Nonlocal constitutive relation for steady granular flow

Absorbing-state transitions in particulate systems under spatially varying driving

Non-equilibrium transitions into absorbing states are widespread, and amorphous materials under cyclic shear have emerged as useful model systems in which to study their properties. Recent work has focused on homogeneous driving in which the shear amplitude is uniform in space, despite most real world flows involving spatially inhomogeneous conditions that are known to produce qualitatively distinct phenomenology. Here we study the absorbing state transition under inhomogeneous driving using a modified random organization model. For smoothly varying driving the steady state results map onto the homogeneous absorbing state phase diagram, with the position of the boundary between absorbing and diffusive states being insensitive to the driving wavelength. The phenomenology is well-described by a one-dimensional generalized continuum model that we pose. For discontinuously varying driving the position of the absorbing phase boundary and the exponent characterising the fraction of active particles are altered relative to the homogeneous case.

Transient stress and fabric model for quasi-static granular flows in three dimensions

We present and validate a general three-dimensional continuum model for predicting the coupled fabric and stress transient response in 3D dense granular flows for the quasi-static regime. The model is inspired by isotropic and kinematic hardening theory, which is widely applied to plastic loading cycles in metals, which constitutes a connection between two different flowing materials through the same plastic modeling framework. The first part of the model consists of a differential evolution equation for the fabric tensor, which incorporates a new parameter called contact persistence to model the capacity of the fabric network to keep its contacts according to the relative direction of the shear-rate. The second part of the model is an expression for the shear stress comprised of a backstress, proportional to the fabric tensor, and a term proportional to the shear-rate direction. This shear stress decomposition was obtained from DEM data extracted within a 3D Couette cell during unsteady processes wherein the shear-rate direction rotates instantaneously with respect to the axis perpendicular to the walls of the cell. The results of the model are compared with DEM simulations for different changes in shear orientation, achieving a good agreement for the evolution of the fabric and deviatoric stress tensors. The model is shown to be compatible with the second law of thermodynamics, revealing that the origin of the backstress flow resistance in granular media is distinct from the cause of backstresses in metals; rather than arising from stored defect energy, it arises from the dependence of dilatancy on the alignment of the fabric and flow-rate.

Propagation of internal deformations in dense granular flows

This paper analyses how local deformation develops in dense granular flows. Local kinematic fields including particle velocity fluctuations, local strain, and non-affine deformation are measured in simulated homogeneous shear flows at different inertial numbers I, prescribing the shear strain rate  and the normal stress. Results evidence that these fields are unsteady and spatially correlated, as previously observed in a wide range of soft materials. They reveal a mechanism of propagation of local deformation by which seed events of particle rearrangements trigger further rearrangements in their vicinity. It appears that this mechanism is strongly dependent on the inertial number, with cluster size and propagation velocity increasing as a power law of I when I → 0. This mechanism can help understand and model some behaviours of granular flows such as non-locality and transient rheology.

Granular Temperature Controls Local Rheology of Vibrated Granular Flows

We use numerical simulations to demonstrate a local rheology for dense granular flows under shear and vibration. Granular temperature has been suggested as a rheological control but has been difficult to isolate. Here, we consider a granular assembly that is subjected to simple shear and harmonic vibration at the boundary, which provides a controlled source of granular temperature. We find that friction is reduced due to local velocity fluctuations of grains. All data obey a local rheology that relates the material friction coefficient, the granular temperature, and the dimensionless shear rate. We also observe that reduction in material friction due to granular temperature is associated with reduction in fabric anisotropy. We demonstrate that the temperature can be modeled by a heat equation with dissipation with appropriate boundary conditions, which provides complete closure of the system and allows a fully local continuum description of sheared, vibrated granular flows. This success suggests local rheology based on temperature combined with a diffusion equation for granular temperature may provide a general strategy to model dense granular flows.

The yielding of granular matter is marginally stable and critical

Amorphous materials undergo a transition from liquid-like to solid-like states through processes like rapid quenching or densification. Under external loads, they exhibit yielding, with minimal structural changes compared to crystals. However, these universal characteristics are rarely explored comprehensively in a single granular experiment due to the added complexity of inherent friction. The discernible differences between static configurations before and after yielding are largely unaddressed, and a comprehensive examination from both statistical physics and mechanical perspectives is lacking. To address these gaps, we conducted experiments using photoelastic disks, simultaneously tracking particles and measuring forces. Our findings reveal that the yielding transition demonstrates critical behavior from a statistical physics standpoint and marginal stability from a mechanical perspective, akin to the isotropic jamming transition. This criticality differs significantly from spinodal criticality in frictionless amorphous solids, highlighting unique characteristics of granular yielding. Furthermore, our analysis confirms the marginal stability of granular yielding by assessing the contact number and evaluating the balance between weak forces and small gaps. These factors serve as structural indicators for configurations before and after yielding. Our results not only contribute to advancing our understanding of the fundamental physics of granular materials but also bear significant implications for practical applications in various fields.

