Identification of rolling resistance as a shape parameter in sheared granular media

Characterizing sliding and rolling contacts between single particles

Contacts between particles in dense, sheared suspensions are believed to underpin much of their rheology. Roughness and adhesion are known to constrain the relative motion of particles, and thus globally affect the shear response, but an experimental description of how they microscopically influence the transmission of forces and relative displacements within contacts is lacking. Here, we show that an innovative colloidal-probe atomic force microscopy technique allows the simultaneous measurement of normal and tangential forces exchanged between tailored surfaces and microparticles while tracking their relative sliding and rolling, unlocking the direct measurement of coefficients of rolling friction, as well as of sliding friction. We demonstrate that, in the presence of sufficient traction, particles spontaneously roll, reducing dissipation and promoting longer-lasting contacts. Conversely, when rolling is prevented, friction is greatly enhanced for rough and adhesive surfaces, while smooth particles coated by polymer brushes maintain well-lubricated contacts. We find that surface roughness induces rolling due to load-dependent asperity interlocking, leading to large off-axis particle rotations. In contrast, smooth, adhesive surfaces promote rolling along the principal axis of motion. Our results offer direct values of friction coefficients for numerical studies and an interpretation of the onset of discontinuous shear thickening based on them, opening up ways to tailor rheology via contact engineering.

Rheology of nonconvex granular flows based on particle rotational characteristics

Particle shape has a profound impact on the flow behaviors of granular materials, yet effectively incorporating the role of particle shape into granular rheology remains challenging. In this study, we employ three representative types of nonconvex particles generated through the multisphere approach and identify a consistent one-to-one relationship between the rescaled friction coefficient and the inertial number I across both inertial and quasistatic flow regimes. However, variations in particle shape cause notable deviations in rheological data compared to their spherical counterparts. Based on the observed dependence of rheological data on I for various nonconvex particles and their convergence at high I, we propose an inertial number I_{s} to effectively capture the impact of particle shape on flow states. The model parameters defining I_{s} are shown to be nearly independent of flow states and configurations, with physical interpretations related to particle rotational characteristics during shear. For practical application, we propose an empirical formula to capture the dependence of model parameters on particle geometrical shapes. The robustness of the proposed model is validated by predicting flow in an inclined flow configuration and applying it to additional nonconvex particles with more irregular and asymmetric features. This establishes a crucial foundation for extending the application of this generalized rheological model to other complex granular flows.

Modeling of the Particle Abrasion Process and a Discrete Element Method Study of Its Shape Effect

This study introduces a novel method for particle abrasion derived from fundamental natural phenomena and mechanical principles, allowing precise control over the degree of abrasion and more accurately mimicking natural processes. The method's validity is confirmed using a specific shape index. Through conventional triaxial tests, the mechanical behavior of granular aggregates with varying degrees of abrasion was analyzed. The findings indicate that increased particle abrasion leads to a decrease in the average coordination number and sliding amount, while the rotation amount increases. This suggests an inverse relationship between the degree of abrasion and the structural stability and interlocking of the particle aggregate. The fabric anisotropy of the system is mainly attributed to the anisotropy of the contact normal force, which decreases as particle abrasion increases. The partial stress ratio of the particle system is influenced by fabric anisotropy and remains independent of particle shape. Additionally, the internal friction angle may be overestimated in conventional triaxial tests.

A Discrete Elements Study of the Frictional Behavior of Fault Gouges

A series of discrete elements simulations is presented for the study of fault gouges' frictional response. The gouge is considered to have previously undergone ultra-cataclastic flow and long-time consolidation loading. We explore the effect of different particle characteristics such as size, polydispersity, and also shearing velocities on gouge's response under the conditions met in the seismogenic zone. Monte-Carlo analyses suggest that the local stick-slip events disappear when averaging over a large number of numerical samples. Moreover, the apparent material frictional response remains almost unaffected by the spatial randomness of particles' position and by the particle's size distribution. On the contrary, the mean particle size controls the formation and thickness of the observed shear bands, which appear after the peak friction is met. Furthermore, the apparent friction evolution fits well to an exponential decay law with slip, which involves a particle size dependent critical slip distance. For the studied conditions and depth, the shearing velocity is found to play a secondary role on the apparent frictional response of the gouge, which highlights the importance of analyses involving multiphysics for studying the rheology of fault gouges. Besides improving the understanding of the underlying physics of the problem, the above findings are also useful for deriving pertinent constitutive models in the case of modeling with continuum theories.

