Breaking down chipping and fragmentation in sediment transport: the control of material strength

. As rocks are transported, they primarily undergo two breakdown mechanisms: chipping and fragmentation. Chipping occurs at relatively low collision energies typical of bed-load transport, and involves shallow cracking; this process rounds river pebbles in a universal manner. Fragmentation involves catastrophic breakup by fracture growth in the bulk — a response that occurs at high collision energies such as rock falls — and produces angular shards. Despite its geophysical signiﬁcance, the transition from chipping to fragmentation is not well studied. Indeed, most models implicitly assume that impact erosion 5 of pebbles and bedrock is governed by fragmentation rather than chipping. Here we experimentally delineate the boundary between chipping and fragmentation by examining the mass and shape evolution of concrete particles in a rotating drum. Attrition rate should be a function of both impact energy and material strength; here we keep the former constant, while systematically varying the latter. For sufﬁciently strong particles, chipping occurred and was characterized by the following: daughter products were signiﬁcantly smaller than the parent; attrition rate was independent of material strength; and particles 10 experienced monotonic rounding toward a spherical shape. As strength decreased, fragmentation became more signiﬁcant: mass of daughter products became larger and more varied; attrition rate was inversely proportional to material strength; and shape evolution ﬂuctuated and became non monotonic. Our results validate a previously proposed probabilistic model for impact attrition, and indicate that bedrock erosion models predicated on fragmentation failure need to be revisited. We suggest that the shape of natural pebbles may be utilized to deduce the breakdown mechanism, and infer past transport environments.

of impact attrition, each collision produces a shower of fine particles -meaning that there is a localized near-surface region where yield is exceeded.
Unlike chipping, catastrophic fragmentation occurs when high-energy collisions cause cracks to propagate radially into the bulk. These radial cracks can split the parent rock into irregularly-shaped daughter particles of significant size in addition to smaller fragments (Perfect, 1997;Kun and Herrmann, 1999;Salman et al., 2004;Grady, 2010). At intermediate impact 5 energies, fatigue failure occurs as fractures grow into the rock as a result of repeated impacts (Bitter, 1963;Moss et al., 1973).
We note that this fatigue failure mechanism forms the basis for widely used bedrock erosion models Dietrich, 2001, 2004).
Since breakdown mechanism and the resulting particle shape vary depending on transport energy and material strength, particle shape can be used to infer transport history. Previous studies on attrition by chipping have shown that pebble shape 10 can be used to determine transport distance in fluvial environments (e.g., Attal and Lavé, 2009;Szabó et al., 2013Szabó et al., , 2015. Similar studies have examined attrition mechanisms in a dune field (e.g., Jerolmack et al., 2011) and on a clastic beach (e.g. Bertoni et al., 2016). However, fewer studies have investigated particle shape change in high energy environments (e.g., Bernd et al., 2010;Arabnia and Sklar, 2016) or examined the transition between chipping and fragmentation at intermediate energies (e.g., Moss et al., 1973;Adams, 1979). Novák- Szabó et al. (2018) proposed that there is a critical energy associated with this 15 transition, and that natural rock materials in bed load are far below this value. This study utilizes laboratory experiments to examine chipping and fragmentation in order to better understand the connection between breakdown mechanism and shape evolution. As a result, particle shape can be more accurately used to determine past transport conditions and environments.

Attrition and the Shape of River Rocks
As sediment is produced in upland regions and makes its way downstream in rivers, particles transform from blocky, angular 20 clasts to smooth, ellipsoidal pebbles (Krumbein, 1941;Kuenen, 1956;Parker, 1991;Kodama, 1994;Attal and Lavé, 2009;Domokos et al., 2014). This shape change is associated with attrition, and is expected to result in an exponentially-decreasing pebble volume, V , with distance downstream, x (Kodama, 1994;Lewin and Brewer, 2002), as expressed in the volumetric version (Novák-Szabó et al., 2018) of Sternberg's Law (Sternberg, 1875): 25 where V o is initial particle volume at the upstream boundary, and γ is a diminution coefficient. If one assumes that particle diameter D ∝ V 1 3 -an assumption that is likely violated for attrition of angular particles ) -Sternberg's Law can also be cast in terms of particle diameter (Sternberg, 1875;Kodama, 1994;Lewin and Brewer, 2002;Attal and Lavé, changes in pebble shape alone may be used to isolate the contribution of impact attrition to downstream fining -assuming that all mass loss results from chipping Novák-Szabó et al., 2018).

