This paper reports annular shear cell measurements granular flows with an eye towards experimentally confirming the flow regimes laid out in the elastic theory of granular flow. Tests were carried out on four different kinds of plastic spherical particles under both constant volume flows and constant applied stress flows. In particular, observations were made of the new regime in that model, the elastic-inertial regime, and the predicted transitions between the elastic-inertial and both the elastic-quasistatic and pure inertial regimes.

Early studies of granular flows focused on two distinctive flow regimes, the quasi-static and rapid flow regimes. Quasi-static flows theory focused on flows that have relatively small flow speed where the developed stresses are independent of flow rate. Analyses of quasi-static flows were based on metal plasticity theories that basically assumed the granular bulk behave like plastic solids that yield at a specified stress state such as that given by a Mohr-Coulomb criterion. Despite the discrete nature of the granular materials, in these models, the granular bulk was treated as continuous media. These models have been used successfully in modeling the small soil deformations caused by structural loading in civil engineering. They have been reasonably successful in modelling large deformation hopper flows (Brennen and Pearce, 1978; Johnson et al., 1990; and Brown and Richards, 1970). Jackson (1983) examined some early plasticity theory based continuum models used to describe granular flows. However, it was found that these models over predicted the dilation of granular materials (Drucker and Prager, 1952 and Jenike and Shield, 1959).

In contrast to the quasi-static flow regime, rapid-flow theory studies granular flows at high shear rates (and thus large deformations) where the stresses depend quadratically on the shear rate (Bagnold, 1954). Unlike the continuum based quasi-static theory, rapid-flow theory develops Navier-Stokes like continuum models from the point view of individual particles. It is assumed that any such flow contains a random field of particle motion analogous to the thermal motion of molecules and that the particles interact with each other through instantaneous collisions. Rapid flow theory (Haff, 1983; Savage and Jeffery, 1981; Jenkins and Savage, 1983; Lun et al., 1984; Jenkins and Richman, 1985; and Jenkins, 2006) was developed based on models that were initially derived for the kinetic theory of gases (Chapman and Cowling, 1964), and granular materials in the rapid flow regime are also referred to as “granular gases.” There are several problems with rapid flow theory that should be noted. First, although there had been many attempts (Goldhirsch et al., 2005; Kumaran, 2004; and Kumaran, 2006), friction still cannot be incorporated perfectly into models. Secondly, rapid flow models are based on the kinetic theory of gases, which assumes collisions between gas molecules are perfectly elastic and energy-conserving. When applied to granular flows, inelastic collisions and the resulting energy dissipation need to be accounted for, and large energy dissipation greatly complicates the models.

Quasi-static and rapid flow regimes are two extreme cases of granular flows. It is impossible to incorporate one theory into the other, due to their contradictory assumptions. In order to draw the entire flowmap for granular flows, it is necessary to identify boundaries for these different flow regimes and to fill in the gaps between.

Based on computer simulations, Campbell ( 2002 ; 2005 ) introduced a new way to bridge the gap between quasi-static and rapid flow regimes. The simulation used a standard spring-dashpot contact model (Cundall, 1974), and by incorporating contact stiffness k into the model was able to draw out the entire granular flowmap. The map divided granular flows into two global regimes, called the elastic and inertial regimes. An example is shown in Figure 1.

FIG. 1.

A flowmap showing constant volume granular flows of particles with surface friction μ = 0.5 divided into four regimes as a function of solid concentration ν and k*. Reproduced with permission from Campbell, C. S., “Granular shear flows at the elastic limit,” J. Fluid Mech. 465, 261–291 (2002). Copyright 2002 Cambridge University Press.

FIG. 1.

A flowmap showing constant volume granular flows of particles with surface friction μ = 0.5 divided into four regimes as a function of solid concentration ν and k*. Reproduced with permission from Campbell, C. S., “Granular shear flows at the elastic limit,” J. Fluid Mech. 465, 261–291 (2002). Copyright 2002 Cambridge University Press.

