24, 25], as well as impact loading superimposed on statically preloaded specimen [26] or rotating bending test specimen [27]. From this wide range of options the concept of automatized Hopkinson bar was selected for the basis of current work. As discussed later in this paper, the Hopkinson bar technique, in general, offers both good control of the loading imposed to the specimen as well as precise measurement of the specimen response. Furthermore, due to the relatively simple basic structure of the test setup, simple but effective numerical modeling approaches can be used in the design phase of the experiments. Previous reports [14, 17,18,19] have demonstrated that loading frequencies of up to 10–20 impacts per second, Fig. 1b), can be achieved via automatization of the test setup. However, as demonstrated in this paper, there is still room for improvement both in the way the cyclic operation is introduced and how the load train is constructed.

This paper describes an automatized Hopkinson bar device, which is capable of producing well defined compressive impact loadings on the specimen at a rate of up to 10 impacts per second. The device is based on the modification of a traditional Hopkinson bar setup, in which the striker is pneumatically driven. It is demonstrated in this paper that relatively straightforward additions to the setup facilitate high speed cyclic operation, well controlled loading of the specimen as well as good measurement diagnostics that provide data and test results which can be readily compared with data from traditional (quasi-static) cyclic experiments. Thus, the technique presented in this paper can be readily adopted in existing test setups to facilitate experimental campaigns on the effects of loading rate in cyclic loading. The feasibility of the approach is demonstrated by providing novel test results for the cyclic impact loading response of Balmoral Red granite loaded in the Brazilian disc configuration. According to a recent review [28], the fatigue lifetime of rock is affected by loading frequency at low loading rates, and on the other hand, very high loading rate sensitivity under monotonic impact loading [29,30,31,32,33] has been observed. However, until now, cyclic impact tests on rocks have typically been carried out manually [29, 34, 35], which has notably restricted the maximum amount of loading cycles in the test campaigns.

Fig. 1
figure 1

Illustration of the load-time histories characteristic to a conventional and b impact cyclic loading

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Theoretical Background and Conceptual Design

In the following, the conceptual design of the test setup and the necessary theoretical background are presented. The test setup is based on a compressive Hopkinson bar, in which compressed air is used to accelerate the striker to the impact velocity. Modifications were introduced to the setup so that compressed air is used between the impacts to rapidly reset the components of the setup back to their original positions. In order to support the design of the setup and selection of proper operation parameters, analytical and numerical models of the experiment were developed. These are presented in the following.

In the cyclic impact experiment each loading cycle can be divided into two distinct steps with notably different time scales: the first step is the impact loading event, which takes place within ∼ 1 ms after the striker hits the first bar in the setup (in this case the shank). During the impact loading event, the time histories of local loads and particle velocities in the setup are governed by the propagation of elastic stress waves, which originate from the impact surface and are multiplied by reflections at the various interfaces of the setup. The second step of the loading cycle has a length corresponding to the loading frequency, i.e., ∼ 100 ms, and it entails the resetting of the test setup prior to the next impact loading event. During this step the components of the test setup are assumed to be in rigid body motion and the main source of loads in the setup is the pressure from compressed air used to reset the components to their starting positions and propel the striker to the impact velocity. Due to these differences between the steps, two different types of analysis methodologies are needed. These are presented in the following. However, as discussed later in detail, the two steps are interconnected through the component velocities.

Impact Loading Event

As noted above, during the impact loading event the local loads and particle velocities are governed by the propagation of elastic stress waves. In the family of Hopkinson bar techniques [36], which are based on the use of slender linear elastic bars, the stress state is essentially uniaxial and a simple set of linear equations can be used to calculate the axial load (F) and velocity (v) of a given bar cross-section:

$$F\left( {x,t} \right) = E_{b} A_{b} \varepsilon \left( {x,t} \right)$$
(1)
$$v\left( {x,t} \right) = - C_{{0b}} \varepsilon \left( {x,t} \right)$$
(2)
$$C_{{0b}} = \sqrt {\frac{{E_{b} }}{{\rho _{b} }}}$$
(3)

In Eqs. 1, 2, 3, variable ε denotes the axial elastic strain of the bar, which is a function of the axial coordinate (x) and time (t). Parameters Eb and Ab are the bar Young’s modulus and cross-sectional area, respectively, whereas parameter C0b is the axial elastic wave speed, which is a function of the bar Young’s modulus and density (ρb). In the case of overlapping waves linear superposition applies, whereas wave transmission and reflection at a boundary (e.g. change of cross-sectional area or material properties) can be solved by assuming velocity continuity and force balance at the boundary. Initial rigid body velocities are introduced to the analysis by adding constant terms to Eq. 2.

The relative simplicity offered by the Hopkinson bar technique is commonly utilized in the field of monotonic high strain rate testing in the so-called Split Hopkinson Bar (SHB) configuration illustrated by Fig. 2a. In this configuration the specimen is placed between two bars, the input bar and the output bar, and the loading is generated by the striker impacting the input bar. As can be seen in the Lagrange diagram depicted by Fig. 2a, three clearly separable waves (usually denoted the incident, reflected and transmitted) are introduced to the system. Each wave is associated with a rising edge and a decreasing edge, as denoted by the lines in Fig. 2a. The effectiveness of the SHB technique in monotonic testing comes from the fact that these three waves can be readily measured with strain gauges. This data is then used to calculate the time-histories of force and velocity at the specimen/bar-interfaces, which facilitates the calculation of the high strain rate stress-strain curve of the test material. However, Fig. 2a also demonstrates the main challenge of the SHB configuration in terms of automated cyclic operation: after the first loading of the specimen, residual wave motion takes place in the test setup. On the output bar side a third bar, so-called trap bar, can be used to guide the transmitted wave away from the main setup, but on the input bar side efficient wave trapping is notably more challenging: typically high-amplitude waves continue traveling back-and-forth in the bar causing continued motion of the input bar [36,37,38,39]. This introduces two challenges for the cyclic test. Firstly, the specimen is likely reloaded, which complicates cycle counting because the level of loading can be notable, but not the same as during the original loading event. Secondly, the residual waves in the bars, in general, cause excessive motion in the test setup. As illustrated by Fig. 2b, in the experiments studied in the work, i.e., the Brazilian disc test, the excessive bar motion can easily lead to unwanted rotation of the specimen. That is, during the impact loading event the high interfacial load and friction in the specimen/bar-interfaces is able to prevent specimen rotation and the specimen slides forwards. In contrast, during the resetting phase the interfacial loads are too low to counteract the friction between the specimen and the supporting plate and thus the specimen tends to rotate as it is moved back to the original position.

Based on the literature review, the challenge of unwanted specimen movement has been typically solved by fixing the specimen or one of the bars to a rigid foundation [9,10,11,12,13,14,15,16,17,18,19]. In some cases [12, 19] a load cell connected to the specimen has been used to measure the specimen response, as depicted by Fig. 2c. Although this approach simplifies the experimental configuration and can be justified in the reported cases, the reflection of the loading wave from the rigid support can affect the measurement of the specimen loading and does not prevent reloading of the specimen. In order to address these challenges, an approach illustrated by Fig. 2d is proposed in this study. In this approach the input bar of the SHB configuration is replaced by a short bar (length comparable to the striker), so-called “shank”, which, as discussed later, is required for the operation of the pneumatic system. However, since the shank is short compared to the output and trap bars, back-and-forth wave motion in the shank takes place quickly. Thus the initial momentum of the striker is quickly and fully transmitted through the specimen to the output bar and finally to the trap bar. As shown later in the Results section, the net effect is that the specimen is loaded with a single pulse which has an initial high amplitude portion followed by incrementally decaying tail. Furthermore, at the end of the impact loading event only the trap bar is moving, which notably simplifies the cyclic operation; in practice there can also be residual motion of the striker in the reversed direction, but the main principle still holds; the specimen and the output bar are virtually stationary after a single loading.

It should be noted that the approach proposed here is suitable without modifications only for compressive loading; other loading modes, such as tension, likely require further development work. The approach comes also with the expense of losing the data from the input bar side, since the short shank does not permit the measurement of the incident and reflected waves as in the conventional SHB experiment. Therefore, relative bar end displacement (specimen strain) cannot be directly determined using the wave records. In the experiments presented and discussed in this paper, this was not a major challenge, since the analysis was based on specimen load and qualitative assessment of high-speed footage. However, if data on the specimen displacement (strain) is needed in future test campaigns, additional techniques, such as Digital Image Correlation (DIC), have to be applied.

Fig. 2
figure 2

Illustrations of a a conventional SHB test, b rotation of the Brazilian disc specimen in a cyclic test if excessive motion takes place, c a setup in which output bar has been replaced by a load cell and rigid support, and d the configuration used in this study. The Lagrange diagrams in a, c, and d illustrate the propagation of the elastic waves within the setups

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Pneumatic-Mechanical Model for the Striker Motion

During the cyclic operation, compressed air is used to accelerate the striker to the impact velocity and, after impact, reload the striker. Similarly, pneumatic stopper/actuator at the end of the bar assembly is used to push the bars and the specimen back to their initial positions. As discussed later in the Experimental section, the pneumatic system is operated in open-loop control, i.e., the pneumatic valves controlling the air flow are opened and closed in predetermined sequence. This approach offers simplicity of construction and needs less instrumentation than a closed-loop control, but on the other hand, is prone to unstable operation, if the operation parameters are not correctly set. In order to support the design of the related components and setting the correct operation parameters, a pneumatic simulation model for the striker motion was constructed. Here the approach presented by Richer and Hurmuzlu [40] for a high-speed pneumatic actuator was adopted in simplified form. Figure 3 presents the main features and parameters of the 1D-model. The striker is located in a constant diameter launch tube and surrounded by two air chambers, whose volume (V1 for front and V2 for back chamber) depends on the axial coordinate of the striker (xstr):

$$V_{1} = x_{{str}} A_{{str}} + V_{{01}}$$
(4)
$$V_{2} = \left( {L_{{tube}} - L_{{str}} - x_{{str}} } \right)A_{{str}} + V_{{02}}$$
(5)

