Rheology and structure of polydisperse three-dimensional packings of spheres

D. Cantor, E. Azéma, P. Sornay and F. Radjai

We use three-dimensional contact dynamics simulations to analyze the rheology of polydisperse packings of spherical particles subjected to simple shearing. The macroscopic and microstructural properties of several packings are analyzed as a function of their size span (from nearly monodisperse to highly polydisperse). Consistently with previous two-dimensional simulations, we find that the shear strength is independent of the size span despite the increase of packing fraction with size polydispersity. At the micro-scale, we analyze the particle connectivity, force transmission and the corresponding anisotropies of the contact and force networks. We show that force distributions become increasingly broader as the size span increases. In particular, stronger forces are captured by larger particles, which are also better connected compared to smaller ones. As a consequence, the contact anisotropy declines with size span, whereas the force anisotropy increases. The additive compensation of these two opposite trends leads to the independence of the shear strength from size polydispersity.

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Visco-inertial regimes of immersed granular flows

L. Amarsid, J.-Y. Delenne, P. Mutabaruka, Y. Monerie, F. Perales and F. Radjai

By means of extensive coupled MD-LBM numerical simulations, allowing for grain dynamics and full sub-particle resolution of the fluid phase, we analyze steady inertial granular flows driven by a viscous fluid. We show that, for a broad range of system parameters (shear rate, confining stress, fluid viscosity and relative fluid-grain density), the frictional strength and packing fraction can be described by a single ