Dynamic Compressive Strength and Fragmentation in Sedimentary and Metamorphic Rocks

Brittle deformation at high strain rates results in intense fragmentation and rock pulverisation. For rocks, the critical strain rate at which this behaviour occurs is ~102 s-1. The mechanical properties of rocks at these strain rates can also be very different from their quasi-static properties. Deformation of rocks at these strain rates is uncommon in nature but can occur during fault rupture, landslide events, and meteorite impacts. In this study, we present results of high strain rate mechanical tests to determine the characteristic strain rate for rate-dependent brittle failure, and the fragment size and shape distributions that result from failure at these conditions. We investigated sandstone, quartzite, limestone, and marble and considered whether the fragment characteristics can be used as diagnostic indicators of loading conditions during brittle failure. We find that the characteristic strain rates, where the dynamic strength is twice the quasi-static strength, range between ~150 and 300 s-1 for rate-dependent brittle failure in the investigated lithologies. Furthermore, we use our results to demonstrate an empirical inverse power-law relationship between fragment size and strain rate for dynamic failure under uniaxial compression. On the other hand, we show that fragment shape is independent of strain rate under dynamic uniaxial loading.


Introduction
Fragmentation of rocks occurs during a variety of geological processes, including seismogenic fault rupture , gravitational mass movements (De Blasio and Crosta, 2014), and meteorite impacts . In all these events, brittle failure may occur under dynamic, high strain rate conditions. At these conditions, the mechanical properties of rocks , the behaviour of individual fractures (Fineberg et al., 1991;Ravi-Chandar and Knauss, 1984a;Sharon and Fineberg, 1999;Zhang et al., 1999), and the interaction between growing fractures (Ramesh et al., 2015) can be strongly rate-dependent and deviate from quasistatic brittle failure.
The results of mechanical experiments demonstrate that rocks behave with a strength that is near constant at low strain rates before increasing markedly beyond a threshold strain rate of ~10 0 -10 3 s -1 Ramesh et al., 2015;Zhang and Zhao, 2014, and refs. therein). A number of analytical and micromechanical models have been developed that demonstrate this increase in strength (Bhat et al., 2012;Hild et al., 2003;Paliwal and Ramesh, 2008). In a study that explored one of these micromechanical models (Paliwal and Ramesh, 2008),  demonstrated that the behaviour of rocks from quasi-static to high strain rates can be described by a universal scaling relationship: where $ and ̇ are the compressive strength and strain rate, respectively, and the material parameters, % and %, are the characteristic strength and characteristic strain rate, respectively.
The characteristic strength is equivalent to the quasi-static uniaxial compressive strength and the characteristic strain rate corresponds to the strain rate at which strength is double the value of the quasi-static strength.
The dependence of rock strength on strain rate is fundamentally linked to the nucleation and growth of fractures. Fractures propagate at a finite velocity: during failure at low rates, the weakest available flaw in a material can initiate a single fracture that can grow fast enough to accommodate the applied loading; at high rates, that single flaw cannot develop a single fracture before other increasingly strong flaws are activated Ramesh et al., 2015). Furthermore, it has been demonstrated that the behaviour of individual fractures may change as a result of high-rate loading; increasing fracture toughness and producing hierarchical fracture branches (Ravi-Chandar and Knauss, 1984a;Sharon and Fineberg, 1999;Zhang et al., 1999). The consequence of these processes is that rock failure at high strain rates leads to intense fracturing and fragmentation.
Upper crustal fault zones generally consist of a fault core and a surrounding damage zone. The fault core contains highly comminuted material that accommodates most of the cumulative shear strain while the damage zone consists of fragmented and brecciated rock with little or no shear displacement (Faulkner et al., 2003). Brittle deformation in the damage zone is generated by coseismic transient loading conditions; either by rapid reduction of normal stress (Brune et al., 1993), local strain near fracture tips , and/or the propagation of shock waves during supershear rupturing . Coseismic fracturing and fragmentation, the extent and magnitude of which is controlled by the loading rate, may be an important component of the energy budget of rupture events Ghaffari et al., 2019;Rockwell et al., 2009;Wilson et al., 2005). The most extreme deformation caused by coseismic dynamic fracturing is thought to be expressed in so-called pulverised rocks, which have extremely high fracture densities and low shear strain. These rocks can be found tens to hundreds of meters from their fault core (Dor et al., 2006a;Fondriest et al., 2015;Mitchell et al., 2011;Rempe et al., 2013).
The close link between the variation of strength with strain rate and fragmentation behaviour has been demonstrated with a variety of dynamic uniaxial compression experiments Doan and Billi, 2011;Ghaffari et al., 2019;Yao et al., 2020;. Furthermore, fragmentation laws have been proposed that describe the variation of fragment size in an expanding shell as a function of the applied strain rate (Glenn and Chudnovsky, 1986;Grady, 1982;Levy and Molinari, 2010;Zhou et al., 2006aZhou et al., , 2006b, all of these models consistently predict that, at high strain rates, fragment size has an inverse power-law relationship with strain rate with an exponent of 2/3. This exponent arises as a consequence of the equilibrium conversion of kinetic energy released in fragmentation ( & ∝ '( , where is the fragment size) to fracture surface energy ( ) ∝ ( ) (Grady, 1982). Direct comparisons between fragmentation models and the products of compressive failure are problematic because the fragmentation of an expanding shell is a tensile process. However, in general, experimental studies have shown that the fragmentation models overestimate average fragment sizes and/or do not follow a power law with the expected exponent (Ghaffari et al., 2019;Hogan et al., 2013Hogan et al., , 2012Lankford and Blanchard, 1991;Wang and Ramesh, 2004). Nevertheless, the combination of experimental and/or theoretical fragmentation models with field observations of naturally fragmented rock masses provides an important opportunity to determine transient loading conditions during coseismic deformation or other high-strain rate geoprocesses (Rowe and Griffith, 2015).
In this study, we aim to investigate the fragmentation behaviour of rocks under dynamic loading, characterising how fragment size and shape varies as functions of strain rate, therefore providing potential diagnostic indicators of transient loading conditions in naturally deformed rocks. To achieve this, we conduct quasi-static and dynamic uniaxial loading experiments on rock materials, determining dynamic rock mechanical properties and performing post-mortem analysis of the generated rock fragments. For this study, we decided to limit our investigation to sedimentary and metamorphic rocks: sandstone, quartzite, limestone, and marble. These rock types were chosen because, first, pulverisation has been widely reported in association with faults hosted in sandstones (Dor et al., 2006a(Dor et al., , 2006bKey and Schultz, 2011;Peppard et al., 2018) and carbonates (Agosta and Aydin, 2006;Fondriest et al., 2015;Sagy and Korngreen, 2012;Schröckenfuchs et al., 2015), and second, it allowed us to investigate the effect of porosity on fragmentation and dynamic material properties in lithologies with similar mineralogies.
The Seeberger Sandstone is composed of 89 % quartz, ~10 % phyllosilicates (mostly clay minerals and subordinate micas), and minor accessory minerals (Ebert et al., 2014). Grain sizes range between ~50-150 μm. Macroscopically, weakly developed bedding and Liesegang banding can be observed, but do not cause significant mechanical anisotropy. The bulk density and porosity of the Seeberger Sandstone was determined by He-pycnometry to be 2130 ± 21 kg m -3 and 20.0 ± 0.8 % respectively. The Taunus Quartzite is composed of 91 % quartz, ~8 % phyllosilicates, and minor accessory minerals (Ebert et al., 2014). Grain sizes range from ~100-400 μm. The quartzite contains minor joints with an approximate spacing of 10 cm, the joints are commonly mineralized with chlorite. The bulk density and porosity of the Taunus Quartzite was determined by Hepycnometry to be 2645 ± 6 kg m -3 and 0.6 ± 0.1 % respectively. The Savonnières Limestone is an oolitic grainstone with partial sparitic cement. Vacuolar ooids are common and the rock contains occasional shell fragments. Mineralogically, the rock is composed of close to 100 % calcite. The ooids typically range in size from 400 to 750 μm. The bulk density and porosity of the Savonnières Limestone was determined by He-pycnometry to be 1881 ± 31 kg m -3 and 31.4 ± 1.1 % respectively. The Carrara Marble is composed of 98 % calcite, and minor quantities of quartz, mica, dolomite, epidote, and pyrite (Pieri et al., 2001). Grain sizes range from 200 -400 μm.
Calcite grains are commonly twinned. In hand specimen, the marble is massive and homogeneous.
The bulk density and porosity of the Carrara Marble was determined by He-pycnometry to be 2711 ± 6 kg m -3 and 0.3 ± 0.0 % respectively.
The samples used for rock deformation experiments were cored from blocks of each lithology. Typical samples were ~41 mm in diameter, however cores of the Taunus Quartzite were ~35 mm in diameter for a number of reasons: firstly, the Taunus Quartzite is very strong, narrower diameter cores made it easier to achieve failure with the SHPB. Secondly, the joints within the Taunus Quartzite made drilling challenging, reducing core diameter made it easier to produce appropriate samples. Additionally, cores of the Savonnières Limestone were ~54 mm in diameter for quasi-static uniaxial compressive strength (UCS) testing due to the material's extreme weakness; larger diameter samples ensured that a larger force could be applied to the samples and that a greater amount of data could be recorded in the elastic regime. The lengths of the samples were maintained such that samples had ~1:1 aspect ratios for SHPB experiments  and ~2:1 aspect ratios for quasi-static UCS testing

Quasi-static
Uniaxial compression experiments under quasi-static conditions were performed with a servo-controlled Form+Test Alpha 2-3000 S hydraulic press in the Department of Geology at the University of Freiburg. Longitudinal and transverse strains in the samples were measured using displacement gauges. Experiments were run under constant loading-rate conditions, which, within the elastic loading regime leads to constant strain rates. Experiments were run at strain rates between 5.3 × 10 -6 and 3.4 × 10 -5 s -1 . Between 3 and 8 quasi-static experiments were carried out for each lithology. and  . Samples were recovered from the SHPB by collection in a secure box lined with a polyethylene bag.