Interplay between wall slip and shear banding in a thixotropic yield stress fluid

We study the local dynamics of a thixotropic yield stress fluid that shows a pronounced non-monotonic flow curve. This mechanically unstable behavior is generally not observable from standard rheometry tests, resulting in a stress plateau that stems from the coexistence of a flowing band with an unyielded region below a critical shear rate c. Combining ultrasound velocimetry with standard rheometry, we discover an original shear-banding scenario in the decreasing branch of the flow curve of model paraffin gels, in which the velocity profile of the flowing band is set by the applied shear rate  instead of c. As a consequence, the material slips at the walls with a velocity that shows a non-trivial dependence on the applied shear rate. To capture our observations, we propose a differential version of the so-called lever rule, describing the extent of the flowing band and the evolution of wall slip with shear rate. This phenomenological model holds down to very low shear rates, at which the dimension of the flowing band becomes comparable to the size of the individual wax particles that constitute the gel microstructure, leading to cooperative effects. Our approach provides a framework where constraints imposed in the classical shear-banding scenario can be relaxed, with wall slip acting as an additional degree of freedom.

Scaling Description of Frictionless Dense Suspensions under Inhomogeneous Flow

Predicting the rheology of dense suspensions under inhomogeneous flow is crucial in many industrial and geophysical applications, yet the conventional "μ(J)" framework is limited to homogeneous conditions in which the shear rate and solids fraction are spatially invariant. To address this shortcoming, we use particle-based simulations of frictionless dense suspensions to derive new constitutive laws that unify the rheological response under both homogeneous and inhomogeneous conditions. By defining a new dimensionless number associated with particle velocity fluctuations and combining it with the viscous number, the macroscopic friction, and the solids fraction, we obtain scaling relations that collapse data from homogeneous and inhomogeneous simulations. The relations allow prediction of the steady state velocity, stress, and volume fraction fields using only knowledge of the applied driving force.

Finite size effects during the penetration of objects in a granular medium

In many industrial or geotechnical applications, objects move through a granular medium and an important issue is the prediction of the force that develops during the motion of the intruder. In this paper, we experimentally study the vertical penetration of intruders into granular media and analyze both the average force and the fluctuations during motion. We investigate configurations where the size of the intruder becomes close to a few grain sizes, a regime that has not been studied before. Finite size effects are observed, showing that both the mean force and the fluctuations significantly increase when decreasing the ratio of the intruder size to the particle size, and scaling laws are identified to characterize this effect. The role of a conical tip in front of the cylinder to facilitate the penetration is also studied, showing that it is more efficient when the aspect ratio between the intruder size and the grain size is low.

Down-hill creep of a granular material under expansion/contraction cycles

We investigate the down-hill creep of an inclined layer of granular material caused by quasi-static oscillatory variations of the size of the particles. The size variation is taken to be maximum at the surface and decreasing with depth, as it may be argued to occur in the case of a granular soil affected by atmospheric conditions. The material is modeled as an athermal two dimensional polydisperse system of soft disks under the action of gravity. The slope angle is below the angle of repose and therefore the system reaches an equilibrium configuration under static external conditions. However, under a protocol in which particles slowly change size in a quasistatic oscillatory way, the system is observed to creep down in a synchronized way with the oscillation. We measure the creep advance per cycle as a function of the slope angle and the degree of change in particle size. We also find that the creep rate is maximum at the surface and smoothly decreases with depth, as it is observed to occur in the field.

Microscopic origin of granular fluidity: An experimental investigation

Granular fluidity has been central to the development of nonlocal constitutive equations, which are necessary for characterizing nonlocal effects observed in experimental granular flow data. However, validation of these equations has been largely computational due to challenges in laboratory experiments. Specifically, the origin of the fluidity on a microscopic, single-particle level is still unproven. In this work, we present an experimental validation of a microscopic definition of granular fluidity, and show the importance of basal boundary conditions to the validity of the theory.

Force on a sphere suspended in flowing granulate

We investigate the force of flowing granular material on an obstacle. A sphere suspended in a discharging silo experiences both the weight of the overlaying layers and drag of the surrounding moving grains. In experiments with frictional hard glass beads, the force on the obstacle was practically flow-rate independent. In contrast, flow of nearly frictionless soft hydrogel spheres added drag to the gravitational force. The dependence of the total force on the obstacle diameter is qualitatively different for the two types of material: It grows quadratically with the obstacle diameter in the soft, low-friction material, while it grows much weaker, nearly linearly with the obstacle diameter, in the bed of glass spheres. In addition to the drag, the obstacle embedded in flowing low-friction soft particles experiences a total force from the top as if immersed in a hydrostatic pressure profile, but a much lower counterforce acting from below. In contrast, when embedded in frictional, hard particles, a strong pressure gradient forms near the upper obstacle surface.

Microrollers flow uphill as granular media

Pour sand into a container and only the grains near the top surface move. The collective motion associated with the translational and rotational energy of the grains in a thin flowing layer is quickly dissipated as friction through multibody interactions. Alternatively, consider what will happen to a bed of particles if one applies a torque to each individual particle. In this paper, we demonstrate an experimental system where torque is applied at the constituent level through a rotating magnetic field in a dense bed of microrollers. The net result is the grains roll uphill, forming a heap with a negative angle of repose. Two different regimes have been identified related to the degree of mobility or fluidisation of the particles in the bulk. Velocimetry of the near surface flowing layer reveals the collective motion of these responsive particles scales in a similar way to flowing bulk granular flows. A simple granular model that includes cohesion accurately predicts the apparent negative coefficient of friction. In contrast to the response of active or responsive particles that mimic thermodynamic principles, this system results in macroscopic collective behavior that has the kinematics of a purely dissipative granular system.

Rheology of a 2D granular film

We study experimentally the rheology of a macroscopic particle-laden soap film, designated as a "Granular Film", in the simple shear configuration. Macroscopic particles are dispersed in a soap film, while being large enough that they bridge both fluid interfaces. We simultaneously perform macroscopic rheological measurements with a classical rheometer and investigate interactions at the particle scale with a camera underneath the film. The determination of the velocity field of the grains reveals the presence of an inhomogeneous shear within the granular film. Trying to correlate both measurements unveils the non-locality of the rheology of the granular film: similar to what has been observed in a dry granular material, we find an highly-sheared zone close to the moving wall contrasting with a large quasistatic area. This behavior can be accounted for through extended kinetic theory and correlated with a transition in the dominant component of the stress.

Viscoelastic response of confined powder under large strain oscillations, characterized by its noise temperature

We report a study on granular matter with and without small additions of silicon oil, under low-frequency and large amplitude oscillatory shear strain under constant normal pressure, by running experiments with a rotational rheometer with a cup-and-plate geometry. We analysed the expansion with the Chebyshev polynomials of the orthogonal decomposition of stress-strain Lissajous-Bowditch loops. We found the onset of the strain amplitude for the yielding regime indicated a regime change from filament-like structures of grains to grain rearrangements for the dry granulate and from oscillations to the breaking and regeneration of liquid bridges for wet granulates. We have shown that this viscoelastic dynamics can be characterized by a noise temperature following Sollich et al. (Phys Rev Lett https://doi.org/10.1103/PhysRevLett.78.2020 , 1997). The analysis of the first harmonics of the Chebyshev expansion showed that the state of disorder of dry and wet granular matter in pre-yielding and yielding regimes involved ensembles of different inherent states; thus, each of them was governed by a different noise temperature. The higher-order harmonics of the Chebyshev expansion revealed a proportionality between the viscous nonlinearity and the variation in the elastic nonlinearity induced by the deformation, which shows the coupling between the elastic deformation and the viscous flow of mesoscopic-scale structures.

Controlling rheology via boundary conditions in dense granular flows

Boundary shape, particularly roughness, strongly controls the amount of wall slip in dense granular flows. In this paper, we aim to quantify and understand which aspects of a dense granular flow are controlled by the boundary conditions, and to incorporate these observations into a cooperative nonlocal model characterizing slow granular flows. To examine the influence of boundary properties, we perform experiments on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall; the latter is selected among 6 walls with various roughnesses, local concavity, and compliance. We find that we can successfully capture the full flow profile using a single set of empirically determined model parameters, with only the wall slip velocity set by direct observation. Through the use of photoelastic particles, we observe how the internal stresses fluctuate more for rougher boundaries, corresponding to a lower wall slip, and connect this observation to the propagation of nonlocal effects originating from the wall. Our measurements indicate a universal relationship between dimensionless fluidity and velocity.

Frictional Weakening of Vibrated Granular Flows

We computationally study the frictional properties of sheared granular media subjected to harmonic vibration applied at the boundary. Such vibrations are thought to play an important role in weakening flows, yet the independent effects of amplitude, frequency, and pressure on the process have remained unclear. Based on a dimensional analysis and DEM simulations, we show that, in addition to a previously proposed criterion for peak acceleration that leads to breaking of contacts, weakening requires the absolute amplitude squared of the displacement to be sufficiently large relative to the confining pressure. The analysis provides a basis for predicting flows subjected to arbitrary external vibration and demonstrates that a previously unrecognized second process that is dependent on dissipation contributes to shear weakening under vibrations.

Quantitative Understanding of the Onset of Dense Granular Flows

The question of when and how dense granular materials start to flow under stress, despite many industrial and geophysical applications, remains largely unresolved. We develop and test a simple equation for the onset of quasistatic flows of granular materials which is based on the frictional aging of the granular packing. The result is a nonmonotonic stress-strain relation which-akin to classical friction-is independent of the shear rate. This relation suffices to understand the quasistatic deformations of aging granular media, and its solid-to-liquid transition. Our results also elucidate the (flow) history dependence of the mechanical properties, and the sensitivity to the initial preparation of granular media.

Flow reversal triggers discontinuous shear thickening response across an erodible granular bed in a Couette-Poiseuille-like flow

Granular rheology is experimentally investigated in a vertical Couette-Poiseuille-like channel flow of photoelastic disks, where an erodible bed is sheared intermittently by an upward-moving shear band and a gravity-induced reverse flow. The shear band conforms to the existing nonlocal Eyring-like rheology but the bed exhibits discontinuous shear thickening from the Bagnold inertial regime near the band-bed interface to the Herschel-Bulkley plastic regime near the static wall. This newly discovered bed rheology is rate dependent and is associated with the fragility of the contact networks indicated by the statistics of local stress states inferred from the material photoelastic responses.

Quantifying cooperative flow of fat crystal dispersions

We quantify the cooperative flow behaviour of fat crystal dispersions (FCDs) upon varying crystallization conditions. The latter enabled altering the multiscale microstructure of the FCDs, from the nanometer-sized platelets, and the dispersed fractal aggregates, up to the strength of the mesoscopic weak-link network. To the goal of characterizing strongly-confined flow in these optically-opaque materials, we acquire high-resolution rheo-magnetic-resonance-imaging (rheo-MRI) velocimetry measurements using an in-house developed 500 μm gap Couette cell (CC). We introduce a numerical fitting method based on the fluidity model, which yields the cooperativity length, ξ, in the narrow-gap CC. FCDs with aggregates sizes smaller than the confinement size by an order of magnitude were found to exhibit cooperativity effects. The respective ξ values diverged at the yield stress, in agreement with the Kinetic Elasto-Plastic (KEP) theory. In contrast, the FCD with aggregates sizes in the order of the gap size did not exhibit any cooperativity effect: we attribute this result to the correspondingly decreased mobility of the aggregates. We foresee that our optimized rheo-MRI measurement and fitting analysis approach will propel further similar studies of flow of other multi-scale and optically-opaque materials.

Universal features of the stick-slip dynamics of an intruder moving through a confined granular medium

Experiments and simulations of an intruder dragged by a spring through a two-dimensional annulus of granular material exhibit robust force fluctuations. At low packing fractions (ϕ<ϕ_{0}), the intruder clears an open channel. Above ϕ_{0}, stick-slip dynamics develop, with an average energy release that is independent of the particle-particle and particle-base friction coefficients but does depend on the width W of the annulus and the diameter D of the intruder. A simple model predicts the dependence of ϕ_{0} on W and D, allowing for a data collapse for the average energy release as a function of ϕ/ϕ_{0}. These results pose challenges for theories of mechanical failure in amorphous materials.

The effect of grain shape and material on the nonlocal rheology of dense granular flows

Nonlocal rheologies allow for the modeling of granular flows from the creeping to intermediate flow regimes, using a small number of parameters. In this paper, we report on experiments testing how particle properties affect the model parameters used in the Kamrin & Koval cooperative nonlocal model, using particles of three different shapes (circles, ellipses, and pentagons) and three different materials, including one which allows for the measurement of stresses via photoelasticity. Our experiments are performed on a quasi-2D annular shear cell with a rotating inner wall and a fixed outer wall. Each type of particle is found to exhibit flows which are well-fit by nonlocal rheology, with each particle having a distinct triad of the local, nonlocal, and frictional parameters. While the local parameter b is always approximately unity, the nonlocal parameter A depends sensitively on both the particle shape and material. The critical stress ratio μ_s, above which Coulomb failure occurs, varies for particles with the same material but different shape, indicating that geometric friction can dominate over material friction.

Turbulent-like velocity fluctuations in two-dimensional granular materials subject to cyclic shear

We perform a systematic experimental study to investigate the velocity fluctuations in the two-dimensional granular matter of low and high friction coefficients subjected to cyclic shear of a range of shear amplitudes, whose velocity fields are strikingly turbulent-like with vortices of different scales. The scaling behaviors of both the transverse velocity power spectra E_T(k) ∝ k^-α_T and, more severely, the longitudinal velocity power spectra E_L(k) ∝ k^-α_L are affected by the prominent peak centered around k ≈ 2π of the inter-particle distance due to the static structure factor of the hard-particle nature in contrast to the real turbulence. To reduce the strong peak effect to the actual values of α_ν (the subscript 'ν' refers to either T or L), we subsequently analyze the second-order velocity structure functions of S(2)ν(r) in real space, which show the power-law scalings of S(2)ν(r) ∝ r^β_ν for both modes. From the values of β_ν, we deduce the corresponding α_ν from the scaling relations of α_ν = β_ν + 2. The deduced values of α_ν increase continuously with the shear amplitude γ_m, showing no signature of yielding transition, and are slightly larger than α_ν = 2.0 at the limit of γ_m → 0, which corresponds to the elastic limit of the system, for all γ_m. The inter-particle friction coefficients show no significant effect on the turbulent-like velocity fluctuations. Our findings suggest that the turbulent-like collective particle motions are governed by both the elasticity and plasticity in cyclically sheared granular materials.

Critical yielding rheology: from externally deformed glasses to active systems

We use extensive computer simulations to study the yielding transition under two different loading schemes: standard simple shear dynamics and self-propelled dense active systems. In the active systems, a yielding transition toward an out-of-equilibrium flowing state known as the liquid phase is observed when self-propulsion is increased. The range of self-propulsions in which this pure liquid regime exists appears to vanish upon approaching the so-called 'jamming point' at which the solidity of soft-sphere packings is lost. Such an 'active yielding' transition shares similarities with the generic yielding transition for shear flows. A Herschel-Bulkley law is observed along the liquid regime in both loading scenarios, with a clear difference in the critical scaling exponents between the two, suggesting the existence of different universality classes for the yielding transition under different driving conditions. In addition, we present the direct measurements of growing length and time scales for both driving scenarios. A comparison with theoretical predictions from the recent literature reveals poor agreement with our numerical results.