Close modal

In the elastic regimes, particles are in constant contact with their neighbors, and granular flows are dominated by force chains. Force chains are heavily loaded quasi-linear strings of particles that deform elastically at the contact points to support the applied load. The elastic nature of force chains is apparent in that they were discovered using photoelastic techniques (e.g., Drescher and de Josselin de Jong, 1972 and Howell et al., 1999). The elastic regime is further divided into two sub-regimes, the elastic-quasistatic regime and the elastic-inertial regime. At small shear rates, when the flow is constrained to a constant volume, the average force is determined by the degree of compression of the force chains. Force chains are created at rate proportional to the shear rate γ and break apart after a duration proportional to 1/γ. As a result, the averaged stresses are independent of shear rate. This is the elastic-quasistatic regime which encompasses the old quasi-static flows. At larger shear rate, where force chains and particle inertia have comparable effect, the averaged force within the flow will have the form a + , where a is the force from quasi-static behavior, and is the additional part generated by particle inertia. (Thus the stresses rise linearly proportional to the shear rate above their quasistatic values.) This is the elastic-inertial regime. In effect, there is no physical difference between the two subregimes, they are differentiated only by where the inertial effect becomes noticeable (i.e., if a, the flow is elastic-quasistatic and when becomes comparable to a, the flow enters elastic-inertial regime).

In the pure inertial regime, granular flows are free of force chains, and in simple shear flows, the stresses vary quadratically with the shear rate (Bagnold, 1954). Again, the broad inertial regime contains two sub-regimes which are the inertial-collisional regime and inertial-non-collisional regime. They are separated by whether the flow is dominated by binary collisions only (collisional) or filled with clusters of particles (non-collisional) (Campbell, 2002 ; 2011 ).

In the elastic regime, where forces are generated by the elastic deformation of particle contacts, it is natural to scale the stresses as τd/k, where τ is the stress, and d is the particle’s diameter. Since τf/d2 and f = , where f is the contact force, and δ is the particle deformation, the scaled stress can be understood as the ratio of elastic deformation to particle diameter in the direction of the stress, i.e., τd/kδ/d. (A nearly identical parameter is used to represent particle compressibility in Singh et al., 2015.) In the inertial regimes, force chains disappear, and flow follows Bagnold’s Law. As a result, stresses are scaled inertially as τ/ρd2γ2.

A dimensionless stiffness parameter k* = k/ρd3γ2 governs much of the transition between elastic and inertial behavior. There are several ways to interpret this parameter. Noting that k* = (τ/ρd2γ2)/(τd/k), it is the ratio of inertial stress scaling to the elastic stress scaling. Also, k*=k/ρd3γ2k/(mγ2)1/(Tbc2γ2) (where m is the mass of the particle, and Tbcm/k is the binary contact time) can be understood as the square ratio of flow time scale 1/γ governing the rate at which the flow pushes particles together, to the binary collision time Tbc, a scale of the time it takes the elastic forces to drive the particles apart. A third interpretation of this parameter is that if δI is the degree of particle deformation due to inertia (think of the compression of a particle in a collision), then δI/d ∼ (k*)−1/2.

Campbell (2002) studied the behavior of granular flow under controlled volume conditions. A flowmap delineating the four flow regimes is shown in Figure 1. One would assume, at fixed concentration, by increasing the shear rate, the flow would go through all four flow regimes, from quasi-static flow at low shear rate, eventually reaching the rapid flow regime at high shear rate. However, Figure 1 shows that is not the case under controlled volume conditions. At large concentrations ν, force chains exist even at static conditions, and changing the shear rate will not cause the force chains to disappear. Thus at large concentrations, transitions are only possible between elastic-quasistatic to elastic-inertial behaviors solely by changing the shear rate. A transition from elastic-quasistatic to pure inertial behavior can only be achieved by reducing the concentration to the point where force chains disappear. However, the opposite is not true. At smaller solid fractions with the flow in the inertial regime by increasing the shear rate (reducing k), the flow enters the elastic-inertial regime. This can be explained as that, at large shear rate, particles are brought together at a rate (∼γ) comparable to the rate that they are pushed apart (∼1/Tbc). So, force chains may form within the flow at concentrations too small for them to form under static conditions. Ironically, this indicates that rapid flow can only exist at low shear rates under constant volume constraints.

