## Abstract

This paper presents an experimental and numerical study of Kuru grey granite impacted with a seven-buttons drill bit mounted on an instrumented drop test machine. The force versus displacement curves during the impact, so-called bit–rock interaction (BRI) curves, were obtained using strain gauge measurements for two levels of impact energy. Moreover, the volume of removed rock after each drop test was evaluated by stereo-lithography (three-dimensional surface reconstruction). A modified version of the Holmquist–Johnson–Cook (MHJC) material model was calibrated using Kuru granite test results available from the literature. Numerical simulations of the single drop tests were carried out using the MHJC model available in the LS-DYNA explicit finite-element solver. The influence of the impact energy and additional confining pressure on the BRI curves and the volume of the removed rock is discussed. In addition, the influence of the rock surface shape before impact was evaluated using two different mesh geometries: a flat surface and a hyperbolic surface. The experimental and numerical results are compared and discussed in terms of drilling efficiency through the mechanical specific energy.

This article is part of the themed issue ‘Experimental testing and modelling of brittle materials at high strain rates’.

## 1. Introduction

Percussive drilling technology has been widely used to achieve excavation in hard rock formations. During percussive operations, a compressive wave generated by the impact of a hammer moves the drill rod towards its interface with the rock at the bottom of the hole. The rock material under the drill bit indenters is crushed and chips can eventually be formed if side cracks merge between adjacent buttons or with a free border [1]. As the degradation of the rock by crushing requires very high specific energy, the efficiency of the percussive technique mostly relies on this chipping process. Unfortunately, this efficiency is observed to drop drastically when the depth increases [2]. This issue is identified as a key bottleneck for exploration and viable commercialization of, for example, geothermal resources. Indeed, the confining and borehole pressures are the most probable causes for the drop in percussive efficiency of the technique. In this context, an accurate description of the rock behaviour under pressurized and dynamic loadings is needed to support the drill bit design and to optimize the drilling operating conditions.

One of the most important features of rock and soil behaviour is the pressure sensitivity of the yielding. Widely used models, such as e.g. Drucker & Prager [3] and Hoek & Brown [4], give a linear and a power pressure dependence of yielding, respectively. In addition to these models, a cap is generally proposed to handle compressive hydrostatic stress states by closing the yield surface [5]. Gurson's model [6] derived with a Drucker–Prager matrix [7] also provides a closed pressure strengthening the yield surface. However, the behaviour of hard rock under hydrostatic compressive loading has only seldom been investigated experimentally due to the load limitations of conventional testing facilities. Hence, the behaviour of hard rock under triaxial pressure and calibration of the model (e.g. the location of the cap) is usually extrapolated and does not arise from experimental data. Besides, a rate sensitivity of the rock behaviour is often observed [8]. Therefore, various experimental techniques have been developed to characterize the dynamic behaviour of rock materials [9]. The dynamic impacts applied by percussive operations expose the rock material to a large range of strain rates. Hence, the strain-rate sensitivity and eventual coupling with pressure sensitivity are also of special interest. The behaviour of granite rock under dynamic loading conditions has been investigated using dedicated experimental apparatus such as split Hopkinson pressure bars (SHPB) [10,11]. The combined effect of strain rate and confining pressure was originally investigated by mean of rigs added to SHPB specimens, but did not provide a constant confinement during the tests. Recent experimental apparatus, where a pressure chamber was specifically designed to adapt to a SHPB device, enabled one to apply a constant confinement at high strain rates [12].

During dynamic indentation, the pressure and strain-rate sensitivity of the rock material influences the stress and strain fields under the indenter (compressive-based crushing versus tensile-based cracking). A vast state of the art exists on the experimental, analytical and numerical description of brittle rock fracture under indentation and multiple indentation, i.e. with several buttons [13]. As far as the numerical simulations are concerned, the substantial influence of (i) the presence and position of the cap and (ii) the strain rate and viscous effects, on the predicted (iii) stress distribution under the buttons and (iv) force versus displacement curve resulting from the bit–rock interaction (BRI) were enlightened [14]. Moreover, it was observed that the nature of the energy transfer will influence the efficiency of penetration [15]. An impact achieved by rigid-body motion is generally more efficient than a stress wave transmitted from a hammer impacting a rod. The shape and length of the stress wave may also influence this efficiency.