The terms Ltube and Lstr in Eqs. 4 and 5 refer to the launch tube and striker lengths, respectively (Fig. 3). The inner cross-sectional area of the launch tube equals the cross-sectional area of the striker and is denoted by Astr. The constant terms V01 and V02 in Eqs. 4 and 5 simulate the additional volume between the control valve and the air chamber, i.e., the air channel and the rapid exhaust valve (see the description of the test setup below for more details).

The acceleration of the striker (\(\ddot{x}_{{str}}\)) is given by the sum of the forces resulting from the air pressures (Pi) and friction between the striker and the launch tube (given by the coefficient of friction µfric and force exerted by gravity), Eq. 6:

$$\rho _{{str}} L_{{str}} A_{{str}} \ddot{x}_{{str}} = A_{{str}} \left( {P_{1} - P_{2} } \right) - \mu _{{fric}} \rho _{{str}} L_{{str}} A_{{str}} g$$
(6)

In Eq. 6 ρstr denotes the density of the striker and the constant g is the acceleration due to gravity.

Following [40], the time derivative of the chamber pressure (\(\dot{P}_{i}\)) depends on the mass flows in and out of the chamber (\(\dot{m}_{{iin}}\) and \(\dot{m}_{{iout}}\)) as well as from the rate of change of the chamber volume, which is linearly dependent on the striker velocity:

$$\dot{P}_{i} = \frac{{RT}}{{V_{i} }}\left( {\alpha _{{in}} \dot{m}_{{iin}} - \alpha _{{out}} \dot{m}_{{iout}} } \right) - \alpha \frac{{P_{i} }}{{V_{i} }}\dot{V}_{i}$$
(7)

In Eq. 7 T is the temperature, R is the gas constant, and αi is a parameter that describes the heat transfer during the process. Following [40], the parameters were set in the following manner: αin = 1.4, i.e., adiabatic conditions for air flow into the chamber, αout = 1.0, i.e., isothermal conditions for air flow out of the chamber and α = 1.2, i.e., intermediate conditions for air compression/expansion caused by the striker motion in the chamber.

The flow of air in and out of the chamber takes place through a valve, which is modelled as orifice [40]. The mass flow through the orifice (\(\dot{m}_{v}\)) is calculated according to Eq. 8:

$$\dot{m}_{v} = \left\{ {\begin{array}{*{20}c} {C_{f} A_{v} C_{1} \frac{{P_{u} }}{{\sqrt T }}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~if~\frac{{P_{d} }}{{P_{u} }} \le P_{{cr}} } \\ {C_{f} A_{v} C_{2} \frac{{P_{u} }}{{\sqrt T }}\left( {\frac{{P_{d} }}{{P_{u} }}} \right)^{{1/k}} \sqrt {1 - \left( {\frac{{P_{d} }}{{P_{u} }}} \right)^{{\left( {k - 1} \right)/k}} } ~{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} if~\frac{{P_{d} }}{{P_{u} }} > P_{{cr}} } \\ \end{array} } \right.$$
(8)

In Eq. 8, Cf and Av denote the discharge coefficient and cross-sectional area of the orifice, respectively, Pu and Pd denote the pressures upstream and downstream of the orifice, respectively. Equation 8 applies to air flow both in and out of the chamber, when the upstream and downstream pressures are set accordingly. The constants Ci, Pcr and k are selected following [40], see Table 1. The constant Cf for the pneumatic valves is obtained via calibration with experimental data, as discussed later.

It should be noted that compared to the original work by Richer and Hurmuzlu [40], the pneumatic model is simplified in the current work by excluding pressure losses in the air channels and due to leakage flow around the striker as well as using simple on/off type model of the valves. That is, the upstream orifice pressure for flow into the chamber is given directly by the supply pressure and the downstream pressure for outward flow is the normal air pressure. The closing and opening of the valves is modelled by setting the corresponding mass flow to zero or to the value given by Eq. 8, respectively. As shown later in the results, these simplifications do lead to differences between the numerical predictions and experimental data, as the pressure build-up in the chambers is overestimated by the simulations. However, the model was found to grasp the essential physics of the system and thus give predictions with sufficient accuracy.

Fig. 3
figure 3

Illustration of the pneumatic simulation model for the striker motion

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Experimental Setup and Numerical Simulations

In the following the developed cyclic impact loading setup is described. The discussion is concentrated on the key components, i.e., the automated loading and stopping mechanisms as well as the instrumentation needed in cyclic experiments. In addition, the numerical implementation of the models described above is presented.