Stress propagation in locally loaded packings of disks and pentagons

The mechanical strength and flow of granular materials can depend strongly on the shapes of individual grains. We report quantitative results obtained from photoelasticimetry experiments on locally loaded, quasi-two-dimensional granular packings of either disks or pentagons exhibiting stick-slip dynamics. Packings of pentagons resist the intruder at significantly lower packing fractions than packings of disks, transmitting stresses from the intruder to the boundaries over a smaller spatial extent. Moreover, packings of pentagons feature significantly fewer back-bending force chains than packings of disks. Data obtained on the forward spatial extent of stresses and back-bending force chains collapse when the packing fraction is rescaled according to the packing fraction of steady state open channel formation, though data on intruder forces and dynamics do not collapse. We comment on the influence of system size on these findings and highlight connections with the dynamics of the disks and pentagons during slip events.

Effect of Particle Size Distributions and Shapes on the Failure Behavior of Dry Coke Aggregates

Carbon anodes participate in chemical reactions to reduce alumina in the Hall–Héroult process, of which coke aggregates make up a major part. The failure analysis of coke aggregates not only leads to a better understanding of the deformation mechanisms of anode paste under compressive loading but also can identify potential causes of structural defects in carbon anodes, such as horizontal cracks. The coke aggregates are composed of particles with different size distributions and shapes, which may strongly affect the failure behavior of the anode during compaction. In this paper, the effects of particle size distributions and shapes on the mechanical behavior and the failure of coke aggregates are investigated using the discrete element method modeling technique. The numerical results reveal that, although the mechanical behavior of coke mixtures is generally dependent on larger particles, the presence of fine particles in the coke aggregates reduces fluctuations in the stress–strain diagram. In addition, the rolling resistance model is employed as a parameter representing the effect of particle shape. It is shown that the rolling resistance model can be an alternative to the overlapped spheres model, which has a higher computational cost than the rolling resistance model. The second-order work criterion is used to evaluate the stability of the coke aggregates, the results of which indicate that the addition of fine particles as well as increasing the rolling resistance between the particles increases the stability range of the coke aggregates. Moreover, by using the analysis of micro-strain contour evaluations during the compaction process, it is shown that, both by adding fine particles to the coke mixture and by increasing the rolling resistance between the particles, the possibility of creating a compression band in the coke aggregates is reduced. Since the presence of the compaction bands in the anode paste creates an area prone to horizontal crack generations, the results of this study could lead to the production of carbon anodes with fewer structural defects.

Particle dynamics in two-dimensional point-loaded granular media composed of circular or pentagonal grains

Granular packings exhibit significant changes in rheological and structural properties when the rotational symmetry of spherical or circular particles is broken. Here, we report on experiments exploring the differences in dynamics of a grain-scale intruder driven through a packing of either disks or pentagons, where the presence of edges and vertices on grains introduces the possibility of rotational constraints at edge-edge contacts. We observe that the intruder’s stick-slip dynamics are comparable between the disk packing near the frictional jamming fraction and the pentagonal packing at significantly lower packing fractions. We connect this stark contrast in packing fraction with the average speed and rotation fields of grains during slip events, finding that rotation of pentagons is limited and the flow of pentagonal grains is largely confined in front of the intruder, whereas disks rotate more on average and circulate around the intruder to fill the open channel behind it. Our results indicate that grain-scale rotation constraints significantly modify collective motion of grains on mesoscopic scales and correspondingly enhance resistance to penetration of a local intruder.

The effect of particle shape on discharge and clogging

Granular flow is common across different fields from energy resource recovery and mineral processing to grain transport and traffic flow. Migrating particles may jam and form arches that span constrictions and hinder particle flow. Most studies have investigated the migration and clogging of spherical particles, however, natural particles are rarely spherical, but exhibit eccentricity, angularity and roughness. New experiments explore the discharge of cubes, 2D crosses, 3D crosses and spheres under dry conditions and during particle-laden fluid flow. Variables include orifice-to-particle size ratio and solidity. Cubes and 3D crosses are the most prone to clogging because of their ability to interlock or the development of face-to-face contacts that can resist torque and enhance bridging. Spheres arriving to the orifice must be correctly positioned to create stable bridges, while flat 2D crosses orient their longest axes in the direction of flowlines across the orifice and favor flow. Intermittent clogging causes kinetic retardation in particle-laden flow even in the absence of inertial effects; the gradual increase in the local particle solidity above the constriction enhances particle interactions and the probability of clogging. The discharge volume before clogging is a Poisson process for small orifice-to-particle size ratio; however, the clogging probability becomes history-dependent for non-spherical particles at large orifice-to-particle size ratio and high solidities, i.e., when particle–particle interactions and interlocking gain significance.

