## Abstract

In 2011, Holmquist and Johnson presented a model for glass subjected to large strains, high strain rates and high pressures. It was later shown that this model produced solutions that were severely mesh dependent, converging to a solution that was much too strong. This article presents an improved model for glass that uses a new approach to represent the interior and surface strength that is significantly less mesh dependent. This new formulation allows for the laboratory data to be accurately represented (including the high tensile strength observed in plate-impact spall experiments) and produces converged solutions that are in good agreement with ballistic data. The model also includes two new features: one that decouples the damage model from the strength model, providing more flexibility in defining the onset of permanent deformation; the other provides for a variable shear modulus that is dependent on the pressure. This article presents a review of the original model, a description of the improved model and a comparison of computed and experimental results for several sets of ballistic data. Of special interest are computed and experimental results for two impacts onto a single target, and the ability to compute the damage velocity in agreement with experiment data.

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

## 1. Introduction

In 2011, Holmquist and Johnson [1] presented a computational constitutive model for glass subjected to large strains, high strain rates and high pressures (going forward this will be referred to as the ‘original’ model). This model included many features to describe the complex responses observed in glass (the reader is referred to [1] for a thorough discussion of the complex behaviour of glass and modelling capabilities). These features included a material strength that was dependent on the location and/or condition of the material. Provisions were made for the strength to be dependent on whether it was in the interior, on the surface (different surface finishes could be accommodated), adjacent to failed material, or if it was failed. The intact and failed strengths were also dependent on the pressure and strain rate. Thermal softening, damage softening, time-dependent softening and the effect of the third invariant were also included. The shear modulus could be constant or variable. The pressure–volume relationship included permanent densification and bulking. Damage was accumulated based on plastic strain, pressure and strain rate. Although this model was capable of reproducing much of the laboratory data (tension, compression, plate impact, spall, interface defeat, various surface conditions, etc.) it produced computed ballistic results that were severely mesh dependent, converging to a solution that was much too strong.

This article presents a discussion as to why the original model was mesh dependent and presents an improved model that significantly reduces this dependency. The improved model maintains all the capabilities of the original model, but uses a new approach for the treatment of the interior and surface strengths. The improved model also includes two new features that provide more flexibility in defining damage and variations in the shear modulus. The remainder of this article presents a review of the original model, a description of the improved model and computed results for several ballistic impact configurations that demonstrate convergence, accuracy and robustness.

## 2. Review of the original glass model

The original model was presented in 2011 [1] and is reproduced in figure 1. The model has three components: one to describe the strength, one to describe the damage and one to describe the pressure–volume behaviour. This model was presented in detail in [1] and will not be represented here, but a short overview will be provided to add rationale and context for the development of the improved model. This overview will also identify the cause of the mesh dependency.

The strength portion of the model is presented in figure 1*a*. The strength is a function of the pressure, *P*, the strain rate, the location of the material (interior, surface or adjacent to failed material), and the damage, *D*. For undamaged material *D* = 0, for partially damaged material 0 < *D* < 1 and for failed material *D* = 1.

There are three curves, which represent the intact material. The reference intact strength represents the strength for material that is adjacent to failed material. This is the minimum intact strength and effectively represents a material with a very rough surface. The surface strength defines the strength of the surface, used to represent the condition of the surface. The interior strength is the maximum strength and represents material that is not on the surface or adjacent to failed material.