Test Setup

The test setup was built in the IMPACT laboratory of Tampere University by modifying an existing Hopkinson bar setup, in which ⌀22 mm × 1200 mm bars and ⌀22 mm strikers with typical length between 100 and 200 mm are used. The bars and the strikers are made of tempering steel EN 42CrMo4. The elastic waves in the bars are measured with pairs of strain gauges attached at the bar center points and connected via half-Wheatstone bridge to a Kyowa CDV-700 A strain gauge amplifier and a Yokogawa DL708E digital oscilloscope recording typically between 1 MHz and 10 MHz. In monotonic testing a relatively long launch tube (∼ 2 m) is used to achieve a wide range of striker velocities using compressive air.

Figure 4 shows an overview of the cyclic impact loading setup and its key components, whereas Fig. 5 presents schematically the main systems of the setup, i.e., mechanical, control electronics and measurement instrumentation, as well as the pneumatic system. Figure 6 illustrates the rigid-body motion of the test setup components as well as the air pressure of the various chambers in the test setup during one loading cycle. As discussed earlier, the input bar is replaced by a short shank (length 100 mm, tempering steel, see Online resource 1), which seals the end of the launch tube pneumatically but transmits the dynamic loading created by the striker to the rest of the setup. This concept is well-known in percussive drilling. As seen in Fig. 4a, the shank housing includes the pneumatic air connection for the air chamber in front of the striker, an entry for inductive position sensor as well as a small chamber at the front section, a so-called air spring, which returns the shank to its original position after impact. Teflon rings around the shank are used as pneumatic seals and slide bearings. The launch tube in the cyclic loading setup is 320 mm long, i.e., notably shorter than in monotonic testing so that high enough loading frequency can be achieved. The compressive air flow for the acceleration of the striker is supplied by an air channel at the back end of the launch tube. As shown by Fig. 5, the flow of the compressed air to the chambers in front of and behind the striker is controlled by a combination of an on-off high speed electropneumatic valve (Festo MH4-series) and a mechanical rapid exhaust valve (Festo SE-series). The chamber pressures are monitored by electronic pressure transducers (Festo SPTW-series).

The pneumatic stopper shown in Fig. 4c has a two-piston structure with the smaller, so-called inner piston surrounded by the larger semi-hollow outer piston. Both pistons are made of aluminum alloy and compressive air is fed to the backside of the inner piston via channel in the outer piston. In addition, the two pistons are held apart with a spring, as shown in Fig. 4c. This design was selected, because the small inner piston and the spring represent only a small change in impedance compared to the trap bar, which ensures that initial reflection of the incoming loading wave takes place as if the bar end was free, i.e., the travel of the loading wave in the trap bar is disturbed as little as possible by the stopper so that wave propagation in the setup corresponds to Fig. 2d. The actual stopping of the trap bar takes then place gradually upon the build-up of forces due to the compression of the spring and the air in the chamber. Thus, with the current stopper design a constant supply pressure can be maintained in the stopper chamber, as seen in Figs. 5 and 6. This simplifies the overall design notably, as there is no need to actively synchronize the striker and stopper air flows. In contrast, if a large piston was in direct contact with the trap bar, this would represent a boundary with a sudden increase in mechanical impedance and thus part of the compressive loading wave would reflect back as a wave of compression, which could possibly reload the specimen. It should be noted that detailed pneumatic-mechanical calculations of the stopper were not carried out, because first trial experiments showed the feasibility of the approach (example data is provided later in the Results). Furthermore, the two-piston design enabled the utilization of the existing pneumatic stopper, which has been successfully used in experiments involving larger kinetic energies and also in low-frequency cyclic fatigue tests, in which the stopping mechanism acts also as a push/pull actuator [41] (in fact, the spring loaded flange shown in Fig. 4c is related to the push/pull mechanism, which was, however, not used in the current experiments).

The electropneumatic valves of the striker launch mechanism are operated in open-loop control so that they open and close synchronously (i.e., at a given time one valve is open while the other one is closed). As shown in Fig. 5, the control signal is provided by a control unit (imc CS-7008-FD), which also records the measurement signals from the pressure transducers, the striker position sensor and the strain gauge amplifier. The maximum sampling rate of the control unit is 100 kHz, which is somewhat low for Hopkinson bar measurements, but on the other hand, the unit incorporates real-time analysis of the input signals. This feature is used in the setup to detect specimen failure by tracking the maximum force (strain) imparted to the output bar during each cycle. That is, the start of a new loading cycle is detected from the striker position sensor reading, after which a search window is set for the output bar strain. If the peak value of the strain drops below a preset threshold, the unit stops the output to the pneumatic valves. At the same time the unit sends a TTL trigger signal to the high-speed oscilloscope, which has been continuously recording the strain gauge amplifier readings to a buffer. After receiving the trigger, the oscilloscope stops recording and saves the data. Thus the final cycles prior to specimen failure, 5…10 depending on the loading frequency, are recorded at high temporal resolution. The control unit can also be programmed to send the trigger signal after certain number of loading cycles.

