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Keywords:

  • Discrete Element Method;
  • bedrock incision;
  • debris flow;
  • granular flow;
  • segregation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] Field data and laboratory experiments suggest that bedrock wear from debris flows is largely due to particle–bed impacts, rather than solely due to abrasion by sliding, and that the associated bedrock erosion rates are dependent on the particle size distribution in the debris flow. Here we use Discrete Element Method (DEM) simulations with an established contact mechanics model to explore grain-size influences on contact forces associated with particle–bed impacts in sheared granular mixtures. We first compare DEM simulations with experimental observations obtained from shallow granular flows in rotating drums of diameters 0.56 m and 4.0 m. Our simulations reproduce, without parameter tuning, experimentally measured segregation, boundary pressures, and height profiles. We perform additional simulations systematically varying particle size distributions in binary mixtures. We show that local time-averaged boundary pressures in thin flows are essentially the normal component of the weight of the flow, independent of particle size distribution. However, other statistical measures of boundary forces scale with mass-averaged particle size. We demonstrate that this is because individual particle–bed impacts, rather than impacts from multiple particle collisions, dominate the largest contact forces. We show that these largest impact forces vary as the square of grain size and the 1.2 power of impact velocity as predicted from the contact mechanics model underlying the DEM. These results support the particle size dependence of a recently proposed bedrock incision model and suggest that next steps for a predictive bedrock incision model require the statistics of the largest impact velocities.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] When debris flows sweep down steep canyons they often entrain most sediment in their paths and scour to underlying bedrock. Inspection of the bedrock after such a flow event indicates that the bedrock is also worn and cut by the flow [Wieczorek et al., 2000; Stock and Dietrich, 2006].

[3] Considerable progress has been made toward understanding bedrock incision by fluvial processes [e.g., Whipple and Tucker, 2004; Sklar and Dietrich, 2004, 2008; Turowski et al., 2008; Johnson et al., 2009; Johnson and Whipple, 2010; Lague, 2010]. However, there is significant evidence that the process by which bedrock is incised in the steepest hillslopes is fundamentally different than the processes associated with fluvial bedrock incision. For example, the common practice of plotting local channel slope as a function of cumulative drainage area along a river longitudinal profile [e.g., Whipple and Tucker, 1999; Sklar and Dietrich, 1998; Whipple and Tucker, 2004] typically reveals a distinct scaling break at slopes below which mass flows typically do not entrain sediment or travel [Stock and Dietrich, 2003, 2006]. Experiments using flowing mixtures of gravel, sand, fine particles, and mud, here called granular-fluid flows, have provided further evidence for the ability of debris flows comprised of these materials to incise bedrock in a manner distinctly different from that by fluvial processes [Hsu et al., 2007, 2008; Hsu, 2010]. Relatively little work has been done on granular-fluid flow incision mechanics and theory, leaving a significant gap in the understanding of landscape evolution in steep upland regions [e.g., Stock and Dietrich, 2003; Stock et al., 2005; Stock and Dietrich, 2006].

[4] To develop a foundation for modeling bedrock incision by debris flows, Stock and Dietrich [2006] reasoned that the incision process is driven largely by the collisional forces associated with the relatively fluid-poor coarse front, or “snout” of such flows. They suggested that an expression for collisional stresses first proposed by Bagnold [1954] in simple sheared granular-fluid flows

  • display math

could be used to estimate the relevant collisional bed stresses resulting from debris flows. In this equation, ρ is the density of the material comprising the particles, d is particle size, and inline image is the shear rate. The parameter λ is what Bagnold called the linear concentration (of solids), where he showed that this parameter could be expressed as:

  • display math

Here, νs is the volumetric solids concentration of the granular material, and νs,max is the maximum solids fraction, well-known for uniform spheres to be approximately 0.74. However, while some work has been done to understand packing limits for certain mixtures (e.g., theoretical work by Rodine and Johnson [1976] suggests νs,max ≈ 1 for multisize mixtures typical of natural debris flows), there is no general expression predicting νs,max for a mixture of different-sized particles. Based in part on equation (1), Stock and Dietrich [2006] proposed an expression for longterm bedrock incision rate dz/dt due to debris flows (where z is measured normal and outward from the free surface):

  • display math

which depends on scaling of the excursion stresses with the average inertial stress (K0), the relationship of rock resistance to incision rate (K1), the inverse of the resistance of the bedrock to erosion (R), the frequency of occurrence of debris flows (f), the length of the snout (L), and a modified form of Bagnold's expression for collisional stress (bracketed term) raised to an empirically determined power, n. The modifications to Bagnold's collisional stress are as follows: the cosϕ term is introduced to account for bed slope effects, νs (the volumetric solids concentration) replaces λ2, De is a representative grain scale in the coarse front of a debris flow, and w is an empirical exponent. Stock and Dietrich [2006] derived expressions for the coefficients of equation (3) in terms of physical parameters such as the mechanical property of the bedrock and crack spacing to develop a geomorphic transport law for debris flow incision.

[5] In developing equation (3), Stock and Dietrich [2006] recognized challenges to applying equation (1) to natural flows composed of particle mixtures in water or muddy fluids. Most notably, debris flows are composed of particles of a wide size distribution for which the appropriate choice of d is not clear. This choice is complicated by the fact that particles tend to self-segregate by particle size [e.g., Oyama, 1939; Gray and Thornton, 2005], as well as by other particle properties such as density and shape [e.g., Kakhakhar et al., 1997; Plantard et al., 2006; Hill et al., 2010], and there is substantial evidence that the constitutive behavior of even dry particle flows (granular flows) depends nonlinearly on the components in the mixture [e.g., Rognon et al., 2007; Hill and Zhang, 2008; Yohannes and Hill, 2010; Hill and Yohannes, 2011]. Some have suggested that for predicting the dynamics of a granular-fluid flow in general, or a debris flow in particular, a particle size representative of a debris flow or of a region of interest be used [Iverson, 1997; Stock and Dietrich, 2006; Hsu, 2010]. For mixtures of different sized particles, the expression for λ (equation (2)) is also ambiguous, as νs,max is extremely sensitive to the particle size distribution [e.g., Rodine and Johnson, 1976; Yohannes and Hill, 2010; Hill and Yohannes, 2011]. Dimensional analysis by Iverson [1997] suggests it is appropriate to replace λ with νs/(1 − νs). Others [e.g., Stock and Dietrich, 2006; Takahashi, 2007] have used νs as a substitute for λ2.

[6] An additional uncertainty in the application of equation (1) to debris flows involves the functional dependence of the normal stress (σBag in equation (1)) on inline image, in the presence of interstitial liquids. Many [e.g., Hanes and Inman, 1985; Hsiau and Jang, 1998], including Bagnold [1954] himself, suggested that for particle-fluid mixtures the power law dependence of stresses on inline image, is somewhat less than quadratic. Hsiau and Jang [1998] suggested that the dependence of stress on inline image varies if the granular temperature (or the variance of the particle velocities in a granular flow or granular-fluid flow) is non-constant, which is true for segregating mixtures [Hill and Zhang, 2008]. Stock and Dietrich [2006] addressed this uncertainty in the dependence of stress on inline image by allowing dz/dt to vary as inline image, where w must be determined by the nature of the flow.

[7] Data from physical experiments using small and large drums [Hsu et al., 2008; Hsu, 2010] support the bedrock incision relationship proposed by Stock and Dietrich [2006]. In particular, Hsu et al. [2008] showed that incision rates into erodible basal panels in a 0.56 m diameter drum depended on grain size and bed material tensile strength in a manner similar to that predicted by Stock and Dietrich [2006]. In a 4 m diameter instrumented drum, Hsu [2010] simultaneously monitored the mean and dynamic loads on the bed and bedrock incision, from which she concluded that impact wear dominates incision in both dry granular flows and liquid-poor granular-fluid flows. Further, Hsu [2010] showed this wear is associated with large excursion forces and that the local variance of the force plate outputs increased roughly as the square of D84. In the presence of moderate-to-high concentrations of interstitial fluids such as water and mud, Hsu [2010] found that sliding wear also becomes important in the erosion of relatively low tensile-strength bedrock substrates. In this paper, we focus on the dynamics of liquid-poor granular-fluid flows, typical of debris flow fronts, that we consider well-represented by dynamics in granular flows. We consider primarily those dynamics relevant to the incision of high tensile-strength bedrock found in nature and for which equation (3) is most relevant.

[8] To summarize, two key assumptions of the Stock and Dietrich [2006] debris flow incision model (equation (3)) supported by experimental investigations [Hsu et al., 2008; Hsu, 2010] are 1) excursions in collisional stresses at the coarse front drive bedrock incision for the majority of rock strengths found in nature, and 2) these collisional stress excursions scale with the square of a representative grain size in the coarse front. Although the drum experiments by Hsu [2010] support these two assumptions, the data do not reveal how, theoretically, to calculate excursion stresses or the grain size dependency of these stresses. We propose that an important next step toward building a mechanistic and predictive model for bedrock incision by rocky debris flows, debris flow snouts, and other fluid-starved granular flows involves determining how the excursion stresses and the collisional forces scale with grain size distribution and flow dynamics.