In this study, we present the experimental results and numerical predictions of a rigid-body impact on a Kuru granite block. Section 2 sums up experimental data available from the literature on Kuru granite (material testing under various loading conditions and drop tests performed with a seven-button drill bit). Section 3 describes the material model and the identification of material parameters on available experimental data. Numerical simulations of impact tests are presented in §4, completed with several parametrical studies. Finally, conclusions and perspectives for further work are drawn in §5.

## 2. Experimental tests

### (a) Material tests on Kuru granite

Experimental material tests were performed on Kuru grey granite specimens to capture the behaviour of the rock: Brazilian disc and triaxial compression cylinder specimens were used to obtain the tensile and compressive properties of the rock, respectively. The specimens were loaded under quasi-static and dynamic loading conditions to give information about the strain-rate sensitivity of the tensile and compressive strengths. For a complete description of the experimental procedure and results, the reader is referred to [16] for quasi-static confined compression and Brazilian disc tests, [17] for dynamic confined compression tests with SHPB, and [10] for dynamic Brazilian disc tests with SHPB. Figure 1 summarizes these experimental data in terms of differential stress versus pressure at peak stress. Note that the specimen size may differ from one test to another due to testing apparatus constraints. A size effect is expected [18] and observed, but will be neglected here to limit the scope of the study.

### (b) Impact tests with seven-button drill bit

Impact tests with the seven-button drill bit were performed to assess the dependence of the impact energy and indexation angle on the BRI behaviour. Impacts on a Kuru granite block were carried out with an instrumented Instron drop tower system CEAST 9350 (figure 2*a*) and a drill bit with a diameter of 33 mm and seven tungsten carbide inserts (figure 2*b*). The rotation of the bit between each impact was enabled by a specially built rotating apparatus, which was integrated into the drop test machine (figure 2*c*). The impact energy *e*_{i} and impact velocity *v*_{i} are related to the total mass of the system *m* = 6.383 kg through , and are controlled by adjusting the initial height of the dropping assembly before the drop test. One advantage of such rigid-body impacts is the controllability of parameters such as impact velocity, energy and rebound, the possibility to use drill bits of any size, and the straightforward post-treatment of experimental data. The specifications of the drop machine and experimental apparatus are provided with more details in [19]. The tests were organized in three series of sequential drops with an indexation angle Δ*θ* of the rock block between each drop. The number of drop tests carried out in each series is specified in table 1, together with the average test results. The impact energy and the indexation (multiple of 10°) were kept constant during each test series. The test series 1, consisting of 17 successive drops, was carried out at impact energy *e*_{i} = 13 J and indexation Δ*θ* = 10° between each drop. The series 2 was carried out at higher impact energy *e*_{i} = 34 J, with similar indexation Δ*θ* = 10° between each of the 16 successive drops. The last series 3 was also carried out at *e*_{i} = 34 J but with an indexation angle Δ*θ* = 20° between each of the 10 successive drops. The influence of indexation is not investigated here, so the results of the series 2 and 3 (with *e*_{i} = 34 J) will be averaged in the discussion. The first impact of the first series was recorded after a number of approximately 30 impacts. These preliminary impacts enabled one to have the drill bit fully engaged in the rock block, i.e. no intact rock remained on the impacted surface. The results from these preliminary impacts are not presented here because they did not represent a stabilized drilling regime. Then, the three test series were carried out successively.

A strain gauge located approximately 5 cm above the embedded drill bit recorded the strain during the impact with a sampling frequency of 1 MHz. The force is calculated directly from the strain measurement through stress computation. The displacement is then calculated using Newton's law, by double integration of the force in the time domain and using the impact velocity computed from the height of drop (and considering friction). These calculations were done by the software Visual IMPACT and provided the resulting force versus displacement curves for each drop test. Figure 3*a* provides the force versus displacement curves obtained from the test series 3, performed with an impact energy *e*_{i} = 34 J and indexation angle Δ*θ* = 20° between each of the 10 impacts. A shift of origin has been applied to each BRI curve to get an optimal superposition of the peak forces. To fit these experimental BRI curves, trilinear laws have been calibrated for the three test series, but the reader is referred to [19] for further details. From these trilinear laws, two important outputs are reported in table 1, namely the peak force *F*_{p} and the indentation displacement (displacement of the drill bit reached at loss of contact between the bit and the rock).