Grain-scale DEM study of open-ended pipe pile penetration in granular soils

Open-Ended Pipe Piles (OEPP) are particularly popular in offshore engineering. An important feature of the installation of these piles is the rate with which soil enters the pile from the bottom, and its interaction with the internal pile shaft. The response of OEPP crucially depends on the occurrence of soil plugging, which can make the behavior of an OEPP similar to a pile of solid cross section. Plugging is generally attributed to arching effects in the soil; therefore, understanding this phenomenon requires an investigation at the grain scale. This is precisely the objective of this study, where the Distinct Element Method (DEM) is used to study the installation of an Open-Ended Pipe Pile in a Virtual Calibration Chamber comprising 128000 grains, under constant horizontal stress. Despite the relatively small number of particles, this numerical model is found to be able to reproduce several aspects of the mechanisms actually observed in thefield. The results are compared to those obtained from actual experiments of miniature pile penetration tests. Stress and strainfields that develop in the soil inside and outside the pile provide interesting data and shed light on the mechanisms at play during OEPP installation, especially as for the influence of grains interlocking.

Shape or friction? Which of these characteristics drives the shear strength in granular systems?

The shape of the particles and local friction, separately, are known to strongly affect the macroscopic properties of an assembly of grains. But the combined effects of these two parameters still remain poorly described. By means of extensive two dimensional contact dynamics simulations, we perform a systematic analysis of the interplay between friction and shape on strength properties of granular systems. The shape of the particles is varied from disks to triangles, while the friction is varied from 0 to 0.7. We find that the macroscopic friction first increases with angularity, but it may decline (for low friction values), saturate (for intermediates friction values), or continue to increase (for large friction values) for the most angular shapes. In other words, the effect of the particle’s angularity on the shear strength depends on the level of sliding friction. In contrast, the effect of local friction on the shear strength does not depend on the specific properties of shape. The results presented here highlight the subtle coupling existing between shape and friction effects.

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.

Combined effects of contact friction and particle shape on strength properties and microstructure of sheared granular media

We present a systematic numerical investigation concerning the combined effects of sliding friction and particle shape (i.e., angularity) parameters on the shear strength and microstructure of granular packings. Sliding friction at contacts varied from 0 (frictionless particles) to 0.7, and the particles were irregular polygons with an increasing number of sides, ranging from triangles to disks. We find that the effect of local friction on shear strength follows the same trend for all shapes. Strength first increases with local friction and then saturates at a shape-dependent value. In contrast, the effect of angularity varies, depending on the level of sliding friction. For low friction values (i.e., under 0.3), the strength first increases with angularity and then declines for the most angular shapes. For high friction values, strength systematically increases with angularity. At the microscale, we focus on the connectivity and texture of the contact and force networks. In general terms, increasing local friction causes these networks to be less connected and more anisotropic. In contrast, increasing particle angularity may change the network topology in different directions, directly affecting the macroscopic shear strength. These analyses and data constitute a first step toward understanding the joint effect of local variables such as friction and grain shape on the macroscopic rheology of granular systems.

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.

Shear Thickening and Jamming of Dense Suspensions: The "Roll" of Friction

Particle-based simulations of discontinuous shear thickening (DST) and shear jamming (SJ) suspensions are used to study the role of stress-activated constraints, with an emphasis on resistance to gearlike rolling. Rolling friction decreases the volume fraction required for DST and SJ, in quantitative agreement with real-life suspensions with adhesive surface chemistries and "rough" particle shapes. It sets a distinct structure of the frictional force network compared to only sliding friction, and from a dynamical perspective leads to an increase in the velocity correlation length, in part responsible for the increased viscosity. The physics of rolling friction is thus a key element in achieving a comprehensive understanding of strongly shear-thickening materials.

Microstructural analysis of sheared polydisperse polyhedral grains

This article presents an analysis of the shear strength of numerical samples composed of polyhedra presenting a grain size dispersion. Previous numerical studies using, for instance, disks, polygons, and spheres, have consistently shown that microstructural properties linked to the fabric and force transmission allow granular media to exhibit a constant shear resistance although packing fraction can dramatically change as a broader grain-size distribution is considered. To have a complete picture of such behavior, we developed a set of numerical experiments in the frame of the discrete element method to test the shear strength of polydisperse samples composed of polyhedral grains. Although the contact networks and force transmission are quite more complex for such generalized grain shape, we can verify that the shear strength independence still holds up for 3D regular polyhedra. We make a particular focus upon the role of different contact types in the assemblies and their relative contributions to the granular connectivity and sample strength. The invariance of shear strength at the macroscopic scale results deeply linked to fine compensations at the microstructural level involving geometrical and force anisotropies of the assembly.