[9] To address this, we use Discrete Element Method (DEM) simulations (first proposed by Cundall and Strack [1979]) to explore controls on boundary forces generated by granular flows. This approach is well-suited for dry granular flows, which we suggest represents salient features of the coarse front of even muddy debris flows, and has the distinct advantage over continuum models in explicitly predicting the dynamics of individual particles. There are several benefits to using this type of simulation. Segregation dynamics that concentrate large particles toward the flow front emerge spontaneously in the simulations, so there is no need to impose an empirical model for segregation. Further, the collective and individual impact processes of the particles on the boundaries can be directly monitored and related to other variables such as local and system-wide measures of particle size distribution [e.g., Alam and Luding, 2003; Rognon et al., 2007; Hill and Zhang, 2008; Yohannes and Hill, 2010; Hill and Yohannes, 2011].

2. Computational Simulations and Experimental Comparisons

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

2.1. Discrete Element Method (DEM) Simulations

[10] For our computational simulations (henceforth referred to as simply “simulations”) we use the Discrete (or Distinct) Element Method (DEM) [Cundall and Strack, 1979] in 3-d. Our particle–particle and particle–wall contact forces are represented using the following well-established non-linear force model:

  • display math
  • display math

as previously derived based in part on Hertzian and Mindlin contact theories [e.g., Hertz, 1882; Mindlin and Deresiewicz, 1953; Johnson, 1985], the Coulomb friction law, and damping (as in Cundall and Strack [1979]) linked to the coefficient of restitution by Tsuji et al. [1992]. Fn and Ft are the magnitudes of the contact forces in the directions normal and tangential, respectively, to the plane of contact between two spheres. δn and δt are the modeled deformations in the normal and tangential directions, represented, as is common practice, as overlap between contacting spheres. inline image and inline image. The parameters kn, the normal stiffness coefficient, kt, the tangential stiffness coefficient, ηn, the normal damping coefficient, and ηt, the tangential damping coefficient, are calculated according to material properties of our particles (e.g., modulus of elasticity E and Poisson's ratio ν) and the size of the particles d [Hertz, 1882; Mindlin and Deresiewicz, 1953; Johnson, 1985; Tsuji et al., 1992]. μ is the coefficient of friction between particles. We chose material properties of quartz glass for the simulations (Table 1) to best match materials readily available for our physical experiments (henceforth referred to as simply “experiments”) described below. We also chose the sizes of our simulated particles to correspond to those readily available for our experiments: d = 13.8 mm, 25.0 mm, and 50.0 mm. The physical particles vary in size by ∼10% around their mean value, so we gave our simulated particles a similar ∼10% polydispersity. We provide model parameters for mixtures of 13.8 mm and 50.0 mm particles in Table 2. The number of particles in the simulations reported here varied from ∼200 for the runs of only 25.0 mm particles to ∼1200 for the runs of only 13.8 mm particles.

Table 1. Material Properties of Simulated Particlesa
SymbolPropertyValue
ρmaterial density2650 kg/m3
Emodulus of elasticity29 GPa
νPoisson's ratio0.15
μcoefficient of friction (between particles)0.5
ecoefficient of restitution0.89
Table 2. DEM Force Model Coefficients for a Mixture of 13.8 mm and 50.0 mm Particlesa
Parameter13.8 and 13.8 (mm)13.8 and 50 (mm)50 and 50 (mm)
  • a

    Parameters are given according to contacts between two particles of d = 13.8 mm, one particle of d = 13.8 mm and one particle of d = 50 mm, and two particles of d = 50 mm, as noted in the first line of the table.

kn(N/m3/2)1.16 × 1091.45 × 1092.21 × 109
kt(N/m3/2)1.60 × 1092.00 × 1093.05 × 109
ηn(Ns/m5/4)1.02 × 1021.60 × 1029.69 × 102
ηt(Ns/m5/4)1.20 × 1021.87 × 1021.14 × 103

[11] At each time step of a simulation, we calculate all forces on each particle (due to weight and particle–particle and/or particle–wall contacts). Based on this, we determine the translational and rotational accelerations of the particles. Then, we calculate particle velocities and displacements at each time step by numerically integrating the translational and rotational accelerations using the fourth-order Runge-Kutta method.

2.2. Experimental Drums and Simulated Drums

[12] We performed the experiments and simulations we describe here in rotating drums. Admittedly, the boundary conditions of a drum are somewhat different from those of a typical debris flow in the field; specifically, the base of a model debris flow in a drum is curved so the effective slope changes relatively quickly compared to a debris flow in the field. We consider the front of the flow in the drum most similar to the front of the flow of a debris flow in the field. Distinct advantages of using drums arise from the fact that the material is recirculated, facilitating continuous and reproducible measurements of dynamic quantities including boundary pressures and detailed characterization of grain scale segregation.

[13] We performed our experiments in either a “small physical drum” of (diameter D = 0.56 m; width W = 0.15 m), described in detail by Hsu et al. [2008], or a “large physical drum” (D = 4 m; W = 0.8 m), described in detail by Hsu [2010] (Table 3). We modeled most of our simulations after experiments in the small physical drum for which the essential details are as follows. The front vertical wall is made of clear acrylic, and the surface profile and particle locations adjacent to the wall can be captured with a digital camera. (We use ImageJ [Abramoff et al., 2004] to determine the position of particles in the digital images.) The vertical sidewalls are relatively smooth. For these experiments, the drum bed was roughened with sandpaper to reduce sliding of the particles along the bed. The main advantage of using this drum regards the small number of (spherical) particles used in the experiments which allows us to reproduce the experimental conditions in our simulations and complete the simulations in a reasonable amount of time.

Table 3. Experimental and Simulated Drums
DrumD × WaOther Remarks
  • a

    Drum diameter (D) and width (W) are given in meters.

small physical drum0.56 × 0.15sandpaper on bed; no instrumentation
large physical drum4 × 0.825 mm × 25 mm ridges span bed; 15 cm (4.3°) × 15 cm force plate
smooth computational drum0.56 × 0.15particle–bed coefficient of friction μpb = 0.7
bumpy computational drum0.56 × 0.157 mm semi-cylindrical ridges span bed

[14] In the large physical drum, the vertical walls are smooth, but the drum bed over which the particles move is equipped with treads of a 25 mm square cross-section that span the width of the drum. The treads were installed at 6° intervals (i.e., every 21 cm) with one exception in each quadrant. In three quadrants there is a 60 cm span without treads where samples of bedrock substrate are housed for incision experiments. In the fourth quadrant, there is a span missing three treads in the immediate vicinity of a 15 cm × 15 cm steel load plate attached at the center of the drum bed to an Interface Force Model SWP10-5KB000 Precision Force Transducer. The instrumented system was designed in-house by engineers at the University of Minnesota St. Anthony Falls Laboratory. The force recorded by the force sensor at any given time is the sum of all forces associated with particle–sensor contacts. The force is recorded at a frequency of 200 kHz; these data are averaged over 1 ms intervals. Because this force is typically due to several simultaneous particle–bed contacts over the sensor area, we report this information in terms of a pressure, calculated by dividing the measured force by the area of the sensor. Because of the size of the drum and the amount of material needed for experimental flows, we performed experiments in this large physical drum using naturally occurring aspherical gravel particles rather than manufactured spheres.

[15] We performed simulations with one of two computational drums designed to reproduce the salient boundary conditions of the physical experiments (Table 3). Our model for the small physical drum, the “smooth computational drum,” has the same dimensions as the small physical drum. The bed of the model drum is physically smooth, and the particle–bed frictional coefficient μpb = 0.7 is slightly greater than that between particles to account for the additional friction associated with the sandpaper in the small physical drum. Snapshots from a simulation with this drum are in Figures 1a and 1c.

image

Figure 1. Snapshots showing the two simulated drums used for the computational results presented in this paper. Details of each are given in Table 3. (a) A snapshot of the simulations representing the small physical drum, where the walls are physically smooth. For this snapshot 3.13 kg of d = 13.8 mm particles are rotated in the drum at a rotation speed Ω = 12 rotations per minute (rpm). (b) A snapshot of the simulations representing the large physical drum, where the outer walls, or bed, have periodic treads as described in the text. For this snapshot 1.6 kg of d = 10.0 mm particles are rotated in the drum at a rotation speed of 6 rpm. (c) Oblique view of the simulation shown in Figure 1a. (d) Oblique view of the simulation shown in Figure 1b.

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[16] In our model for the large physical drum, the “bumpy computational drum,” we “installed” treads and a force sensor along the boundary, similar to those in the large physical drum. However, precise DEM simulations of the large physical drum are prohibitively computationally expensive due to the asphericity of the particles used for these experiments and the sheer number of particles. The duration of a simulation increases by approximately NlnN, where N is the number of particles, and, although there are a number of methods available for simulating aspherical particles, these typically take significantly more time [Thomas and Bray, 1999; Ting et al., 1995]. Therefore, we use spherical particles in the bumpy computational drum, and the drum has the dimensions of the small physical drum (0.56 m × 0.15 m). The simulated treads are comprised of 7-mm-diameter semi-circular cylinders that span the bed, scaled down from the height of the treads in the large physical drum according to the ratio of the drum diameters. As in the large physical drum, the treads are fixed to the bed at regular 6° intervals, except in the region of the simulated force plate. Force sensor data are produced from the simulations by summing the forces associated with particle–bed contacts in the 4.3° × 15 cm region lacking treads (Figure 1b). Snapshots from a simulation using the bumpy computational drum are shown in Figure 1b and Figure 1d. We address the issues of scaling the simulations to physical experiments in section 2.3.

