The behaviour of dense assemblies of dry grains submitted to continuous shear deformation has been the subject of many experiments and discrete particle simulations. This paper is a collective work carried out among the French research group Groupement de Recherche Milieux Divises (GDR MiDi). It proceeds from the collection of results on steady uniform granular flows obtained by different groups in six different geometries both in experiments and numerical works. The goal is to achieve a coherent presentation of the relevant quantities to be measured i.e. flowing thresholds, kinematic profiles, effective friction, etc. First, a quantitative comparison between data coming from different experiments in the same geometry identifies the robust features in each case. Second, a transverse analysis of the data across the different configurations, allows us to identify the relevant dimensionless parameters, the different flow regimes and to propose simple interpretations. The present work, more than a simple juxtaposition of results, demonstrates the richness of granular flows and underlines the open problem of defining a single rheology.

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On dense granular flows.

A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes. However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate. One difficulty is that grains can behave like a solid (in a sand pile), a liquid (when poured from a silo) or a gas (when strongly agitated). For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows, and soil mechanics for slow plastic flows. However, the intermediate dense regime, where the granular material flows like a liquid, still lacks a unified view and has motivated many studies over the past decade. The main characteristics of granular liquids are: a yield criterion (a critical shear stress below which flow is not possible) and a complex dependence on shear rate when flowing. In this sense, granular matter shares similarities with classical visco-plastic fluids such as Bingham fluids. Here we propose a new constitutive relation for dense granular flows, inspired by this analogy and recent numerical and experimental work. We then test our three-dimensional (3D) model through experiments on granular flows on a pile between rough sidewalls, in which a complex 3D flow pattern develops. We show that, without any fitting parameter, the model gives quantitative predictions for the flow shape and velocity profiles. Our results support the idea that a simple visco-plastic approach can quantitatively capture granular flow properties, and could serve as a basic tool for modelling more complex flows in geophysical or industrial applications.

https://hal.archives-ouvertes.fr/hal-00118375/document

A constitutive law for dense granular flows

Discrete particle simulation of particulate systems: A review of major applications and findings

[1] Field observations of mature crustal faults suggest that slip in individual events occurs primarily within a thin shear zone, <1–5 mm, within a finely granulated, ultracataclastic fault core. Relevant weakening processes in large crustal events are therefore suggested to be thermal, and to involve the following: (1) thermal pressurization of pore fluid within and adjacent to the deforming fault core, which reduces the effective normal stress and hence also the shear strength for a given friction coefficient and (2) flash heating at highly stressed frictional microcontacts during rapid slip, which reduces the friction coefficient. (Macroscopic melting, or possibly gel formation in silica-rich lithologies, may become important too at large enough slip.) Theoretical modeling of mechanisms 1 and 2 is constrained with lab-determined hydrologic and poroelastic properties of fault core materials and lab friction studies at high slip rates. Predictions are that strength drop should often be nearly complete at large slip and that the onset of melting should be precluded over much (and, for small enough slip, all) of the seismogenic zone. A testable prediction is of the shear fracture energies that would be implied if actual earthquake ruptures were controlled by those thermal mechanisms. Seismic data have been compiled on the fracture energy of crustal events, including its variation with slip in an event. It is plausibly described by theoretical predictions based on the above mechanisms, within a considerable range of uncertainty of parameter choices, thus allowing the possibility that such thermal weakening prevails in the Earth.

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Heating and weakening of faults during earthquake slip

We study the plane shear flow of a dense assembly of dissipative disks using discrete simulation and prescribing the pressure and the shear rate. Those shear states are steady and uniform, and become intermittent in the quasistatic regime. In the limit of rigid grains, the shear state is determined by a single dimensionless number, called the inertial number I , which describes the ratio of inertial to pressure forces. Small values of I correspond to the quasistatic critical state of soil mechanics, while large values of I correspond to the fully collisional regime of kinetic theory. When I increases in the intermediate dense flow regime, we measure an approximately linear decrease of the solid fraction from the maximum packing value, and an approximately linear increase of the effective friction coefficient from the static internal friction value. From those dilatancy and friction laws, we deduce the constitutive law for dense granular flows, with a plastic Coulomb term and a viscous Bagnold term. The mechanical characteristics of the grains (restitution, friction, and elasticity) have a small influence in the dense flow regime. Finally, we show that the evolution of the relative velocity fluctuations and of the contact force anisotropy as a function of I provides a simple explanation of the friction law.