Mass Loss as a Result of Attrition
It is common to cast the attrition process in terms of the mass lost per collision, ∆m. This mass loss is proportional to collision energy, ∆E, such that: where A is a material susceptibility parameter (Anderson, 1986) that Miller and Jerolmack (2020) called the "Attrition Number" [s 2 m −2 ], and C 1 is an experimentally determined constant. Since collision energy is a function of the mass of the particle, m, and impact velocity, v i , where ∆E = 1 2 mv 2 i , we expect that mass should decrease exponentially with number of impacts (Novák- Szabó et al., 2018). The Attrition Number, A, incorporates various aspects of material strength that determine suscep-10 tibility to attrition, such as hardness, fracture toughness, yield strength, or Young's modulus (e.g., Ghadiri and Zhang, 2002;Sklar and Dietrich, 2004;Wang et al., 2011).
When the stress produced by a sufficiently powerful impact on a sufficiently weak material exceeds the breaking strength of that material, a crack tip forms. The energy from the surrounding strain can then cause the crack tip to spread through the material and split the rock (Bond, 1952(Bond, , 1955. If the energy imparted by the collision is insufficient, repeated impacts can 15 cause the crack tips to propagate into the material through fatigue failure (Bitter, 1963;Moss et al., 1973). Fractures formed by brittle failure tend to be large-scale, developing through the volume of the particle as a result of elastic deformation (Evans and Wilshaw, 1977;Ghadiri and Zhang, 2002). It has been proposed that susceptibility to brittle fracture, called the brittle Attrition Number, A b , depends on a material's ability to store energy elastically (Engle, 1978;Wang et al., 2011). From mechanical considerations and dimensional analysis, several studies arrived at a similar parameter (Sklar and Dietrich, 2004;Wang et al., 20 2011;Miller and Jerolmack, 2020): where σ s is yield strength [Pa], Y is Young's modulus [Pa], and ρ s is the density of the material [kg m −3 ]. While the parameter A b formally relates to crack growth in purely brittle materials under fragmentation, it was found to describe mass loss due to shallow chipping of natural (and possibly semi-elastic) rocks (Miller and Jerolmack, 2020). For this reason we will use A b to 25 characterize material strength in our study. The applicability of A b may imply that chipping of natural rocks is also a form of brittle fracture, but that its depth is limited; the locally shattered regions produced by impacts may be Hertzian fracture cones (Greeley and Iversen, 1987;Wang et al., 2017;Miller and Jerolmack, 2020). Impact experiments on concrete and rocks indicate that this picture is applicable for most materials (Momber, 2004a). It is important to note, however, that plastic deformation may be relevant for impact attrition -even for very brittle materials (Rhee et al., 2001 (Momber, 2004b). Indeed, for chipping of semi-elastic materials, Ghadiri and Zhang (2002) proposed an alternative Attrition Number that depends on H and K c , rather than Y and σ s . All of these parameters may be correlated with each other, depending on the deformation mechanism (Shipway and Hutchings, 1993;Rhee et al., 2001;Emmerich, 2007;Mohajerani and Spelt, 2010); describing these different relations is beyond the scope of the present paper.