[17] To produce thin flows characteristic of debris flows, we fill only a small percentage of our drums with particles (e.g., Figure 1). The particle movement in our drums, described in detail by Hsu [2010], is similar to that at the front or snout of debris flows in the field as reported by Swartz and McArdell [2005]. Particle trajectories in both the physical and computational drums follow a similar 3-dimensional circulation pattern: particles at the bed and walls are dragged upstream and then flow down the top surface of the flow toward the front. When viewed from the side, the flow exhibits a shear zone between the upstream and downstream moving particles, and the shear rate is non-zero throughout the entire depth. When viewed from the top, the particles on top move toward the sidewalls at the front of the flow. In a mixture of different sized particles, individual particle trajectories vary with particle size giving rise to an overall segregation of the particles from the front to the back of the flow. Large particles tend to circulate near the flow front, and the larger the particles, the more their trajectories are restricted to the front. Smaller particles tend to circulate near the middle and back of the flow.

[18] The flow of particles does not vary significantly from one drum to the next: The gravel flows in the large physical drum are similar to flows of spherical particles in the small physical drum and the two simulated drums; The flow depth to drum diameter ratio is the same when the fill fraction is the same, and casual observation of the particles in motion in the small and large physical drums indicate that the average particle motion is not affected much by the treads. However, there are some minor differences that are worth noting among the different systems. The simulation results indicate that the treads in the bumpy computational drum reduce the slip velocity adjacent to the bed and give rise to a flatter surface profile (e.g., compare Figures 1a and 1b). Additionally, since the diameter of the large physical drum is nearly an order of magnitude larger than that of the small drums, and we used similar sized particles in all drums, the flow depth to particle size ratio was much greater in the experiments we performed in the large physical drum than in any of the smaller drums.

2.3. Quantitative Comparisons of the Dynamics in the Physical and Simulated Drums

[19] We compare results from two sets of experiments and simulations whose parameters are summarized in Table 4.

Table 4. Parameters of the Systems Used for the Comparison Studiesa
Figure, Run TypePhysical ExperimentsNumerical SimulationsΩb (rpm)
Drum D × Wc (m × m)ParticlesDrum D × Wc (m × m)Particles
Massd (kg)Mate, Intf (mm, mm)Massd (kg)Mate, Intf (mm, mm)
  • a

    Entries correspond to simulations and physical experiments performed to compare the simulation results with similar experiments (section 2.3). Some results are shown in Figure 2 as noted. In all cases except one, we used spherical particles; for the experiments these were glass spheres and for the simulations these were spherical particles whose properties are described in section 2.1 and listed in Table 1. For the physical experiments in the large drum (Figures 2c and 2d), we used gravel particles of median grain size 10 mm, whereas for the corresponding simulations we used spherical particles.

  • b

    Rotation speed of the drum in rotations per minute (rpm).

  • c

    Drum diameter (D) and width (W).

  • d

    Total mass of all particles in each experiment or simulation.

  • e

    Matrix particles that comprise the majority of the particles in each experiment or simulation.

  • f

    Single intruder particle where applicable.

  • g

    The segregation experiments using a single large intruder particle among a matrix of smaller particles.

2a, height profile0.56 × 0.153.1313.8, –0.56 × 0.153.1313.8, –12
2b top, seg.g0.56 × 0.153.1313.8, 250.56 × 0.153.1313.8, 2512
2b bottom, seg.g0.56 × 0.153.1313.8, 500.56 × 0.153.1313.8, 5012
2c and 2d, force plate4.0 × 0.845410, –0.56 × 0.151.6710, –6

[20] From the first set of runs, we compare surface profiles and segregation behavior measured in the small physical drum with those in the smooth computational drum. Figure 2a shows instantaneous surface profiles measured from experiments (top) and simulations (bottom) for 13.8 mm particles. Each trace in both figures represents the surface profile for one of several drum rotations. The similarity of the experimental and computational profiles indicates that the internal stresses of the bulk flow are likely well-represented by the DEM simulations. Figure 2b shows some results from simple segregation experiments and simulations using systems of primarily 13.8 mm spherical “matrix” particles and a single larger “intruder” particle. Figure 2b illustrates typical longitudinal positions of relatively large intruder particles using a probability distribution (pdf) of the intruder particle in each case. The plots indicate that, for all cases, the large particles spend the majority of their time toward the flow front, though this tendency decreases for a smaller intruder particle. Such behavior of large particles is similar to that observed in large-scale debris flows having a much wider particle size distribution [e.g., Suwa, 1988; Iverson, 1997]. Despite close similarities, there are a few noteworthy differences between the experimental and computational results in Figure 2b. The experimental results are somewhat noisier and the peaks are broader than the simulation results. We propose that these differences are primarily due to the fact the simulations cannot perfectly reproduce the behavior of particles that are slightly aspherical and have asperities. Other potential causes for differences between experiments and simulations involve slight imperfections in the experimental drum, including an imperfect circular boundary. However, the simulations appear to adequately reproduce the segregation trends observed in the physical experiments.

image

Figure 2. Results from analogous physical experiments (exp) and computational simulations (sim) as noted in Table 4. (a) Height profiles of the granular flows. (b) Probability distribution function (pdf) of the location of a large intruder particle (size noted in plots) among smaller matrix particles. (c) From top to bottom in each column: instantaneous pressure (p4), locally averaged pressure ( inline image), and local standard deviation of the bed pressure (p4,σ) all plotted as functions of drum bed location (θ). (d) The inline image vs. θ normalized by the weight and sensor size (equation (A1)).

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[21] We use the second set of runs to compare boundary pressures in the large physical drum with those in the bumpy computational drum (Figures 2c and 2d). We calculate the simulated force plate measurements by considering the particle–bed forces over a time interval of 1 ms, which corresponds to analogous measurements by the physical force sensor. We report the results in terms of a pressure p4, where the subscript “4” refers to the approximate angle over which the sensor extends (≈4.3°) and thus the angle over which the averaging takes place (calculation details are shown in Appendix A.) All data shown in Figure 2c corresponds to one pass of the force sensor along the bottom of the debris flow. The top panels contain the “raw data”. The middle panels of Figure 2c contain the data from the plots of p(θ) averaged over 1° bins, inline image. (Here and henceforth, a line over any variable q, such as inline image indicates a temporal average as indicated in the calculations.) The bottom panels in Figure 2c contain the standard deviation of p(θ) for each 1° bin, which we denote p4,σ. Despite the differences between the results from the simulations and the physical experiments, the profile of the mean normal stress inline image is convincingly similar. This is particularly apparent when these pressure data are scaled by the total mass (M) in each drum and the area (A) of the relevant sensor: inline image (Figure 2d).

[22] Certain differences in the two data sets appear related to the size difference of the two drums. For example, the standard deviations of the normal boundary stresses in each 1° bin are similarly noisy, but the noise does not scale with M. The measures of the stress fluctuations are relatively larger in the smaller simulated drum than in the larger experimental drum. We attribute this difference in stress fluctuations to the difference in the sensor size relative to the particle size (smaller in the simulations than in the physical experiments). Specifically, the scale of particle size to the force plate influences the number of contacts on the plate and therefore the force distribution [e.g., Jalali et al., 2006]. Furthermore, the larger the force sensor relative to the particle size, the less the force sensor is capable of capturing individual particle–bed collisions. Thus, increasing the size of the force sensor relative to the particle size reduces the effect of individual impacts on the average pressure, effectively smoothing the effect of large (or small) impacts on the pressure calculations. This discrepancy would likely have been alleviated in part had we used a larger area for the force sensor in the simulations, but this would cause other problems. A model sensor length equivalent to that in the experiment (15 cm) would span 31° of the simulated drum. As we will show in section 3.1, the particle size distribution changes substantially over this angular distance, so the results would not capture local effects so much as global system effects. Thus we accept the different scale of the measured fluctuations in the physical experiments and simulations and use the simulations to understand how measures of contact forces such as local pressures, mean contact forces, and contact force variances vary relative to one another with change in particle size distribution rather than deriving an absolute measure of the fluctuations.

[23] Additional scaling effects associated with the drum width and tread height may also influence the simulation results. For example, the scaled down treads in the simulations relative to the particle size could allow more slip of the particles relative to the bed. In particular, if the treads are too small compared to the particle size, they do not effectively reduce particle slip as the particles “see” them essentially as negligible roughness elements. However, in our simulations the treads were sufficiently large to reduce boundary slip, so we assume the simulated ridges have the same effect on the flows as the ridges in the physical experiments.

[24] Having confirmed that the dynamics of the particles in the physical drums (e.g., height profiles, segregation details, and boundary forces) are represented relatively well by those in the simulations, we next systematically investigate the effects of particle size distribution on boundary stresses using our simulations.

3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

3.1. Simulated Mixtures

[25] To systematically investigate the effect of particle size distribution on particle–bed forces, we performed several simulations in the smooth computational drum using different combinations of the same particle sizes used in the validation experiments and simulations. In each case ∼4.3 kg of particles were rotated at 6 rpm in the smooth computational drum. The simulations were run for three minutes (of computational time), and the contact force data were obtained at a frequency of 75 Hz. The results do not change significantly when the simulations are run for a longer period of time. We focus first on two monosized systems (13.8 mm particles, and 25.0 mm particles) and four binary mixtures comprised primarily of 13.8 mm particles with different numbers of larger particles (25.0 mm or 50.0 mm). We refer to the mixtures with one, four and eight 50.0 mm particles as Mix150, Mix450, and Mix850, respectively, and the mixture with ninety-nine 25.0 mm particles as Mix9925 (Table 5).