Rheophysics of dense granular materials: Discrete simulation of plane shear flows

We show that the rheological properties of dry granular materials, as well as foams and emulsions, are similar to typical thixotropic fluids: under a sufficiently strong shear the viscosity decreases in time, leading to a hysteresis in an up-and-down stress ramp. This leads to a viscosity bifurcation around a critical stress: for smaller stresses, the viscosity increases in time and the material eventually stops flowing, whereas for slightly larger stresses the viscosity decreases continuously with time and the flow accelerates. These results show that all jammed systems exhibit strong mechanical similarities around the transition between a ‘‘fluid’’ and a ‘‘solid’’ state, and that the transition between these states is discontinuous. This similarity is further emphasized by the fact that both a simple model for the dynamics of a grain on a sandpile @Quartier et al., Phys. Rev. E 62, 8299 ~2000!# and a simple model for the thixotropic behavior of ~colloidal! pastes @Coussot et al., Phys. Rev. Lett. 88, 175501 ~2002!# extrapolated to granular flows qualitatively predict this viscosity bifurcation.

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Viscosity bifurcation in granular materials, foams, and emulsions.

Using molecular dynamic simulations, we investigate the characteristics of dense flows of model cohesive grains. We describe their rheological behaviour and its origin at the scale of the grains and of their organization. Homogeneous plane shear flows give access to the constitutive law of cohesive grains which can be expressed by a simple friction law similar to the case of cohesionless grains, but intergranular cohesive forces strongly enhance the resistance to the shear. Then we show the consequence on flows down a slope: a plugged region develops at the free surface where the cohesion intensity is the strongest. Moreover, we measure various indicators of the microstructure within flows which evidence the aggregation of grains owing to cohesion and we analyse the properties of the contact network (force distributions and anisotropy). This provides new insights into the interplay between the local contact law, the microstructure and the macroscopic behavior of cohesive grains. Movies are available with the online version of the paper.

/pdf/dense-flows-of-cohesive-granular-materials-1m3kfkq7wm.pdf

Dense flows of cohesive granular materials

Using discrete simulations, we investigate the behavior of a model granular material within an annular shear cell. Specifically, two-dimensional assemblies of disks are placed between two circular walls, the inner one rotating with prescribed angular velocity, while the outer one may expand or shrink and maintains a constant radial pressure. Focusing on steady state flows, we delineate in parameter space the range of applicability of the recently introduced constitutive laws for sheared granular materials (based on the inertial number). We discuss the two origins of the stronger strain rates observed near the inner boundary, the vicinity of the wall and the heteregeneous stress field in a Couette cell. Above a certain velocity, an inertial region develops near the inner wall, to which the known constitutive laws apply, with suitable corrections due to wall slip, for small enough stress gradients. Away from the inner wall, slow, apparently unbounded creep takes place in the nominally solid material, although its density and shear to normal stress ratio are on the jammed side of the critical values. In addition to rheological characterizations, our simulations provide microscopic information on the contact network and velocity fluctuations that is potentially useful to assess theoretical approaches.

https://hal.archives-ouvertes.fr/hal-00353055/document

Annular shear of cohesionless granular materials: from the inertial to quasistatic regime.

The collisional flow of a slightly inelastic granular material down a rough inclined plane is usually described by kinetic theories. We present an experimental study aimed at analysing the assumptions and the quantitative predictions of such theories. A two-dimensional channel coupled to a model granular material and image analysis allow detailed and complete measurement of the kinematics and structure of the flows. We determine the range of inclination and particle flux for which the flow is stationary and uniform. The characteristic profiles of solid fraction, mean velocity and granular temperature are systematically measured. Both the true collisional and the dilute kinetic regimes are examined. We show that a quasi-hydrodynamic description of these regimes seems relevant, and that the pressure and the viscosity terms are in good qualitative agreement with the prediction of the kinetic theory. The profiles are well described by the kinetic theory near the top of the flow, at low solid fraction. Conversely there are large discrepancies near the rough plane, where the material is structured in layers.

François Chevoir

Papers

Rheophysics of dense granular materials: Discrete simulation of plane shear flows

Viscosity bifurcation in granular materials, foams, and emulsions.

Dense flows of cohesive granular materials

Annular shear of cohesionless granular materials: from the inertial to quasistatic regime.

Experimental study of collisional granular flows down an inclined plane