Friction of a slider on a granular layer: nonmonotonic thickness dependence and effect of boundary conditions

Roughness-Induced Friction on Liquid Foams

Complex liquids flow is known to be drastically affected by the roughness condition at the interfaces. We combined stresses measurements and observations of the flow during the motion of different rough surfaces in dry liquid foams. We visually show that three distinct friction regimes exist: slippage, stick-slip motion, and anchored soap films. Our stress measurements are validated for slippage and anchored regimes based on existing models, and we propose a leverage rule to describe the stresses during the stick-slip regime. We find that the occurrence of the stick-slip or anchored regimes is controlled by the roughness factor, defined as the ratio between the size of the surface asperities and the radius of curvature of the Plateau borders.

Effective drag of a rod in fluid-saturated granular beds

We measure the drag encountered by a vertically oriented rod moving across a sedimented granular bed immersed in a fluid under steady-state conditions. At low rod speeds, the presence of the fluid leads to a lower drag because of buoyancy, whereas a significantly higher drag is observed with increasing speeds. The drag as a function of the depth is observed to decrease from being quadratic at low speeds to appearing more linear at higher speeds. By scaling the drag with the average weight of the grains acting on the rod, we obtain the effective friction μ_{e} encountered over six orders of magnitude of speeds. While a constant μ_{e} is found when the grain size, rod depth, and fluid viscosity are varied at low speeds, a systematic increase is observed as the speed is increased. We analyze μ_{e} in terms of the inertial number I and viscous number J to understand the relative importance of inertia and viscous forces, respectively. For sufficiently high fluid viscosities, we find that the effect of varying the speed, depth, and viscosity can be described by the empirical function μ_{e}=μ_{o}+kJ^{n}, where μ_{o} is the effective friction measured in the quasistatic limit, and k and n are material constants. The drag is then analyzed in terms of the effective viscosity η_{e} and found to decrease systematically as a function of J. We further show that η_{e} as a function of J is directly proportional to the fluid viscosity and the μ_{e} encountered by the rod.

Geological implication of grain-size segregation in dense granular matter

To the current common belief, grain size segregation in granular matter requires sufficient porosity. Therefore, grain size segregation found in a natural fault gouge could imply elevated fluid pressure and the reduced normal stress on fault, possibly caused by the frictional heat during an earthquake. To clarify whether fluidization is essential to grain size segregation, we conduct numerical simulation on a simple model of fault gouge in a plane shear geometry under constant volume condition: the volume fraction is fixed at 0.6, at which the granular system possesses yield stress. We observe apparent grain size segregation at this volume fraction, meaning that grain size segregation alone does not imply fluidization of granular matter. We also show that segregation is driven by the nonlinear velocity profile, and that the gravity is not essential to segregation. The physical condition tested here may be relevant to earthquake faults: the normal stress of 1 MPa, the sliding velocity of 1 m s^-1, and the duration of 0.1 s.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Detection of particle motion using image processing with particular emphasis on rolling motion

Image-processing has been used in granular systems for detecting particle positions and motion near optically accessible surfaces like sediment flow and bedload transport. We review the image-processing techniques used for single and multiple particles. To enhance reliability in particle recognition, tools like Canny edge and Hough transform are intensively used. We show exemplarily how they can be applied to detect not only particle positions but also rotatory motion. The different steps are described in detail and the algorithm is applied to different examples, which are discussed in view of the obtained accuracy.

Unexpected friction behaviours due to capillary and adhesion effects

In this present work, we show that pouring water into the interface of SiC sliding against the same SiC material makes the friction coefficient far exceed that governing initial dry friction, and a subsequent load increase causes a dramatic abnormal friction force decrease. With the load further increasing to a critical value, friction force sharply increases again. The small amounts of water in the interface generates a large capillary force, the capillary bridges break to cause micro-separation of interface increase to allow bulk water to enter the friction interface, and a squeezing out of the water film to form a strong adhesion force between SiC-SiC materials, are considered to be responsible for the three stages, respectively.