Size
On average, we were able to recover 88.2% of the mass of each sample from the SHPB experiments, recovery was generally poorer in the sandstone and limestone, with average recoveries of 83.4% and 84.4% respectively, while 95.5% and 88.6% recoveries were achieved with the quartzite and marble respectively (see GitHub repository for further details). Fragment size distributions were obtained by sieving the fragmented rock masses resulting from the experiments. Each sample of the same lithology were passed through identical sieve stacks, although sieve stacks were varied between lithology. Each sieve stack had at least 7 sieves ranging from 16 mm to 0.125 mm square apertures. Sieving was carried out dry and shaking was carried out by hand for a duration of one minute per sample.
A consequence of using sieving to determine fragment size distributions is that it results in mass-size distributions, as opposed to number-size distributions. A variety of statistical distributions can be used to fit fragment size distributions from failure events (see Grady, 2010).
Here, we have chosen to use cumulative Weibull distributions: where ( ) is the cumulative mass of fragments that have a size less than ; . is the total sample mass; and λ and k are the scale and shape parameters of the distribution respectively. There is no theoretical basis for our usage of the Weibull distribution which we chose only for its simplicity, general usage (Grady, 2009), and approximate match to the shape of our data. Non-linear least squares fitting was used to determine the Weibull distributions. Other distributions could have been used to fit our fragment size data, e.g., Rayleigh (Levy and Molinari, 2010), log-normal (Wang and Ramesh, 2004), or generalized extreme value (Hogan et al., 2012). The median of a Weibull distribution, ̅ , can be calculated as: In the case of the quartzite, limestone, and marble, no more than 26% and on average 2%, 8%, and 16% respectively of the total mass of the samples passed the finest sieve. Thus, for these lithologies, the median fragment size is an interpolated value within the distribution. However, the sandstone produced substantially finer fragments such that on average 42% of the total mass of the samples passed the finest sieve. In 5 sandstone samples, the mass percentage passing through the finest sieve exceeded 50%, and therefore the calculated median fragment size is extrapolated below the size of the finest sieve size. We were unable to extend these distributions to finer fragment sizes due to unreliability of dry sieving through finer sieves. Additionally, we attempted to use laser diffraction granulometry (Malvern MasterSizer 3000) on the fine fragments but were unable to achieve consistent overlap between sieving and laser diffraction analyses and between different runs of the laser diffraction granulometer on the same sample to extend the fragment size distributions to smaller fragment size. Our problems of integrating fragment size distributions across different methodologies are similar to those reported by Rockwell et al. (2009) on pulverized rocks from the San Andreas and Garlock faults in southern California.

Shape
The shapes of fragments generated by the experiments were determined by image analysis.
We focussed on two size fractions: fragments larger than 2 mm, and fragments from 0.5 to 2 mm in size. These size fractions had to be imaged separately and followed different image processing procedures for a variety of practical reasons (for details, see Supplementary Material). Images were processed using the Fiji software package ; see Supplementary Material for details) and fragment shape parameters were determined for each fragment. Here, we have considered two shape factors, circularity, C, and axial ratio, AR (Heilbronner and Barrett, 2014): where A is the area of the fragment, P is the perimeter of the fragment, w is the minor axis length of the best-fitting ellipse, and l is the major axis length of the best-fitting ellipse. C has a value of 1 for a perfect circle and approaches 0 as the shape's perimeter increases relative to its area. AR has a value of 1 for a perfectly equant shape, i.e. any regular polyhedra, and approaches 0 as the shape becomes increasingly elongate.

Fragment Shape Distributions
Overall, the shape distributions of fragments do not vary as functions of strain rate or with lithology ( Figures 5 and 6). For the fragments larger than 2 mm and across all lithologies, the average circularity is 0.733 ± 0.087 and the average axial ratio is 0.627 ± 0.146 ( Table 2). No individual distribution, regardless of strain rate or lithology, is an outlier of these average values.
For the fragments that range between 0.5 and 2 mm and across all lithologies, the average circularity is 0.598 ± 0.180 and the average axial ratio is 0.642 ± 0.152 ( Table 2). The average circularity of these smaller fragments is less than the average circularity of the larger fragments while axial ratios remain similar, additionally the standard deviations of the distributions of these finer fragments is generally larger. We largely attribute these variations to differences in method between larger and smaller fragments (see Supplementary Material). However, we observe similar trends in the fragment shape distributions from sample to sample between the different fragment size fractions (Figures 5 and 6), demonstrating the overall robustness of the fragment shape analysis.    . suggesting a characteristic strain rate less than 345 but greater than ~10 1 s -1 . Again, this is lower than the value of the characteristic rate reported in this study, however again, our study is based on a larger data set, and provides more constraint on the value of the characteristic strain rate.
