We conducted the binary collision experiments on a total of five sets of bricks, two sets of quartz diorite, and one set each for the sandstone, schist, and volcaniclastic rock. Throughout the course of each experimental run, the initially cuboid rocks would quickly lose their sharp corners and then slowly become rounder without any major fragmentation. There were two exceptions to this case. First, for the sandstone at around 20 000 collisions, a large piece roughly 2 cm long and 1 cm wide broke off one of the corners, exposing a reddish-orange oxidized surface. Second, with the schist on three occasions, the entire block more or less split in two, fracturing at weathering planes. In both cases, fracturing occurred at a pre-existing weak region of the rock that appeared to be associated with chemically weathered surfaces. Furthermore, for both sandstone and schist we observed that immediately following the large fracture events, the mass loss of the parent grain would increase as the freshly exposed rough surface of the grain smoothed.

We define cumulative mass loss as $M={\mathrm{\Sigma }}_{n=\mathrm{1}\mathrm{\dots }N}\mathrm{\Delta }M$, where N is the cumulative number of collisions, and equivalently cumulative impact energy as $E={\mathrm{\Sigma }}_{n=\mathrm{1}\mathrm{\dots }N}\mathrm{\Delta }E$. Plots of cumulative mass loss against cumulative impact energy for all rock types show two distinct patterns: an initial rapid phase of mass loss that is similar for all lithologies and impact energies, followed by a transition to a slower, linear mass loss curve whose slope varies with rock type (Fig. 5a). To verify the functional relationship between mass loss and energy while controlling for material properties, we performed experiments with three different masses of brick spanning a range of collision energies of 0.04–0.22 J. Mass loss curves for all experiments are in good agreement with each other and with a single linear trend (Fig. 6). Linear fits were then made to the second slower phase of all mass loss curves, resulting in the relation

To test the robustness of the linear fit, we generated a plot of $M/M_{\mathrm{0}}-b$ versus kE, where b would be dimensionless and k would have units of inverse energy. The quantity kE is analogous to $E/E_s$, where E_s is hypothesized to be a critical energy for chipping or fragmentation to occur. The collapse of data for all experiments shows that a linear relation is reasonable, but as anticipated it fails to fit the initially steep portion of the mass loss curve (Fig. 5b). We want to relate the two parameters in the linear fit (Eq. 7) to physically meaningful quantities. We turn first to the slope k, which controls the long-term attrition rate (dM $/$ dE) for a given energy and should be controlled by material properties – and hence be related to the attrition number A. We approximate the long-term attrition rate, dM $/$ dE, using the total mass loss divided by the cumulative impact energy, $M/E$. Data indicate that the fitting parameter k is directly proportional to the long-term attrition rate (Fig. 7a), with a slope of 1. This direct relationship indicates that the slope to the linear fit data, k, reasonably characterizes the long-term attrition rates for all of the lithologies explored in this study.

Figure 5Attrition mass loss curves. (a) Plot of total cumulative mass loss versus cumulative impact energy for each set of rocks. (b) Plot of total cumulative mass loss minus y intercept, b (Eq. 7), from linear fits to raw data in (a) versus cumulative impact energy multiplied by the value of fit slope. Insets for both (a) and (b) display plots with log–log axes.

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Figure 6Attrition rate for bricks with different collision energies. Plot of total cumulative mass abraded versus cumulative impact energy for three sets of brick with different masses. The inset displays a plot of average mass abraded per impact versus average energy per impact. Each data point corresponds to a separate set of bricks.

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We can then examine the relation between the attrition numbers and the long-term attrition rates for each lithology. The brittle attrition number A_b is plotted against long-term attrition rates $M/E$ for all samples (Fig. 7b) and demonstrates good correlation with some scatter, likely due to uncertainty in material property measurements, indicating that the brittle attrition number incorporates appropriate material properties to describe the long-term attrition of different lithologies. Although same order of magnitude as A_b, the semi-brittle attrition number A_s varies widely and does not correlate strongly with observed attrition rates $M/E$ (Fig. 7b, inset); we do not consider this parameter further in our analysis.