Between each drop of the three test series, a measurement of the accumulated volume of the removed rock, *V*_{rem}, was obtained using a rubber replica of the rock surface. The rubber specimens were measured with two cameras connected to a stereo-lithography reconstruction system (ATOS/GOM) that provided a discretization of the bottom-hole surface into a mesh of size 0.1 mm. The volume under the reconstructed surfaces was then computed with a zero level reference captured on the imprint surface corresponding to intact rock (around the impacted hole). The effective penetration achieved at each impact, *u*_{eff}, can be obtained by dividing the volume measurement *V*_{rem} by the area of the hole of diameter 33 mm, *A*_{drill}. The penetration evolution during the three test series is reported in figure 3*b*, where the fit on each test series provides , the average effective penetration per impact in millimetres. Note that the initial penetration is not zero because 30 impacts were performed before the first test series in order to get the impacted rock surface fully damaged before the comparative series. The penetration is approximately twice as high when the impact energy is increased from 13 J to 34 J, and is significantly smaller than the indentation displacement computed from the BRI laws, , corresponding to the penetration experienced by the drill bit. The relation between these two displacement measurements is further discussed in [19]. These results can be interpreted in terms of mechanical specific energy (MSE)
2.1
where is the average volume of rock removed at each impact. The specific energies, reported in table 1, are of the same order as the uniaxial compressive strength of the rock (235 MPa [16]), suggesting a reasonable excavation efficiency [20]. The efficiency of the rock excavation is higher, i.e. MSE is smaller, when the impact energy is smaller (*e*_{i} = 13 J). Rock powder and small debris were observed during the series 1 (at *e*_{i} = 13 J) while some chips were created during the series 2 and 3 (at *e*_{i} = 34 J), but not at every impact. The collection of debris was not ensured consistently throughout the test series, so these observations are only qualitative. Nevertheless, it shows that the chipping process observed during the impacts at *e*_{i} = 34 J does not effectively increase the efficiency. Therefore, it suggests that if the chip formation is not substantial, impacting the rock with lower energy may turn out to be more efficient.

## 3. Modelling of rock material

The material model chosen to describe the behaviour of the Kuru granite rock is inspired from the Holmquist–Johnson–Cook (HJC) model [21] and its modified version (MHJC) [22]. This model can take the hydrostatic stress, the third stress invariant, the strain-rate sensitivity and the damage into account, and has proven accurate for large-scale computations of impact tests. The model is described here, followed by the identification of the parameters.

### (a) Description of the material model

The model, described with positive values representing compressive states, is an elasto-viscoplastic model coupled with isotropic damage, with separation of hydrostatic and deviatoric contributions. The Cauchy stress tensor **σ**, the von Mises equivalent stress *σ*_{eq}, the rate-of-deformation tensor **d** and conjugate equivalent strain rate are defined by
3.1
where **σ′** and **d′** are the deviatoric stress and rate-of-deformation tensors, respectively, *P* = tr(**σ**)/3 is the hydrostatic pressure and *µ* = tr(**d**) is the volumetric strain. The volumetric strain rate is decomposed into its elastic and plastic parts, , and so is the deviatoric rate-of-deformation tensor **d′** = **d′**^{e} + **d′**^{p}, with equivalent plastic strain rate conjugated to the von Mises stress defined as
3.2
where is the Jaumann derivative of the stress deviator and *G* is the elastic shear modulus. The MHJC model states that the deviatoric response is determined by the following constitutive equation:
3.3
Here *B* and *N* are the pressure hardening parameters, *C* is the strain-rate sensitivity exponent and *S*_{max} is the normalized maximum strength that the material can develop. The variables , , and are normalized with *σ*_{C}, the quasi-static uniaxial compressive strength, and , the reference strain rate, and *σ*_{3T} is the maximum hydrostatic tension that the material can withstand. The damage lies between *D* = 0 (no damage) and *D* = 1 (complete damage), and is split into shear and compressive damages, *D*_{S} and *D*_{C}, with evolution laws
3.4
where and are the plastic equivalent and volumetric strain rate, respectively, is the plastic volumetric strain in the fully compacted granular material and is the plastic strain at failure, where *α* and *β* are material parameters and the threshold to fracture the material. In addition, a tensile damage variable *D*_{T} is defined by
3.5
where is the minimum value of the volumetric strain reached at step *n* of the loading history and *µ*_{0} is the volumetric tensile strain threshold for crack formation. This tensile contribution is not coupled to the constitutive equations. The elements are eroded when one of the following conditions is fulfilled:
3.6
where and are the strain to failure under uniaxial compression and tension, respectively. Finally, the relation between hydrostatic quantities *µ* and *P*, given with more details in [21], can be summarized as
3.7
where *K* is the elastic bulk modulus, *H* is the volumetric strain hardening modulus and *K*_{1}, *K*_{2} and *K*_{3} are related to the fully compacted behaviour of the rock. The model was firstly implemented in [22] in the finite-element code LS-DYNA [23] and slightly modified to fit the model presented above.