Table 5. Particle Mixtures Used for the Results Presented in Section 3
Mixture NameType 1Type 2
Size (mm, ±)Mass (kg)Size (mm, ±)Mass (kg)
13.813.8, 10%4.30
25.025.0, 10%4.30
Mix15013.8, 10%4.1350, 0%0.17
Mix45013.8, 10%3.6150, 0%0.69
Mix85013.8, 10%2.9150, 0%1.39
Mix992513.8, 10%2.1525.0, 10%2.15

[26] Snapshots of simulations of the 13.8 mm particles alone, Mix150, Mix450, and Mix850 (Figure 3) reveal some basic details of the flow. In all four simulations the larger particles spend most of the time at the very front of the flow. The flow is shallow enough so that the large particles protrude from the top. As the number of 50.0 mm particles increases both the local average particle size and the length of the region frequented by 50.0 mm particles increases.

image

Figure 3. Snapshots from simulations of granular flows in a rotating drum obtained using systems of particles of the same total mass (4.3 kg) rotated in the smooth computational drum (D = 0.56 m, w = 0.15 m) at a rotational speed of 12 rpm. (a) 13.8 mm particles alone, (b) Mix150 (one 50.0 mm particle among many more 13.8 mm particles), (c) Mix450 (four 50.0 mm particles among many more 13.8 mm particles), and (d) Mix850 (eight 50.0 mm particles among many more 13.8 mm particles). System properties are given in Table 5.

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[27] The mass-averaged local particle size inline image (please see Appendix A for calculations) illustrates quantitatively the variability of the particle size distribution along the bed (Figure 4). Admittedly, compared to some other measures of representative grain size (e.g., the median grain size) the use of inline image has the disadvantage that there is typically no representative particle size of exactly that value in our mixtures. On the other hand, the local average energy and impulse imparted on the bed (hypothesized as important for bedrock incision by Sklar and Dietrich [2004] and Hsu [2010]) scale with the local average particle size. In Figure 4 we plot inline image for the four systems illustrated in Figure 3. As indicated in the snapshots in Figure 3, 50.0 mm particles congregate toward the front of the flow, rasing the mean size there. This region of increased inline image increases with increasing numbers of 50.0 mm particles in the mixture, corresponding to a longer region frequented by the larger particles. For all cases, the maximum value of inline image is located near the front of the flow and decreases with distance toward the tail until a point is reached where no 50.0 mm particles travel during the course of the simulation. We note that there is a distinct oscillation in inline image vs. θ, particularly for Mix850. The wavelength of the oscillations is ∼50 mm indicating some degree of ordering of the 50.0 mm particles relative to one another, a unique feature of relatively uniform particles.

image

Figure 4. Mass-averaged particle size, inline image, as a function of θ, or position along the bed, for simulations shown in Figure 3. As described in the text, the data are averaged over 1° bins. Lines are included in the plots to guide the eye.

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3.2. Normal Boundary Stress (Pressure)

[28] To determine how the boundary pressure varies with local particle size distribution, we calculate the local average pressure inline image at 1° intervals along the drum bed. As detailed in Appendix A, we do so by summing the normal contact forces within a 1° bin centered at θ and dividing by the area of that degree bin. Plots of inline image (Figure 5) for the four systems simulated systems shown in Figure 3 are essentially the same, except for a few bumps near the front that correspond to locations where the larger 50.0 mm particles spend more of their time. These data indicate that the local time-averaged pressure is independent of the local particle size distribution and likely dependent only on the distribution of total mass. To test this interpretation, we calculate the normal stress associated with the weight of the particles, inline image, analogous to the hydrostatic or lithostatic pressure in a continuous system (as detailed in Appendix A).

image

Figure 5. Plots of average pressure, inline image, measured at the bed, inline image (×) and average pressure, inline image, calculated from the weight of particles (lines) versus location along the bed (θ) from simulations shown in Figure 3. In each case, the relevant system is noted in the upper right hand corner of each plot: (a) 13.8 mm particles alone, (b) Mix150 (one 50.0 mm particle among many more 13.8 mm particles), (c) Mix450 (four 50.0 mm particles among many more 13.8 mm particles), and (d) Mix850 (eight 50.0 mm particles among many more 13.8 mm particles). System properties are given in Table 5.

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[29] For all mixtures inline image along the entire length of the flow with some relatively minor exceptions (Figure 5). The similarity in the two sets of data for each mixture extends to the degree of bumpiness corresponding to the average location of 50 mm particles. Comparing these results to the inline image data in Figure 4, we conclude that the average normal stress at any point along the boundary is nearly independent of the local and global particle size distribution, except as the particle size distribution governs the distribution of the system mass.

[30] It is interesting to consider Bagnold's classic model for collisional stress (equation (1)) in light of these results. Bagnold [1954] suggested that the mean normal stress inline image is equal to the mean collisional stress. In free surface flows such as ours, the mean normal stress may also be determined from the lithostatic pressure. Considering this in equation (1), implies that the shear rate, inline image, is determined by inline image and d. So long as the particle mass is distributed similarly from one system to the next, then, it makes sense that inline image does not change from one system to the next. Then, if Equation 1 holds, as particle size d increases inline image should decrease. For deep flows, this decrease in inline image has been reported, for example, in experiments by Hill et al. [2003] and simulations by Rognon et al. [2007].

[31] Although the variation in inline image does not correlate with a noticeable variation in inline image in the simulations, there is credible experimental evidence that a higher fraction of larger particles increases the rate of bedrock incision by granular flows [e.g., Hsu et al., 2008; Hsu, 2010]. Based on these results, we hypothesize that incision by granular flows is not directly associated with the average boundary pressure, but instead is related to the cumulative effects of individual particle–bed collisions as suggested for bedrock incision by fluvial processes by Sklar and Dietrich [2001, 2004]. Therefore, we focus next on statistics of the individual particle–bed contact forces and how they depend on local and global particle size distribution.

3.3. Spatial Variability of Collisional Contact Forces: Probability Distribution Functions

[32] The probability distribution function (pdf) of the forces provides a general picture for the variability of normal particle–bed contact forces. In Figure 6 we plot pdfs from a few distinct regions of the bed, as noted, for the 13.8 mm uniform system (Figure 6a) and the Mix850 system (Figure 6b). For both systems, there is a higher probability of large forces toward the front of the flow (e.g., from θ = − 10° to −5°) than toward the back (e.g., from θ = 30° to 35°), though the difference is much smaller in the 13.8 mm system (Figure 6a) than in the Mix850 system (Figure 6b). This indicates that the presence of large particles among the small particles has a significant effect on the variability of the forces along the boundary. The shape of the pdfs cannot be described by a simple function, such as a single power law or exponential. Therefore, for simple measures of how the forces vary with particle size distribution, we consider certain basic statistics of the forces associated with particle–bed contacts and how these vary in our simulated mixtures in the next subsection.

image

Figure 6. Probability distribution functions of the boundary forces at particular locations along the boundary for (a) 13.8 mm particles and (b) Mix850 (eight 50.0 mm particles among many more 13.8 mm particles; Table 5). The lines indicate the weight of a 13.8 mm particle (dashed line) and a 50 mm particle (dash-dotted line).

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3.4. Local Force Variability and Local Average Particle Size

[33] We calculate three basic statistics of the particle–bed contact forces within each 1° bin: the mean contact force, inline image, the standard deviation of the force per contact, fσ(θ), and a measure of the maximum contact force, fmax(θ). The calculations for inline image and fσ(θ) are detailed in Appendix A. For a representative local maximum contact force fmax(θ) we average the top 0.1% of forces within each degree bin. We performed the same calculation for percentages ranging between 0.1–5% and found that as the percentage value increases, the representative maximum force calculated in this way approaches inline image, but the functional dependence of the representative force on mean particle size is unchanged.

[34] In Figures 7a7c we show that inline image, fσ, and fmax vs. θ, all peak near the flow front and decrease toward the tail. However, unlike inline image, the scaling varies considerably from one mixture to the next, particularly near the flow fronts where inline image varies most dramatically from one mixture to the next. For the 13.8 mm monosized system, there is a sharp peak in each of inline image, fσ(θ), and fmax(θ) associated with the flow front (θ < − 10° to −5°). For Mix150, Mix450, and Mix850, the peaks in inline image, fσ, and fmax near the flow front are progressively larger as the number of 50.0 mm particles in the mixture increases. Additionally, the peaks are broader and extend farther back in the flow with increasing numbers of 50.0 mm particles. The values of each measure of the contact forces for the different mixtures eventually drop to the values of the 13.8 mm system. In each case this happens at essentially the same position at which inline image drops to 13.8 mm (See Figure 4).

image

Figure 7. Details of forces along the bed as functions of location (θ) for the simulations shown in Figure 3, averaged over 1° bins. (a) Local average contact force ( inline image). (b) Local standard deviation of the contact force (fσ). (c) Local maximum of the contact forces (fmax) Lines are included in the plots to guide the eye. Table 5 contains particle system details.

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[35] We suggest that there are two possible causes for the variability of the contact force statistics. The first involves the dynamics at the flow front. In both the physical experiments and the simulations, the flow is noticeably more disordered and highly collisional at the front and more fluid-like and frictional toward the tail. We hypothesize that the effect of multiple particles colliding violently at the flow front results in larger contact forces, an effect described previously in the literature [e.g., Iverson, 1997; Stock and Dietrich, 2006; Berger et al., 2011]. The second factor evident in the contact force statistics (Figure 7) is the simultaneous variation of the forces and the local average particle size inline image. To isolate the variability of the force statistics associated with inline image from those associated with change in phase (collisional to frictional) or slope, we consider data in the central region of the flows (40° > θ > − 10°). Specifically, we do not consider the data near the flow front (θ < − 10°) where the flow is clearly influenced by the disordered, gas-like dynamics and the flow height is highly spatially variable. Also, we do not consider the data too near the rear of the flow (θ > 40°) beyond which there are no large particles ( inline image mm), and the dynamics change due to an unreasonably high bed slope.