Most granular flows are not confined to be a certain volume, but rather under controlled stress conditions like chute flows, landslides, and hopper flows. Campbell (2005) examined granular flow under stress controlled situations from the elastic flow point of view. In controlled stress conditions, the material can expand or contract locally in order to support an applied load at a given shear rate. By increasing the shear rate from zero, a granular flow under constant applied stress will expand as the shear rate increases and thus experience the expected progression from elastic-quasistatic to elastic-inertial regimes and eventually end up in the pure inertial regimes.

One of the referees has asked us to include a discussion of the models popular over the last decade for which the primary parameter is the inertial number, I=γd/P/ρ, where P is the pressure (or 1/3 the trace of the stress tensor, GDR MiDi (Groupement De Recherche Milieux Divisés), 2004; Jop et al., 2006; and Pouliquen et al., 2006). I had not mentioned this in the original manuscript because inertial number models only apply to a portion of the flowmap and cannot account for the flow regime transitions that are the focus of this paper. Note that I is essentially the inverse square root of the inertial stress scaling. It is often incorrectly stated that these models are a new rheology while actually just a clever way of computing Bagnold rheology and is thus limited to the inertial regimes of the flowmap shown in Figure 1. It is arguable about how accurately these models can handle the inertial/quasistatic boundary because as I → 0. μ(I) → const., indicative of critical state flow. But critical state is a quasistatic and thus elastic flow and Campbell (2005) shows that there is an elastic-inertial regime between the purely inertial flow of I significantly greater than zero and the elastic flow of I = 0 (and physically, one expects that between a purely inertial regime where all forces are supported inertially and a regime where all forces are supported elastically, there must be a regime where elastic and inertial effects are of comparable importance). Furthermore, the inertial number models can at best handle the low-stress critical state and fail when particle compressibility becomes important (Campbell, 2005 and Singh et al., 2015). Considering Figure 1, it is clear that the inertial number models cannot handle any of the inertial/elastic-inertial boundaries as the physics of elastic flows and even the appropriate dimensional quantities do not exist in these models. Also, as the pressure P is a component of I, these models are limited to situations where P can be approximately known a priori. Otherwise the pressure becomes a function of the pressure. (As if to highlight this point, Bouzid et al. (2015) derive a constant volume rheology from an inertial number model and end up with a Bagnold rheology in which the inertial number I does not appear.) And this is why the models are inherently approximate, as one typically knows only one component of the normal stress, say from the applied force to a boundary or a hydrostatic overburden, which will be different from the actual pressure P due to normal stress difference effects. (See the discussion in Campbell, 2011.) More recently there has been an attempt to add nonlocality to the inertial number models (Bouzid et al., 2013; 2015 and Kamrin and Koval, 2012). It is somewhat questionable to complicate approximate models in these ways, especially when the complications do nothing to remove the approximations. But these papers do point out deficiencies in the inertial number models but it is not clear that their non-local solutions. In fact, in one particular concern of all three papers, it is clear they do not. Of the ten examples of non-local behavior in the three papers, seven involve explaining the curvature seen in velocity profiles near flow boundaries. But that has long been understood as the result of particle rotations and couple stresses (Campbell, 1993a), physics that are completely missing from even the extended inertial number models.

The experiments were performed in an annular shear cell such have been used in many previous studies (Savage and Sayed, 1984; Wang and Campbell, 1992; Daniels and Behringer, 2005; and Daniels and Behringer, 2006). Figure 2(a) shows a cross section of the annular shear cell used in these tests. The whole apparatus consists of two concentric circular aluminum disks mounted on the same vertical shaft. The lower disk assembly is mounted so that it rotates with the shaft. The annular trough on this bottom disk has an inner diameter of 351.4 mm and a width of 142 mm. The depth of the trough is 55.7 mm. The upper disk is mounted along the shaft through a combined radial and linear ball bearing assembly, which allows the upper disk move freely in both rotational and axial directions. But the rotation of the upper disk is restrained mechanically by a stop bar. The center shaft is driven by a 10 hp AC motor through a 10:1 worm gear speed reducer. The speed of the motor is adjusted by computer through a variable frequency drive. The apparatus is designed to perform both controlled volume and controlled stress experiments. In controlled volume cases, the position of the top plate is locked by a collar mounted on the center shaft.

FIG. 2.

(a) A schematic of the annular shear cell used in this study. (b) A three component force sensor is mounted on the top surface of the test channel.

FIG. 2.

(a) A schematic of the annular shear cell used in this study. (b) A three component force sensor is mounted on the top surface of the test channel.

Close modal

In controlled stress situations, the top plate is allowed to move freely in the axial direction. The mass of the top plate is 62 kg, which is balanced by a set of counterweights for controlled stress studies. By adjusting the counterweights, various level of normal stress can be applied. A pair of pneumatic cylinders is also used to apply additional force when needed.

Figure 2(b) shows a detail of the sensor area. A Kistler type 9317B three component (x,y,z) force sensor and a Type K thermal couple are connected to a sensor plate mounted at the center of the top surface of the shear cell. The area of the sensor plate is 48.57 cm2, over which stresses are averaged. A displacement transducer is used to measure the vertical position of the top plate thus determine the depth of the annular opening. An encoder is connected to the shaft measuring the rotational speed of the cell. The data are transmitted to a computer through a dual mode amplifier.

For each test, a small number of the test particles were glued onto the contact plates in order to create roughened boundary conditions (Campbell, 1993b). On the top contact surface in the test channel, each row of particles contains the same number of particles. At the center row of the ring, particles are spaced by two diameters from center to center. The distance between two adjacent particles in different rows is two particle diameters as well. The pattern follows such a rule and grows from center row until reaching edges of the ring. On the bottom surface, glued particles are arranged in similar ways. The only difference is that center-to-center spacing is four particle diameters in the center row and between adjacent particles in different rows. The use of different spacing was the result of computer simulation studies that showed that particles pressed against the channel walls (which rotate with the lower disc) and produced a stronger coupling between the particles and the rotating lower disc. The tighter spacing on the upper surface helped balance this by creating an effectively rougher surface thus increasing the coupling with the upper surface.

For fixed concentration tests, a typical experiment started with filling the trough in the lower disk by a certain mass of particles. Different masses resulted in different concentrations. Then the upper disk is lowered and locked to a preset height, which separates the top and bottom surfaces of the trough by six particle’s diameters. (Computer simulations showed that this depth resulted in roughly linear velocity profiles in the center of the channel.) Stresses were recorded at various shear rates and averaged for a long enough period of time. At small and moderate shear rates, a data point was usually averaged for 20 s, which would allow the shear cell to rotate at least 6 times at the smallest shear rate and more than 20 revolutions at a moderate shear rate. However, at a large shear rate, in order to prevent particles getting softened from frictionally generated heat, a data point was averaged for a shorter time period which was at least 10 s (at such large shear rates, the shear cell could rotate more than 15 times within 10 s).

Fixed stress experiments are similar to the fixed concentration ones. They start with filling the test cell with a certain mass of particles (corresponding to 60% at H = 6d). Then the top disk was lowered but not locked in the vertical direction, so the material could expand or contract in order to balance the applied load. At small stress levels, the load was applied by removing weight from the counterweight. At large stresses, the load was generated by adding extra weights on top of the upper disk or by using a pair of pneumatic cylinders to add further downward force on the disk. Stresses and displacement of the upper disk were recorded at a set of pre-selected values of k*. A threshold value for displacement was set that would stop the motor before the top cleared the wall of the trough and allowed particles to escape.

The choice of possible test material is very limited. Since the elastic nature of granular flow is the subject of present study, soft spherical particles are preferred. As the particles deform elastically, they must be softer than the material from which the shear cell is constructed, otherwise the elastic deformation of the shear cell walls will bias the measurements. This precludes using stiff materials like glass beads, in favor of softer materials like plastics. They also had to be relatively cheap as these experiments are hard on the particles, and the test materials had to be replaced often. Furthermore, to get to small values of k* = k/ρd3γ2, particles of large diameter d are preferred. Four different kinds of particles were tested. They were of 2 different sizes, 6 mm and 8 mm in diameter and three different densities. All the materials are pellets used in air-soft games. Pictures of test particles are shown in Figure 3, and their properties are listed in Table I. The test particles’ stiffness was measured in an INSTRON 5567 uniaxial compression test device with an insulated cage within which the temperature can be carefully controlled. Measurements were done at various temperatures. Figure 4 shows the compression test result at room temperature for all four kinds of test particles. For all these materials the contact force can be described by f = knδn, where f is the normal contact force; kn, n are two constants, and δ is the deformation at the contact. Ideally for elastic spheres, at a small deformation δ, the interparticle contact force in the normal direction can be described by a Hertzian Law, f = kHδ3/2, where kH is the Hertzian pre-coefficient. Notice that kH0.5, where k = ∂f/∂δ is the stiffness. It can be seen in Figure 4 that the 0.12 g 6 mm, 0.2 g 6 mm, and 0.27 g 8 mm particles followed Hertzian behavior. However, the force-deformation relation for 0.3 g 6 mm particles followed a different power law which was found to be f = k1.15δ1.15, and as a result k1.150.15. We will see later that this difference in materials elastic property will be reflected in the final results.

FIG. 3.

Particles used in the test. (a) 6 mm particles with singe particle mass 0.12 g, 0.2 g, and 0.3 g. (b) 8 mm particles with single particle mass 0.27 g.

FIG. 3.

Particles used in the test. (a) 6 mm particles with singe particle mass 0.12 g, 0.2 g, and 0.3 g. (b) 8 mm particles with single particle mass 0.27 g.

Close modal
TABLE I.

Properties of different particles used in experiment.

Mass of particle (g)Diameter (mm)Measured density (kg/m3)Static friction coefficientBehavior at contactMeasured coefficient kmManufacturer
0.12 1033 0.24 Hertzian kn = kH, n = 1.5 2.63 × 10+08 N/m1.5 CHIMEI, Taiwan 
0.2 1841 0.26 Hertzian kn = kH, n = 1.5 2.83 × 10+08 N/m1.5 CHIMEI, Taiwan 
0.3 2593 0.3 Power law with kn = k1.15, n = 1.15 1.02 × 10+07 N/m1.5 CHIMEI, Taiwan 
0.27 1034 0.24 Hertzian kn = kH, n = 1.5 1.85 × 10+08 N/m1.5 MARUSHIN, Japan 
Mass of particle (g)Diameter (mm)Measured density (kg/m3)Static friction coefficientBehavior at contactMeasured coefficient kmManufacturer
0.12 1033 0.24 Hertzian kn = kH, n = 1.5 2.63 × 10+08 N/m1.5 CHIMEI, Taiwan 
0.2 1841 0.26 Hertzian kn = kH, n = 1.5 2.83 × 10+08 N/m1.5 CHIMEI, Taiwan 
0.3 2593 0.3 Power law with kn = k1.15, n = 1.15 1.02 × 10+07 N/m1.5 CHIMEI, Taiwan 
0.27 1034 0.24 Hertzian kn = kH, n = 1.5 1.85 × 10+08 N/m1.5 MARUSHIN, Japan 
FIG. 4.

Compression test results for all four kinds of particles at room temperature. 0.12 g 6 mm, 0.2 g 6 mm, and 0.27 g 8 mm particles exhibited Hertzian behavior, f = kHδ3/2, while the test result for 0.3 g 6 mm particles followed a power law with an exponent of 1.15, f = k1.15δ1.15. kH has the unit of N/m1.5 and k1.15 has the unit of N / m1.15.

FIG. 4.

Compression test results for all four kinds of particles at room temperature. 0.12 g 6 mm, 0.2 g 6 mm, and 0.27 g 8 mm particles exhibited Hertzian behavior, f = kHδ3/2, while the test result for 0.3 g 6 mm particles followed a power law with an exponent of 1.15, f = k1.15δ1.15. kH has the unit of N/m1.5 and k1.15 has the unit of N / m1.15.