In the original implementation, the interior material is separated from failed material by nominally one finite element of reference strength material (material adjacent to failed material). This is the cause of the severe mesh dependency. As the mesh is refined, the volume of reference material is reduced. In the limiting case (as the mesh refinement approaches infinity), the volume of reference material approaches zero, effectively resulting in all the intact material being represented by the interior strength. This effect is demonstrated in the mesh refinement study presented in figure 2. A simple three-dimensional ballistic impact configuration is used where experimental data are available [2]. A 0.375 cal conical steel projectile impacts a 15.75 mm thick glass target at *V*_{s} = 600 m s^{−1} and the residual velocity is used to assess the effect of the mesh. The mesh resolution is defined by the number of composite elements used across the projectile radius. A composite element consists of 24 tetrahedral elements oriented in a hexahedral configuration. Figure 2*a* provides an example mesh resolution for four composite elements across the radius of the projectile (similar size elements are used in the target). Figure 2*b* presents the computed normalized residual velocity as a function of the mesh refinement using preliminary constants for a borosilicate glass. The normalized residual velocity is defined as , where *V*_{r} is the residual velocity and *V*_{s} is the impact velocity. Also included are test data from Anderson *et al*. [2], shown for an impact velocity of *V*_{s} = 667 and 546 m s^{−1}. The computed results from the original model are represented by the red line, a least-squares fit to nine different mesh resolutions. Also shown are the computed deformed geometries for a mesh resolution of three and nine elements. It is clear that the results are converging to a solution that is much too strong. Another possible approach, to limit the mesh sensitivity, is to define a thickness, or ‘*distance*’, for the next to failed material such that it is no longer a function of the mesh. Here a *distance* = 5 mm is used (approximately the projectile radius) and the computed results are shown with the blue curve in figure 2*b* (a least-squares fit to seven different mesh resolutions). This approach produces a converged solution that is less mesh dependent and closer to the experimental data. It may be possible to develop constants for this approach that provide a good correlation with the laboratory data and with ballistic impact data. The concern with this approach is that the ‘*distance*’ is not a material property and will be problem/geometry dependent. Because of this concern an alternative approach was developed as presented in the next section.

## 3. Description of the improved glass model

A description of the improved glass model is presented in figure 3. Only the strength and damage portions are presented because the pressure–volume behaviour is identical to that of the original model. The improved model is a result of incorporating three modifications into the original model. The first modification is the most important and involves modifying the internal and surface strengths, the second is a new feature that decouples the damage model from the strength model, and the third is a new feature that provides for a variable shear modulus that is dependent on the pressure. Each modification will be discussed in turn.

The strength portion of the model is shown in figure 3*a*. The intact strength is defined by a single curve with minimum values for the interior strength, , and the surface strength, This formulation allows for high internal tensile strengths to be developed including variations in the surface strength, as did the original model, but now there is significantly less strength in the lower pressure and tension region (for pressures less than the Hugoniot elastic limit, HEL). The authors are not aware of any test data that provide the internal strength at lower pressures other than spall data and the HEL. This formulation assumes that the internal and surface strengths have minimum values that are not pressure or strain rate dependent in the low-pressure and tension region, but become pressure and rate dependent as the pressure becomes more compressive. This form allows for high internal tensile strengths to be developed, as exhibited in plate-impact spall experiments [3–6], and for high internal compressive strengths to be developed, as exhibited in confined compression tests [7] and plate-impact tests [8]. And most importantly, this formulation produces computed solutions that are much less sensitive to the mesh without the need for the *distance* parameter.

The damage model was also slightly modified and is presented in figure 3*b*. Damage, *D*, is accumulated in the same manner as the original model
3.1
where _{p} is the increment of equivalent plastic strain during the current cycle of integration and is the plastic strain to failure under the current dimensionless pressure, *P** = *P*/σ_{max}. The general expression for the failure strain is
3.2
where *D*_{1}, *N* and *C*_{f} are dimensionless constants and The strain rate constant *C*_{f} provides an increase in the strain to failure for elevated strain rates and *P*_{plastic} is the pressure at which plastic strain begins to accumulate. For pressures less than *P*_{plastic}, no plastic strain can be developed, resulting in failure (*D* = 1.0) when the yield surface is reached.

The addition of *P*_{plastic} decouples the damage model from the strength model, providing more flexibility in describing damage. In the original model, the pressure at which plastic strain begins to accumulate is defined by *T*, which is used to determine the reference strength as shown in figure 1.

Lastly, a new feature was added that provides for a variable shear modulus, *G*, that is dependent on the pressure, expressed as
3.3
where *P* is the pressure (compression is positive), *G*_{o} is the shear modulus at *P *= 0 and *α* is a constant with dimensions area/force. *G* can be increasing (if *α* is positive) or decreasing (if *α* is negative). There is also a provision that limits the magnitude of the shear modulus defined by an additional constant, *G*_{limit}. Although the original model provided for a variable shear modulus it was restricted to be proportional to the current bulk modulus, based on the assumption of a constant Poisson's ratio. This assumption was found to be incorrect for some glasses as demonstrated by Zha *et al.* [9], who measured the shear and bulk modulus as a function of pressure for SiO_{2} glass and showed that Poisson's ratio varied significantly for pressures up to approximately 10 GPa. The form presented in equation (3.3) provides the ability to reproduce the response exhibited by SiO_{2} and other glasses.

## 4. Example computations for high-velocity impact conditions

The following computed results are for high-velocity impact conditions. The computed results use the improved glass model and constants for a specific form of borosilicate glass, Borofloat® 33, manufactured by Schott AG. The determination of constants is not a straightforward process because some of the constants cannot be determined explicitly from the test data. The general approach to determine constants was presented by Holmquist & Johnson [1]. For Borofloat 33, all of the constants are determined directly from data in the literature [7,8,10–12], except for the damage and failed-strength constants which are inferred from dwell and penetration data [13,14]. Time-dependent softening is not used because it has been shown to inhibit the propagation of failure [11,15], in disagreement with experimental data.

The objective of this section is to demonstrate the ability of the improved glass model to produce computed results that are in good agreement with experimental data, for a wide range of impact conditions, using one set of constants. Of particular interest is the ability to produce: a converged solution that is in agreement with ballistic data; the propagation of damage/failure that is in good agreement with an edge-on-impact experiment; and multiple impacts onto the same target. All the computations were performed with the 2016 version of the EPIC code [16] using one-, two- and three-dimensional finite elements. For the two- and three-dimensional examples involving severe distortions the finite elements are automatically converted to meshless particles during the course of the computation [17].

### (a) Convergence

Figure 4 presents a comparison of computed results as a function of mesh refinement using the original model and the improved model. Also shown are test data provided by Anderson *et al*. [2]. The red line is a least-squares fit to the computed results using the original model, clearly converging to a solution that is much too strong. The green line is a least-squares fit to the computed results using the improved model; the results are much less mesh sensitive and converge to a solution that is within the experimental data. The computed results for a mesh refinement of five and nine elements are also shown and their responses are very similar.

### (b) Plate-impact spall

A characteristic of glass is that it can develop high internal tensile stresses when not near a surface. This has been demonstrated by several researchers using plate-impact spall experiments [3–6], and more recently using laser shock techniques [10]. Figure 5 presents computed results for a plate-impact spall configuration demonstrating the ability to produce high internal tensile stresses. The computation is performed in one-dimensional uniaxial strain geometry, with a mesh resolution of 0.02 cm per element. For this example, the internal tensile strength is assumed to be and *P*_{plastic} = 0. Figure 5*a* presents the geometry where a 1.0 cm glass impactor strikes a 3.0 cm glass target at *V* = 310 m s^{−1}. This configuration will produce maximum tension inside the target located 1.0 cm from the rear target surface (spall plane). Figure 5*b* and 5*c* present the net *z*-stress (along the target) at various times after impact. The elastic compressive stress at *t* = 4.90 µs is shown in black, propagating from left to right, and has not yet reached the rear surface of the target. At *t* = 6.52 µs (red) the leading compressive wave has reflected off the rear surface, travelling from right to left, relieving the target from compression. At *t* = 7.33 and 7.52 µs (green and blue), the target begins to go into tension, located at the spall plane, reaching a magnitude of 1.3 GPa (gold). *σ*_{spall} = 1.3 GPa is the maximum tensile stress allowed (under uniaxial strain conditions) where Figure 5*c* shows the axial stress, and damage, during the failure process. Failure occurs very quickly because the material can develop no plastic strain in tension because *P*_{plastic} = 0. At *t* = 7.62 µs (gold), the maximum internal tensile stress cannot be maintained and quickly attenuates to 70 MPa (black/red/blue in figure 5*c*). This is due to the formulation of the model. When failure occurs at the spall plane, the material is failed. The material next to the spall plane is now governed by the intact strength, which can only maintain a small tensile stress (70 MPa for this example). This 70 MPa propagates to the rear surface, producing a small pull-back signal as shown in figure 5*d*. This is an interesting result; the actual spall stress is 1.3 GPa, but, due to the rapid stress attenuation after failure, the stress that reaches the rear surface is only 70 MPa. The authors are not aware of any experimental data that can validate this response, but it is consistent with experimental plate-impact results that monitor the rear surface that show either complete elastic release or a very low (or zero) spall strength [3,4]. It should be noted that the experimental tensile spall behaviour (full elastic release or zero spall stress) is usually explained by the presence of a failure wave. Although the model presented here can produce failure/damage velocities in good agreement with an edge-on-impact experiment (discussed in §4f) it is not known if the model can reproduce the complex failure wave phenomenon. This will require further research.