Shape Evolution in the Chipping Regime
In the limit of pure chipping, where the fragments produced by each impact are suitably small compared to the parent clast, 5 the shape evolution of a particle can be modeled purely geometrically. Essentially, areas that protrude from the pebble have a positive curvature and are more likely to strike another particle or the bed surface and chip off. Thus, particles undergo curvature-driven attrition that evolves their shape toward a sphere (Firey, 1974;Domokos et al., 2009;Várkonyi and Domokos, 2011;Domokos et al., 2014). In the typical case of bed load with gravels impacting a streambed, the situation is close to the purely curvature-driven limit (Szabó et al., 2013Novák-Szabó et al., 2018). The particle shape evolution resulting from 10 attrition by chipping occurs in two phases . The first phase involves an angular clast quickly becoming round as protruding edges are removed without a significant change in axis dimensions (Krumbein, 1941;Kuenen, 1956;Adams, 1979;Domokos et al., 2014). The second phase occurs once a particle is entirely convex as the axis dimensions are slowly reduced . Field studies indicate that two-phase attrition applies to sediment in both fluvial and aeolian environments (e.g., Szabó et al., 2013;Miller et al., 2014;Novák-Szabó et al., 2018). It has been found that circularity 15 and aspect ratio are two convenient shape parameters that effectively characterize this shape evolution (Miller et al., 2014); we adopt them in this study.
In order to observe the shape changes in particles undergoing attrition on a spectrum from chipping to fragmentation, one can vary either material strength or impact energy (Eq. 2). In this study, concrete particles with the same initial mass were repeatedly dropped from the same height in a rotating drum to simulate transport. Thus, initial impact energy remained constant 20 and experiments were conducted for varying material strength (A b ). Mechanically strong particles experienced chipping with nearly constant mass loss per impact on their trajectory toward a spherical shape. Weak particles would fragment into large, irregular pieces after as few as 5 impacts. Concrete particles with intermediate strengths experienced both chipping and fatigue failure, becoming more rounded but with occasional fragmentation events that prevented the particles from following the expected "universal rounding" curve associated with chipping.

Methods
Experiments were conducted by placing concrete blocks of varying strength in a rotating drum. The concrete blocks were created by pouring a mixture of concrete mix and sand into 6-cm cubical molds (Fig. 3a). While concrete mixes meant for and stronger particles (66.7-80 % VCM) remained in the drum for intervals ranging from 50 to 500 drops. After each rotation interval, the particle was removed from the drum, weighed, and photographed before being returned to the drum to undergo another series of rotations. 20 At the conclusion of the rotating drum experiments, all images were analyzed using ImageJ. The image processing program converted the original photograph into a binary image in order to isolate the shape of the particle and measure shape parameters, including area, perimeter, circularity, and aspect ratio (Fig. 4). To verify the circularity measurements calculated by the image processing program, the shape measurement algorithm was applied to synthetic circles and squares of known shape. While measured circularity was found to be resolution dependent, the maximum error was 10% over the resolution range that is 25 relevant for our experiments.
An Instron Universal Testing System was utilized to conduct uniaxial compression tests of the concrete particles. Similar to the rounding experiments, 10 mixtures of concrete mix and sand were used with 5 particles created from each mixture.
A 25 kN load cell was used for the particles made of 12.5% to 50% VCM and a 150 kN load cell was used for particles with 66.7%, 75%, and 80% VCM. During the compression tests, the upper plate was driven down at a constant rate of 3 mm  Figure 4. A visual depiction of the ImageJ macro used to determine shape parameters. First, a photograph was taken of the particle at a fixed distance over an LED light table. The macro (a) cropped the image to include only the particle, (b) converted the original image into a binary image, then (c) filled any holes within the particle shape. The macro then (d) measured the area (A), perimeter (P ), major axis (a), and minor axis (b) of the particle. These measurements were used to calculate circularity (R) and aspect ratio (AR), parameters that were used to quantify shape change over the course of the rounding experiments. Circularity measures how closely a shape approaches that of a circle, where R = 1 indicates a perfect circle and R < 1 indicates deviations from a circle. Part (e) shows circularity measurements for particles of different shapes. however, were highly variable; especially for weaker materials that did not follow classic brittle failure (Fig. 5a). Accordingly, here we use ultimate strength-the greatest stress withstood by a material-similar to some previous studies (e.g., Sklar and Dietrich, 2001;Miller and Jerolmack, 2020). It is assumed that ultimate strength is proportional to yield strength. Young's modulus is typically determined from a linear fit to the stress-strain plot -i.e., in the elastic regime before failure. Due to the variable shapes of our stress-strain curves, however, we estimated Young's modulus as the ultimate strength divided by the 5 associated strain (Fig. 5a), in order to avoid ambiguity of how to choose an approximately linear regime over which to fit.