Furthermore, we note that our study is consistent with the results of Millon et al. (2016)  Other studies of dynamic strength in sedimentary and metamorphic rocks are generally consistent with the results of our study. Green et al. (1972) conducted one of the earliest studies on the dynamic strength properties of rocks and found a transition to dynamic behaviour in Solnhofen Limestone at ~10 2 s -1 . Howe et al. (1974) found dynamic strength increases in the anisotropic Yule Marble at rates > 10 s -1 . In a study on the Berea Sandstone and Indiana Limestone, Blanton (1981) demonstrated a lack of dynamic strength effects at strain rates up to 10 s -1 . On the same limestone, Frew et al. (2001) found DIFs of up to 1.7 at rates of ~100 s -1 . Rosakis (1999) reported dynamic strength properties of Dionysus-Pentalicon Marble (see Bhat et al., 2012), demonstrating DIFs of 2 at ~400 s -1 . In sandstones, Alam et al. (2015) found to measure within a real rock (Housen and Holsapple, 1999), and therefore, it may be the case that the Carrara Marble simply has an unusual distribution of flaws to explain the discrepancy in characteristic strain rate. Nevertheless, the scaling relationship of  assumes that the mode-I fracture toughness is a constant value; but fracture toughness is known to increase as a function of strain rate (Bhat et al., 2012;Ravi-Chandar and Knauss, 1984a;. Consequently, the behaviour of the Carrara Marble may be a consequence of a decreased sensitivity of fracture toughness with strain rate in comparison to other rocks.  Atkinson (1979) and Meredith et al. (1984) 0.76 ± 0.16 with the same type of distribution, its shape will depend on the method used to obtain that distribution (e.g. sieving, sedimentation, laser diffraction). Finally, there is no single common method of determining the "characteristic" size of a fragment size distribution. We arbitrarily chose to use the median of the fragment mass-size distribution (also known as D50) as the "characteristic" size of each distribution.

Granite
Despite these complexities, we find that the fragmentation behaviour of the Taunus Quartzite is very similar to the behaviour of felsic crystalline rocks , producing average fragment sizes of 40-1 mm at strain rates of 30-300 s -1 (Figure 4). Somewhat expectedly, the Seeberger Sandstone produces considerably finer fragments at equivalent strain rates, which we attribute to the relative ease of fracturing and separation of grains in a porous material. By comparison, the Savonnières Limestone produces coarser fragments than the Seeberger Sandstone but finer fragments than the Taunus Quartzite and other non-porous crystalline rocks (Figure 4).
We suggest that this may be a consequence of the relative grain sizes of the lithologies; the texture of the Savonnières Limestone is dominated by 0. and 345 s -1 respectively for the Seeberger Sandstone, and D50 values between 21.4 and 0.46 mm at strain rates between 20 and 515 s -1 respectively for the Savonnières Limestone. These fragment sizes are generally consistent with the results of our study except for the average fragment size of the Seeberger Sandstone at large strain rates where we produced considerably finer fragments. In a study of the dynamic failure properties of the Carrara Marble, Doan and Billi (2011) found that pulverisation of marble was "easier" than pulverisation in granite. Our study is consistent with this observation, where, at the same strain rate, the Carrara Marble produces fragments at least one order of magnitude finer than the fragmentation of felsic crystalline rocks reported by  and the quartzite of this study. The Carrara Marble even produces finer fragments than the Savonnières Limestone, which may be a consequence of its finer grain size.
A number of models have been proposed to determine average fragment size as a function of strain rate (Glenn and Chudnovsky, 1986;Grady, 1982;Levy and Molinari, 2010;Zhou et al., 2006aZhou et al., , 2006b. These models are principally concerned with tensile fragmentation of an expanding shell. While each model has important differences, all models predict that in the dynamic regime, fragment size is an inverse power law function of strain rate with an exponent of 2/3, a value that arises as a consequence of the assumption of equilibrium conversion of the kinetic energy released in fragmentation to fracture surface energy (Grady, 1982). All of the rocks in this study demonstrate an inverse power law relation between median fragment size under uniaxial compressive failure and strain rate (Figure 4). However, the exponents of those power-laws vary, depending on lithology, between 0.65 +/-0.23 and 2.90 +/-0.63 (Table 1). Ghaffari et al. (2019) also reported an inverse power-law relationship between fragment size and strain rate in uniaxial compression in Westerly Granite, however their exponent of 0.42 is considerably lower than a value of 2/3, unlike our results which produced exponents that were generally greater.