### (c) Interface defeat and high-velocity penetration

Figures 6 and 7 present computed results for a gold rod impacting a borosilicate cylinder with a copper buffer attached to the impact surface. The buffer is included to attenuate the impact shock and produce gradual loading on the glass. The computations are performed in two-dimensional axisymmetric geometry using five composite elements (four crossed triangular elements in a composite element) across the projectile radius and similar size elements in the target. Figure 6 presents the computed results for an impact velocity of *V* = 800 m s^{−1} including the initial geometry. Results are shown at *t* = 0, 30, 60 and 90 µs after impact and produce interface defeat (no glass penetration). Also shown is damage in the target at *t* = 90 µs, which shows a crack that runs to the periphery of the target and a zone of damaged material below the glass surface, directly under the projectile. This zone of damaged material has been observed in targets subjected to indentation experiments and interface defeat tests. Figure 7 presents the computed results for an impact velocity of *V* = 900 m s^{−1} where prompt penetration occurs (no dwell). These results are in good agreement with test data provided by Anderson *et al.* [13].

Behner *et al.* [14] performed experiments using no buffer (gold rods impacting bare glass), producing penetration at much lower impact velocities. For these experiments, the penetration velocities were determined (using a series of flash X-rays) and are presented as a function of impact velocity in figure 8*a*. Also shown are the computed penetration velocities for impact velocities of *V* = 400, 500, 1500 and 2400 m s^{−1}. The computed penetration velocities are presented for both two-dimensional axisymmetric geometry (red) and three-dimensional half-symmetry geometry (yellow). The three-dimensional results are slightly higher than the two-dimensional results for *V* = 1500 and 2400 m s^{−1} but straddle the experimental results. Figure 8*b* presents the computed three-dimensional results for the three impact velocities. The three-dimensional results produce a slightly higher penetration velocity (than the two-dimensional results) probably due to the development of radial failure. The overall computed results are in good agreement with the test data provided by Behner *et al*. [14].

### (d) Steel projectiles impacting thin glass plates

Anderson *et al*. [2] performed experiments using conical-nosed steel projectiles (Rc 53) impacting thin plates of borosilicate glass at three different scale sizes. For the largest scale (scale = 1.0), the projectile was 12.7 mm in diameter and 38.1 mm long (0.50 cal) and the target was 21.0 mm thick and 457.2 mm square. For the smaller scales (scale = 0.75 and 0.44), both the projectile and target plates were scaled accordingly. The experimental results produced some significant findings: the targets appeared stronger as the scale was reduced; glass is very strong, producing significant projectile erosion; there were spall cones produced on the rear of the plates; the glass plates were not capable of stopping the projectile, even at very low impact velocities (no *V*_{50} could be determined); as the impact velocity increased so did the amount of glass damage/cracks; there is a discontinuity in the *V*_{s}–*V*_{r} curve at approximately *V*_{s} = 500–600 m s^{−1} (this is also where the maximum projectile erosion occurred, and where the *V*_{50} occurred when a polycarbonate back plate was included). The computations focus only on the scale = 0.75 as it appears to be representative of the other scales (the scale effect is not investigated here). For the scale = 0.75 the projectile is 9.6 mm in diameter (0.375 cal), 28.6 mm long, with a mass of 11.50 g (the same projectile as presented in figure 2). The glass plates are 342.9 mm square and 15.75 mm thick.

Figure 9 presents a comparison of the computed and experimental results for the 0.375 cal steel projectile impacting a 15.75 mm thick borosilicate plate. All the computed results use the 0.375 cal projectile presented in figure 2. The glass plates were reduced to 160 mm square (the experimental targets were 342.9 mm square) to reduce the computational time and they were performed in half-symmetry with unrestrained (free) boundaries. The mesh resolution uses five composite elements across the projectile radius with similar size elements throughout the entire target (this will be the mesh resolution used for all the remaining computations). Figure 9 presents computed results for *V*_{s} = 300, 500, 600 and 950 m s^{−1} showing damage (the particles are not shown so that target fragmentation can be better visualized), and a comparison of the computed and experimental residual velocities. The computed results are in good agreement with the test data for impact velocities of *V*_{s} < 700 m s^{−1} and slightly overpredict target resistance for the higher impact velocities. Noteworthy results include the ability to produce: spall cones; perforation at very low impact velocities (*V*_{r} = 12 m s^{−1} at *V*_{s} = 200 m s^{−1}); projectile damage and erosion; an increase in damage/cracks as the impact velocity increases; and a discontinuity in the perforation response in the range *V*_{s} = 500–600 m s^{−1}. The discontinuity is a result of transitioning from a rear surface cone formation to a perforation of the cone.