Close modal

From Table I, 0.12 g (6 mm) and 0.27 g (8 mm) particles have the same density, and it is reasonable to assume they are made from the same material. However, it is not clear why 8 mm particles have a smaller stiffness (according to Hertz theory, it should be larger). It was also noted during the experiments that 0.2 g (6 mm) and 0.3 (6 mm) particles have larger friction coefficients than the other two kinds of particles. Static friction coefficients were roughly measured for all four kinds of particles by gluing particles of each type to the corners of two identical square plates. One plate was placed on one another, making only four point-contacts at the tips of 4 pairs of particles. The plates were then tilted until sliding begins, and the static friction coefficient is computed as the tangent of the angle at which slide commenced. For 0.12 g (6 mm) and 0.27 g (8 mm) particles, μs ∼ 0.24; for 0.2 g (6 mm) μs ∼ 0.26 and 0.3 g (6 mm) particles, μs ∼ 0.3.

Several problems related to the material properties had been apparent in the experiment. Since tests were performed at large solid concentrations ν ≥ 48%. Heat was generated quickly inside the test cell at large shear rates, and the particles became soft with increasing temperature. This was alleviated by monitoring the temperature with the Type K thermocouple mounted on the sensor plate and stopping the experiment before the temperature rose enough to affect the elastic properties (roughly room temperature plus 10 °C for which the tests showed little change in the elastic properties). This meant that, particularly for tests at large concentrations and shear rates, that often only a single data point could be taken before the shear cell was stopped and allowed to cool for several hours. Another problem was the generation of dust. After some period, dust generated due to surface abrasion would accumulate and disturb test results. So before that occurred, the test material was discarded and replaced with fresh material.

Figure 5 shows unscaled raw normal stress τ data from the experiment plotted against shear rate γ on log-log plots. Slope lines were added on the graph in order to help identify the flow behavior. In the elastic inertial regime, the stresses are linearly dependent on the shear rate and should plot with a slope of 1. In the pure inertial regime, they should plot with a slope of 2.

FIG. 5.

Normal stress τ as a function of shear rate γ at various concentrations. (a) ν = 0.48, (b) ν = 0.54, (c) ν = 0.56, (d) ν = 0.58, (e) ν = 0.60.

FIG. 5.

Normal stress τ as a function of shear rate γ at various concentrations. (a) ν = 0.48, (b) ν = 0.54, (c) ν = 0.56, (d) ν = 0.58, (e) ν = 0.60.

Close modal

At small concentrations, as shown in Figure 5(a), flows of all four particles stayed in the inertial regime. By increasing the solid concentration, the 0.3 g particles enter the elastic-inertial regime first at ν = 0.54 in Figure 5(b) while the other particles stay in the inertial regime. Note that at low shear rate the 0.3 g particles show inertial behavior (slope of 2) but with increasing shear rate, the slope drops to one indicating elastic-inertial behavior. So this is an example of the elastic-inertial to pure inertial transition seen towards the left of Figure 1 at low concentrations. Note from Table I that the 0.3 g particles have the largest surface friction which makes them easier to form force chains and explains why they are the first to transition.

At ν = 0.56 in Figure 5(c), flows of 0.12 g particles and 0.27 g particles were in the inertial regime while flows of 0.2 g particles and 0.3 g particles were in the elastic-inertial regime. Again from Table I, these two particles have the largest surface friction and thus are expected to transition before the others. At ν = 0.58 in Figure 5(d), the 0.12 g particles and 0.27 g particles are in the elastic-inertial regime, and the 0.2 g particles show a slope slightly less than 1, indicating perhaps a transitional behavior between elastic-inertial and elastic-quasistatic regimes (a slope between 0 and 1), while the 0.3 g particles show a transition from elastic-quasistatic to elastic-inertial behavior seen at high concentrations in Figure 1. Finally, at ν = 0.6, all four kinds of particles exhibit quasi-static behavior at low shear rates and transition to elastic-inertial as the shear rate is increased.

Since the contact properties are different for different particles, we cannot simply scale the stress and shear rate as τd/k and k/ρd3γ2. From Figure 3, it can be seen that the contact force is non-linear and may be expressed generically as f = knδn or k=dfdδ=nknδn1 so that the stress can be written as τknδn/d2. For 0.3 g 6 mm particles, n = 1.15, kn = k1.15, while for the rest three kinds of particles n = 1.5, kn = kH. In order to take the nonlinearity at contacts into account, stress and shear rate are scaled as τd2n/kn1/n and (kn/ρd4−nγ2)1/n, where τd2n/kn1/nδ/d and (kn/ρd4−nγ2)1/nδI/d, following the prescription of Campbell (2002). Here δI is the particle deformation generated by particle inertia. Thus dimensional analysis dictates

τd2nkn=fν,knρd4nγ2,ε,μ,other material parameters,

where ε is the coefficient of restitution (or some other dimensionless measure of the energy dissipation at contacts), μ is the particle surface friction coefficient, and “other material parameters” can include a myriad of dimensionless possibilities such as a difference between dynamic and static friction, rolling friction, the parameters needed to model the velocity dependence of the coefficient of restitution (Raman, 1918), work modification of the particle surface (Mullier et al., 1991), etc.