### (b) Model parameter identification for Kuru granite

Among the material parameters summarized in table 2, 10 were calibrated on the experimental data presented in §2 and in [16] (*E*, *ν*, *σ*_{C}, *σ*_{3T}, *B*, *N*, *C*, , *β*, ), eight were taken from the literature (*ρ*, *S*_{max}, *K*_{1}, *K*_{2}, *K*_{3}, , *α*, ), six were computed from the other parameters (*K*, *G*, *µ*_{C}, *P*_{C}, *P*_{L}, *µ*_{0}) and one was obtained by inverse calibration on indentation tests (). The large number of parameters provides flexibility to the MHJC model, but the inverse calibration process on indentation remains challenging. The stress states under the indenter are very inhomogeneous [16] and the effect of each model parameter cannot be isolated easily. It was therefore chosen to calibrate all possible parameters on material test data presented in §2*a*, to use literature data for the remaining parameters, and to inverse calibrate the parameter that was mostly influencing the force versus displacement curve, namely .

The elastic parameters *E* and *ν* are calibrated on the elastic part of axial stress versus axial and radial strain curves obtained for confined compression tests under quasi-static loading conditions [16]. The uniaxial compressive strength *σ*_{C} is identified from the average peak load obtained for quasi-static uniaxial compression tests, also reported in [16]. The peak stresses obtained from the various confining pressure tests performed under quasi-static and dynamic loading conditions enable one to identify the pressure sensitivity parameters *σ*_{3T}, *B* and *N*, and the strain-rate sensitivity parameter *C* (the reference strain rate is chosen from quasi-static loading conditions). The calibrated yield functions under quasi-static and dynamic strain rates are illustrated in figure 1, together with experimental data. Two of the three parameters involved in the shear damage *D*_{S}, namely *β* and , are calibrated on the post-peak part of the differential stress versus axial strain curve obtained for quasi-static compressive tests performed at a confining pressure of 10 MPa. The influence of these two parameters on the post-peak prediction is illustrated in [22]. Otherwise, the tests necessary for the identification of some parameters were not performed in our experimental investigation, and their values were consequently taken from the literature. This is the case for the normalized shear strength threshold and hydrostatic parameters, which require high level confining pressure tests. The hydrostatic parameters were chosen from [24], calibrated for Iddefjord granite rock (for which the composition, grain size and porosity of 0.64% are close to those for the Kuru granite rock studied in this paper). The third parameter for shear damage, *α*, was also taken from [24] in order to limit the number of free parameters to calibrate. Several other model parameters were computed from others and should, as such, not be considered as free parameters of the model. This is the case for the elastic parameters *K* and *G*, for the crushing limit (*µ*_{C} and *P*_{C}) and *P*_{L} (deduced from *σ*_{C} and ) and for the tensile threshold *µ*_{0} obtained using equation (3.5) proposed in [24]. With all the above-mentioned parameters fixed, the parameter driving the tensile damage, , was obtained by inverse calibration on quasi-static indentation force versus displacement curves to reach a realistic peak force. Contrary to their influence on the BRI curves, the damage and volumetric parameters were found to have an important influence on the damage field under the indenters. It was observed that the chosen values for these parameters (from calibration and literature) led to some chipping prediction. Even though the volume of damaged area was overestimated (see results in §4*c*), it was chosen to keep these values in order to focus the analysis on the chipping process, which plays a major role in the efficiency of percussive drilling. It can also be emphasized that the data obtained from the literature were obtained under quasi-static loading conditions.