### (e) Steel projectiles impacting thin glass plates with a Lexan® substrate (single and multiple impacts)

Anderson *et al*. [2] performed experiments using conical-nosed steel projectiles impacting thin plates of borosilicate glass with a Lexan® substrate. The experimental projectile and target geometry are identical to those presented in the previous subsection, except the target now includes a Lexan plate bonded to the rear surface of the glass.

Figures 10 and 11 present the computed and experimental results for a 0.375 cal steel projectile impacting a 15.75 mm borosilicate plate backed by a 9.52 mm Lexan plate bonded together with a 1.27 mm layer of polyurethane. The computational geometry is identical to the experimental geometry tested by Anderson *et al*. with the exception of the lateral dimensions of the target. The experimental targets were 342.9 mm square, the computed targets were 160 mm square (using a plane of symmetry). The initial geometry for the computations is shown in figure 10*a* and figure 10*b* presents computed results for impact velocities of *V*_{s} = 500, 700, and 950 m s^{−1} (showing both material and damage).

Figure 11 presents a comparison of the computed and experimental results for both single and multiple impacts (the multiple impacts will be discussed later). The single-impact experiments are shown with solid black circles and the computations are shown with the solid red circles. The dashed red line provides an approximate fit to the computed results. The computed results produce a ballistic limit (*V*_{50}) that is slightly lower than the experimental *V*_{50} = 657 m s^{−1} provided by Anderson *et al.* [2]. At the higher impact velocities, the correlation is improved. It is not clear why the computed ballistic limit is less than the experiment, but it may be due to inaccuracies in the polyurethane model or limitations in the glass model. Recent computed results (not included here) were in good agreement with ballistic impact experiments against Lexan-only plates provided by Anderson *et al*. [2].

Anderson *et al.* [2] also performed multiple impacts onto a single target. The target geometry was identical to that used in the single-impact experiments, but now the target is impacted five separate times. The first impact was always in the target centre, while the other four impacts were in the centre of the four quadrants. Thus, the second to fifth impacts were all located the same distance from the first impact, approximately 120 mm. The order of the impacts was as follows: (i) target centre, (ii) upper left quadrant, (iii) upper right quadrant, (iv) lower right quadrant, and (v) lower left quadrant. An interesting experimental result was that there was always degraded target performance for the second impact, but there was no additional degradation for the third to fifth impacts. This allowed the second to fifth impacts to be grouped together as shown in figure 11, represented with the open circles. This indicates that damage does not propagate effectively through damaged glass.

Figures 11–13 present computed results for two impacts onto a single target. The initial geometry is presented in figure 12*a*. The target size is reduced from the experimental dimension (342.9 mm square) to 292 × 80 mm (using half-symmetry), which greatly reduces the time required to complete the computations. Although the target is significantly smaller than the tested target, it is large enough that valid comparisons can be made. The initial geometry shows the second impact located 120 mm from the first impact (same as the experiment) and occurs 100 µs after the first impact. One hundred microseconds was chosen because it is sufficiently long to ensure that the plastic work in the target is complete. Additionally, the first impact is located 86 mm from the target edge, a sufficient distance to eliminate edge effects. The computed results are presented in figure 12*b*, which shows both material and damage (red is failed). For this computation, the first projectile impacts at *V*_{1s} = 800 m s^{−1} and the second projectile impacts at *V*_{2s} = 500 m s^{−1}. At *t* = 100 µs, the first projectile has perforated the target (*V*_{1r} = 497 m s^{−1}), generating significant damage. Although the damage does not extend to the location of the second impact, it is sufficient to influence (degrade) the target resistance, as demonstrated by the second projectile perforating at *V*_{2r} = 281 m s^{−1} (an undamaged target stops this impact velocity as shown in figure 10). Another indication that target resistance has been significantly reduced is the small amount of damage produced in the second projectile, as compared to the damage produced from an undamaged target (as shown in figure 10).