Additive rheology of complex granular flows

Granular flows are omnipresent in nature and industrial processes, but their rheological properties such as apparent friction and packing fraction are still elusive when inertial, cohesive and viscous interactions occur between particles in addition to frictional and elastic forces. Here we report on extensive particle dynamics simulations of such complex flows for a model granular system composed of perfectly rigid particles. We show that, when the apparent friction and packing fraction are normalized by their cohesion-dependent quasistatic values, they are governed by a single dimensionless number that, by virtue of stress additivity, accounts for all interactions. We also find that this dimensionless parameter, as a generalized inertial number, describes the texture variables such as the bond network connectivity and anisotropy. Encompassing various stress sources, this unified framework considerably simplifies and extends the modeling scope for granular dynamics, with potential applications to powder technology and natural flows.

Granular materials are abundant in nature, but we haven’t fully understood their rheological properties as complex interactions between particles are involved. Here, Vo et al. show that granular flows can be described by a generalized dimensionless number based on stress additivity.

Rheology of granular materials composed of crushable particles

We investigate sheared granular materials composed of crushable particles by means of contact dynamics simulations and the bonded-cell model for particle breakage. Each particle is paved by irregular cells interacting via cohesive forces. In each simulation, the ratio of the internal cohesion of particles to the confining pressure, the relative cohesion, is kept constant and the packing is subjected to biaxial shearing. The particles can break into two or more fragments when the internal cohesive forces are overcome by the action of compressive force chains between particles. The particle size distribution evolves during shear as the particles continue to break. We find that the breakage process is highly inhomogeneous both in the fragment sizes and their locations inside the packing. In particular, a number of large particles never break whereas a large number of particles are fully shattered. As a result, the packing keeps the memory of its initial particle size distribution, whereas a power-law distribution is observed for particles of intermediate size due to consecutive fragmentation events whereby the memory of the initial state is lost. Due to growing polydispersity, dense shear bands are formed inside the packings and the usual dilatant behavior is reduced or cancelled. Hence, the stress-strain curve no longer passes through a peak stress, and a progressive monotonic evolution towards a pseudo-steady state is observed instead. We find that the crushing rate is controlled by the confining pressure. We also show that the shear strength of the packing is well expressed in terms of contact anisotropies and force anisotropies. The force anisotropy increases while the contact orientation anisotropy declines for increasing internal cohesion of the particles. These two effects compensate each other so that the shear strength is nearly independent of the internal cohesion of particles.

Scaling laws for the mechanics of loose and cohesive granular materials based on Baxter's sticky hard spheres

We have conducted discrete element simulations (pfc3d) of very loose, cohesive, granular assemblies with initial configurations which are drawn from Baxter's sticky hard sphere (SHS) ensemble. The SHS model is employed as a promising auxiliary means to independently control the coordination number z_{c} of cohesive contacts and particle volume fraction ϕ of the initial states. We focus on discerning the role of z_{c} and ϕ for the elastic modulus, failure strength, and the plastic consolidation line under quasistatic, uniaxial compression. We find scaling behavior of the modulus and the strength, which both scale with the cohesive contact density ν_{c}=z_{c}ϕ of the initial state according to a power law. In contrast, the behavior of the plastic consolidation curve is shown to be independent of the initial conditions. Our results show the primary control of the initial contact density on the mechanics of cohesive granular materials for small deformations, which can be conveniently, but not exclusively explored within the SHS-based assembling procedure.

Inertial shear flow of assemblies of frictionless polygons: Rheology and microstructure

Motivated by the understanding of shape effects in granular materials, we numerically investigate the macroscopic and microstructural properties of anisotropic dense assemblies of frictionless polydisperse rigid pentagons in shear flow, and compare them with similar systems of disks. Once subjected to large cumulative shear strains their rheology and microstructure are investigated in uniform steady states, depending on inertial number I, which ranges from the quasistatic limit ([Formula: see text]) to 0.2. In the quasistatic limit both systems are devoid of Reynolds dilatancy, i.e., flow at their random close packing density. Both macroscopic friction angle [Formula: see text], an increasing function of I , and solid fraction [Formula: see text], a decreasing function of I, are larger with pentagons than with disks at small I, but the differences decline for larger I and, remarkably, nearly vanish for [Formula: see text]. Under growing I , the depletion of contact networks is considerably slower with pentagons, in which increasingly anisotropic, but still well-connected force-transmitting structures are maintained throughout the studied range. Whereas contact anisotropy and force anisotropy contribute nearly equally to the shear strength in disk assemblies, the latter effect dominates with pentagons at small I, while the former takes over for I of the order of 10^-2. The size of clusters of grains in side-to-side contact, typically comprising more than 10 pentagons in the quasistatic limit, very gradually decreases for growing I.