Material Properties
An analysis of uniaxial compression test results revealed that particles with a higher percentage of concrete mix have a greater material strength and stiffness, as measured through ultimate strength and Young's Modulus (Fig. 5a). Mechanically strong particles could withstand loads ranging from 3.0e6 to 5.4e6 N m −2 , although the strongest particle (66.7% VCM) held up to 5 1.2e7 N m −2 . Mechanically weaker particles could hold loads ranging from 1.0e5 to 1.2e6 N m −2 , although the weakest material (12.5% VCM) withstood as little as 4.3e4 N m −2 before failing (Fig. 5b) (Fig. 5c). Despite variation in the measured Young's Moduli, the general trend indicates that materials with a higher percentage of concrete mix were more resistant to deformation. 15 The density of each type of concrete particle was also calculated from measured weight, and volume for each cube estimated

Mass Loss
The mass of each particle decreased as that particle rotated in the metal drum. To quantify and characterize mass loss, we utilize the following parameters: mass fraction (M ), cumulative mass loss (µ), and fractional mass loss per impact (∆m * ).
Mass fraction is defined as the ratio of the mass of the particle during a given rotation to the initial particle mass (M = m i /m o ).
Cumulative mass loss is the ratio of remaining mass to the initial particle mass (µ = 1 − M ). Fractional mass loss per impact 25 is the ratio of the mass lost during a given impact to the particle mass just prior to that impact (∆m * = (m i−1 − m i )/m i−1 = ∆m/m i−1 ).
As particles experienced transport in the metal drum, mass was reduced most rapidly at the beginning of each experiment (Fig. 6a, c, e). For mechanically strong particles, fractional mass loss per impact was approximately constant (Fig. 6b, d, f). A line of best fit was used to determine the Young's Modulus for each particle composition and can be described through the following equation: Y = 1.32e6Cm, where R 2 = 0.67 In order to visualize mass loss distribution, a series of histograms were generated for each particle strength (Fig. 7). These histograms indicate that mechanically strong particles tended to produce chips with a narrow distribution of masses that are small in comparison to the mass of the initial particle. Mechanically weak particles tended to produce fragments with a wide distribution of masses that could be a significant proportion of the initial particle mass.
We now test the control of material strength on mass loss across the chipping to fragmentation transition. We find first that 5 our data do not follow the inverse-square relation between mass loss and ultimate strength proposed by Dietrich (2001, 2004). Instead, we find an exponent of less than (but close to) one, indicating that ultimate strength alone is insufficient to describe the control of material on attrition rate (Fig. 8). To make contact with the proposed relations Eqs. 1 and 2, we first fit an exponential Sternberg-like relation to the curves of fractional mass loss against rotation number, and verify that the data are reasonably well fit (Fig. 6). This supports the notion that mass lost per unit energy is constant (Eq. 2) -even if highly variable 10 for weaker materials. We then examine Eq. 2 and rearrange to solve for k = AC 1 = 2∆m * /v 2 i to empirically determine k for each material. We anticipate that the empirical mass-loss parameter k scales linearly with the brittle Attrition number A b ; indeed the data are consistent with such a relation (Fig. 8b) and with recent experiments on binary collisions of real rocks Miller and Jerolmack (2020), though the experimental parameter C 1 determined here is three orders of magnitude larger than that study. A secondary pattern is that, for the strongest materials, mass loss is independent of material strength (Figs. 7, 8). continued to be placed in the drum until very little of the original particle remained. Additionally, Sternberg's Law (Eq. 1)-denoted by the cyan dashed line-was fit to each plot of fractional mass loss with the form M = mi/mo = e −k Cm N R , where kC m is a material parameter that differs for each concrete mixture. We can see that fractional mass loss scales exponentially with the number of impacts in the rotating drum.

Particle Shape
Rounding experiments showed that particles of differing mechanical properties experienced different shape evolution trajectories. Mechanically strong particles with a greater proportion of concrete mix evolved from a cubical block toward a sphere.
The mechanically weak particles began with the same cube-like shape, but fragmented into several angular pieces during their time in the rotating drum (Fig. 9). The strong particles could withstand thousands of impacts (66.7-80 % VCM), while the 5 weakest particles experienced 10-20 impacts before disintegrating (12.5-16.7 % VCM). Particles of intermediate strength (20-50 % VCM) evolved from a cubical block toward a spherical shape, but experienced several large breakage events along the trajectory toward a sphere. These intermediate particles remained in the rotating drum for several hundred impacts.
Aspect ratio, a ratio of the major and minor axes, was also used to visualize particle shape change over the course of the 5 experiments. An aspect ratio of 1 indicates that the length and width of a particle are equal. Since all particles were formed in cubical molds, the aspect ratio of the particles before the tumbling experiments was close to 1. Since the mechanically strong particles evolved toward a sphere, the aspect ratio remained close to 1 for the duration of the experiment. The aspect ratio of intermediate and weak particles deviated from 1 over the course of the experiments, indicating that fragmentation events were occurring (Fig. 10b).