Drag law for an intruder in granular sediments

We investigate the drag experienced by a spherical intruder moving through a medium consisting of granular hydrogels immersed in water as a function of its depth, size, and speed. The medium is observed to display a yield stress with a finite force required to move the intruder in the quasistatic regime at low speeds before rapidly increasing at high speeds. In order to understand the relevant time scales that determine drag, we estimate the inertial number I given by the ratio of the time scales required to rearrange grains due to the overburden pressure and imposed shear and the viscous number J given by the ratio of the time scales required to sediment grains in the interstitial fluid and imposed shear. We find that the effective friction μ_{e} encountered by the intruder can be parametrized by I=sqrt[ρ_{g}/P_{p}]v_{i}, where ρ_{g} is the density of the granular hydrogels, v_{i} is the intruder speed, and P_{p} is the overburden pressure due to the weight of the medium, over a wide range of I where the Stokes number St=I^{2}/J≫1. We then show that μ_{e} can be described by the function μ_{e}(I)=μ_{0}+αI^{β}, where μ_{0}, α, and β are constants that depend on the medium. This formula can be used to predict the drag experienced by an intruder of a different size at a different depth in the same medium as a function of its speed.

Increasing 'ease of sliding' also increases friction: when is a lubricant effective?

We investigate experimentally the effective Coulomb friction exerted by a granular medium on a shearing plate, varying the medium depth. The plate is driven by a spring connected to a motor turning at a constant speed and, depending on the system configuration, performs continuous sliding or stick and slip in different proportions. We introduce an order parameter which discriminates between the different regimes expressing the fraction of time spent in slipping. At low driving speed, starting from zero layers of interstitial granular material, the average friction coefficient decreases when a few layers are added, while the order parameter stays close to zero. By further increasing the granular depth, the friction undergoes a sudden increase but the order parameter does not change notably. At an intermediate driving speed, however, both the friction and the order parameter undergo a sudden increase, which for the order parameter amounts to several orders of magnitude, indicating that the plate is more braked but nevertheless keeps sliding more easily. For medium-high driving speeds, full sliding is obtained for only one layer of interstitial matter, where friction has a minimum, and is maintained for all increasing depths while friction increases. These observations show that the ease of slipping is not determined by friction alone, rather by the highly complex interplay between driving velocity, friction, and the depth of the medium.

Continuum modeling of secondary rheology in dense granular materials

Recent dense granular flow experiments have shown that shear deformation in one region of a granular medium fluidizes its entirety, including regions far from the sheared zone, effectively erasing the yield condition everywhere. This enables slow creep deformation to occur when an external force is applied to a probe in the nominally static regions of the material. The apparent change in rheology induced by far-away motion is termed the "secondary rheology," and a theoretical rationalization of this phenomenon is needed. Recently, a new nonlocal granular rheology was successfully used to predict steady granular flow fields, including grain-size-dependent shear-band widths in a wide variety of flow configurations. We show that the nonlocal fluidity model is also capable of capturing secondary rheology. Specifically, we explore creep of a circular intruder in a two-dimensional annular Couette cell and show that the model captures all salient features observed in experiments, including both the rate-independent nature of creep for sufficiently slow driving rates and the faster-than-linear increase in the creep speed with the force applied to the intruder.

Sliding friction on wet and dry sand

We show experimentally that the sliding friction on sand is greatly reduced by the addition of some-but not too much-water. The formation of capillary water bridges increases the shear modulus of the sand, which facilitates the sliding. Too much water, on the other hand, makes the capillary bridges coalesce, resulting in a decrease of the modulus; in this case, we observe that the friction coefficient increases again. Our results, therefore, show that the friction coefficient is directly related to the shear modulus; this has important repercussions for the transport of granular materials. In addition, the polydispersity of the sand is shown to also have a large effect on the friction coefficient.

