Shear strength of wet granular materials: Macroscopic cohesion and effective stress : Discrete numerical simulations, confronted to experimental measurements.

Rheometric measurements on assemblies of wet polystyrene beads, in steady uniform quasistatic shear flow, for varying liquid content within the small saturation (pendular) range of isolated liquid bridges, are supplemented with a systematic study by discrete numerical simulations. The numerical results agree quantitatively with the experimental ones provided that the intergranular friction coefficient is set to the value [Formula: see text], identified from the behaviour of the dry material. Shear resistance and solid fraction [Formula: see text] are recorded as functions of the reduced pressure [Formula: see text], which, defined as [Formula: see text], compares stress [Formula: see text], applied in the velocity gradient direction, to the tensile strength [Formula: see text] of the capillary bridges between grains of diameter a, and characterizes cohesion effects. The simplest Mohr-Coulomb relation with [Formula: see text]-independent cohesion c applies as a good approximation for large enough [Formula: see text] (typically [Formula: see text]. Numerical simulations extend to different values of µ and, compared to experiments, to a wider range of [Formula: see text]. The assumption that capillary stresses act similarly to externally applied ones onto the dry granular contact network (effective stresses) leads to very good (although not exact) predictions of the shear strength, throughout the numerically investigated range [Formula: see text] and [Formula: see text]. Thus, the internal friction coefficient [Formula: see text] of the dry material still relates the contact force contribution to stresses, [Formula: see text], while the capillary force contribution to stresses, [Formula: see text], defines a generalized Mohr-Coulomb cohesion c, depending on [Formula: see text] in general. c relates to [Formula: see text] , coordination numbers and capillary force network anisotropy. c increases with liquid content through the pendular regime interval, to a larger extent, the smaller the friction coefficient. The simple approximation ignoring capillary shear stress [Formula: see text] (referred to as the Rumpf formula) leads to correct approximations for the larger saturation range within the pendular regime, but fails to capture the decrease of cohesion for smaller liquid contents.

Reply to "Comment on 'Flow of wet granular materials: A numerical study' ".

In his Comment on our paper [Phys. Rev. E 92, 022201 (2015)10.1103/PhysRevE.92.022201], Chareyre criticizes, as inaccurate, the simple approach we adopted to explain the strong enhancement of the quasistatic shear strength of the material caused by capillary cohesion. He also observes that a similar form of the "effective stress" approach, accounting for the capillary shear stress, which we neglected, results in a quantitatively correct prediction of this yield stress. We agree with these remarks, which we deem quite relevant and valuable. We nevertheless point out that the initial approximation, despite ∼25% errors on shear strength in the worst cases, provides a convenient estimate of the Mohr-Coulomb cohesion of the material, which is directly related to the coordination number. We argue that the effective stress assumption, despite its surprising success in the range of states explored in Khamseh et al. [Phys. Rev. E 92, 022201 (2015)10.1103/PhysRevE.92.022201], is bound to fail in strongly cohesion-dominated material states.

Flow of wet granular materials: A numerical study.