[36] Parametric plots of the force statistics as a function of grain size from Figures 4 and 7 for − 10° < θ < 40° (Figure 8) show significant overlap between the data from the Mix450 and Mix850 systems. These results indicate that the local average particle size inline image plays a significant role in determining the force statistics inline image, fσ, and fmax. In all cases, there is a clear monotonic relationship between local average particle size and local measures of the forces between particles and the boundary. The data for the average and maximum forces inline image and inline image are nearly equivalently well-fit with exponential and power law relationships, whereas the data for inline image are better fit with an exponential curve, using linearized least squares analysis (Table 6). Both fitting results indicate that inline image varies approximately linearly with inline image; fmax varies somewhat faster with inline image, and fσ varies much more slowly with inline image.

image

Figure 8. Parametric plots of local average contact force, inline image (circles), local standard deviation of contact forces fσ (squares), and local maximum contact forces fmax (diamonds) as a function of local average particle size inline image for each 1° bin from the data in Figures 4 and 7 for Mix450 (filled symbols) and Mix850 (open symbols). (Mix450 refers to a 4.3 kg mixture of four 50.0 mm particles among many more 13.8 mm particles, and Mix850 refers to a 4.3 kg mixture of eight 50.0 mm particles among many more 13.8 mm particles) (details in Table 5). The lines represent exponential functions fit to the data, y = a1exp inline image where y represents the quantities inline image, fσ, and fmax. The coefficients found from fitting these exponential functions to the data a1 and b1 are given in Table 6. System properties are given in Table 5.

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Table 6. Fitting Parameters for inline imagea, fσb, and fmaxc as Functions of inline imaged
Mixturey (Force Measure)a1e (N)b1e (1/m)equation imageb2f (−)
  • a

    Local average contact force between particles and drum bed.

  • b

    Local standard deviation of contact force between particles and drum bed.

  • c

    Local maximum contact force between particles and drum bed.

  • d

    Local mass-averaged particle size.

  • e

    Refers to an exponential function fitted to the data: y = a1exp inline image.

  • f

    Refers to a power law function fitted to the data: inline image.

  • g

    From data shown in Figures 7 and 8.

  • h

    From data shown in Figures 9 and 10.

Mix450 and Mix850g inline image0.2250.0390.0261.00
Mix450 and Mix850gfσ0.8930.0080.5780.203
Mix450 and Mix850gfmax2.5900.0570.0881.526
Mix9925h inline image0.2270.0550.0340.998
Mix9925hfσ0.2880.0730.0251.301
Mix9925hfmax2.1700.0820.1461.450

[37] We note that it is not surprising to find a monotonically increasing dependence of the average and maximum contact forces on the local average particle size. For instance, we can relate the force inline image associated with a particular collision between a particle and the boundary to the impulse or average rate of momentum change of the particle inline image: inline image, where m is the particle mass, and inline image is the difference between the velocity before and after the collision. This suggests that as long as Δt (the collisional contact time) does not vary too much with particle size, the impulse should have a monotonically increasing dependence on d (considering, for example, that particle mass m ∼ d3). Thus, increasing the number of large particles in a mixture increases the number of large impulses, subsequently increasing average particle–contact force (explored in more detail in section 4).

[38] We now examine the influence of the globally-averaged particle size compared to the influence of the locally averaged particle size on the particle–bed contact force statistics. To do so, we look at the variability of the force statistics in two pairs of mixtures with similar globally-averaged particle sizes, dsys (Appendix A). In particular, for both Mix9925 and Mix450, dsys ≈ 20 mm. For Mix850, dsys ≈ 25 mm, comparable to the system of 25.0 mm particles alone. The front-to-tail sorting varies from one mixture to the next, even for mixtures of equal average particle sizes (Figure 9a). Specifically, Mix450 and Mix850 have a more significant variability of inline image than does Mix9925 (and trivially, there is no size variation for the system comprised solely of 25.0 mm particles). On the other hand, inline image is greater farther back along the bed for Mix9925 and the system of 25.0 mm particles. As a result inline image, fmax(θ), and fσ(θ) (Figures 9b9d) are all high farther back in the flow for both Mix9925 and the 25.0 mm monosized system compared with those of Mix450 and Mix850. In contrast, relatively high values of inline image correspond to large values of fmax at the flow fronts for Mix450 and Mix850.

image

Figure 9. Mass averaged grain size and force statistics plotted as functions of bed location, θ, (plots analogous to those shown in Figures 4 and 7) for two sets of mixtures, each having similar average system sizes: (1) Mix9925 and Mix450 and (2) 25 and Mix850 (Mix9925 is a 4.3 kg mixture of ninety-nine 25 mm particles among many more 13.8 mm particles; Mix450 is a 4.3 kg mixture of four 50.0 mm particles among many more 13.8 mm particles; Mix850 is a mixture of eight 50.0 mm particles among many more 13.8 mm particles; 25 mm is a system of 25.0 mm particles alone). System properties are given in Table 5. In each case, the dependent quantity is averaged over 1° bins. (a) Local average particle size inline image. (b) Local average contact forces inline image. (c) Local standard deviation contact forces fσ. (d) Local maximum contact forces fmax. Lines are included to guide the eye.

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[39] Similar to Mix450 and Mix850, a flow of Mix9925 exhibits clear monotonic relationships between each statistical measure of the contact forces and the local average particle sizes (Figure 10). The various force statistics exhibit both power law and exponential relationships with inline image, similar to those exhibited for the mixtures of 13.8 mm and 50.0 mm particles (Table 6). However, the dependence of the force measure on inline image for Mix9925 is stronger than that for Mix450 and Mix850, as evidenced by the higher values of the parameter in the exponent and the power of these relationships (Table 6). These differences are even more apparent when these two sets of data are plotted side by side (Figure 11). For all values of the local average particle size inline image, inline image and fmax are greater for Mix9925 than for Mix450 and Mix850.

image

Figure 10. Parametric plots of average contact forces inline image (°), standard deviation of the contact forces fσ (solid squares), and maximum contact forces fmax (solid diamonds) as a function of local average particle size inline image for each 1° bin from the data in Figure 9 for a mixture of 13.8 mm particles and ninety-nine 25 mm particles (Mix9925). System properties are given in Table 5.

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image

Figure 11. Parametric plots of (a) local average contact force inline image, and (b) local maximum contact forces fmax as a function of local average particle size inline image for Mix450, Mix850, and Mix9925 over relevant values of inline image for all three mixtures ( inline image). The lines represent the exponential fits for each shown in Table 6. System properties are given in Table 5.

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[40] In summary, local measures of particle–bed forces depend solely on the local average particle size of different mixtures of the same two particle sizes (Figure 8). However, if one of the particle sizes in a mixture is changed, the functional dependence of the local measure of particle–bed forces on local average particle size changes significantly (Figure 11).

3.5. Variations of System Force Statistics With System Particle Size Distribution

[41] Next, we consider how various system-wide force statistics depend on dsys for each mixture or system of particles. We do so for the sets of mixtures described above (Table 5) and other mixtures of 13.8 mm and 25.0 mm particles and of 13.8 mm and 50.0 mm particles, all having a total mass 4.3 kg. For each mixture, we calculate the average bed forces over the whole system, inline image, and the variability in terms of the standard deviation of the forces in the entire system fσ,sys (Appendix A). We also compute fmax,sys, the average of the highest 0.1% contact forces across the entire bed. For each set of mixtures, inline image (Figure 12a), fσ,sys (Figure 12b) and fmax,sys (Figure 12c) increase approximately exponentially with dsys. Similar to the local force statistics (Figure 11), the system-wide force statistics for the 13.8 mm and 25.0 mm mixtures are more sensitive to a change in average particle size than the analogous statistics for the 13.8 mm and 50.0 mm mixtures (Table 7). Furthermore, for the same average mixture size, inline image, fσ,sys and fmax,sys are typically larger for the 13.8 mm and 25.0 mm mixtures than they are for the 13.8 mm and 50.0 mm mixtures (Figure 12).

image

Figure 12. Whole system particle–bed force statistics plotted as a function of system-averaged particle size for 4.3 kg mixtures of 13.8 mm and 50.0 mm particles (circles) and 13.8 mm and 25.0 mm particles (crosses). (a) Average contact force on the bed inline image. (b) Standard deviation of the contact force on the bed fσ,sys. (c) Maximum contact force on the bed fmax,sys. The lines are exponential fits to the data, and the fitting parameters are given in Table 7.

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Table 7. Parameters Obtained When an Exponential Function equation image was Fitted to the Data Shown in Figure 12
Mixturey (Force Measure)a1 (N)b1 (1/m)
  inline image0.2490.038
13.8 mm and 50 mmfσ,sys0.2850.070
 fmax,sys2.8760.064
  inline image0.1090.097
13.8 mm and 25 mmfσ,sys0.1220.125
 fmax,sys0.9980.132

4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[42] To understand the particle size dependence of the boundary forces due to granular flows, we consider the dynamics of the particle–bed impacts at the particle scale. In particular, we focus on the dynamics of an isolated particle impacting a bed with an impact velocity vi. Immediately after the impact, the contact force between the particle and the bed increases and the particle velocity decreases, until the particle stops, reverses direction, and begins to rebound. At that time, the deformation associated with that particular particle–bed interaction, δn in equation (4a), has reached its maximum value δn,max. As the particle rebounds; the particle center moves away from the boundary, and the contact force gradually decreases with δn until δn = 0 and the particle–bed contact is broken.