In addition to the sensors described above, the test setup includes also a high-speed camera (pco dimax HS), which is aimed at the specimen. In order to obtain high frame rate data from the specimen loading events without collecting large amounts of “unnecessary” data during the resetting of the setup, the camera is operated in the following manner: the striker position sensor signal is used to trigger a waveform generator (Keithley 3390), which then waits a predetermined time period (∼ 1–2 ms) corresponding to the time it takes the striker to actually hit the shank after the sensor signal is already up and then sends a burst of TTL trigger signals to the high speed camera. The camera shutter is following this signal, i.e., each trigger signal corresponds to an individual frame. To verify synchronization, the camera sends for each frame a trigger output signal to the control unit, as shown in Fig. 5. As a result, each loading event during the cyclic test is measured at least at 100 kHz and supplemented with synchronized high speed photography of the specimen.

Fig. 4
figure 4

Cyclic impact loading setup: a overview of the setup, the loading mechanism is seen in the foreground, b and c, CAD models (cut view) of the shank housing and pneumatic stopper, respectively

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Fig. 5
figure 5

Main systems diagram of the test device. The upper part of the diagram presents the control and measurement system, in the middle the main mechanical components are illustrated (not in scale) and the lower part of the diagram presents the pneumatic system

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Fig. 6
figure 6

Illustration of the rigid-body motion of the setup components during one loading cycle. The colors are used to illustrate the pressures in the chambers; blue for low, magenta for medium, and red for high pressure, respectively

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Brazilian Disc Experiments on Balmoral Red Granite

In order to demonstrate the capabilities of the test device, a set of cyclic impact tests were carried out on Brazilian disc specimens of Balmoral red granite. The same test material and specimen type were used earlier by Mardoukhi et al. [33], who studied the monotonic mechanical properties under quasi-static and impact loading. The latter experiments were carried out using SHB. In the current work the specimens were studied in the as-received condition. The tensile stress in the specimen along the vertical direction was calculated based on the externally applied force (F) [32] using Eq. 9:

$$\sigma _{T} = \frac{{2F}}{{\pi D_{s} t_{s} }},$$
(9)

where Ds and ts are the diameter and the thickness of the Brazilian disc, respectively.

Figure 7. shows the specimen mounted in the setup. As can be seen, the specimen is placed between the shank and the output bar, which are protected from surface damage by 3 mm thick maraging steel discs. The specimen is sitting on a polymer support plate, which has side rails to prevent excessive specimen motion after failure; during the actual loading the specimen can move freely in the loading direction. As noted above, theoretically specimen movement including rotation should be avoided by the design of test device dynamics, i.e., after the initial loading the stress wave is guided to the trap bar and the rest of the setup should remain stationary. However, in practice there is always some amount of specimen motion per each cycle, which, if left unattended, might cumulate over cycles to cause net rotation of the specimen (Fig. 2b). In the Brazilian disc experiment this would involve a (unwanted) change in the loading direction in the specimen coordinate system. To prevent this, small rubber pads were attached to the specimen with double-sided tape at approximately 5 o’clock and 7 o’clock locations, as illustrated in Fig. 7b). In addition, a support was placed at 2 o’clock location, i.e., on the upper part of the specimen on the side of the shank (Fig. 7). This rubber-padded support prevents vertical specimen movement during the reloading, whereas during the actual impact loading the contact between the pad and the specimen is released by the horizontal movement of the specimen. With this type of mounting it can be ensured that the specimen is held in place during hundreds of consecutive loading cycles, but at the same time the loading conditions correspond to the boundary conditions of a monotonic Brazilian disc experiment, i.e., axial compressive loading with free expansion in the transverse directions. As discussed later in the Results section, in general the specimens failed at the horizontal center line (Fig. 7b), i.e., on the plane of maximum tensile stress, which supports this conclusion.

Fig. 7
figure 7

Brazilian disc specimen a mounted in the test setup and b as seen by the high speed camera. The superimposed illustrations in b indicate the support rubber pads (striped boxes) as well as the horizontal crack formed in the specimen (arrows). It should be noted that in these figures the output bar is on the left side of the specimen and the impact loading is applied from the right

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In addition to impact loading, cyclic experiments were carried out in the quasi-static loading region using a servohydraulic materials testing machine Instron 8800 under closed-loop load control. In the tests sinusoidal loading (Fig. 1a) was applied on the specimen at a frequency of 1 Hz so that the ratio of the minimum and maximum load per cycle was ∼ 0.05, i.e., the specimen was almost but not fully unloaded during the cycle. This ensured that interfacial friction forces between the specimen and the compression anvils were always present, which prevented the movement of the specimen during the test.