## 4. Numerical simulations of bit–rock impact tests

### (a) Numerical model

Finite-element analyses (FEA) are carried out using three-dimensional solid elements in the LS-DYNA explicit solver (see meshes in figure 4). The parameters of the linear elastic material model chosen for the bit correspond to steel and have negligible influence on the results. The density of the drill bit is adjusted to apply impact velocity and energy that correspond to the experimental values: *e*_{i} = 13 J and *v*_{i} = 2 m s^{−1}. Two different shapes are chosen to represent the initial impacted rock surface: a flat surface (FS) and a hyperbolic surface (HS). During impact on the FS, only the two central indenters impact the rock. The hyperbolic function used for HS is chosen so that the contact occurs simultaneously between the seven buttons and the rock specimen. During sequential drop tests carried out in the laboratory, the impacted surfaces are necessarily uneven due to previous impacts, but the choice of these two idealized surfaces for simulations is believed to bound realistic impact conditions.

A penalty method is used to solve the contact between the drill bit and the rock surface so that the friction coefficient is set to *γ* = 0.5 and is found to have negligible influence on the numerical predictions. To avoid wave reflection artefacts, the displacements constrained at the end of the rock block are supplemented with a non-reflecting boundary condition. The elements, whose erosion criterion as given in equation (3.6) is reached, are eroded. This potentially leads to force drops during the impact simulations. This paper presents the predictions obtained with four FEAs, described in table 3, to evaluate the influence of impact energy, confining pressure and rock shape surface.

### (b) Bit–rock interaction curve analysis

The force versus displacement curves obtained by FEA are presented in figure 5. The effect of the impact energy on the predicted curves is highlighted in figure 5*a*, and compared with trilinear BRI laws calibrated on experimental results [19]. The peak forces predicted by FEA overestimate the experimental values of 5–10% for *e*_{i} = 13 J and *e*_{i} = 34 J, respectively. The residual displacement predicted by FEA (i.e. displacement after the unloading) is much lower than in the experiments. However, a direct comparison between trilinear experimental BRI laws and FEA predictions should be avoided: the first loading stage observed on experimental BRI curves may be interpreted as testing artefacts such as, for instance, small debris on the impacting surface, or non-simultaneous contact of the buttons. The real impacted surface during sequential drop tests is also damaged from previous impacts. This is not accounted for in the numerical simulations. Thus, if considering only the second loading and unloading slopes of the trilinear BRI laws, the residual displacements obtained with FEA after unloading are comparable with experimental values. Besides, the strain rate encountered in the impacted area is of the order of , so the strain-rate sensitivity has an influence on the impact prediction. A simulation with *C* = 0 showed that the peak force and loading slope of the BRI curves were increased by strain hardening of only a few per cent. However, the strain-rate sensitivity of the rock plays only a negligible role in the effect of impact energy on BRI loading slopes here, because the local strain rates encountered with *e*_{i} = 13 J and *e*_{i} = 34 J remain of the same order.

Figure 5*b* highlights the effect of confining pressure and shape of the rock surface on the BRI curves. For the drop test simulation against HS, the displacement is approximately two times smaller and the peak force is almost two times larger than for the simulation against FS (at same impact energy *e*_{i} = 13 J). Besides, the effect of confinement on the force versus displacement curve is evaluated by applying a confining pressure *p* = 20 MPa, corresponding to a depth of 2 km. The effect of confinement is of the order of 10% increase in peak force and 10% decrease in indentation displacement, which is comparable to the predictions reported in the literature [14,16]. The effect of confinement is much smaller than the effect of the rock surface shape here. The link between the BRI curves and the excavation efficiency is not straightforward because the indentation displacement *u*_{ind} cannot be directly linked to the effective displacement *u*_{eff}. However, the dynamics of the drilling system may be strongly influenced by the peak force and penetration behaviour at the tool–rock interaction. From this point of view, the evaluation of the influence of the number of drill bit buttons in contact with the rock surface on the BRI curves is of importance.