Shear strength and microstructure of polydisperse packings: The effect of size span and shape of particle size distribution

By means of extensive contact dynamics simulations, we analyzed the effect of particle size distribution (PSD) on the strength and microstructure of sheared granular materials composed of frictional disks. The PSDs are built by means of a normalized β function, which allows the systematic investigation of the effects of both, the size span (from almost monodisperse to highly polydisperse) and the shape of the PSD (from linear to pronouncedly curved). We show that the shear strength is independent of the size span, which substantiates previous results obtained for uniform distributions by packing fraction. Notably, the shear strength is also independent of the shape of the PSD, as shown previously for systems composed of frictionless disks. In contrast, the packing fraction increases with the size span, but decreases with more pronounced PSD curvature. At the microscale, we analyzed the connectivity and anisotropies of the contacts and forces networks. We show that the invariance of the shear strength with the PSD is due to a compensation mechanism which involves both geometrical sources of anisotropy. In particular, contact orientation anisotropy decreases with the size span and increases with PSD curvature, while the branch length anisotropy behaves inversely.

Contact angle entropy and macroscopic friction in noncohesive two-dimensional granular packings

We study the relationship between the granular contact angle distribution and local particle friction on the macroscopic friction and bulk modulus in noncohesive disk packings. Molecular dynamics in two dimensions are used to simulate uniaxial loading-unloading cycles imposed on the granular packings. While macroscopic Mohr friction depends on the granular pack geometric details, it reaches a stationary limit after a finite number of loading-unloading cycles that render well-defined values for bulk modulus, grain coordination, porosity, and friction. For random packings and for all polydispersities analyzed, we found that as interparticle friction increases, the bulk modulus for the limit cycle decreases linearly, while the mean coordination number is reduced and the porosity increased, also as approximately linear functions. On the other hand, the macroscopic Mohr friction increases in a monotonous trend with interparticle friction. The latter result is compared to a theoretical model that assumes the existence of sliding planes corresponding to definite Mohr-friction values. The simulation results for macroscopic friction are well described by the theoretical model that incorporates the local neighbor angle distribution that can be quantified through the contact angle entropy. As local friction is increased, the limit entropy of the neighbor angle distribution is reduced, thus introducing the geometric component to granular friction. Surprisingly, once the limit cycle is reached, the Mohr friction seems to be insensitive to polydispersity as has been recently reported.

Numerical study of one-dimensional compression of granular materials. I. Stress-strain behavior, microstructure, and irreversibility

The behavior of a model granular material, made of slightly polydisperse beads with Hertz-Mindlin elastic-frictional contacts, in oedometric compression (i.e., compression along one axis, with no lateral strain) is studied by grain-level numerical simulations. We systematically investigate the influence of the (idealized) packing process on the microstructure and stresses in the initial, weakly confined equilibrium state, and prepare both isotropic and anisotropic configurations differing in solid fraction Φ and coordination number z. Φ (ranging from maximally dense to moderately loose), z (which might vary independently of Φ in dense systems), fabric and force anisotropy parameters, and the ratio K_{0} of lateral stresses σ_{2}=σ_{3} to stress σ_{1} in the compression direction are monitored in oedometric compression in which σ_{1} varies by more than three orders of magnitude. K_{0} reflects the anisotropy of the assembling process and may remain nearly constant in further loading if the material is already oedometrically compressed (as a granular gas) in the preparation stage. Otherwise, it tends to decrease steadily over the investigated stress range. It is related to force and fabric anisotropy parameters by a simple formula. Elastic moduli, separately computed with an appropriate matrix method, may express the response to very small stress increments about the transversely isotropic well-equilibrated states along the loading path, although oedometric compression proves an essentially anelastic process, mainly due to friction mobilization, with large irreversible effects apparent upon unloading. While the evolution of axial strain ε_{1} and solid fraction Φ (or of the void ratio e=-1+1/Φ) with axial stress σ_{1} is very nearly reversible, especially in dense samples, z is observed to decrease (as previously observed in isotropic compression) after a compression cycle if its initial value was high. K_{0} relates to the evolution of internal variables and may exceed 1 in unloading. The considerably greater irreversibility of oedometric compression reported in sands, compared to our model systems, should signal contact plasticity or damage.

Effect of friction on dense suspension flows of hard particles

We use numerical simulations to study the effect of particle friction on suspension flows of non-Brownian hard particles. By systematically varying the microscopic friction coefficient μ_{p} and the viscous number J, we build a phase diagram that identifies three regimes of flow: frictionless, frictional sliding, and rolling. Using energy balance in flow, we predict relations between kinetic observables, confirmed by numerical simulations. For realistic friction coefficients and small viscous numbers (below J∼10^{-3}), we show that the dominating dissipative mechanism is sliding of frictional contacts, and we characterize asymptotic behaviors as jamming is approached. Outside this regime, our observations support the idea that flow belongs to the universality class of frictionless particles. We discuss recent experiments in the context of our phase diagram.