Strain localization in planar shear of granular media: the role of porosity and boundary conditions

Shear strain localization into shear bands is associated with velocity weakening instabilities and earthquakes. Here, we simulate steady-state plane-shear flow of numerical granular material (gouge), confined between parallel surfaces. Both constant shear stress and constant strain-rate boundary conditions are tested, and the two types of boundary conditions are found to yield distinct velocity profiles and friction laws. The inertial number, I, exerts the largest control on the layers' behavior, but additional dependencies of friction on normal stress and thickness of the layer are observed under constant stress boundary condition. We find that shear-band localization, which is present in the quasistatic regime ([Formula: see text]) in rate-controlled shear, is absent under stress-controlled loading. In the latter case, flow ceases when macroscopic friction coefficient approaches the quasistatic friction value. The inertial regime that occurs at higher inertial numbers ([Formula: see text]) is associated with distributed shear, and friction and porosity that increase with shear rate (rate-strengthening regime). The finding that shear under constant stress boundary condition produces the inertial, distributed shear but never quasistatic, localized deformation is rationalized based on low fluctuations of shear forces in granular contacts for stress-controlled loading. By examining porosity within and outside a shear band, we also provide a mechanical reason why the transition between quasistatic and inertial shear coincides with the transition between localized and distributed strain.

Emergence of structure in columns of grains and elastic loops

It is possible to build free-standing, load-bearing structures using only rocks and loops of elastic material. We investigate how these structures emerge, and find that the necessary maximum loop spacing (the critical spacing) is a function of the frictional properties of the grains and the elasticity of the confining material. We derive a model to understand both of these relationships, which depends on a simplification of the behavior of the grains at the edge of a structure. We find that higher friction leads to larger stable grain-grain and grain-loop contact angles resulting in a simple function for the frictional critical spacing, which depends linearly on friction to first order. On the other hand, a higher bending rigidity enables the loops to better contain the hydrostatic pressure of the grains, which we understand using a hydroelastic scale. These findings will illuminate the stabilization of dirt by plant roots, and potentially enable the construction of simple adhesion-less structures using only granular material and fiber.

Interplay between hysteresis and nonlocality during onset and arrest of flow in granular materials

The jamming transition in granular materials is well-known for exhibiting hysteresis, wherein the level of shear stress required to trigger flow is larger than that below which flow stops. Although such behavior is typically modeled as a simple non-monotonic flow rule, the rheology of granular materials is also nonlocal due to cooperativity at the grain scale, leading for instance to increased strengthening of the flow threshold as system size is reduced. We investigate how these two effects - hysteresis and nonlocality - couple with each other by incorporating non-monotonicity of the flow rule into the nonlocal granular fluidity (NGF) model, a nonlocal constitutive model for granular flows. By artificially tuning the strength of nonlocal diffusion, we demonstrate that both ingredients are key to explaining certain features of the hysteretic transition between flow and arrest. Finally, we assess the ability of the NGF model to quantitatively predict material behavior both around the transition and in the flowing regime, through stress-driven discrete element method (DEM) simulations of flow onset and arrest in various geometries. Along the way, we develop a new methodology to compare deterministic model predictions with the stochastic behavior exhibited by the DEM simulations around the jamming transition.

Efficacy of simple continuum models for diverse granular intrusions

Granular intrusion is commonly observed in natural and human-made settings. Unlike typical solids and fluids, granular media can simultaneously display fluid-like and solid-like characteristics in a variety of intrusion scenarios. This multi-phase behavior increases the difficulty of accurately modeling these and other yielding (or flowable) materials. Micro-scale modeling methods, such as DEM (Discrete Element Method), capture this behavior by modeling the media at the grain scale, but there is often interest in the macro-scale characterizations of such systems. We examine the efficacy of a macro-scale continuum approach in modeling and understanding the physics of various macroscopic phenomena in a variety of granular intrusion cases using two basic frictional yielding constitutive models. We compare predicted granular force response and material flow to experimental data in four quasi-2D intrusion cases: (1) depth-dependent force response in horizontal submerged-intruder motion; (2) separation-dependent drag variation in parallel-plate vertical-intrusion; (3) initial-density-dependent drag fluctuations in free surface plowing, and (4) flow zone development during vertical plate intrusions in under-compacted granular media. Our continuum modeling approach captures the flow process and drag forces while providing key meso- and macro-scopic insights. The modeling results are then compared to experimental data. Our study highlights how continuum modeling approaches provide an alternative for efficient modeling as well as a conceptual understanding of various granular intrusion phenomena.

Coupling non-local rheology and volume of fluid (VOF) method: a finite volume method (FVM) implementation

Additional to a behavior switching between solid-like and liquid-like, dense granular flows also present propagating grain size-dependent effects also called non-local effects. Such behaviors cannot be efficiently modeled by standard rheologies such as µ(I)-rheology but have to be dealt with advanced non-local models. Unfortunately, these models are still new and cannot be used easily nor be used for various configurations. We propose in this work a FVM implementation of the recently popular NGF model coupled with the VOF method in order to both make non-local modeling more accessible to everyone and suitable not only for single-phase flows but also for two-phase flows. The proposed implementation has the advantage to be extremely straightforward and to only require a supplementary stabilization loop compared to the theoretical equations. We then applied our new framework to both single and two-phase flows for validation.