We now turn to the intercept (Fig. 7a). We find that the value b in Eq. (7) is related to the quantity of pebble mass that is lost before attrition reaches the slower, linear portion of the mass loss curve (Fig. 7c). In other words, it is the amount of attrition that occurs in the rapid first portion. The parameter b is related to the initial mass loss of each particle, with an average value of $b=M/M_{\mathrm{0}}=\mathrm{0.0018}$, and is approximately constant for all experiments (Fig. 7c). This result suggests that all particles transition to the slower, linear portion of the mass loss curve when they have lost a certain fraction of mass. Since collision energies and rock strengths are different, the only factor common to all experiments is particle shape; all particles were initially cuboids. To test whether b is related to shape, we plot the evolution of corner curvature and mass against cumulative energy (Fig. 7c) for the subset of samples for which shape was measured; results show that the former tracks the latter and becomes approximately constant when rock mass $M/M_{\mathrm{0}}\ge \mathrm{0.0018}$. This value is the same as b (Fig. 7a), meaning that curvature of corners becomes constant when the fraction of mass lost is equal to $b=M/M_{\mathrm{0}}$.

Figure 7Attrition numbers. (a) Plot of slope (k) from linear fits to normalized mass curves multiplied by initial mass versus total mass loss over total energy for all samples. (b) Plot of dM $/$ dE versus brittle attrition number. Each data point represents a different sample; VC is volcaniclastic, QD is quartz diorite, and SS is sandstone. For lithologies with multiple samples, colored data points are the average attrition rate, with grayed data points showing results for individual samples. The inset is a plot of calculated semi-brittle attrition number. (c) Plot showing change in mass fraction (left axis) and maximum curvature (right axis) versus cumulative impact energy for quartz diorite and sandstone samples. They both transition from a high rate of change to a slower one at an average intercept value of $M/M_{\mathrm{0}}=\mathrm{0.0018}$.

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By putting together the brittle attrition number and initial mass corresponding to k and b, the attrition relation for mass loss versus impact energy is

where

The data collapse in Fig. 5b justifies this equation. For the case when M≫0.0018M₀, the attrition relation reduces to

This brittle attrition relation suggests that when the sharp edges are worn away, the attrition rate is directly proportional to the brittle attrition number multiplied by the constant C₁.

The SEM images show a considerable amount of damage in the region near the edge of the grains (Fig. 4b, c). This damage is characterized by large cracks that span parallel to the collision surface with smaller cracks branching perpendicular to them. In some instances, these cracks produced from impact intersect inherent cracks or grain boundaries of the material, extending the damage zone further into the interior of the grain. The results of the damage zone length measurements are plotted in Fig. 8. Note that the measured distributions of crack lengths from the SEM images are unreliable in the small length limit due to image resolution. On the other hand, the large length limit is an order of magnitude larger than the smallest resolvable length, so these measurements are dependable. Further, for the thin sections imaged, we were only examining a single two-dimensional plane of a three-dimensional object. Therefore, a measured crack length is a result of both the actual crack length and its orientation, as the length of the cracks running obliquely to the thin section plane will be underestimated. The tail of the distribution of lengths shows power-law scaling with exponents that range in value from −1 to −2.3. We observe convergence of all distributions for each lithology in the large length limit, where the cracks are easiest to discern and measure. However, in the lower length limit, the distributions separate from one another as the length measurements become less reliable due to the resolution of the images.

Figure 8SEM results. (a) Plot of the distribution of the length of damage within abraded rocks from SEM images of thin sections. (b) Damage lengths plotted in the form of Eq. (6) with corresponding power-law fits. (c–f) SEM images of the largest crack length for each rock type outlined in red. (c) Brick. (d) Quartz diorite. (e) Sandstone. (f) Volcaniclastic rock.

Figure 9Grain size distributions of products of attrition. (a) Plot of the number distribution of grain size from the particle analyzer (solid circles) and sieving (circles with black outlines) methods. (b) Number distribution of grain size plotted in the form of Eq. (6) and normalized by the mean value. Data are combined for all lithologies. Data believed to be affected by the measuring technique are excluded from the plot (<1 and >200 µm from the particle analyzer). Data are fit with a power-law function with exponent of −2.5. The solid blue line denotes the expectation from brittle fragmentation: power law with exponent −2.