Stochastic disks that roll

We study a model of rolling particles subject to stochastic fluctuations, which may be relevant in systems of nano- or microscale particles where rolling is an approximation for strong static friction. We consider the simplest possible nontrivial system: a linear polymer of three disks constrained to remain in contact and immersed in an equilibrium heat bath so the internal angle of the polymer changes due to stochastic fluctuations. We compare two cases: one where the disks can slide relative to each other and the other where they are constrained to roll, like gears. Starting from the Langevin equations with arbitrary linear velocity constraints, we use formal homogenization theory to derive the overdamped equations that describe the process in configuration space only. The resulting dynamics have the formal structure of a Brownian motion on a Riemannian or sub-Riemannian manifold, depending on if the velocity constraints are holonomic or nonholonomic. We use this to compute the trimer's equilibrium distribution with and without the rolling constraints. Surprisingly, the two distributions are different. We suggest two possible interpretations of this result: either (i) dry friction (or other dissipative, nonequilibrium forces) changes basic thermodynamic quantities like the free energy of a system, a statement that could be tested experimentally, or (ii) as a lesson in modeling rolling or friction more generally as a velocity constraint when stochastic fluctuations are present. In the latter case, we speculate there could be a "roughness" entropy whose inclusion as an effective force could compensate the constraint and preserve classical Boltzmann statistics. Regardless of the interpretation, our calculation shows the word "rolling" must be used with care when stochastic fluctuations are present.

Constant Stress and Pressure Rheology of Colloidal Suspensions

We study the constant stress and pressure rheology of dense hard-sphere colloidal suspensions using Brownian dynamics simulation. Expressing the flow behavior in terms of the friction coefficient-the ratio of shear to normal stress-reveals a shear arrest point from the collapse of the rheological data in the non-Brownian limit. The flow curves agree quantitatively (when scaled) with the experiments of Boyer et al. [Phys. Rev. Lett. 107, 188301 (2011)]. Near suspension arrest, both the shear and the incremental normal viscosities display a universal power law divergence. This work shows the important role of jamming on the arrest of colloidal suspensions and illustrates the care needed when conducting and analyzing experiments and simulations near the flow-arrest transition.

Effects of shape and size polydispersity on strength properties of granular materials

By means of extensive contact dynamics simulations, we analyze the combined effects of polydispersity both in particle size and in particle shape, defined as the degree of shape irregularity, on the shear strength and microstructure of sheared granular materials composed of pentagonal particles. We find that the shear strength is independent of the size span, but unexpectedly, it declines with increasing shape polydispersity. At the same time, the solid fraction is an increasing function of both the size span and the shape polydispersity. Hence, the densest and loosest packings have the same shear strength. At the scale of the particles and their contacts, we analyze the connectivity of particles, force transmission, and friction mobilization as well as their anisotropies. We show that stronger forces are carried by larger particles and propped by an increasing number of small particles. The independence of shear strength with regard to size span is shown to be a consequence of contact network self-organization, with the falloff of contact anisotropy compensated by increasing force anisotropy.

Bonded-cell model for particle fracture

Particle degradation and fracture play an important role in natural granular flows and in many applications of granular materials. We analyze the fracture properties of two-dimensional disklike particles modeled as aggregates of rigid cells bonded along their sides by a cohesive Mohr-Coulomb law and simulated by the contact dynamics method. We show that the compressive strength scales with tensile strength between cells but depends also on the friction coefficient and a parameter describing cell shape distribution. The statistical scatter of compressive strength is well described by the Weibull distribution function with a shape parameter varying from 6 to 10 depending on cell shape distribution. We show that this distribution may be understood in terms of percolating critical intercellular contacts. We propose a random-walk model of critical contacts that leads to particle size dependence of the compressive strength in good agreement with our simulation data.

Internal friction and absence of dilatancy of packings of frictionless polygons

By means of numerical simulations, we show that assemblies of frictionless rigid pentagons in slow shear flow possess an internal friction coefficient (equal to 0.183±0.008 with our choice of moderately polydisperse grains) but no macroscopic dilatancy. In other words, despite side-side contacts tending to hinder relative particle rotations, the solid fraction under quasistatic shear coincides with that of isotropic random close packings of pentagonal particles. Properties of polygonal grains are thus similar to those of disks in that respect. We argue that continuous reshuffling of the force-bearing network leads to frequent collapsing events at the microscale, thereby causing the macroscopic dilatancy to vanish. Despite such rearrangements, the shear flow favors an anisotropic structure that is at the origin of the ability of the system to sustain shear stress.