We simulate dense assemblies of frictional spherical grains in steady shear flow under controlled normal stress P in the presence of a small amount of an interstitial liquid, which gives rise to capillary menisci, assumed isolated (pendular regime), and attractive forces, which are hysteretic: Menisci form at contact, but do not break until grains are separated by a finite rupture distance. The system behavior depends on two dimensionless control parameters, inertial number I and reduced pressure P*=aP/(πΓ), comparing confining forces ∼a2P to meniscus tensile strength F0=πΓa, for grains of diameter a joined by menisci with surface tension Γ. We pay special attention to the quasistatic limit of slow flow and observe systematic, enduring strain localization in some of the cohesion-dominated (P*∼0.1) systems. Homogeneous steady flows are characterized by the dependence of internal friction coefficient µ* and solid fraction Φ on I and P*. We also record normal stress differences, fairly small but not negligible and increasing for decreasing P*. The system rheology is moderately sensitive to saturation within the pendular regime, but would be different in the absence of capillary hysteresis. Capillary forces have a significant effect on the macroscopic behavior of the system, up to P* values of several units, especially for longer force ranges associated with larger menisci. The concept of effective pressure may be used to predict an order of magnitude for the strong increase of µ* as P* decreases but such a crude approach is unable to account for the complex structural changes induced by capillary cohesion, with a significant decrease of Φ and different agglomeration states and anisotropic fabric. Likewise, the Mohr-Coulomb criterion for pressure-dependent critical states is, at best, an approximation valid within a restricted range of pressures, with P*≥1. At small enough P*, large clusters of interacting grains form in slow flows, in which liquid bonds survive shear strains of several units. This affects the anisotropies associated with different interactions and the shape of function µ*(I), which departs more slowly from its quasistatic limit than in cohesionless systems (possibly explaining the shear banding tendency).

Discrete simulation of dense flows of polyhedral grains down a rough inclined plane.

The influence of grain angularity on the properties of dense flows down a rough inclined plane are investigated. Three-dimensional numerical simulations using the nonsmooth contact dynamics method are carried out with both spherical (rounded) and polyhedral (angular) grain assemblies. Both sphere and polyhedra assemblies abide by the flow start and stop laws, although much higher tilt angle values are required to trigger polyhedral grain flow. In the dense permanent flow regime, both systems show similarities in the bulk of the material (away from the top free surface and the substrate), such as uniform values of the solid fraction, inertial number and coordination number, or linear dependency of the solid fraction and effective friction coefficient with the inertial number. However, discrepancies are also observed between spherical and polyhedral particle flows. A dead (or nearly arrested) zone appears in polyhedral grain flows close to the rough bottom surface, reflected by locally concave velocity profiles, locally larger coordination number and solid fraction values, smaller inertial number values. This dead zone disappears for smooth bottom surfaces. In addition, unlike sphere assemblies, polyhedral grain assemblies exhibit significant normal stress differences, which increase close to the substrate.

Shear flow of dense granular materials near smooth walls. I. Shear localization and constitutive laws in the boundary region.

We report on a numerical study of the shear flow of a simple two-dimensional model of a granular material under controlled normal stress between two parallel smooth frictional walls moving with opposite velocities ± V. Discrete simulations, which are carried out with the contact dynamics method in dense assemblies of disks, reveal that, unlike rough walls made of strands of particles, smooth ones can lead to shear strain localization in the boundary layer. Specifically, we observe, for decreasing V, first a fluidlike regime (A), in which the whole granular layer is sheared, with a homogeneous strain rate except near the walls, then (B) a symmetric velocity profile with a solid block in the middle and strain localized near the walls, and finally (C) a state with broken symmetry in which the shear rate is confined to one boundary layer, while the bulk of the material moves together with the opposite wall. Both transitions are independent of system size and occur for specific values of V. Transient times are discussed. We show that the first transition, between regimes A and B, can be deduced from constitutive laws identified for the bulk material and the boundary layer, while the second one could be associated with an instability in the behavior of the boundary layer. The boundary zone constitutive law, however, is observed to depend on the state of the bulk material nearby.

MRI investigation of granular interface rheology using a new cylinder shear apparatus.

The rheology of granular materials near an interface is investigated through proton magnetic resonance imaging. A new cylinder shear apparatus has been inserted in the magnetic resonance imaging device, which allows the control of the radial confining pressure exerted by the outer wall on the grains and the measurement of the torque on the inner shearing cylinder. A multi-layer velocimetry sequence has been developed for the simultaneous measurement of velocity profiles in different sample zones, while the measurement of the solid fraction profile is based on static imaging of the sample. This study describes the influence of the roughness of the shearing interface and of the transverse confining walls on the granular interface rheology.

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

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.

Rheophysics of dense granular materials: discrete simulation of plane shear flows.

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.