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The results from the characterization of the GSD of attrition products are shown in Fig. 9. The plot combines the full measurements from the laser particle analyzer and sieving methods. Distributions from all lithologies and experimental runs show the same functional form. However, the full distributions display artifacts of the measuring techniques in both the fine and coarse tails of the distributions. For the fine tail, the distributions drop off rapidly, presumably due to the combined effects of the low-end measuring limit of the particle analyzer and the loss of material during the collection of attrition products. For the coarse end of the Coulter counter data, sieving produces artifacts in the grain size distributions as the particle size approaches the sieve diameter, as is evident by the erratic fluctuations in the grain size distributions on approach to d=0.5 mm. Ignoring Coulter counter data over the range 0.2–1.0 mm, we observe consistent and smooth grain size distributions from 1 µm to the maximum observed size from sieve analysis for all rock types. To determine the functional form of the grain size data, we remove the unreliable data points that are biased by the measurement method; for the fine tail, this includes grain sizes less than 1 µm, and for the coarse tail this includes particle analyzer data greater than 200 µm. We normalize each curve by its mean value, collapsing all curves onto each other so that we may fit one function to the entire dataset for all lithologies. We then solve for the best-fit power law to all data points (Eq. 6). The fit shows an exponent of −2.5, which is slightly higher than the expectation of −2 for full fragmentation, but still follows a Weibull distribution with very good agreement (Fig. 9).

While a linear relation between mass loss and impact energy has been shown to reasonably model aeolian erosion (Anderson, 1986) and has been inferred in models of bedrock erosion (Sklar and Dietrich, 1998, 2004), our experiments definitively demonstrate that this linear relation is applicable for energies associated with fluvial bed-load transport over a wide range of rock strengths. There is an intriguing shape dependence of the initial attrition rate. Indeed, data seem to indicate that these initially very angular cubes all erode at the same rate regardless of energy or strength until the corners are suitably rounded such that energy and rock strength become important. We surmise that in this region the corners are so sharp that virtually any impact can remove mass because the yield stress will be locally exceeded in the limit of infinite curvature (Ghadiri and Zhang, 2002a). However, as observed in this study, rocks achieve the secondary linear mass loss curve quickly while their shapes are still very close to cuboids. Thus, for natural streams it is likely a reasonable assumption that b may be neglected; therefore, the relation $M/\left(M_{\mathrm{0}}E\right)\simeq k=C_{\mathrm{1}}A$ is the applicable one to examine attrition in natural streams.

The slope k has units of 1$/$energy, and thus $\mathrm{1}/k$ may be generically interpreted as a critical energy associated with breakage for each material. How energy relates to breakage depends on the failure mechanism, in particular how elastic or plastic the deformation associated with collision is (Momber, 2004b). We examined two different formulations for the attrition number, A. It appears that our data are reasonably well described by $A_{\mathrm{b}}=\frac{\mathit{\rho }Y}{{\mathit{\sigma }}^{\mathrm{2}}}$ and not by $A_{\mathrm{s}}=\frac{\mathit{\rho }DH}{K_{\mathrm{c}}^{\mathrm{2}}}$, indicating that material failure may be considered to be in the brittle regime. While previous work showed that bedrock erosion rate depends on the inverse square of the tensile strength (Sklar and Dietrich, 2004), our experiments elucidate clearly and simply which rock material properties need to be taken into account through the development and verification of A_b. A similar attrition parameter was proposed by Wang et al. (2011) for the erosion of yardangs by windblown sand, but the material control on attrition rate was not isolated from collision energy in their work. Moreover, here we verify the concept for energies relevant to fluvial transport. Wang et al. (2011) noted that the parameter A_b can be considered to be the elastic potential energy per unit volume at the yield point. We note, however, the existence of the prefactor C₁, which at present is an empirical parameter derived from our particular experimental setup. The physical meaning of C₁ likely combines a few factors, certainly including the details of the collision itself, the impact angle, rotation speed of the impactor, and other aspects of the collision geometry (Wang et al., 2011). The value of C₁ may also be related to particle shape, although experiments by Domokos et al. (2014) show that dM $/$ dE is constant for a given particle over nearly the entire evolution from cuboid to sphere, suggesting perhaps that C₁ is independent of shape. Our data tentatively suggest that C₁ is independent of material properties, since it is (roughly) constant across a range of material properties. Regardless, the brittle attrition number A_b appears to be a useful similarity criterion for comparing laboratory and field attrition rates; however, rates determined from our experiments may not yet be directly scalable to the field due to uncertainty in the controls on C₁.