[43] To illustrate typical force scales of this interaction, we plot the temporal dependence of the contact force for a collision between a 25 mm particle and the fixed boundary for three different values of vi (Figure 13). The total duration of each collision is on the order of 10−4 seconds, indicative of our use of relatively hard particles [Yohannes and Hill, 2010; Hill and Yohannes, 2011]. We use the maximum contact force fc,max as a representative force scale for each collision and plot fc,max as a function of vi from simulations using three different sized particles, 13.8 mm, 25.0 mm, and 50.0 mm (Figure 14). We find that for a given particle size fc,max ∼ vi1.2, and for a given value of vi we find that fc,max ∼ d2.

image

Figure 13. The force between a particle 25.0 mm diameter particle and a fixed boundary as a function of time after initial contact (tc) for specific impact velocities, vi, as indicated in the legend.

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image

Figure 14. The maximum particle–bed force fc,max as a function of impact velocity vi for isolated particle–bed collisions for four different particle sizes: 13.8 mm (circles), 25 mm (squares) and 50 mm (diamonds) particles. The peak force increases as a power law function of the impact velocity with a power of 1.2, and for a given value of vi, fc,max ∼ d2.

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[44] The contact model (equation (4a)) provides a means for obtaining a relatively simple theoretical expression for the maximum contact force: fc,max,cm. Neglecting the damping force and the gravitational force, the maximum force per contact is fc,max,cm = knδn, max3/2. To determine how this expression varies with d and vi, we consider separately the functional forms of the force parameter kn and the maximum deformation from the contact model δn,max,cm. According to Hertzian contact theory, for contact between a particle of diameter d and a flat bed,

  • display math

provided that both the bed and particle have the same modulus of elasticity E and Poisson's ratio ν.

[45] To find δn,max,cm, we equate the kinetic energy loss of a particle of mass m and impact velocity vi to the energy required for the particle–bed deformation from the time of initial contact to the time of maximum deformation, neglecting the damping force and the gravitational force in equation (4a). In this case, for a particle–bed overlap (or deformation) equal to x, the energy associated with an additional incremental deformation of dx is knx3/2dx. Equating the kinetic energy loss during the entire deformation process to the energy associated with the total deformation gives:

  • display math

where δn,max,cm refers to the maximum overlap predicted by the contact model. Integrating and rearranging this expression gives an explicit function for δn,max,cm:

  • display math

Using m = ρπd3/6 and kn from equation (5), we find

  • display math

Considering equations (5) and (8), the maximum force per contact predicted by the contact model fc,max,cm, may be written in terms of the particle material properties and diameter as:

  • display math

We note that the power law dependencies of fc,max,cm on vi and d are the same as those obtained from the simulation data (Figure 14).

[46] This expression for fc,max,cm can be used to estimate the scaling of the average system boundary stresses with average particle size in a binary mixture of particles of two sizes d1 and d2. We simplify the problem by assuming that the number of contacts by each type of particle is exactly proportional to its number representation in the mixture and that all of the particles have the same impact velocity. From equation (9), we can express a theoretical system average force inline image based on only the maximum forces per contact as

  • display math

where α relates the time-averaged force for a particular particle–bed contact to the maximum force of that contact (assumed here to be a constant parameter); N1 and N2 are the assumed numbers of contacts associated with two types of particles, proportional to the number of particles of each type. We calculate the results for several different mixtures consisting of one of the two sets of particles: (1) d1 = 13.8 mm and d2 = 25.0 mm and (2) d1 = 13.8 mm and d2 = 50.0 mm. We use the same physical details as we use in our simulations: E = 29 GPa, ν = 0.15, ρ = 2650 kg/m3, and total mass = 4.3 kg. We use an impact velocity of 0.14 m/s, which corresponds to the impact velocity of a particle released from a distance 1 mm above a horizontal surface, chosen somewhat arbitrarily based on the scale of the particles. The functional form of these results do not depend on this value. Finally, we use α = 1, though in reality the number should be significantly smaller than one. We plot the results in Figure 15.

image

Figure 15. Theoretical system-averaged particle–bed force inline image as a function of system average particle size inline image (equation (10)) for two different sets of mixtures: 4.3 kg of different fractions of 13.8 mm and 25.0 mm spheres and 4.3 kg of different fractions of 13.8 mm and 50.0 mm spheres (denoted “25” and “50” in the legend, respectively).

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[47] Despite the many simplifications used to reach these theoretical results, they exhibit many of the same features as the results from the computational simulations (compare Figure 15 with Figure 12). inline image depends monotonically and nonlinearly on dsys as was the case for inline image and fmax,sys. Further, for a particular value of dsys, inline image is higher for the mixture with 25.0 mm particles than that with the 50.0 mm particles. From the calculations described above, this is apparently due to the fact that each particle–bed contact force scales with the size of the particle making contact with the bed. For any particular mixture, it takes considerably more 25.0 mm particles in the mixture of 13.8 mm and 25.0 mm particles than it does 50.0 mm particles in the mixture of 13.8 mm and 50.0 mm particles to give rise to the same averaged particle size inline image mm. For example, for a mixture comprised of 13.8 mm and 25.0 mm particles where inline image = 24 mm, approximately 180 out of 260 (63%) of the total number of particles are 25 mm particles, whereas for a mixture comprised of 13.8 mm and 50.0 mm particles where inline image = 24 mm, approximately 7 out of 854 (1%) of the total number of particles are 50.0 mm particles. The dominance of 25.0 mm particles in the first mixture outweighs the effect of the greater force per impact of the 50.0 mm particles in the second mixture. Consequently the mean force inline image is greater in the mixture of 13.8 mm and 25.0 mm particles. We hypothesize that this is also true with more realistic systems where particle–bed contacts are also influenced by collisions involving multiple particles simultaneously. It is not immediately obvious how to generalize these results to quantify the influence of particle mixtures using a more general description of particle size distribution. Instead, here we emphasize our result that a single measure of particle size for a mixture is not enough information to predict a mean particle–bed force for a mixture. Notable quantitative differences between the plots of inline image vs. inline image (Figure 15) compared with inline image and fmax,sys vs. inline image (Figure 12) pertain primarily to the magnitude difference between the theoretical and computational results. ( inline image and fmax,sys are much smaller than inline image for a particular value of inline image.) This is likely at least in part due to our use of α = 1 as a placeholder in equation (10), and our choice of a constant value for vi.

[48] Comparing the scaling results from isolated particle–bed collisions with analogous quantities from the rotating drum simulations provides additional information regarding the relative importance of collective interparticle interactions on the statistics of the particle–bed forces. To do so, we calculate the maximum force and impact velocity associated with each contact between a particle and the drum from a few cases reported in section 3: Mix150, Mix850, and 25.0 mm particles alone. For each case we plot the maximum force for each contact as a function of the normal component of the impact velocity, vi, (Figure 16) and superpose the relevant results from Figure 14 onto these results. From these plots we see that the highest values of fc,max from the computational results exhibit the same power law dependence on vi as do isolated particle–bed collisions and contact model predictions (∼ vi1.2d2). Furthermore, when we consider the largest values of fc,max for Mix150 and Mix850 (Figures 16a and 16b, respectively), we note that the similarity between the results from these two systems indicates that the statistics of the largest forces considered in this way are essentially independent of interactions between multiple large (50 mm) particles. For the mixtures of 13.8 mm and 50.0 mm particles, Mix150 and Mix850, there are two lines of data at higher values of vi and fc,max, one corresponding to each of the two particle sizes. The points on the branch corresponding to the 13.8 mm particles have considerably lower values of fc,max for the same values of vi, consistent with the relationship fc,max ∼ d2. For the 25.0 mm system (Figure 16c) there is one line of data extending away from the origin consistent with the single size of particles in the system. For the smallest values of fc,max and vi there is no clean functional relationship between fc,max and vi which indicates that these forces are largely associated with multiple particle collisions. The absence of points significantly below the power law relationships plotted in Figure 16 indicates that, for nearly all values of vi, multiple particle interactions serve to enhance, not buffer or decrease, the value of fc,max for a particular value of vi. The spatial variation of these force statistics is shown for one mixture (Mix850) in Figure 17. These results show that near the front of the flow where the particle–bed forces are the largest, the relationship fc,max ∼ vi1.2 for the 50.0 mm particles is the strongest; multiple-particle dynamics appear more dominant farther back in the flow where the particle–bed forces are relatively small.

image

Figure 16. The maximum force, fc,max, for each individual particle–bed contact as a function of the impact velocity vi for three granular systems: (a) Mix150 (one 50.0 mm particle among many more 13.8 mm particles), (b) Mix850 (eight 50.0 mm particles among many more 13.8 mm particles), and (c) 25.0 mm particles alone. System properties are given in Table 5. The solid lines represent the maximum contact forces for a single particle impacting a horizontal flat boundary. In Figures 16a and 16b, the lines represent this information for a 50.0 mm particle and a 13.8 mm particle, and in Figure 16c, the line represents this information for a 25.0 mm particle as noted. System properties are given in Table 5.

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image

Figure 17. The maximum force for each individual particle–bed contact as a function of the impact velocity for Mix850 (eight 50.0 mm particles among many more 13.8 mm particles) for specific locations along the drum bed as noted in the insets. The solid lines represent the maximum contact forces from a single 50 mm particle (top left) and a 13.8 mm particle (bottom right) when impacting a horizontal flat boundary. System properties are given in Table 5.