Discussion
Tests indicate that material strength and resistance to deformation increased with the amount of concrete mix in the particle's composition. However, there was significant variation in the ultimate strengths and Young's Moduli recorded for each set of particles, as well as the shapes of the stress-strain curves (Fig. 5a). Typical stress-strain curves show an elastic region, where the curve follows a linear regime, and a plastic region, where the curve rounds and eventually reaches the fracture point. Some of 5 our curves do not follow this pattern, but instead display a series of peaks in the transition from the elastic to the plastic regime ( Fig. 5a). This variation in particle strength and behavior, in response to compressional stress, is likely due to inconsistencies 13 https://doi.org /10.5194/esurf-2021-17 Preprint. Discussion started: 9 March 2021 c Author(s) 2021. CC BY 4.0 License. inherent in the process of making these concrete particles. It is possible that inhomogeneities produced during particle creation make the curves deviate from industrially produced concrete (e.g., Lan et al., 2010). Furthermore, molds used to create the concrete particles varied in shape from 6 cm to 8 cm, and asperities would form on the concrete along the open side of the mold. These asperities may also have influenced compression testing by reducing the ultimate strength of each particle. The compressing plates should come into contact with flat surfaces; asperities would cause the force to load unevenly and may 5 result in premature failure and the observed peaks in the stress-strain curves (e.g., Vasconcelos and Lourenço, 2009).
In addition to potentially causing premature failure or unusual patterns of strain in particles, the variation in initial particle shape may have impacted the rounding experiments. Ideally, particle shape should not impact the outcome of rounding experiments because particle shape does not play a key role in setting attrition rate (Várkonyi and Domokos, 2011;Szabó et al., 2013;Bertoni et al., 2016). Particle composition, on the other hand, can influence shape evolution (Sklar and Dietrich, 2004). 10 The high composition of sand in mechanically weak particles frequently caused fragments to disintegrate on impact, producing a large population of sand and fine concrete fragments that may contribute to the bimodal distribution seen in the mass loss distributions of some intermediate strength particles (Fig. 7). The population of fine fragments was not collected or measured, as this pattern of disintegration differs from fragmentation in natural materials.
Nonetheless, we can identify a clear control of material properties on the attrition rate, that is consistent with the brittle At-15 trition Number, A b , being the relevant material grouping. The findings are in agreement with recent results on binary collisions of natural rock materials Miller and Jerolmack (2020). The large difference of the experimental parameter C 1 between that study and ours is not entirely understood. This parameter likely encodes details of the collision process (impact angle, rotation, etc.) but also, by definition, includes details of how material properties were measured. Our findings reinforce the idea that the brittle Attrition Number is a useful material parameter for determining relative susceptibility to attrition, but uncertainties in 20 C 1 limit our ability to directly extrapolate experimental findings to the field. Additionally, results indicate that there is an upper limit where material strength no longer influences particle breakdown. Our data indicate that the brittle Attrition Numbers below 0.014 s 2 m −2 are approaching this limit. Both circularity and mass loss become nearly constant for particles with brittle Attrition Numbers lower than the limit (Fig. 8a, 11).
The results of the rounding and compression experiments confirm the existence of a pure chipping regime. Other studies 25 have utilized both experiments and field observations from rivers, beaches, and dune fields to predict the shape evolution of particles undergoing attrition by chipping (e.g., Szabó et al., 2013;Miller et al., 2014;Bertoni et al., 2016;Novák-Szabó et al., 2018). The circularity of mechanically strong particles produced by this study increases smoothly and monotonically with increasing mass loss. Additionally, circularity as a function of cumulative mass loss for our relatively strong particles plot very close to the universal curve found from experiments and field observations on natural materials taken in diverse environments 30 (Fig. 12). Deviations from this universal curve in the initial stage of mass loss are likely due to the different initial conditions; natural rock fragments have common initial shapes (Domokos et al., , 2020, while the particles produced in this study are initially cubes with variable asperities on their faces. Additionally, particles with VCM < 50 % begin to show significant deviations from the universal curve; circularity grows slowly and more erratically. We view this departure, which increases as the material strength weakens, as a direct signature of the prevalence of attrition by fragmentation. Particles with VCM > 50 For the attrition of materials, the two end-members can readily be defined. In the limit of pure chipping, repeated low-energy impacts result in small-scale shattering that is confined to the near surface of the impacted particle (Miller and Jerolmack, 2020).
This manifests in the patterns of rounding observed in river rocks transported by bed load. In the limit of pure fragmentation 5 (catastrophic failure), where a single high-energy impact is capable of propagating cracks throughout the body of a rock, bulk breakup produces large angular fragments from the initial particle. The intermediate regime, however, is more difficult to determine. Here, fatigue failure occurs as multiple impacts are required to propagate cracks through the bulk of the particle. By performing experiments that span these regimes, we can populate the theoretical phase-space for attrition conceptualized by Zhang and Ghadiri (2002) (Fig. 1). By setting a threshold for fragmentation based on fractional mass loss per impact, we can 10 then approximate the number of impacts required for a material to fragment (Fig. 13). In defining this threshold as a significant fractional mass loss per impact (e.g., 10 %) (Shipway and Hutchings, 1993), then only the weakest material (12.5 % VCM) that breaks with nearly every impact can be categorized in the catastrophic failure regime. Alternately, if we define the threshold between chipping and fragmentation as a low fraction of the mass lost (e.g., 1 % or 0.5 %), then the four weakest concrete mixes (12.5 %, 14.3 %, 16.7 %, and 20 % VCM) fall within the catastrophic failure limit. 15