Tensile failure, as modelled in the expanding shell problem, is fundamentally different from compressive failure . The creation, activation, and growth of internal defects that occurs during compressive failure can make a material quite different from its pristine condition (Hogan et al., 2016), and large amounts of strain energy can be stored in brittle materials in compression which, when released, can generate very fine fragments (Ramesh et al., 2015).
Consequently, compressive fragmentation cannot be directly compared to tensile fragmentation models. Hogan et al. (2016) and Ramesh et al. (2015) have developed a method to convert compressive strain rates to equivalent tensile strain rates such that compressive failure could be compared to tensile fragmentation models. In a previous study , we found that applying this method did result in fragmentation from multiple lithologies collapsing onto a single power-law relationship, however the exponent of that relationship remained significantly different from the expected exponent of 2/3. The results presented here could be analysed with the same method to produce a similar result; however, we have decided to take a more phenomenological approach and derive a simple empirical relationship between strain rate and fragment size. To achieve this, we first normalise the compressive strain rate by the characteristic strain rate for each lithology. Then we attempted to find a normalisation factor for the average fragment size that caused the results to collapse onto a single power law. Noting that the method of Hogan et al. (2016) and Ramesh et al. (2015) achieved the collapse of the data to a single power-law, we decided to attempt a normalisation factor akin to the characteristic length used in their method.
That characteristic length, s0, is determined by considering the characteristic time, t0, for the growth of a cohesive crack under external loading and the sound velocity of the material (Camacho and Ortiz, 1996;Drugan, 2001): where σt is the tensile strength of the material. Noting that the tensile strength of rocks is typically some constant fraction of the compressive strength (e.g. Hogan et al., 2016), we reduce the characteristic length scale to remove the constant terms, i.e.: While HI values for the Carrara Marble have been determined experimentally (Atkinson, 1979;Meredith et al., 1984). HI is unknown for the other lithologies, instead, we take the average value of HI from similar lithologies and include the effect of considerably larger uncertainties ( Table 4).
Normalisation of the strain rates and fragment sizes for all the experiments in this study and the experimental results of  are shown on Figure 7. We notice that this normalisation results in the collapse of all non-porous lithologies to a single inverse power-law relation, and the porous lithologies (Seeberger Sandstone, φ = 20.0 ± 0.8%; Savonnières Limeston, φ = 31.4 ± 1.1%) to a parallel trending inverse power-law. The general form of this power law is expressed as: We fit the data for non-porous and porous rocks using non-linear least squares fitting in log-log space to determine that, for non-porous rocks, the exponent, μ, is 1.93 ± 0.14, and the constant k is . 8 . : . . The power law for porous rocks has an exponent, μ, of 1.78 ± 0.40, and a constant, k, of . 8 .

Fragment shapes
Our results demonstrate that fragment shape is independent of both strain rate and lithology (Figures 5 and 6). For all of the lithologies in this study, average fragment circularity and axial ratios at all strain rates are both ~0.6. Our results therefore suggest that fragment shape cannot be used as a fingerprint for the conditions at which deformation and, more specifically, fragmentation occurred. Nevertheless, we note that our experiments were only conducted under uniaxial compression; it remains to be seen whether fragment shape in dynamic failure is dependent on the state of stress, whether simply confined or under truly triaxial conditions.
To understand why fragment shape may be independent of strain rate during dynamic fragmentation, we consider the geometric fragmentation of a unit area/volume. At low strain rates, an area/volume will be fragmented by a small number of fractures, producing a small number of large fragments. At high strain rates, the area/volume will be fragmented by a large number of fractures, producing a large number of small fragments. The exact distribution of fragment sizes is fundamentally linked to the method by which fractures are constructed (Grady and Kipp, 1985).
Here, we implement a variety of construction algorithms to randomly fragment a two-dimensional area with the aim of seeing how fragment shape changes as a function of the number of fractures (i.e. strain rate). We chose three geometric fragmentation construction algorithms as described by Grady and Kipp (1985): Random Lines, Random Line Segments, and Voronoi Segmentation.
The Random Lines algorithm is the simplest of the three. Here, a specified number of randomly distributed points are placed within a unit area. Each point is then assigned a random orientation (between 0 and 180), and a fracture (straight line) is extended from that point such that it extends from one edge of the unit area to another, passing through the point at the assigned orientation.
This process is conducted for all points simultaneously. The Random Line Segments algorithm has several similarities to the Random Lines algorithm but is more complex, and realistic. Here, each fracture is added sequentially with the condition that the fracture must terminate against any pre-existing fracture. Voronoi segmentation is conceptually rather different to the previous algorithms but also produces random fragmentation of a unit area. Here, a specified number of random points ("seeds") are chosen, and the unit area is divided into segments which contain all of the locations closest to a single seed. For each algorithm, 10 images were generated, each with varying numbers of lines or seeds. Those images were then analysed to characterise the shapes (circularity and axial ratio) of the fragments, excluding those at the edge of the unit area, as a function of the number of lines/seeds.