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[49] The importance of isolated pure particle–bed impacts in determining the largest boundary forces gives some indication of why our simple theoretical model for average particle–bed forces has some qualitative success in predicting observed trends in inline image vs. inline image for different mixtures. Our model would be improved with information on how the distribution of the impact velocities depends on system parameters.

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[50] Our results in sections 3 and 4 provide some clear direction for bedrock incision models associated with debris flows. First, the temporally-averaged boundary normal stress profile of a granular flow depends primarily on the distribution of mass along the boundary and is essentially unrelated to the particle size distribution. Since experimental and field data show that the rate of bedrock incision by debris flows is dependent on particle size distribution of the particles comprising a debris flow [Stock et al., 2005; Hsu et al., 2008; Hsu, 2010], we conclude that the pressure associated with debris flow does not control the bedrock incision except as it influences the distribution of mass along the boundary.

[51] Second, in contrast with the pressure results, the statistics of normal forces associated with individual particle–bed collisions do scale with particle size in a granular flow. Specifically, we have found that increasing the number of larger particles in a binary mixture for mixtures of the same particle sizes and total mass, gives rise to an increase in the mean force per contact, in the standard deviation of the mean force per contact, and in a representative maximum contact force. These relationships are qualitatively the same whether we consider local relationships between particle size and forces in small regions along the drum bed or the system-wide measures of the forces and particle size. We have found that the size-dependence of the contact forces may be modeled by considering the contact mechanics of the particles. The associated scaling of the maximum particle–bed forces with particle size fc,max ∼ d2 is consistent with a recently proposed bedrock incision model [Stock and Dietrich, 2006]. We conclude from this that the particle–bed contact force statistics rather than the pressure associated with the contact forces provides a more reasonable basis on which to build a mechanistically-based bedrock incision model for debris flows.

[52] Finally, we consider the next steps for using our results from DEM simulations of granular flows as a guide in model development for bedrock incision from debris flows. In developing equation (3), Stock and Dietrich [2006] included an assumption that excursion forces that incise bedrock scale with a measure of the average collisional stress, specifically inline image. Our simulation results suggest that this model could be linked to the mechanics of the particle dynamics by using the scaling for the largest impact forces. These scale with the square of the size of the largest particles as does the incision rate in the Stock and Dietrich [2006] model. On the other hand, compared to a shear rate dependence in the Stock and Dietrich [2006] model, our findings show that boundary forces scale non-linearly with impact velocity. In this context, an important next step for building an impact-based model for bedrock incision involves the development of a predictive form of a probability distribution function of impactor velocities as they depend on particle size distribution and other factors including debris flow depth and shear rate. For an alternative model for bedrock incision by debris flows, it is instructive to consider a model for bedrock incision into exposed bedrock in riverbeds proposed by Sklar and Dietrich [2004] due to saltating bedload over exposed bedrock. They suggested that in this context, the bedrock incision rate can be expressed simply as the product of three terms: the average volume of rock detached per particle impact, the rate of particle impacts per unit area per unit time, and the fraction of the river bed made up of exposed bedrock. We could apply our results to a stochastic form of this model, but to do so we would need to develop a predictive form for the statistical distribution of impact velocities as well, particularly those sufficiently large to give rise to impact forces capable of removing bedrock.

[53] We conclude this section with two cautionary notes. First, we note that we have focused on quantities such as the statistics of inline image and inline image but high values of either does not necessarily correlate with high incision rates. More information is needed about the strength of the bed material before conclusions can be drawn. For example, while inline image and inline image are typically larger for the mixtures of 13.8 mm and 25.0 mm particles than equivalent mixtures of 13.8 mm and 50.0 mm particles, the 50.0 mm particles still give rise to individual particle–bed forces considerably larger than the 25.0 mm particles. If the strength of the bed material is sufficiently large, bedrock incision would still be greater for the mixtures of 13.8 and 50.0 mm particles than for mixtures of 13.8 and 25.0 mm particles. Fatigue due to repeated small collisions may also play a significant role. In other words, in addition to the force statistics one needs information about the material properties of the bedrock itself to fully develop a model for bedrock incision. Our second cautionary note involves the material properties of the particles themselves. The kinematics of softer particles in a granular flow are not the same as for harder particles in a granular flow. For example, the power law relationship between p and inline image is reduced [Silbert et al., 2001], and it is likely that the force statistics change accordingly [e.g., Yohannes and Hill, 2010; Hill and Yohannes, 2011, and references therein]. Therefore it is likely that our results would change for flows of softer particles.

6. Summary and Future Work

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[54] We used DEM simulations of dry granular flows in a drum to gain insight on bedrock incision associated with debris flows, particularly incision due to dynamics in the fluid-starved front of debris flows. As we have demonstrated, Discrete Element Method (DEM) simulations are capable of producing, with no calibration, physically reasonable particle size distributions and boundary stresses associated with thin granular flows. This indicates that it is a reasonable tool for developing the distribution of impact velocities, collisional frequencies, impact forces, and other dynamics necessary for a mechanistic-based model for impact-driven bedrock erosion.

[55] In particular, we used DEM simulations to investigate the parameters that determine normal stress and other statistical measures of the normal forces associated with sheared granular mixtures for insight into mechanisms associated with bedrock incision due to debris flows. We found that the normal stress or pressure profile along the boundary of a granular flow depends primarily on the distribution of mass along the boundary and is essentially unrelated to the particle size distribution. In contrast, we found that statistical measures of the contact forces (the mean impact forces, the standard deviation of the mean impact forces, and the maximum impact forces) depend strongly on the mean particle size. This is true for both the local dynamics and the global dynamics of a granular flow. Since experimental and field data show that the rate of bedrock incision by debris flows is dependent on particle size distribution of the particles comprising a debris flow [Stock et al., 2005; Hsu et al., 2008; Hsu, 2010], we conclude that the impact forces rather than the pressure (mean impact force normalized by area) must be mechanistically linked to bedrock incision rates associated with collisional wear by debris flows.

[56] We developed a simple model that suggests that, even though the particles in a granular flow behave in many ways like a continuous flowing mass, the statistics of the particle–bed contact forces are controlled primarily by isolated particle–bed collisions rather than cooperative collisions involving, simultaneously, many contacting particles and the bed. Two pieces of evidence support this conjecture: (1) A mechanistic-based model for the force associated with isolated particle–bed collisions replicates the size dependence of the contact force statistics associated with bulk granular flow in a drum, and (2) The largest contact forces from the granular flow simulations follow the same dependence on particle size and impact velocity predicted from the mechanistic-based model for isolated particle–bed collisions. Impacts have similarly been found important for governing other details of granular flows [Menon and Durian, 1997; Jalali et al., 2002].

[57] Finally, we suggest that for a mechanistic-based model of bedrock incision due to debris flows, two key steps need to be taken next: (1) Obtain the relationship between the volume of rock detached per particle impact and impact force for bedrock commonly found in the field and (2) Obtain impact velocity statistics as they depend on particle size distribution and system dynamics such as local system mass and depth-averaged velocity to input into equation (10) for the statistics of the largest impact forces. The first would likely require additional laboratory experiments, though Sklar and Dietrich [2004] suggest a method for deducing this that is likely applicable for certain debris flow incision problems as well. The second would be nearly inaccessible experimentally for bulk granular flows, but could be determined from additional DEM simulations such as those we have presented in this paper that have a track record of representing physical velocities, forces and other dynamics in physical granular flows.

Appendix A:: Calculations for Particle Size and Force Statistics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[58] In this Appendix we demonstrate how we calculate the statistics presented in the text.

[59] As mentioned in section 2.2, to compare force data from the simulations to that from the experiments in the big drum, we consider the sum of all forces normal to the bed in the region where there are no treads, which we label the force sensor in Figure 1b. To compute the force that corresponds to an individual force sensor readout at drum position θ (reference angle shown in Figure 2a), we add all of the normal forces between particles and the drum bed within the 15 cm × 2.1 cm region corresponding to the force sensor while the drum is rotated by an angle Δθ =  Ω /Γ, where Ω is the drum rotation speed and Γ is the data output frequency (1 kHz). We divide this net force by the number of discrete time steps in the simulation that occur over a time interval. Since the force sensor output from the experiments and simulations results from force distributed over the area of the sensor, we report the measurements in terms of a normal boundary stress, or pressure p4:

  • display math

where the subscript “4” (in p4 and Nc,4) is approximately equal to the angle over which the sensor extends (here, ≈4.3°) and thus the angle over which the averaging takes place. i refers to the ith contact made between a particle and the part of the wall corresponding to the simulated sensor at time τ of which there are Nc,4(τ). Θ refers to the angle over which the sensor extends (≈ 4.3°). θi is the azimuthal location of that contact and at time t must satisfy θs(t) −  Θ/2 < θi < θs(t) +  Θ/2 where θs(t) is the angular position of the center of the sensor at time t. ti refers to the time at which that contact is recorded and must satisfy to + 1/2Γ  > ti > to − 1/2Γ, where to refers to the time at which the middle of the simulated sensor passes θ. A is the area of the sensor. Nτ is the total number of time steps.

[60] p4,σ denotes the standard deviation of p4(θ) for each 1° bin. Specifically, for a particular degree bin containing N normal stress measurements whose mean is inline image, inline image. (Please see Figure 2c.)

[61] For a representative measure of the local particle size, we consider the mass- averaged local particle size inline image.

  • display math

Here, mj(θjτ) and dj(θjτ) are the mass and diameter, respectively, of the jth particle at time τ whose center is at a location θj, where inline image. Np(τ) is the number of such particles at time τ.