Jamming transition in a two-dimensional open granular pile with rolling resistance

We present a molecular dynamics study of the jamming/unjamming transition in two-dimensional granular piles with open boundaries. The grains are modeled by viscoelastic forces, Coulomb friction and resistance to rolling. Two models for the rolling resistance interaction were assessed:  one considers a constant rolling friction coefficient, and the other one a strain dependent coefficient. The piles are grown on a finite size substrate and subsequently discharged through an orifice opened at the center of the substrate. Varying the orifice width and taking the final height of the pile after the discharge as the order parameter, one can devise a transition from a jammed regime (when the grain flux is always clogged by an arch) to a catastrophic regime, in which the pile is completely destroyed by an avalanche as large as the system size. A finite size analysis shows that there is a finite orifice width associated with the threshold for the unjamming transition, no matter the model used for the microscopic interactions. As expected, the value of this threshold width increases when rolling resistance is considered, and it depends on the model used for the rolling friction.Received: 14 June 2014,  Accepted: 7 October 2014; Reviewed by: R. Arévalo, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore; Edited by: L. A. Pugnaloni; DOI: http://dx.doi.org/10.4279/PIP.060007Cite as: C F M. Magalhaes, A P F Atman, G Combe, J G Moreira, Papers in Physics 6, 060007 (2014) 

Effect of size polydispersity versus particle shape in dense granular media

We present a detailed analysis of the morphology of granular systems composed of frictionless pentagonal particles by varying systematically both the size span and particle shape irregularity, which represent two polydispersity parameters of the system. The microstructure is characterized in terms of various statistical descriptors such as global and local packing fractions, radial distribution functions, coordination number, and fraction of floating particles. We find that the packing fraction increases with the two parameters of polydispersity, but the effect of shape polydispersity for all the investigated structural properties is significant only at low size polydispersity where the positional and/or orientational ordering of the particles prevail. We focus in more detail on the class of side/side contacts, which is the interesting feature of our system as compared to a packing of disks. We show that the proportion of such contacts has weak dependence on the polydispersity parameters. The side- side contacts do not percolate but they define clusters of increasing size as a function of size polydispersity and decreasing size as a function of shape polydispersity. The clusters have anisotropic shapes but with a decreasing aspect ratio as polydispersity increases. This feature is argued to be a consequence of strong force chains (forces above the mean), which are mainly captured by side-side contacts. Finally, the force transmission is intrinsically multiscale, with a mean force increasing linearly with particle size.

Mesoscale and macroscale kinetic energy fluxes from granular fabric evolution

Recent advances in high-resolution measurements means it is now possible to identify and track the local "fabric" or contact topology of individual grains in a deforming sand throughout loading history. These provide compelling impetus to the development of methods for inferring changes in the contact forces and energies at multiple spatiotemporal scales, using information on grain contacts alone. Here we develop a surrogate measure of the fluctuating kinetic energy based on changes in the local contact topology of individual grains. We demonstrate the method for dense granular materials under quasistatic biaxial shear. In these systems, the initially stable and solidlike response eventually gives way to liquidlike behavior and global failure. This crossover in mechanical behavior, akin to a phase transition, is marked by bursts of kinetic energy and frictional dissipation. Mechanisms underlying this release of energy include the buckling of major load-bearing structures known as force chains. These columns of grains represent major repositories for stored strain energy. Stored energy initially accumulates at all of the contacts along the force chain, but is released collectively when the chain overloads and buckles. The exact quantification of the buildup and release of energy in force chains, and the manner in which force chain buckling propagates in the sample (i.e., diffuse and systemwide versus localized into shear bands), requires detailed knowledge of contact forces. To date, however, the forces at grain contacts continue to elude measurement in natural granular materials like sand. Here, using data from computer simulations, we show that a proxy for the fluctuating kinetic energy in dense granular materials can be suitably constructed solely from the evolving properties of the grain's local contact topology. Our approach directly relates the evolution of fabric to energy flux and makes possible research into the propagation of failure from measurements of grain contacts in real granular materials.

Packings of irregular polyhedral particles: strength, structure, and effects of angularity

We present a systematic numerical investigation of the shear strength and structure of granular packings composed of irregular polyhedral particles. The angularity of the particles is varied by increasing the number of faces from 8 (octahedronlike shape) to 596. We find that the shear strength increases with angularity up to a maximum value and saturates as the particles become more angular (below 46 faces). At the same time, the packing fraction increases to a peak value but declines for more angular particles. We analyze the connectivity and anisotropy of the microstructure by considering both the contacts and branch vectors joining particle centers. The increase of the shear strength with angularity is shown to be due to a net increase of the fabric and force anisotropies but at higher particle angularity a rapid falloff of the fabric anisotropy is compensated by an increase of force anisotropy, leading thus to the saturation of shear strength.