### (c) Rock damage

The effect of impact energy, confining pressure and rock surface shape can also be analysed in terms of damaged field. The spatial distributions of the damage, given in figure 6, show that: (i) chipping is expected to occur without confining pressure, flat rock surface and low impact energy (see FS-13-0 in figure 6*a*); (ii) the impact energy is prone to chipping but also to the development of subsurface cracks (see FS-34-0 in figure 6*b*); (iii) less chipping is expected with confining pressure of *p* = 20 MPa (see FS-13-20 in figure 6*c*); and (iv) no chipping is expected when the seven buttons are impacting the rock simultaneously (with HS rock surface) because the damaged zones under each button do not connect (see HS-13-0 in figure 6*d*). The volume of damaged elements (reaching *D*_{C} = 1) is computed and corresponding values of effective penetration and MSE are reported in table 3 for each numerical simulation (see also figure 5 for MSE). Note that the volume of the elements belonging to a chip cannot be quantified here, as the chips are not formed in volume. Therefore, the chip was not accounted for in the evaluation of the volume of damaged rock. However, as a general trend, the MSE predicted numerically are lower than the experimental values (up to seven times lower for the configuration FS-13-0). A reason for this deviation is that the damage parameters were chosen from the literature. A new calibration of these parameters could lead to volumes of damaged rock closer to experimental values. However, this should be carefully carried out to keep good predictions of BRI curves and chipping process, and is not addressed here to limit the scope of the study.

The predicted efficiency is slightly reduced when the impact energy increases (the MSE increases of approximately 10%). This observation is in line with experimental results and suggests that an optimal impact energy can be determined for a given rock material. It is also observed that the confining pressure, in addition to preventing chipping, also decreases the efficiency of the process (the MSE also increases by approximately 10%). The influence of the rock shape surface on the MSE is substantial, with an increase of 25% when the seven drill bit buttons are in contact with the rock. This is in line with (i) the decrease in indentation displacement observed on the BRI curves, and (ii) the decrease of excavation efficiency when chipping is not promoted.

## 5. Conclusion and outlooks

This paper presents an experimental and numerical study of drill bit impact by sequential drop tests on a Kuru granite block. The experimental drop test series were carried out with a seven-buttons drill bit at impact energies of 13 J and 34 J and the MSE was found to increase with impact energy (corresponding to a decrease in excavation efficiency). Based on material test data from the literature, the parameters of a pressure-dependent elasto-plastic material model with coupled damage (MHJC) were identified for Kuru granite, and finite-element simulations of the impact were carried out. The predicted BRI curves and volume of damaged rock were compared to the experimental results for the two levels of impact energy. Moreover, the influence of the shape of the rock surface (flat or hyperbolic, corresponding to the impact of two buttons or seven buttons, respectively) and of the confining pressure were evaluated. The main conclusions of this study are that: (i) increasing the impact energy does not necessarily increase the excavation efficiency, both experimentally and numerically; (ii) a confining pressure corresponding to 2 km depth has an effect of approximately 10% increase on peak force, 10% decrease in indentation displacement and volume of rock removed, and prevention of chipping; (iii) the BRI curves and excavation efficiency predictions are more affected by the number of buttons impacting the rock (i.e. the rock shape surface) than by the confining pressure; and (iv) the predicted volume of rock damage is, so far, overestimated by the MHJC model. For further investigation, the influence of the damage parameters should be quantified, in particular for their effect on the volume of damaged rock and MSE evaluation. This is also the case for the hydrostatic response parameters, taken from the literature in this study. Furthermore, the damage of rock prior to impact could be addressed by repeated impact simulations for instance. Besides, studies of drop tests on intact rock could be pursued in order to compare to the present numerical results. Finally, the choice of a three-dimensional button configuration (with three buttons simultaneously impacting the rock for instance) could enable one to account for the chip volume in the total rock removal numerical prediction.

## Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

## Authors' contribution

M.F. contributed to the paper with drop tests, model choice, parameter identification, numerical simulations and paper redaction. A.K. contributed to the paper by supporting all experimental tests (material and drop tests), support in numerical simulations and participation in the redaction. M.H. participated with dynamic experimental tests and support for paper redaction.

## Competing interests

We have no competing interests.

## Funding

This study is part of the research activities of the Knowledge-building project for industry INNO-Drill, funded by the Research Council of Norway (grant no. NFR254984) and an industrial consortium.

## Acknowledgement

The authors would like to acknowledge Hieu N. Hoang for his contribution to the numerical model as well as Prof. Emmanuel Detournay and Alexandre Depouhon for the design of the experimental set-up for the drop test.

## Footnotes

One contribution of 15 to a theme issue ‘Experimental testing and modelling of brittle materials at high strain rates’.

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3587081.

- Accepted November 3, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.