With the Random Lines algorithm, we find that axial ratio is constant as a function of the number of lines while circularity increases (Figure 8a). With the Random Line Segments and Voronoi algorithms, we found no variation of either circularity or axial ratio with increasing number of lines/seeds (Figures 8b and 8c). Of the three algorithms, Voronoi segmentation achieves the closest match to our experimental results with average circularities of ~0.7 and average axial ratios of ~0.6. We emphasise though, that none of the implemented algorithms are particularly representative of the real process of dynamic fragmentation as they ignore all of the physical and dynamic processes of fracture nucleation and growth. Nevertheless, they demonstrate that uniform fragment shapes as a function of strain rate is a consequence of fragmentation as a stochastic process where the growth of fractures is limited by the presence of other growing fractures. A further implication of our results is that the behaviour of fractures did not change over the conditions explored in our study, i.e. uniaxial compression at rates between ~30 and 300 s -1 .
Experimental studies have shown that fractures, particularly Mode-I fractures, in the dynamic regime may bifurcate or branch under increasing loading rates (Ravi-Chandar and Knauss, 1984b;Zhang et al., 1999), this would be expected to lead to the production of more elongate and less circular fragments as a function of increasing rate. However, our results show no variation of fragment shape in common with the simplified geometric models of fragmentation where each fracture behaves the same, regardless of the "strain rate". Segments, c) Voronoi Segmentation.

Conclusions
In this study, we have shown that the characteristic strain rates for rate-dependent strength and brittle behaviour under uniaxial compressive loading in sandstone, quartzite, limestone, and marble ranges between ~144-322 s -1 . The degree of fragmentation increases with strain rate over the full range of dynamic rates investigated (~30-350 s -1 ). Our results demonstrate that compressive fragmentation cannot be described by tensile fragmentation models, and instead we describe an empirical fragmentation relationship for uniaxial compression that describes average fragment size as a function of strain rate. This relation has the form of an inverse power-law with an exponent of 1.93 +/-0.14 and a constant that varies with porosity, or some material property correlated with porosity. The applicability of this relation at strain rates at greater or lesser strain rates remains to be seen. The results of our study also demonstrate that fragment shape during dynamic failure is independent of strain rate and lithology, at least for uniaxial compression and the homogeneous lithologies investigated here.
The clear implication of this study is that fragment size may be used as a diagnostic indicator of the strain rate at failure while fragment shape cannot be used. Nevertheless, we note that our experiments were conducted under uniaxial conditions, confinement or truly triaxial stress states during fragmentation may cause significant changes to the process of dynamic fragmentation (Liu et al., 2019;Liu and Zhao, 2021;) and the fragmentation relation described here. Furthermore, our experiments were conducted on intact rocks, it remains unclear how fragments generated by repeated low-rate events (Aben et al., 2016) can be differentiated from single high-rate events.

Introduction
A GitHub repository that corresponds to this manuscript has been made publicly available (https://github.com/ASprae/Dynamic-Strength-and-Fragmentation). The repository contains all the data acquired, used, and plotted in this study: -Petrophysical properties of all lithologies in this study and that of .
-Summaries of all mechanical experiments, quasi-static and dynamic, including all the results from  used in this study.
-Raw data for the SHPB experiments and a script for processing the data.
-All sieve data in this study and .
-All images used to derive fragment shape distributions, including the starting image and the final thresholded image, data outputs from analysis of those images from Fiji, and python scripts for the analysis of that data.
-Python scripts that implement the geometric fragmentation algorithms described in the text and analyse the fragment shape distributions of those geometric fragmentation patterns.
The methodology used in this study is very similar to a previous study by the same authors . The following text (Supplementary Text 1) is adapted from the Supplementary Material of that paper. Several sections, where we could not improve the clarity of the writing, have been directly copied from .

Supplementary Text 1: Split Hopkinson Pressure Bar Methodology
Split-Hopkinson Pressure Bars (SHPBs) consist of three units: a striker with acceleration apparatus, an incident bar, and a transmission bar ( Figure S1). The sample is placed between the incident and transmission bars. In this study, titanium alloy (EB = 110 GPa, ρB = 4.43 g cm -3 ) bars were used for each bar and the striker. The bars and striker were 5 cm in diameter. All experiments used a 25 cm-long cylindrical striker. The pulse generated by the direct impact of this striker on the incident bar would be inappropriate for the testing of brittle materials. Brittle materials must be loaded at a steady rate to ensure uniform stress throughout the sample, i.e. a so-called "stress equilibrium" or "dynamic force balance" . This "triangular" pulse shape was achieved by the use of a pulse shaper placed at the front end of the incident bar. We used the same type of pulse shaper used by ; pre-hit cuboids of aluminium foam. The aluminium foam had an original thickness of 2 cm, and was cut such that the foam covered the entire 5 cm diameter incident bar. The aluminium foam has an original bulk density of ~300 kg m -3 and porosity of ~90%. The behaviour of the pulse-shaper can be modified to be appropriate for the testing of the sample material by pre-hitting the pulse-shaper prior to the experiment. All pulse shapers were pre-hit by the striker at a velocity of ~10 m s -1 , resulting in a final thickness of ~7.5 mm, consistent with the pulse shapers used by . To avoid damage to the bars during the experiment, 2 mm thick titanium alloy plates were attached to the bar ends with silicon grease, and a thin sheet of Teflon is placed between the sample and the titanium plates. The sample was then placed between the incident and transmission bars, within a secure box lined with a polyethylene bag so that the sample could be recovered. We used a cradle of adhesive tape within the box to keep the sample in place prior to the experiment.