Shear-induced diffusion in dense granular fluids

Granular materials are comprised of solid, athermal grains. Whilst immune to thermal motion, these grains move and diffuse when they undergo shear deformation. Here we introduce this process of shear-induced diffusion with a focus on dense flows. The goal is to present the established scaling laws for continuum diffusivity and to relate them to the micro-mechanisms of a granular random walk. We then suggest how this knowledge may help advance our understanding of granular rheology and diffusion in other soft-materials.

Inertial Force Transmission in Dense Granular Flows

Dense granular flows are well described by several continuum models; however, their internal dynamics remain elusive. This study explores the contact force distributions in simulated steady and homogenous shear flows. The results demonstrate the existence of high magnitude contact forces in faster flows with stiffer grains. A proposed physical mechanism explains this rate-dependent force transmission. This analysis establishes a relation between contact forces and grain velocities, providing an entry point to unify a range of continuum models derived from either contact forces or grain velocity.

Mechanical properties of 2D aggregates of oil droplets as model mono-crystals

We investigate the elastic and yielding properties of two dimensional defect-free mono-crystals made of highly monodisperse droplets. Crystals are compressed between two parallel boundaries of which one acts as a force sensor. As the available space between boundaries is reduced, the crystal goes through successive row-reduction transitions. For small compression forces, the crystal responds elastically until a critical force is reached and the assembly fractures in a single catastrophic global event. Correspondingly there is a peak in the force measurement associated with each row-reduction. The elastic properties of ideal mono-crystal samples are fully captured by a simple analytical model consisting of an assembly of individual capillary springs. The yielding properties of the crystal are captured with a minimal bond breaking model.

Viscosity of cohesive granular flows

Cohesive granular materials such as wet sand, snow, and powders can flow like a viscous liquid. However, the elementary mechanisms of momentum transport in such athermal particulate fluids are elusive. As a result, existing models for cohesive granular viscosity remain phenomenological and debated. Here we use discrete element simulations of plane shear flows to measure the viscosity of cohesive granular materials, while tuning the intensity of inter-particle adhesion. We establish that two adhesion-related, dimensionless numbers control their viscosity. These numbers compare the force and energy required to break a bond to the characteristic stress and kinetic energy in the flow. This progresses the commonly accepted view that only one dimensionless number could control the effect of adhesion. The resulting scaling law captures strong, non-Newtonian variations in viscosity, unifying several existing viscosity models. We then directly link these variations in viscosity to adhesion-induced modifications in the flow micro-structure and contact network. This analysis reveals the existence of two modes of momentum transport, involving either grain micro-acceleration or balanced contact forces, and shows that adhesion only affects the latter. This advances our understanding of rheological models for granular materials and other soft materials such as emulsions and suspensions, which may also involve inter-particle adhesive forces.

Analytical solutions for dense, inclined, granular flow over a rigid, bumpy base

We first phrase a boundary-value problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over in inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow. We compare these with those obtained in numerical solutions of the exact equations.

Analytical nonlocal model for shear localization in wall-bounded dense granular flow

This work employs a Landau-Ginzburg-type nonlocal rheology model to account for shear localization in a wall-bounded dense granular flow. The configuration is a 3D shear cell in which the bottom bumpy wall moves at a constant speed, while a load pressure is applied at the top bumpy wall, with flat but frictional lateral walls. At a fixed pressure, shear zones transit from the top to the bottom when increasing lateral wall friction coefficient. With a quasi-2D model simplification, asymptotic solutions for fluidization order parameters near the top and bottom boundaries are sought separately. Both solutions are the Airy function in terms of a depth coordinate scaled by a characteristic length which measures the width of the corresponding shear zone. The theoretical predictions for the shear zone widths against lateral wall friction coefficient and load pressure agree well with data extracted from particle-based simulation for the flow.

Flows of cohesive granular media

Cohesive granular media have broad applications in industries. However, our understanding of their flow behavior is still limited compared to dry granular media, although rich knowledge about their static and plastic properties has been gained. In this paper, we provide some insights into the flow behavior of cohesive granular media from our recent numerical studies using an inclined plane and a plane shear cell. We evidence that the cohesive nature of flows is significantly affected by material properties of the particles like stiffness and inelasticity in addition to the inter-particle adhesion and introduce the concept of “effective” adhesion, which incorporates the effects of these three variables. We propose constitutive relations involving dimensionless inertial number and “effective” cohesion number, based on the “effective” adhesion to describe the rheology. We also show that increasing adhesion increases the hysteresis in granular media, evidencing the existence of a prominent shear weakening branch in the friction coefficient versus inertial number rheological curve. Moreover, we reveal that this increasing hysteresis gives rise to the increasing occurrence of shear banding instability, pointing to the increasing possibility of jamming in cohesive granular media. Finally, we present a promising experimental approach to investigate the flow behavior of cohesive granular materials, based on a simple method of preparing a long time stable medium with a controlled adhesion between particles.

Microscopic Picture of Erosion and Sedimentation Processes in Dense Granular Flows

Gravity-driven flows of granular matter are involved in a wide variety of situations, ranging from industrial processes to geophysical phenomena, such as avalanches or landslides. These flows are characterized by the coexistence of solid and fluid phases, whose stability is directly related to the erosion and sedimentation occurring at the solid-fluid interface. To describe these mechanisms, we build a microscopic model involving friction, geometry, and a nonlocal cooperativity emerging from the propagation of collisions. This new picture enables us to obtain a detailed description of the exchanges between the fluid and solid phases. The model predicts a phase diagram including the limits of erosion and sedimentation, in quantitative agreement with experiments and discrete-element-method simulations.