Figure 6 shows the relation between elastically scaled normal stress τd2n/kn1/n and (kn/ρd4−nγ2)1/n at different solid concentrations on logarithmic plots. When scaled in this way, quasi-static behavior appeared as horizontal lines, elastic-inertial behavior corresponded to lines with a slope of −1/2, and lines with slope of −1 indicate inertial behavior.

FIG. 6.

Elastically scaled normal stress τd2n/kn1/n as a function of (kn/ρd4−nγ2)1/n at various concentrations. (a) ν = 0.48 (b) ν = 0.54 (c) ν = 0.56 (d) ν = 0.58 (e) ν = 0.60. Notice that for 0.3 g 6 mm particles, n = 1.15, kn = k1.15, while for the rest three kinds of particles n = 1.5, kn = kH.

FIG. 6.

Elastically scaled normal stress τd2n/kn1/n as a function of (kn/ρd4−nγ2)1/n at various concentrations. (a) ν = 0.48 (b) ν = 0.54 (c) ν = 0.56 (d) ν = 0.58 (e) ν = 0.60. Notice that for 0.3 g 6 mm particles, n = 1.15, kn = k1.15, while for the rest three kinds of particles n = 1.5, kn = kH.

Close modal

Note that one cannot expect these scalings to completely collapse the data for all four particle types. At a minimum, the differences in the surface friction prevent that. Furthermore, as pointed out in Campbell (2002), one should not expect collapse for different powers n, except in the elastic-quasistatic regime. Still it is clear that scaling the data in this fashion collapses the curves together.

As discussed in Campbell (2005), it is difficult to identify the regimes in controlled stress flows. This is because the applied stress pretty much controls the measured stresses so that the measured stress changes little with the shear rate. Campbell (2005) used several indirect means to determine the flow regime. One was that the stress ratio τxy/τyy is a constant in the elastic-quasistatic regime and rises in the elastic-inertial regime, making the knee in the diagram an indication of the regime transition. However, for reasons that are unclear the same behavior is not seen in the shear cell results (or for that matter in simulations of the shear cell). Thus we are limited in what can be obtained from controlled stress measurement to giving the change in concentration as a function of applied load and shear rate.

Those results for 0.12 g (6 mm) particles are shown with inertial scaling in Figure 7. Here τ is the applied stress. Note also that each line in the figure corresponds to different values of k*. These experiments were performed by applying a load to the cell, and then changing the shear rate until it corresponded to the desired value of k*, at which point the concentration was recorded. Then the top loading was changed, and the process repeated.

FIG. 7.

Controlled stress experiment data for 0.12 g (6 mm) particles. Solid concentration ν is plotted against inertially scaled normal stress τ/ρd2γ2.

FIG. 7.

Controlled stress experiment data for 0.12 g (6 mm) particles. Solid concentration ν is plotted against inertially scaled normal stress τ/ρd2γ2.

Close modal

Figure 7 shows that for much of the stress range, the solid fraction does not change but stays around ν ∼ 0.6. This behavior corresponds to a state called critical state which was observed in earlier studies (Hvorslev, 1937 and Schofield and Wroth, 1968). Note that at the higher stress levels, there is a slight increase in the concentration due to the compressibility of the particles—also a characteristic of the critical state. At lower stress/high-shear levels the concentration drops indicating a transition from the elastic-quasistatic regime to elastic-inertial regime and eventually to the full inertial regime occur. Similar behavior was observed in the simulations of Aharonov and Sparks (1999), and this is generally the behavior expected from inertial number models (GDR MiDi, 2004) except here τ is the applied stress which is different from the pressure P in the inertial number due to normal stress difference effects.