Numerical Simulations

With reference to the discussion presented above, the cyclic impact experiment was modelled in two steps. Firstly, linear elastic explicit finite element method (FEM) simulations of a single impact loading event were carried out using the commercially available software package Dassault Systemes Simulia Abaqus 2017. The simulations included a three-dimensional ¼-model of those components which are involved in the dynamic loading event, i.e., the striker, shank, specimen, output bar and the trap bar. The effect of the pneumatic stopper on the impact loading event was assumed to be negligible. The components were meshed with reduced integration brick C3D8R-elements (striker, bars and the specimen) and modified tetrahedral C3D10M-elements (shank). Typical element size (edge length) was ∼ 3 mm for the striker, shank and the bars and ∼ 1 mm for the specimen. The boundary conditions involved symmetry along the main axis (simulating the stanchions supporting the bars and the housing for the striker and the shank), whereas the initial conditions involved only the initial velocity of the striker (with other components being stationary). The simulation data was evaluated in terms of the elastic wave records on the bars (simulating strain gauge measurements) and component velocities at the end of the loading event. In practice the simulations were ran so long that the rigid body velocities could be estimated based on the average nodal velocities.

The second type of simulations involved the numerical implementation of the pneumatic-mechanical model for the striker motion in Mathworks Matlab R2021a. The simulation included the variables illustrated in Fig. 3 and the corresponding set of equations (Eqs. 4, 5, 6, 7, 8) were solved by explicit time integration. These simulations involved several loading cycles. The impact of the striker against the shank (i.e., xstr = 0 in Fig. 3) was taken into account in the following manner: upon impact, striker location and acceleration were forced to zero, and the (residual) striker velocity was set to a value given by the impact simulation; due to the design of the test setup, the striker either stops completely upon impact or bounces back with a velocity dependent on the impact velocity.

The main simulation parameters are given in Table 1. As indicated, most of the parameters were selected based on literature or using nominal dimensions. The remaining unknowns, i.e., the valve discharge coefficients (Cf,in/out) were selected by comparing the simulated pressure histories with experimental records (see below Fig. 8 for example); the value of 0.25 used in [40] was found suitable for the flow into the chamber. For the outward flow a higher value was found, which is in agreement with the lesser restriction caused by the rapid exhaust valve.

Table 1 Main dimensions of the test setup and parameters used in the experiments and in the simulations
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Results and Discussion

In the following experimental results obtained with the developed test setup are presented and discussed starting with the operation characteristics of the setup followed by the results of an experimental campaign carried out on Brazilian disc specimens made of Balmoral red granite. Examples of typical video footage of the experiments is provided as electronic supplementary data (Online resources 2 and 3). It should be noted that unless otherwise stated, the experimental results are obtained under cyclic operation mode, i.e., the results are to be considered representative for an experiment involving up to hundreds of cycles.

Operation Characteristics

Figure 8a shows example data from a cyclic experiment in terms of measured chamber air pressures and striker position sensor reading with corresponding simulation predictions. In general, satisfactory agreement between the experiments and simulation predictions was found. The largest discrepancy is seen in the time-histories of the chamber air pressures; as noted above, the pneumatic system is simplified in the pneumatic model and therefore the rate of pressure build-up is overestimated by the simulations. However, the striker velocity for given operating pressure can be well predicted, especially for the 200 mm striker at low operating pressures, as seen in Fig. 8b. For higher pressures, the impact velocity of the shorter 100 mm striker is overestimated by the simulations, but the overall velocity vs. pressure trend is well represented. This good prediction capability was found especially useful during the design phase of the system, when the dimensions of the system were decided. For example, the launch tube length affects both the air chamber volumes and hence the required air flow rates but also the maximum distance the striker can travel during the cycle. All these factors affect the achievable striker velocity and loading frequency.

Fig. 8
figure 8

a Example of cyclic operation at 10 Hz with 200 mm striker and 3.1 bar nominal operating pressure; comparison of simulation predictions with experimental data: air over-pressure in front of and behind the striker as well as striker position sensor reading (value of “1” indicates that the striker is within a distance of 9 mm from the shank) and b experimentally measured and numerically simulated striker velocity vs. nominal operating pressure-relationship for 100 mm and 200 mm strikers, 10 Hz impact frequency