Modeling the Microstructure and Stress in Dense Suspensions under Inhomogeneous Flow

Under inhomogeneous flow, dense suspensions exhibit behavior that violates the conventional homogeneous rheology. Specifically, one finds flowing regions with a macroscopic friction coefficient below the yielding criterion, and volume fraction above the jamming criterion. We demonstrate the underlying physics by incorporating shear rate fluctuations into a recently proposed tensor model for the microstructure and stress, and applying the model to an inhomogeneous flow problem. The model predictions agree qualitatively with particle-based simulations.

Microscopic Origin of Nonlocal Rheology in Dense Granular Materials

We study the microscopic origin of nonlocality in dense granular media. Discrete element simulations reveal that macroscopic shear results from a balance between microscopic elementary rearrangements occurring in opposite directions. The effective macroscopic fluidity of the material is controlled by these velocity fluctuations, which are responsible for nonlocal effects in quasistatic regions. We define a new micromechanically based unified constitutive law describing both quasistatic and inertial regimes, valid for different system configurations.

Particle motion on burned and vegetated hillslopes

Climate change is causing increasingly widespread, frequent, and intense wildfires across the western United States. Many geomorphic effects of wildfire are relatively well studied, yet sediment transport models remain unable to account for the rapid transport of sediment released from behind incinerated vegetation, which can fuel catastrophic debris flows. This oversight reflects the fundamental inability of local, continuum-based models to capture the long-distance particle motions characteristic of steeplands. Probabilistic, particle-based nonlocal models may address this deficiency, but empirical data are needed to constrain their representation of particle motion in real landscapes. Here we present data from field experiments validating a generalized Lomax model for particle travel distance distributions. The model parameters provide a physically intuitive mathematical framework for describing the transition from light- to heavy-tailed distributions along a continuum of behavior as particle size increases and slopes get steeper and/or smoother. We show that burned slopes are measurably smoother than vegetated slopes, leading to 1) lower rates of experimental particle disentrainment and 2) runaway motion that produces the heavy-tailed travel distances often associated with nonlocal transport. Our results reveal that surface roughness is a key control on steepland sediment transport, particularly after wildfire when smoother surfaces may result in the preferential delivery of coarse material to channel networks that initiate debris flows. By providing a first-order framework relating the statistics of particle motion to measurable surface characteristics, the Lomax model both advances the development of nonlocal sediment transport theory and reveals insights on hillslope transport mechanics.

Nonlocal continuum modeling of dense granular flow in a split-bottom cell with a vane-shaped intruder

Shear flow in one spatial region of a dense granular material-induced, for example, through the motion of a boundary-fluidizes the entire granular material. One consequence is that the yield condition vanishes throughout the granular material-even in regions that are very far from the "primary," boundary-driven shear flow. This phenomenon may be characterized through the mechanics of intruders embedded in the granular medium. When there is no primary flow, a critical load must be exceeded to move the intruder; however, in the presence of a primary flow, intruder motion occurs even when an arbitrarily small external load is applied to an intruder embedded in a region far from the sheared zone. In this paper, we apply the nonlocal granular fluidity (NGF) model-a continuum model that involves higher-order flow gradients-to simulate the specific case of dense flow in a split-bottom cell with a vane-shape intruder. Our simulations quantitatively capture the key features of the experimentally observed phenomena: (1) the vanishing of the yield condition, (2) an exponential-type relationship between the applied torque and the rotation rate, (3) the effect of the distance between the intruder and the primary flow zone, and (4) the direction-dependence of the torque/rotation-rate relation, in which the observed relation changes depending on whether the intruder is forced to rotate along with or counter to the primary flow. Importantly, this represents the first fully three-dimensional validation test for a nonlocal model for dense granular flow in general and for the NGF model in particular.

Integration through transients approach to the μ(I) rheology

This work generalizes the granular integration through transients formalism introduced by Kranz et al. [Phys. Rev. Lett. 121, 148002 (2018)10.1103/PhysRevLett.121.148002] to the determination of the pressure. We focus on the Bagnold regime and provide theoretical support to the empirical μ(I) rheology laws that have been successfully applied in many granular flow problems. In particular, we confirm that the interparticle friction is irrelevant in the regime where the μ(I) laws apply.

Power-Law Scaling in Granular Rheology across Flow Geometries

Based on discrete element method simulations, we propose a new form of the constitutive equation for granular flows independent of packing fraction. Rescaling the stress ratio μ by a power of dimensionless temperature Θ makes the data from a wide set of flow geometries collapse to a master curve depending only on the inertial number I. The basic power-law structure appears robust to varying particle properties (e.g., surface friction) in both 2D and 3D systems. We show how this rheology fits and extends frameworks such as kinetic theory and the nonlocal granular fluidity model.