Strain gauges on the incident and transmission bars measure the pulse before and after it has contacted the sample. Each bar has two strain gauges placed diametrically across the bar at the same distance from the sample; the two signals on each bar are averaged to account for any bending of the bar. The signals contain three important components from which the stress and strain history of the sample is determined. The incident pulse, εi, the transmitted pulse, εt, and the reflected pulse (measured in the incident bar), εr. Experiments were always conducted such that the incident and reflected pulses were distinct and did not overlap. Signal processing occurred by the following steps: first, the signals are time-adjusted to account for the distance between the sample and the strain gauges; second, high-frequency noise is filtered; and third, frequency dependent dispersion (Pochhammer-Chree dispersion) is corrected for (See Gama et al., 2004).
Once εi , εt, and εr were corrected, stress, strain-rate, and strain as a function of time were obtained by: where ( ) and ( ) are the stresses at the front and back of the sample respectively, which, in well-carried out experiment should be equal at all times (i.e. stress equilibrium); ̇( ) and ( ) are the longitudinal strain rate and longitudinal strain respectively; , , and are the area of the bar, elastic modulus of the bar, and sound speed of the bar respectively, while and are the area of the sample and length of the sample respectively. Use of equations 1-4 is commonly known as the "3-wave analysis" method of determining stress and strain histories from SHPB experiments (Gray, 2000).
In an SHPB experiment, the sample undergoes a history of strain rate as defined by Equation 3. However, for the purposes of comparing experiments at different rates a "representative" strain rate for the experiment must be chosen . In this study we have used the same definition of the representative strain rate as used by , representative strain rate is defined as the average strain rate between the times where stress is at 25% of the peak stress, and when peak stress occurs. Determining the representative strain rate as an average of the strain rate history is fundamentally similar to the approaches of Yuan et al.
(2011) and . Other methods of defining the representative strain rate include: picking the maximum strain rate prior to yielding , picking the strain rate at the time of peak stress (Zhang et al., 2009), or the "plateau" value of the strain rate history (Xia et al., 2008). Our method of determining the representative strain rate as an average value was chosen for several reasons: first, while strain rate plateaus were easy to identify at small loading rates, at larger rates it becomes increasingly challenging to unambiguously identify the plateau. The chosen method typically picks the plateau in the low rate experiments and avoids subjective choices in the high-rate experiments.
Secondly, picking one value of the strain rate is not representative of the whole experiment; peak rate can be strongly affected by oscillations in the stress wave while the rate at peak stress in low rate experiments, where substantial elastic strain may be recovered by the sample, are typically close to zero. An advantage of our method of picking the representative strain rate is that the variability of strain rate can be represented by the standard deviation of the determined average rate. The choice to start taking the average value from 25% of the peak stress is arbitrary, however this point avoids non-linear, concave-upwards regions of the stress-strain curves at low stresses that result from compaction of silicon grease, Teflon, and air bubbles at the sample bar interfaces .
Strain rate can be varied between experiments by increasing the striker velocity, this was generally possible up to the maximum safety limit of the equipment used, while maintaining stress equilibrium and an appropriate strain rate history during loading. However, if samples are particularly strong, i.e. requiring large striker velocities to cause failure, this may leave a limited amount of strain rate space available to safely operate in. Consequently, subject to maintaining stress equilibrium, avoiding inertial stress generation, and generating an appropriate strain rate history during loading, it can be possible to vary the size of the samples to explore a greater range of conditions. Ideal samples for SHPB experiments have a length-to-diameter ratio between 1:2 and 1:1 and have a cross sectional area close to that of the bars (Gray, 2000;. Consequently, most samples used in our experiments were ~41 mm in diameter and length. However, in a number of cases, changing the size and aspect ratio of the sample was required. First, strain rate can be increased by using a shorter sample, or conversely, strain rate can be decreased by using a longer sample. Secondly, the minimum striker velocity required to cause failure can be decreased by decreasing the cross-sectional area of the sample. In turn this allows a greater range of striker velocities to be used, and therefore a greater range of strain rates. We note that changes in sample size and aspect ratio are expected to have an effect on the material strength , and potentially the fragment size. However, we found no statistically significant systematic change of strength or fragmentation over the sizes used in our experiments. Measurements of dynamic material strength can be significantly affected by the inability of the sample to deform fast enough for the applied stress; this inertia can contribute to the stress (Forrestal et al., 2007;Kolsky, 1949). Inertia can produce additional longitudinal stress, ( ), and confining radial and hoop stresses, ( ) and ( ) respectively: ( 3 )̈ , (6) ( 3 )̈ , (7) where ρ, ν, and a are the density, Poisson's ratio, and radius of the specimen. ̈ is the second derivative of the longitudinal strain with respect to time. A properly conducted SHPB experiment minimizes these effects, primarily due to the use of pulse-shapers. In the experiments conducted in this study, the average maximum longitudinal inertial stress of the experiments in this study was 95.5 kPa, with a maximum value for an experiment being 323.9 kPa (see GitHub Repository), over two orders of magnitude less than the peak stress of the corresponding experiment.