Conclusions
This study connected attrition mechanism to the resulting shape evolution in order to better understand the transition from chipping to fragmentation. By simulating transport over a wide range of material strengths, we were able to populate a phase space for attrition. Concrete particles were rotated in a metal drum to simulate transport, and results indicate that mechanically Fragmentation Threshold 10% mass loss 5% mass loss 2.5% mass loss 1% mass loss 0.5% mass loss Figure 13. Approximate number of impacts required to fragment particles of different material strength. The fragmentation threshold, or boundary between the chipping and fragmentation regimes, is defined by the fraction of mass lost during a given impact (e.g., fragmentation occurs if 10 % of a particle's current mass is lost).
strong particles evolve smoothly and monotonically toward a spherical shape, while weak particles rapidly break into irregular, angular pieces. Intermediate particles undergo a combination of the two breakdown mechanisms and erratically evolve toward a rounded shape. Our finding that the brittle Attrition number is the relevant material grouping governing attrition rates supports recent experimental results from binary collisions of natural rock materials (Miller and Jerolmack, 2020), and indicates that A b is a useful similarity parameter for scaling experiments.

5
Analysis of shape and mass loss parameters indicate that a limit exists where material strength has limited influence on particle breakdown. In the limit determined by this study-a brittle Attrition Number of 0.014 s 2 m −2 or lower-circularity and fractional mass loss approach constant values. This is the pure chipping limit of impact attrition. This condition appears to be common, even universal, for natural rocks undergoing bed-load transport (Novák-Szabó et al., 2018). The agreement of our experiments using concrete blocks, with previously described universal rounding, affirms the robustness of geometric 10 shape evolution by chipping and also the relevance of our experiments to natural pebbles. As material strength decreased in our experiments, the shape evolution of each particle began to deviate from the universal shape evolution curve, with intermediate and weak particles experiencing a greater variation in mass loss per collision and reaching lower circularities over the course of transport in the metal drum. This phase space delineates the "fatigue failure" regime of pebbles, where multiple collisions are required to grow cracks in the bulk to the point where they span the sample an induce fragmentation. While fatigue 15 failure is often invoked to model bedrock erosion due to impact attrition Dietrich, 2001, 2004), it has not to our knowledge been delineated in this manner before. This fatigue failure regime can be thought of as representing a continuous phase transition from pure chipping to pure fragmentation, where the probability of fragmentation per collision increases as material strength weakens (for a constant impact energy). This picture is consistent with the probabilistic conceptual model of Zhang and Ghadiri (2002), where this transition was cast as a function of impact energy (for constant material strength). 20