Figure 13 presents two additional computed results showing damage. The result in figure 13*a* is for a first impact of *V*_{1s} = 800 m s^{−1} and a second impact of *V*_{2s} = 300 m s^{−1}; the second projectile does not perforate the target. The result in figure 13*b* is for a first impact of *V*_{1s} = 800 m s^{−1} and a second impact of *V*_{2s} = 400 m s^{−1}; the second projectile perforates the target at *V*_{2r} = 95 m s^{−1}. Figure 11 presents a comparison of the computed and experimental multiple-impact results. The experiments are shown with the open circles and the computed results are shown with the solid blue circles. The blue dashed line is an approximate fit to the computed results. The computed results are in reasonable agreement with the experimental data and clearly show reduced target resistance for a second impact. It should be noted that the computed crack density, and radial extent of cracking, is less than observed experimentally, particularly when the Lexan back plate is included. It is anticipated that the crack density, and extent of cracks, will increase as the mesh is refined (as was discussed in previous work [15]), which will reduce the glass resistance for a second impact.

### (f) Edge-on impact

E. Strassburger (2016, personal communication) [18] performed edge-on-impact experiments to determine the evolution and velocity of damage in a borosilicate glass target. A steel sphere, 15.87 mm in diameter, was used to impact the edge of a Borofloat glass plate 100 × 100 × 10 mm and high-speed imaging was used to visualize damage and its propagation. Figure 14*a*–*g* presents the initial geometry and a comparison of the computed and experimental result for an impact velocity of *V* = 430 m s^{−1}. The computation is performed in three dimensions with a plane of symmetry, free boundaries and a mesh resolution of 1.0 mm per composite element. Three experimental images are provided by Strassburger [18], presented in figure 14*d*, taken at *t* = 6.8, 14.8 and 24.8 µs after impact. Strassburger used 15 images, over nearly 29 µs, to determine the damage velocity and found that it was relatively constant at *V*_{D} = 2034 m s^{−1}, as shown in figure 14*g*. Computed damage images are presented at the same time (figure 14*e*) as the experimental images (the damage contour legend is the same as presented in figure 9). The computed damage velocity was determined in the same manner as the experiment (linear regression of the propagation distance as a function of time) and was determined to be *V*_{D} = 2043 m s^{−1} as shown in figure 14*g*, in good agreement with the experiment. It should be noted that the computed damage images presented in figure 14*e* show only material that is damaged, thus allowing damage to be visualized through the thickness of the plate, similar to the experiment. If only surface damage is presented, as shown in figure 14*b*,*c*, the extent of damage is under-reported because there is significant interior damage not visualized. Also presented is the computed von Mises stress (figure 14*f*) which propagates at the longitudinal wave velocity (*V*_{L} = 5500 m s^{−1}), clearly exceeding the velocity of damage.

## 5. Summary and conclusion

This article has presented an improved model for glass that uses a new approach to represent the interior and surface strength that is significantly less mesh dependent than a previous model developed by Holmquist & Johnson [1]. This new formulation allows for the laboratory data to be accurately represented and produces converged solutions that are in good agreement with ballistic impact experiments. The model also includes two new features: one that decouples the damage model from the strength model, providing more flexibility in defining the onset of permanent deformation; the other provides for a variable shear modulus that is dependent on the pressure. This article presented a description of the improved model, a demonstration of some of its new features, and a comparison of computed and experimental results for several sets of ballistic impact data. Of special interest are computed and experimental results for two impacts onto the same target that demonstrate the degradation in target resistance for a second impact, and the ability to compute the damage velocity in agreement with an edge-on-impact experiment.

## Authors' contributions

T.J.H. and G.R.J. formulated the model and the logic for implementation. C.A.G. implemented the model into the EPIC code and performed verification computations. T.J.H. evaluated the laboratory data, determined constants, performed validation computations and drafted the manuscript. All authors gave final approval for publication.

## Competing interests

The authors declare they have no competing interests.

## Funding

This work was funded by the US Army Tank Automotive, Research, Development and Engineering Center (TARDEC) in Warren, MI, USA, under contract W56HZV-13-C-0047-WD004.

## Acknowledgements

The authors would like to thank T. Talladay and R. Rickert (TARDEC), P. Patel (ARL) and C. Anderson Jr. (SwRI) for their contributions to this work.

## Footnotes

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

- Accepted September 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.