The same data for all 4 particle types are shown in Figure 8. For each type of particles, k* is varied between 7 × 10+06 and 4 × 10+08. As with the 0.12 g particles in Figure 7, the inertially scaled stress does a good job of collapsing the data. The largest deviation is seen for the 0.3 g particles that, perhaps because of its larger surface friction, deviate from the critical state at smaller dimensionless stress than the other three types. This is perhaps because their larger surface friction makes these particles more dissipative than the others.

FIG. 8.

Comparison of controlled stress data. Solid concentration ν as a function of inertially scaled normal stress τ/ρd2γ2 for different kinds of particles. For each type of particles, k* is varied between 7 × 10+06 and 4 × 10+08. The lines represent least squares fits to the data across the many k*.

FIG. 8.

Comparison of controlled stress data. Solid concentration ν as a function of inertially scaled normal stress τ/ρd2γ2 for different kinds of particles. For each type of particles, k* is varied between 7 × 10+06 and 4 × 10+08. The lines represent least squares fits to the data across the many k*.

Close modal

The goal of this work was to experimentally confirm, as much as possible, the elastic theory of granular flows (Campbell2002 ; 2005 ). And indeed we were able to observe the major flow regimes, elastic-quasistatic, elastic-inertial, and pure inertial, and the transitions between them. In particular, we were able to observe the transition between pure inertial and elastic-inertial at high shear rates and low concentrations when the shear forces particles together faster than the elastic forces pull them apart so that force chains form at low density. Also at large concentrations, we saw the expected transition between elastic-quasistatic and elastic-inertial. All of this was in concert with the flowmap drawn out by Campbell (2005) which is shown in Figure 1. Furthermore, the non-linear elastic scaling proposed in that paper did much to collapse the data even though the contact forces of the various materials exhibited different power law behaviors. (A complete collapse would not be expected due to differences in material properties such as the surface friction.) Attempts to study controlled stress flows were somewhat less successful as peculiarities of the shear cell experiment prohibited distinguishing between elastic-inertial and pure inertial flows. However, the results did show the expected critical state behavior and generally scaled well with an inertially scaled stress.

The work was supported by National Science Foundation under Grant No. CBET-0828514 for which the authors are extremely grateful. We would like to express special thanks to Yunpeng Zhang for his help on the uniaxial compression tests.

A referee has claimed that all these data fall into the paradigm of the inertial number models. Clearly the inertial data must. But the real issue is whether the μ(I) models can explain the elastic regimes for which they do not even contain the necessary dimensional parameters. So in response here are the elastic regime data, expressed in terms of the μ(I) model. There are two parts of the following data that are important. If the μ(I) models are correct then (obviously) μ should be a solely valued function of I. Secondly according to the theory, the solid concentration should also be a solely valued function of I. As the following data show, neither is true.

Figure 9 shows the data solely from the elastic regimes (elastic-quasistatic or elastic-inertial) as a function of I. As you can see. μ is independent of I and that the μ(I) models fail in the elastic regimes, even though the stresses are varying inertially in most of these cases.

FIG. 9.

Plots of the effective friction μ = τxy/τyy as a function of the inertial number I=γd/τyy/ρ. All of these data are taken solely from the elastic regimes (elastic-inertial or elastic-quasistatic). (a) ν = 0.54, (b) ν = 0.56, (c) ν = 0.58, (d) ν = 0.60.

FIG. 9.

Plots of the effective friction μ = τxy/τyy as a function of the inertial number I=γd/τyy/ρ. All of these data are taken solely from the elastic regimes (elastic-inertial or elastic-quasistatic). (a) ν = 0.54, (b) ν = 0.56, (c) ν = 0.58, (d) ν = 0.60.

Close modal

Furthermore the μ(I) models predict that the solid concentration ν is also a function of I, which means that each concentration should collapse to a single inertial number I. Yet I is seen to vary by an order of magnitude or more at constant ν, again indicating that ν is not a function of I. Note also that the ranges of I overlap, so that there are at least four concentrations over a range of 0.06 (a huge variation in the granular world) corresponding to the same value of I. Once again, the μ(I) models fail in the elastic zones.

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