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As indicated in Fig. 8b, cyclic impact loading can be achieved with the test setup up to maximum tested striker speed of 9 m/s and loading frequency of 10 Hz. At higher striker speeds and loading frequencies the limiting factor is related to the pneumatic system, i.e., to the rate of change of the compressed air pressure in front of and behind the striker. For a given striker length and loading frequency striker speed increases both with increasing operating pressure and increasing reloading time (i.e., the time duration the striker is being pushed backwards). An obvious limit condition is the striker hitting the back wall of the launch tube during reloading. Furthermore, the open-loop control of the pneumatic system introduces additional constraint to the valve timings. Based on the FEM simulations, in the Brazilian disc experiment the striker rebounds at a velocity up to 30% of the original impact velocity due to the unloading of the elastically deforming specimen. In order to ensure stable operation of the open-loop control, the striker motion has to be completely stopped after each impact. This is achieved by maintaining pressure in the rear chamber long enough so that the striker is brought to full stop against the shank after initial rebound. In the current system this condition limits the maximum reloading duration to ∼ 30% of the total cycle duration. With this condition full-filled, i.e., loading frequency set to 10 Hz (cycle duration 100 ms) and reloading time to 30 ms, a relative standard deviation of 5…12% in the striker impact velocity was observed in the cyclic Brazilian disc experiments lasting some tens of cycles. When the loading frequency was reduced to 5 Hz (cycle duration thus 200 ms) while maintaining the same reloading time of 30 ms, the relative standard deviation of the striker impact velocity was reduced below 2% in tests lasting up to 1000 cycles.

Figure 9 shows an example of the impact loading on a Brazilian disc specimen. In this particular experiment the strain gauge channel, which is normally reserved for the momentum trap bar, was used to measure the signal from strain gauges attached vertically at the center of the specimen, i.e., at the direction and location of the maximum tensile stress. Figure 9a shows data for 5 consecutive impacts at the rate of 10 Hz, whereas Fig. 9b shows a close-up of one impact event. In Fig. 9c the data shown in Fig. 9a has been manipulated so that the consecutive impact events have been time-shifted to a common starting point and the predictions of the FEM simulations have been superimposed on the figure. The data in Fig. 9 indicates good control of the cyclic loading; the specimen loading history is the same for each cycle. Figure 9c also shows that in general the FEM simulations agree well with the experimental results. The largest discrepancy is seen in the wave motion in the output bar following the initial impact. The FEM simulations predict a full transmission of the loading wave to the trap bar, as depicted in Fig. 2d. However, the simulation geometry involves “perfect” contact between the impacting parts, i.e., the bar ends are completely flat and parallel. In the experiments this cannot be fully achieved despite careful alignment of the setup before the tests. Therefore the transmission of the loading wave from the output bar to the trap bar is not perfect and residual wave motion takes place in the output bar. Second difference is that in the experiments the loading wave is slightly longer than predicted by the simulations. This feature is also related to the imperfect contacts, in this case between the striker and shank, due to which the momentum of the striker is not transmitted to the rest of the system as quickly as predicted (see Appendix for an example test and simulation done without the specimen). It should be noted, however, that in general the residual wave motion in the output bar causes only minor movement of the output bar, and the challenge of specimen rotation (Fig. 2b) is avoided in the setup.

Fig. 9
figure 9

Example of the specimen loading during a cyclic impact experiment at 10 Hz, 200 mm striker, ∼ 2 m/s impact velocity, 5 cycles: a data from strain gauges on the output bar and from vertical gauges at the center of the Brazilian disc specimen, b zoom-in on a single impact event, and c the data shown in a manipulated so that each loading event is time-shifted to a common starting point. In addition, respective data from FEM simulations is superimposed in c

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As seen in Fig. 9b and c, the specimen is loaded only once with high amplitude wave during each striker impact. The initial loading is followed after ∼ 20 ms by a series of secondary loadings (Fig. 9b); these loadings are related to the pneumatic stopper pushing the trap bar and output bar back against the specimen and are of low amplitude compared to the actual loading. Thus it can be concluded that well controlled repeatable impact loading on the specimen can be achieved with the setup.

Cyclic Brazilian Disc Experiments on Balmoral Red Granite

Figure 10 presents the results of the Brazilian disc experiments on Balmoral red granite. Figure 10a and b depict the conventional SN-curve in terms of maximum tensile stress (Eq. 9) vs. number of cycles to failure whereas Fig. 10c shows the maximum tensile stress vs. stress rate (average from 0 to maximum stress). Several points are evident in the data. Firstly, in the impact loading the data is split into two groups (Fig. 10a); at very low cycles to failure, three or less, there is a wide range of maximum tensile stresses carried by the specimen, whereas at higher cycles to failure the data is clustered into a relatively narrow range of stresses. Secondly, the apparent fatigue strength during impact loading seems to be close to the one measured at quasi-static loading. Analysis based on the maximum likelihood fitting of a normal distribution to the probability of survival at 100 cycles resulted in fatigue strengths of 7.4 +/− 1.3 MPa and 7.0 +/− 1.9 MPa for the impact and quasi-static loading, respectively. Thus, the difference in sample means for impact and quasi-static loading is smaller than the sample standard deviation of either one of the two test series. Thirdly, when the test data is plotted in terms of the maximum stress vs. stress rate (Fig. 10b), a notable stress rate sensitivity of the material strength is observed above ∼ 100 GPa/s. This observed stress rate sensitivity is in good agreement with previous reports [30, 32] of monotonic tests on granite. It can also be seen in Fig. 10b that the current data is in good agreement with the monotonic experiments carried out earlier by Mardoukhi et al. [33] on the same material as here. It is also remarkable that all the tests, in which the specimen lasted more than three cycles, are situated at the lower end of the stress rates achieved in the cyclic impact experiments. A straightforward conclusion would be that there is small or negligible apparent stress rate sensitivity of the fatigue strength. This conclusion should, however, be considered tentative, since in this experimental series the increase of loading rate is accompanied by an increase in the loading amplitude. This happens because for a given experimental configuration, the loading rate is controlled mainly by the striker impact velocity, which also directly controls the maximum load imparted during the impact. It cannot be therefore excluded that the test material could show cyclic endurance of more than ~ 3 cycles at higher stress rates, given that the peak load does not increase beyond the monotonic strength. A possible test of this hypothesis would be to repeat the experiments using a short but steep loading pulse which could possibly be obtained with a shorter striker. This kind of study is, however, beyond the scope of this study and will be left for future studies.