Consequently, longitudinal inertial stresses introduce an error of <0.34% of the failure stress, significantly less than the inter-sample variability of the strength of each sample. The average maximum radial and hoop inertial stresses of the experiments in this study were 191.1 kPa and 256.0 kPa, with maximum values of 619.8 kPa and 871.7 kPa, respectively (see GitHub Repository). Assuming a Mohr-Coulomb failure envelope up to these confining pressures and a typical coefficient of friction for rocks between 0.5 and 1.0 , the inertial confining stresses make only a negligible contribution to the failure strength.

Supplementary Text 2: Fragment Image Acquisition and Processing
To determine the shape properties of fragments, we initially chose to manually pick the fragments from each sample, starting from the largest fragments, and lay them upon a piece of black card such that all fragments were fully separated from each other (Supplementary Figure   3). We were able to hand pick all fragments from the selected samples down to fragment sizes (sieve diameters) of 2 mm. Beyond this size, it was impractical to hand pick and separate each fragment. These hand-picked fragments were then imaged with a Canon EOS 1200D camera at a pixel resolution of ~140 pixels/cm.
Fragments smaller than 2 mm were imaged with a slightly different method. In this case, our method was modified due to a number of considerations: first, that fragments would need to range in size from a maximum size of 2 mm (and the images would require a statistically valid number of fragments of this size) down a smallest size as dictated by the requirement that the smallest fragments must be resolved by at least 200 pixels (Kröner et al., 2013), second, that the vast number of fragments and their required dispersal prohibits imaging of the entire sample, and finally, that the fragments could not be manually separated prior to image acquisition. Following these considerations, we used a modified Petrographic Image Capture and Archiving Tool (PICAT; Crawford and Bennight, 2010) with a Canon EOS 5D Mark II Camera, originally designed for the acquisition of whole thin-section images. To acquire images with a sufficient number of ~2 mm fragments, a resolution of ~700 pixels/cm was used. Thus, the minimum particle size that could be imaged with at least 200 pixels was ~0.2 mm. However, due to the practicalities of distinguishing fragment dust and image noise from the fragments themselves and ensuring that the images could be adequately thresholded and segmented, we restricted the minimum fragment size to 0.5 mm. Thus, all fragments between 0.5 and 2 mm of a sample were recombined (after sieving) and mixed. The mixed subsamples were then divided into quarters, and each quarter was scattered over a piece of black card and imaged (Supplementary Figure 4).
Images were processed using the Fiji software package . All images were converted to 8-bit color, thresholded using the Li thresholding algorithm, and a median filter applied. At this point, the image processing method for the fragments smaller than 2 mm differed from the fragments larger than 2 mm. Due to the lack of separation between all fragments less than 2 mm, a distance-transform watershed (MorphoLibJ; Legland et al., 2016) was applied to segment any merged fragments. Borgefors distance weighting was used and the "Dynamic" parameter, which determines the intensity of the search for regional minima of the inverse of the distance transform, was chosen between 2 and 5 such that images were neither systematically oversegmented, nor under-segmented. Following this processing, the areas, perimeters, and best-fitting ellipses of each segmented fragment were determined.
The first stage of post-processing was a further filtering step, where remaining noise and anomalously small fragments (smaller than the expected minimum size of 2 mm or 0.5 mm) and rock dust were removed. These particles could readily be distinguished from the fragments of interest from fragment size distributions, which were distinctly bimodal; with a local minimum in the probability distribution function at particle sizes of ~0.5 mm. All particles finer than that local minimum were eliminated from further analysis. Following this filtering, the shapes of the fragments were determined. Upon checking that the distributions of fragment shapes for the fragments from 0.5 to 2 mm were consistent (similar average, range, and distribution shape) between each of the 4 images for each sample, we combined the data from all images to generate single fragment shape distributions for each sample. Figure S1. Schematic split Hopkinson pressure bar set-up.   for the sample, made by compiling all the data for each of the four quarters above.