Physical interpretation of shear-rate behaviour of soils and geotechnical solution to the coefficient of start-up friction with low inertial number

Shear experiments on soils have revealed the effects of shear rate, confining pressure, and grain size on the residual shear strength, but their nature is not well understood. To interpret these behaviours, a single dimensionless inertial number I from granular physics is introduced. A linear relationship between coefficient of residual friction μ_r and the natural logarithm of I was found by analysing geotechnical test data from other literature and helps to resolve the μ(aI)-rheology, which was proved invalid in the quasi-static regime. A method is proposed that introduces two three-dimensional yield criteria for soils to classify the frictional properties between grains in the quasi-static regime. The empirical coefficient of start-up friction is replaced by strength parameters of the soil. When compliant with the Mohr-Coulomb yield criterion, this coefficient is positively correlated with the internal angle of friction but negatively correlated with the Lode angle. Moreover from further analysis, the calculated strength is smallest in the pure tension state, largest in the pure compression state, and intermediate in the pure shearing state. This result is consistent with the properties of compressive endurable and tensive intolerable for natural geomaterials.

Mitigating memory effects during undulatory locomotion on hysteretic materials

While terrestrial locomotors often contend with permanently deformable substrates like sand, soil, and mud, principles of motion on such materials are lacking. We study the desert-specialist shovel-nosed snake traversing a model sand and find body inertia is negligible despite rapid transit and speed dependent granular reaction forces. New surface resistive force theory (RFT) calculation reveals how wave shape in these snakes minimizes material memory effects and optimizes escape performance given physiological power limitations. RFT explains the morphology and waveform-dependent performance of a diversity of non-sand-specialist snakes but overestimates the capability of those snakes which suffer high lateral slipping of the body. Robophysical experiments recapitulate aspects of these failure-prone snakes and elucidate how re-encountering previously deformed material hinders performance. This study reveals how memory effects stymied the locomotion of a diversity of snakes in our previous studies (Marvi et al., 2014) and indicates avenues to improve all-terrain robots.

Two mechanisms of momentum transfer in granular flows

This Rapid Communication highlights the physical processes at the origin of the constitutive law of dense granular flows. In simulated plane shear flows, we present a micro-mechanical expression for the phenomenological friction law μ(I). The expression highlights two distinct pathways for momentum transport-through either balanced contact forces or grain micro-acceleration. We show that these two rate-dependent processes control and explain the friction law. This understanding may help advance rheological models for granular materials and other soft materials such as emulsions and suspensions.

Differential equation for the flow rate of discharging silos based on energy balance

Since the early work of Hagen [G. H. L. Hagen, Ber. Verhandl. K. Preuss. Akad. Wiss. Berlin 17, 35 (1852)] and Beverloo et al. [W. Beverloo et al., Chem. Eng. Sci. 15, 260 (1961)CESCAC0009-250910.1016/0009-2509(61)85030-6], the flow rate of granular material discharging through a circular orifice from a silo has been described by means of dimensional analysis and experimental fits and explained through the free-fall arch model. Here, in contrast to the traditional approach, we derive a differential equation based on the energy balance of the system. This equation is consistent with the well-known Beverloo rule due to a compensation of energy terms. Moreover, this equation can be used to explore different conditions for silo discharges. In particular, we show how the effect of friction on the flow rate can be predicted. The theory is validated using discrete element method simulations.

A heavy intruder in a locally-shaken granular solid

We experimentally investigate the gravitational-driven motion of a heavy object inside a vertical 2D assembly of identical, plastic cylinders arranged in a regular, triangular lattice. The bottom of the assembly is in contact with a rough plate whose horizontal, sinusoidal motion induces the formation of shear bands in the granular solid, aligned with the edges of the lattice. The intruder sinks when the width of the shear band is as large as its size and halts once the regular configuration of the grains is recovered. The resulting vertical motion of the intruder is random and intermittent, as in disordered granular or colloidal systems near jamming, with alternate flows and blockades. We show, in analogy with earthquakes, that the relation between the size and the duration of the flowing events follows a power-law with an exponent larger than one, and that the statistics of their size is compatible with the Gutenberg-Richter law. We also show that the probability density function of times between flowing events is similar to the Omori law governing the distribution of aftershock sequences following large earthquakes. Finally, the analysis of the velocity fluctuations of the intruder points to a transition from a strong to a weak contact network in the ordered granular assembly, similar to the transition from jammed to fragile states in disordered systems.

Poiseuille Flow of Soft Polycrystals in 2D Rough Channels

Polycrystals are partially ordered solids where crystalline order extends over mesoscopic length scales, namely, the grain size. We study the Poisuielle flow of such materials in a rough channel. In general, similar to yield stress fluids, three distinct dynamical states, namely, flowing, stick-slip, and jammed can be observed, with a yield threshold dependent on channel width. Importantly, the interplay between the finite channel width, and the intrinsic ordering scale (the grain size) leads to a new type of spatiotemporal heterogeneity. In wide channels, although the average flow profile remains pluglike, at the underlying granular level, there is vigorous grain remodeling activity resulting from the velocity heterogeneity among the grains. As the channel width approaches typical grain size, the flowing polycrystalline state breaks up into a spatially heterogeneous mixture of flowing liquid like patches and chunks of nearly static grains. Despite these static grains, the average velocity still shows a parabolic profile, dominated by the moving liquidlike patches. However, the solid-liquid front moves at nearly constant speed in the opposite direction of the external drive.