Cohesive granular materials composed of nonconvex particles

The macroscopic cohesion of granular materials made up of sticky particles depends on the particle shapes. We address this issue by performing contact dynamics simulations of 2D packings of nonconvex aggregates. We find that the macroscopic cohesion is strongly dependent on the strain and stress inhomogeneities developing inside the material. The largest cohesion is obtained for nearly homogeneous deformation at the beginning of unconfined axial compression and it evolves linearly with nonconvexity. Interestingly, the aggregates in a sheared packing tend to form more contacts with fewer neighboring aggregates as the degree of nonconvexity increases. We also find that shearing leads either to an isotropic distribution of tensile contacts or to the same privileged direction as that of compressive contacts.

Rheology of three-dimensional packings of aggregates: microstructure and effects of nonconvexity

We use three-dimensional contact dynamics simulations to analyze the rheological properties of granular materials composed of rigid aggregates. The aggregates are made from four overlapping spheres and described by a nonconvexity parameter depending on the relative positions of the spheres. The macroscopic and microstructural properties of several sheared packings are analyzed as a function of the degree of nonconvexity of the aggregates. We find that the internal angle of friction increases with the nonconvexity. In contrast, the packing fraction first increases to a maximum value but declines as the nonconvexity increases further. At a high level of nonconvexity, the packings are looser but show a higher shear strength. At the microscopic scale, the fabric and force anisotropy, as well as the friction mobilization, are enhanced by multiple contacts between aggregates and interlocking, thus revealings the mechanical and geometrical origins of shear strength.

Nonlinear effects of particle shape angularity in sheared granular media

We analyze the effects of particle shape angularity on the macroscopic shear behavior and texture of granular packings simulated by means of the contact dynamics method. The particles are regular polygons with an increasing number of sides ranging from 3 (triangles) to 60. The packings are analyzed in the steady shear state in terms of their shear strength, packing fraction, connectivity, and fabric and force anisotropies, as functions of the angularity. An interesting finding is that the shear strength increases with angularity up to a maximum value and saturates as the particles become more angular (below six sides). In contrast, the packing fraction declines towards a constant value, so that the packings of more angular particles are looser but have higher shear strength. We show that the increase of the shear strength at low angularity is due to an increase of both contact and force anisotropies and the saturation of the shear strength for higher angularities is a consequence of a rapid falloff of the contact and normal force anisotropies compensated for by an increase of the tangential force anisotropy. This transition reflects clearly the rather special geometrical properties of these highly angular shapes, implying that the stability of the packing relies strongly on the side-side contacts and the mobilization of friction forces.

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.

Force chains and contact network topology in sheared packings of elongated particles

By means of contact dynamic simulations, we investigate the contact network topology and force chains in two-dimensional packings of elongated particles subjected to biaxial shearing. The morphology of large packings of elongated particles in quasistatic equilibrium is complex due to the combined effects of local nematic ordering of the particles and orientations of contacts between particles. The effect of elongation on shear behavior and dilatancy was investigated in detail in a previous paper [Azéma and Radjai, Phys. Rev. E 81, 051304 (2010)]. Here, we show how particle elongation affects force distributions and force-fabric anisotropy via various local structures allowed by steric exclusions and the requirement of force balance. We find that the force distributions become increasingly broader as particles become more elongated. Interestingly, the weak force network transforms from a passive stabilizing agent with respect to strong force chains to an active force-transmitting network for the whole system. The strongest force chains are carried by side-side contacts oriented along the principal stress direction.

Taxonomy of granular rheology from grain property networks

We construct complex networks from symbolic time series of particle properties within a dense quasistatically deforming granular assembly subjected to biaxial compression. The structure of the resulting networks embodies the evolving structural rearrangements in the granular material, in both contact forces and contact topologies. These rearrangements are usefully summarized through standard network statistics as well as building block motifs and community structures. Dense granular media respond to applied compression and shear by a process of self-organization to form two cooperatively evolving structures comprising the major load-bearing columnlike force chains, and the lateral trusslike three-cycle triangle topologies. We construct networks summarizing their individual evolution based on relationships between symbolic time series indicating a particle's chronological force chain and three-cycle membership histories. We test which particle membership histories are similar with respect to each other through the information theory-based measure of Hamming distance. The complex networks summarize the essential structural rearrangements, while the community structures within the networks partition the material into distinct zones of deformation, including interlacing subregions of failure inside the shear band. The taxonomy of granular rheology at the mesoscopic scale distills the inelastic structural rearrangements throughout loading history down to its core elements, and should lay bare an objective and physics-based formalism for thermodynamic internal variables and associated evolution laws.