While the brittle attrition number appears to describe the scaling of mass loss by chipping reasonably well, this still provides an incomplete picture. In particular, the actual value of mass or volume removed per impact must be calibrated with experiment. The SEM images of sectioned rocks show a zone of damage accumulation in a shallow region below the surface (Figs. 4, 8). Our measurements show some isolated, surface-normal cracks that penetrate several hundred microns below the surface (Fig. 8). More common, however, are shattered regions of surface rock that are bounded from below by surface-parallel cracks at depths of a few hundred microns (Figs. 4, 8d, 8e). This is important because lateral cracks are known to produce chipping for natural rocks (Momber, 2004a). Our examination of the damaged rock took place after thousands of collisions, so we do not know what the damage zone from a single impact looks like. Our observations could be explained, however, by the merger of Hertzian fracture cones that often form in ceramics and glasses (Wilshaw, 1971; Greeley and Iversen, 1987; Rhee, 2001; Mohajerani and Spelt, 2010); it has also been claimed that such fracture cones explain the surface texture of sediment grains (Greeley and Iversen, 1987; Johnson et al., 1989). In this model, elastic wave propagation outward from the impact site shatters rock in a small region in which a critical strain is exceeded (Wilshaw, 1971; Rhee, 2001). While still a brittle response, this failure mode is distinct from the cyclic fatigue mechanism that has been proposed to activate slow growth of cracks in fluvial bedrock erosion (Sklar and Dietrich, 1998, 2004). In high-resolution simulations of impacts on silica glass (Wang et al., 2017) – with energies and material properties comparable to our quartz diorite collisions – formation of Hertzian cones produced locally shattered near-surface regions that appear similar in style and scale to our images (Fig. 8). Importantly, this process is known to generate small chips (Rhee, 2001; Mohajerani and Spelt, 2010; Wang et al., 2017). We tentatively suggest that chipping in our experiments – and in pebble rounding and bedrock erosion generally – is dominated by Hertzian fracture. We note that Hertzian fracture has already been proposed as the primary mechanism for rock erosion by aeolian saltation impact (Greeley and Iversen, 1987) but was not examined in detail. Fatigue loading and associated slow fracture growth likely contribute to fragmentation that produces larger particles. In our experiments such fragmentation was rare but did occasionally occur; it is likely to be more common in weaker rocks that are highly weathered (see below) or for higher collision energies.

Maximum measured crack lengths and damage zone depths are comparable to the maximum size of attrition products in our experiments (Figs. 8, 9). Both crack-length and attrition-product size distributions are power laws, though scaling of the former varies among materials and may not be reliable due to measurement limitations. We surmise that the attrition-product GSD is produced directly by localized (Hertzian) impact shattering, but we acknowledge that more work is needed at the individual collision scale – in particular, examining the shattered impact region of a rock in 3D. It is somewhat surprising that maximum crack and attrition-product sizes vary little across all lithologies. Hertzian fracture cone size should depend on material properties such as fracture toughness and Young's modulus, and also on the applied load (Wilshaw, 1971; Rhee, 2001; Mohajerani and Spelt, 2010; Wang et al., 2017). We speculate that more dynamic range is needed in terms of both material properties and impact energies to see the effects of these factors. We also acknowledge our limited measurements due to the experimental challenges, which make our estimated maximum sizes unreliable. Nonetheless, the GSDs of attrition products are well resolved, covering 4 orders of magnitude in size, and their similar form across lithologies demands a generic explanation. A classic model for understanding GSDs resulting from wear is the brittle fracture theory developed by Griffith (1921), who hypothesized that all materials contain pre-existing flaws or cracks. The theory states that when an applied stress exceeds a critical value, the concentrated stress at the tips of these cracks is released as the crack propagates. Growth and intersection of these cracks cause the ultimate failure of the material. In the large energy limit of crushing, whereby complete disintegration of the parent particle occurs, Gilvarry and Bergstrom (1961) showed that the Griffith fracture model implies that the attrition products should have a GSD that follows the form of Eq. (5). More recent numerical simulations and laboratory experiments have shown that the value of the exponent depends on the mechanism of fracture (i.e., grinding, collision, or expansive explosion) and the impact energy (Kun and Herrmann, 1999; Astrom et al., 2004; Kok, 2011). However, none of these studies examined the low-energy limit of chipping that is relevant for bed-load transport. The scaling exponent of −2.5 for the attrition products of these binary collision experiments is surprisingly robust across a range of rock types, indicating a commonality in the failure modes of these different materials under the energies examined. The exponent is also within the range of values reported from studies of brittle fracture fragmentation. These observations support the notion that brittle fracture is the mechanism that creates the products of attrition in our experiments. The large size limit seems governed by the depth of the damage zone. As for the lower size limit, an obvious candidate would be the size of constituent particles in each rock type, i.e., sand grains for the sandstone and clay particles for the brick. Although we could not resolve the finest particles owing to loss, it is clear that fragmentation through constituent particles occurs. The determinant of the lower size limit remains unknown. Nonetheless, chipping robustly produces sand- and silt-sized particles, supporting the proposal that it is an important contributor of sediment to rivers and beaches (Jerolmack and Brzinski, 2010).