Finally, the failure behavior of the specimen is briefly discussed. In general, based on the high-speed footage and post-test inspections, the specimens fractured horizontally near the center plane (see Fig. 7b for an example), as predicted by the theory of Brazilian disc and shown by previous studies [30,31,32,33]. A complicating factor is, however, that as observed both in the impact and quasi-static experiments, the disc specimen is able to carry considerable amount of axial load despite the formation of a centerline crack. Therefore, the in-situ detection of the specimen failure based on the axial load signal is challenging and the specimen is usually reloaded several times after fracture before the test is stopped. In the current test series this challenge did not affect the counting of the cycles to failure, because specimen failure could be readily identified in the post-test analysis of load data and high-speed footage, but recovery of the specimens without post-fracture damage was very challenging. A possible solution to the challenge of failure detection would be the use of specimen mounted strain gauges on each experiment for real-time tracking of specimen compliance. Testing of this idea is, however, left for future studies.

Fig. 10
figure 10

Results of the cyclic Brazilian disc experiments on Balmoral red granite: a maximum tensile stress per cycle versus number of cycles until failure, and b the data of a plotted in terms of maximum tensile stress versus loading rate with data from a previous study [33] added to the figure. The error bars in a indicate the standard deviation of the maximum tensile stress per cycle in the dynamic tests

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Conclusions

This work presents the development of a fully automatized cyclic impact test setup based on the Hopkinson bar technique. The following points are highlighted:

  • The setup is based on impact loading generated by a pneumatically driven striker. A special shank piece (length comparable to the striker length) is placed at the end of the launch tube. The shank transmits the momentum of the striker to the rest of the setup and seals the launch tube pneumatically so that compressed air can be used both for accelerating and reloading the striker.

  • High speed pneumatic valves control the flow of air in front of and behind the striker. The valves are operated with a simple open-loop control which is sufficient for stable operation within the operation range, i.e. loading frequency up to 10 impacts per second and maximum striker velocity of 9 m/s.

  • The impact loading configuration does not involve a long input bar, but the shank is directly in contact with the specimen. With this configuration unwanted specimen loading due to residual wave motion in the input bar is avoided, but on the other hand specimen displacement cannot be measured using the stress waves. To overcome this, synchronized high speed photography is used so that each impact event of the cyclic test is captured, and during the resetting between the impacts the camera is inoperative to reduce the amount of optical data.

  • Specimen load is measured by a sufficiently long output bar placed in contact with the specimen. Residual wave motion in the output bar is removed by using a trap bar. Between the striker impacts, the momentum and kinetic energy imparted to the setup are absorbed by a pneumatic stopper in contact with the trap bar. After each impact the pneumatic stopper resets the bars and the specimen to their initial positions.

The feasibility of the developed test setup is demonstrated by presenting a series of cyclic Brazilian disc experiments on Balmoral red granite supplemented with data from quasi-static cyclic tests carried out with a universal testing machine. Based on the results the following can be concluded:

  • The test method is considered suitable for cyclic Brazilian disc experiments because the specimen fails via cracking on the center plane, as expected, and high-speed footage reveals that specimen rotation during the test is minimal.

  • The test material shows notable stress rate sensitivity of monotonic strength during impact loading. In contrast, the apparent fatigue strength is nearly the same as measured in quasi-static loading. However, the measured impact data might be affected by the fact that in the experiments increasing stress rate, i.e. rise time of the loading pulse, is accompanied with increasing peak load, which leads to monotonic failure at higher stress rates.

  • To overcome the challenge presented above, it is suggested that in future studies impact experiments are carried out using different loading pulse shapes, e.g., quickly rising with low amplitude, which might reveal whether there are considerable stress rate effects also in the cyclic strength. These kinds of pulses can perhaps be achieved by using a very short striker.