[62] To determine how pressure on the boundary, p and other force statistics vary with local particle size distribution, we consider certain statistical measures of the force at 1° intervals along the drum bed. For angular position θ, we consider contact forces within a 1° bin centered at θ. In the following, fj(θjτ) represents the normal force associated with the jth contact at time step τ between a particle and the drum bed at an angle θj satisfying inline image. We use Nc(τ) to represent the number of such contacts at time step τ, of which there are Nτ. We use A1 to represent the area at the drum bed over which the forces between particles and the wall are recorded for each 1° section: A1 = (πD/360) × 0.15 m ≈ 7.33 × 10−4 m2.

[63] Based on these considerations, we calculate a local pressure, the sum of all normal contact forces within a bin divided by the area of that bin:

  • display math

[64] The pressure associated with the component of the gravitational force due to the local mass normal to the drum bed is analogous to the hydrostatic or lithostatic pressure in a continuous system. For this calculation, as above, mj(θjτ) and dj(θjτ) are the mass and diameter, respectively, of the jth particle at time τ whose center is at a location θj, and inline image. Np(τ) is the number of such particles at time τ. Then the “lithostatic” pressure inline image may be calculated according to:

  • display math

[65] As for the spatial variability of inline image and inline image, for the spatial variability of certain basic statistics of the forces associated with particle–bed contacts we consider each particle and each contact between a particle and the boundary within a 1° bin centered at angular position θ. For a central measure of the local contact force, we consider the locally averaged force per contact inline image:

  • display math

where, as above, fj(θjτ) is the normal force at an angle θj at time step τ, Nc,τ is the number of contacts between particles and drum bed at time step τ. For a simple measure of the variability of the contact forces within each degree bin we consider the standard deviation of the force per contact within each bin fσ and a measure of local maximum fmax.

  • display math

[66] As above, for the spatial variability of inline image and inline image, for the spatial variability of certain basic statistics of the forces associated with particle–bed contacts we consider each particle and each contact between a particle and the boundary within a 1° bin centered at angular position θ. For a central measure of the local contact force, we consider the locally averaged force per contact inline image:

  • display math

where, as above, fj(θjτ) is the normal force at an angle θj at time step τ, Nc,τ is the number of contacts between particles and drum bed at time step τ. For a simple measure of the variability of the contact forces within each degree bin we consider the standard deviation of the force per contact within each bin fσ and a measure of local maximum fmax.

  • display math

[67] We consider how certain measures of the force statistics vary for the whole system as the average particle size of the system dsys varies, where:

  • display math

Np is the number of particles in the system, and mi and di represent the particle mass and diameter of the ith particle, respectively. We also consider the measures of forces per contact over the whole system:

  • display math
  • display math

fj(τ) represents the jth particle–bed contact at any point on the wall at time τ (of which there are Nc(τ)).

Notation
A

area of the force sensor (used for calculations for the experimental and simulated drums)

A1

area corresponding to a region of the bed of the drum that spans the width of drum and is 1° long (A1 = (π/180) × D/2 × W)

d

particle size

dsys

system-wide mass-averaged particle size

inline image

local mass-averaged particle size

d50

median particle size, i.e., the particle size compared to which 50% of the particles in the grain size distribution (by mass) are smaller

d84

the particle size compared to which 84% of the particles in the grain size distribution (by mass) are smaller

deff

effective (reduced) particle size with explicit consideration of d of two objects in contact

dj

diameter of the particle j

De

representative particle size (e.g., of the front of a debris flows, as in Stock and Dietrich [2006])

D

diameter of drum

E

modulus of elasticity

Eeff

effective modulus of elasticity with explicit consideration of E and ν of two objects in contact

e

coefficient of restitution

f

the frequency of occurrence of debris flows in a particular location as defined in Stock and Dietrich [2006]

fj

the jth contact between particle and drum bed at one time step

inline image

local average contact force between particles and drum bed

fσ

local standard deviation of contact force between particles and drum bed

fmax

local maximum contact force between particles and drum bed

inline image

the average contact force between particles and drum bed for the entire system

fσ,sys

the standard deviation of contact force between particles and drum bed for the entire system

fmax,sys

the maximum contact force between particles and drum bed for the entire system

fc,max

the maximum contact force for a single contact between a particle and a boundary such as a drum bed as obtained from the computational simulations

fc,max,cm

the maximum contact force for a single contact between a particle and a boundary such as a drum bed as predicted from the contact model

inline image

theoretical value of inline image (equation (10))

Fn

magnitude of the contact force between two particles (or a particle and a wall) in the direction normal to the plane of contact (equation (4a))

Ft

magnitude of the contact force between two particles (or a particle and a wall) in the direction tangential to the plane of contact (equation (4b))

kn

stiffness coefficient relevant to compressive deformation of two particles (or a particle and a wall) in contact

kt

stiffness coefficient relevant to oblique deformation of two particles (or a particle and a wall) in contact

K0

scaling of excursion stresses with average inertial stress associated with dense granular flows over a bed as defined in Stock and Dietrich [2006]

K1

the relationship of rock resistance to incision rate of bedrock material associated with dense granular flows over a bed as defined in Stock and Dietrich [2006]

L

the length of the snout of a debris flow as defined in Stock and Dietrich [2006]

mj

mass of particle j

M

total mass of granular material

n

an empirically determined exponent used by Stock and Dietrich [2006] in their relationship between Bagnold's expression for collisional stress and the incision rate of bedrock due by dense granular flows

Nc

number of contacts between particles and drum bed used in a particular calculation

Np

total number of particles considered in a particular calculation

Nτ

total number of time steps used in a particular calculation

Nc,4

number of contacts between particles and the bed over a 4.3° region

p

normal stress or pressure at a boundary (such as the bed of a drum)

p4

normal stress or pressure at the bed considering contacts over a 4.3° region

inline image

temporally averaged normal stress or pressure at boundary

inline image

p4 locally averaged over a 1° region

inline image

dimensionless p4

inline image

the standard deviation of p4 considering a 1° region

inline image

lithostatic pressure on the drum bed

R

the resistance of bedock to erosion, a consolidation of several parameters contributing to this as detailed in Stock and Dietrich [2006]

tc

duration of a collision

vi

impact velocity (velocity of a particle at initial contact between particle and another object, such as the bed of a drum)

w

an empirically determined exponent used by Stock and Dietrich [2006] in their relationship between shear rate and the incision rate of bedrock by dense granular flows

W

width of drum

α

constant of proportionality relating the time-averaged particle–bed force during a particular particle–bed contact to the maximum force of that contact

inline image

shear rate

Γ

frequency of force data output

δn

overlap (deformation) relevant to the contact between two particles (or a particle and a wall) in the direction normal to the plane of contact (equation (4a))

δt

overlap (deformation) relevant to the contact between two particles (or a particle and a wall) in the direction tangential to the plane of contact (equation (4a))

δn,max

maximum overlap (deformation) between contacting particles (or a particle and a wall) in the normal direction as obtained from the computational simulations

δn,max,cm

maximum overlap (deformation) between contacting particles (or a particle and a wall) in the normal direction as predicted from the contact model

inline image

rate change of δn: d δn/dt

inline image

rate change of δt: d δt/dt

ηn

damping coefficient relevant to compressive deformation of two particles (or a particle and a wall) in contact

ηt

damping coefficient relevant to oblique deformation of two particles (or a particle and a wall) in contact

θ

angular location along the drum bed, measured counter-clockwise from vertically downward (e.g., Figure 2a)

θi

angular location of the ith particle–bed contact considered along the drum bed, measured counter-clockwise from vertically downward (e.g., Figure 2a)

θj

angular location of the jth particle–bed contact considered along the drum bed, measured counter-clockwise from vertically downward (e.g., Figure 2a)

θs

position of the center of the force sensor

Θ

the angle over which the sensor extends along the bed of the drum

λ

linear concentration as in Bagnold [1954]

μ

coefficient of friction

ν

Poisson's ratio

νs

solids volume fraction

νs,max

maximum solids volume fraction (0.74 for monosized spheres)

ρ

density of the material comprising the particles

σBag

collisional stress as defined by Bagnold [1954]

ϕ

angle of inclination of surface over which granular materials flow, e.g., as in equation (3)

Ω

speed of rotation of drum

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

[68] We gratefully acknowledge funding for this research provided by the National Center for Earth Surface Dynamics (NCED) an NSF Science and Technology Center funded under agreement EAR-0120914, and by NSF grants CBET-0932735 and CBET-0756480. We are also thankful for helpful discussions with Jia-Liang Le and Chris Paola at the University of Minnesota and Alex Densmore at Durham University.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Computational Simulations and Experimental Comparisons
  5. 3. Force and Stress Variability Associated With Binary Mixtures and Monosized Systems
  6. 4. Particle Size Dependence of the Boundary Forces: Particle-Scale Mechanics
  7. 5. Discussion
  8. 6. Summary and Future Work
  9. Appendix A:: Calculations for Particle Size and Force Statistics
  10. Acknowledgments
  11. References
  12. Supporting Information
FilenameFormatSizeDescription
jgrf917-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
jgrf917-sup-0002-t02.txtplain text document1KTab-delimited Table 2.
jgrf917-sup-0003-t03.txtplain text document0KTab-delimited Table 3.
jgrf917-sup-0004-t04.txtplain text document2KTab-delimited Table 4.
jgrf917-sup-0005-t05.txtplain text document0KTab-delimited Table 5.
jgrf917-sup-0006-t06.txtplain text document1KTab-delimited Table 6.
jgrf917-sup-0007-t07.txtplain text document0KTab-delimited Table 7.

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