In the limit where k=0, the brittle attrition number, A_b, does not likewise approach zero but is instead associated with A_b=0.25 s²/m² (Fig. 7b). This non-vanishing value of A_b implies that for the range of energies examined in this experiment, there is a limiting rock strength at which little or no attrition occurs. This is similar to the proposed lower limit for collision energy, below which chipping does not occur (Ghadiri and Zhang, 2002a; Novak-Szabo et al., 2018). This result would suggest that some materials should not erode significantly under impact energies representative of bed-load transport. For our experiments, the volcaniclastic rocks are close to this limit. Observations of downstream evolution of pebble shape for volcaniclastic rocks in the Mameyes River in Puerto Rico have shown that significant attrition occurs (Litwin Miller et al., 2014). However, the pebbles from the field were all at least 4 times larger than those used in the laboratory, while estimated collision velocities were comparable. The combined observations of volcaniclastic rocks from experiments and the field suggest the possibility that, as particles lose mass downstream due to chipping, there is a potential lower limit in size that is controlled by rock strength. This idea needs to be explored in more detail.

Although results from these experiments display a steady linear mass loss with impact energy, as evident in the large fracture events with the sandstone and schist, chemical weathering can play an important role in the breakdown of river sediment (Jones and Humphrey, 1997). Howard (1998) observed higher rates of bedrock erosion in regions with more chemical weathering and thereby showed that chemical weathering weakens rocks and reduces material strength. While we find that material properties control attrition rates, chemical weathering can cause a weakening of these material properties. We observe fragmentation events along weathering planes similar to those observed in experiments of Kodama (1994b). In these instances, new angular and rough surfaces produced from the fragmentation process have high attrition rates. On the one hand, chemical weathering appeared to create internal planes of weakness that facilitated failure of large chunks under low-energy attrition. Indeed, these events caused fluctuations in the mass loss curves that were not observed in more structurally sound (stronger) materials. However, when observed over thousands of collisions (i.e., many fracture and/or failure events), the sandstone and schist rocks collapsed onto the same linear curve as other lithologies after accounting for material strength. It appears that mechanical weakening from chemical weathering may be reasonably described with the measured material properties that constitute A_b, so long as tested rock cores are representative of the rocks in question. In a natural setting, we expect that the effects of chemical weathering will be more dominant in transport-limited environments where chemical weathering rates outpace mechanical attrition. On the contrary, where sediment is transported frequently, mechanical wear is actively maintaining fresh unweathered surfaces on rocks, and therefore weathering features are not able to persist. Finally, we note that impact attrition experiments show that water may reduce the strength of silicate materials by half compared to measurements in air (Johnson et al., 1973). This is expected to influence the rate but not the style of attrition and serves as another caveat to applying our measured results to the field.

All data used for plots in this paper are deposited on figshare, a publicly available and open repository: https://doi.org/10.6084/m9.figshare.13169498.v1 (Litwin Miller, 2020).

KLM led the research, performed data analysis, and wrote the paper. DJJ supervised the research, assisted in data interpretation, and edited the paper.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper was edited by Francois Metivier and reviewed by Stephanie Deboeuf, Sebastien Carretier, and Jeffrey P. Prancevic.

This research has been supported by the National Science Foundation, Directorate for Geosciences (grant no. NSF EAR 1331841).