A predictive, size-dependent continuum model for dense granular flows

Dense granular materials display a complicated set of flow properties, which differentiate them from ordinary fluids. Despite their ubiquity, no model has been developed that captures or predicts the complexities of granular flow, posing an obstacle in industrial and geophysical applications. Here we propose a 3D constitutive model for well-developed, dense granular flows aimed at filling this need. The key ingredient of the theory is a grain-size-dependent nonlocal rheology--inspired by efforts for emulsions--in which flow at a point is affected by the local stress as well as the flow in neighboring material. The microscopic physical basis for this approach borrows from recent principles in soft glassy rheology. The size-dependence is captured using a single material parameter, and the resulting model is able to quantitatively describe dense granular flows in an array of different geometries. Of particular importance, it passes the stringent test of capturing all aspects of the highly nontrivial flows observed in split-bottom cells--a geometry that has resisted modeling efforts for nearly a decade. A key benefit of the model is its simple-to-implement and highly predictive final form, as needed for many real-world applications.

From frictional to viscous behavior: three-dimensional imaging and rheology of gravitational suspensions

We probe the three-dimensional flow structure and rheology of gravitational (nondensity matched) suspensions for a range of driving rates in a split-bottom geometry. We establish that for sufficiently slow flows, the suspension flows as if it were a dry granular medium, and confirm recent theoretical modeling on the rheology of split-bottom flows. For faster driving, the flow behavior is shown to be consistent with the rheological behavior predicted by the recently developed "inertial number" approaches for suspension flows.

Spatial distribution functions of random packed granular spheres obtained by direct particle imaging

We measure the two-point density correlations and Voronoi cell distributions of cyclically sheared granular spheres obtained with a fluorescence technique and compare them with random packing of frictionless spheres. We find that the radial distribution function g(r) is captured by the Percus-Yevick equation for initial volume fraction ϕ=0.59. However, small but systematic deviations are observed because of the splitting of the second peak as ϕ is increased toward random close packing. The distribution of the Voronoi free volumes deviates from postulated Γ distributions, and the orientational order metric Q6 shows local order but no long range order. Overall, these measures show significant similarity of random packing of granular and frictionless spheres, but some systematic differences as well.

A non-local rheology for dense granular flows

A non-local theory is proposed to model dense granular flows. The idea is to describe the rearrangements occurring when a granular material is sheared as a self-activated process. A rearrangement at one position is triggered by the stress fluctuations induced by rearrangements elsewhere in the material. Within this framework, the constitutive law, which gives the relation between the shear rate and the stress distribution, is written as an integral over the entire flow. Taking into account the finite time of local rearrangements, the model is applicable from the quasi-static regime up to the inertial regime. We have checked the prediction of the model in two different configurations, namely granular flows down inclined planes and plane shear under gravity, and we show that many of the experimental observations are predicted within the self-activated model.

Effect of noise on solid-to-liquid transition in small granular systems under shear

The effect of noise on the solid-to-liquid transition of a dense granular assembly under planar shear is studied numerically using soft-particle molecular dynamics simulations in two dimensions. We focus on small systems in a thin planar Couette cell, examining the bistable region while increasing shear, with varying amounts of random noise, and determine statistics of the shear required for fluidization. In the absence of noise, the threshold value of the shear stress depends on the preparation of the system and has a broad distribution. However, adding force fluctuations both lowers the mean threshold value of the shear stress and decreases its variability. This behavior is interpreted as thermoactivated escape through a fluctuating barrier.

Physical test of a particle simulation model in a sheared granular system

We report a detailed comparison of a slow gravity-driven sheared granular flow with a discrete-element simulation performed in the same geometry. In the experiments, grains flow inside a silo with a rectangular cross section and are sheared by a rough boundary on one side and smooth boundaries on the other sides. Individual grain position and motion are measured using a particle index-matching imaging technique where a fluorescent dye is added to the interstitial liquid which has the same refractive index as the glass beads. The simulations use a Cundall-Strack contact model between the grains using contact parameters that have been used in many other previous studies and ignore the hydrodynamic effects of the interstitial liquid. Computations are performed to understand the effect of particle coefficient of friction, elasticity, contact model, and polydispersity on mean flow properties. We then perform a detailed comparison of the particle fluctuation properties as measured by the displacement probability distribution function and the mean square displacement. All in all, our study suggests a high level of quantitative agreement between the simulations and experiments.

Rolling friction on a granular medium

We present experimental results for the rolling of spheres on a granular bed. We use two sets of glass and steel spheres with varying diameters and a high-speed camera to follow the motion of the spheres. Despite the complex phenomena occurring during the rolling, the results show a friction coefficient nearly independent of the velocity (0.45-0.5 for glass and 0.6-0.65 for steel). It is found that for a given sphere density, the large spheres reach a longer distance, a result that may also help explain the rock sorting along natural stone accumulations at the foot of mountain slopes.

Transition from rolling to jamming in thin granular layers

We study the granular jamming transition for sheared layers of spherical beads ranging in thickness from 1 to 3 times the grain diameter d. As the layer thickness increases slightly above d, the measured friction jumps discontinuously from 0.02 to >0.1, marking the transition from rolling to jamming. Above a critical layer thickness for jamming, the effective granular pressure displays a power law increase with thickness. For thin layers, friction and P increases as the packing fraction decreases near the jamming transition, in contrast to expectations for bulk granular matter.

Friction and dilatancy in immersed granular matter

The friction of a sliding plate on a thin immersed granular layer obeys Amonton-Coulomb law. We bring to the fore a large set of experimental results which indicate that, over a few decades of values, the effective dynamical friction coefficient depends neither on the viscosity of the interstitial fluid nor on the size of beads in the sheared layer, which bears out the analogy with the solid-solid friction in a wide range of experimental parameters. We accurately determine the granular-layer dilatancy, which dependence on the grain size and slider velocity can be qualitatively accounted by considering the rheological behavior of the whole slurry. However, additional results, obtained after modification of the grain surface by a chemical treatment, demonstrate that the theoretical description of the flow properties of dense granular matter, even immersed, requires the detailed properties of the grain surface to be taken into account.

Velocity correlations in dense granular flows observed with internal imaging

We show that the velocity correlations in uniform dense granular flows inside a silo are similar to the hydrodynamic response of an elastic hard-sphere liquid. The measurements are made using a fluorescent refractive-index-matched interstitial fluid in a regime where the flow is dominated by grains in enduring contact and fluctuations scale with the distance traveled, independent of flow rate. The velocity autocorrelation function of the grains in the bulk shows a negative correlation at short time and slow oscillatory decay to zero similar to simple liquids. Weak spatial velocity correlations are observed over several grain diameters. The mean square displacements show an inflection point indicative of caging dynamics. The observed correlations are qualitatively different at the boundaries.

Stochastic flow rule for granular materials

There have been many attempts to derive continuum models for dense granular flow, but a general theory is still lacking. Here, we start with Mohr-Coulomb plasticity for quasi-two-dimensional granular materials to calculate (average) stresses and slip planes, but we propose a "stochastic flow rule" (SFR) to replace the principle of coaxiality in classical plasticity. The SFR takes into account two crucial features of granular materials-discreteness and randomness-via diffusing "spots" of local fluidization, which act as carriers of plasticity. We postulate that spots perform random walks biased along slip lines with a drift direction determined by the stress imbalance upon a local switch from static to dynamic friction. In the continuum limit (based on a Fokker-Planck equation for the spot concentration), this simple model is able to predict a variety of granular flow profiles in flat-bottom silos, annular Couette cells, flowing heaps, and plate-dragging experiments--with essentially no fitting parameters--although it is only expected to function where material is at incipient failure and slip lines are inadmissible. For special cases of admissible slip lines, such as plate dragging under a heavy load or flow down an inclined plane, we postulate a transition to rate-dependent Bagnold rheology, where flow occurs by sliding shear planes. With different yield criteria, the SFR provides a general framework for multiscale modeling of plasticity in amorphous materials, cycling between continuum limit-state stress calculations, mesoscale spot random walks, and microscopic particle relaxation.

Precursors to avalanches in a granular monolayer

We investigate the stability of a granular monolayer composed of spherical grains on an inclined plate. When the tilt angle alpha increases, some reorganizations are observed throughout the pile. The packing fraction rho of the packing evolves by successive jumps. Those discontinuous events precede the collapse of the pile at a critical angle alphac. The occurrence of precursors before avalanches is modeled by stop-and-go motions of blocks due to the competition between sliding friction and the Janssen effect [J. Durand, (Springer-Verlag, New York, 2000)].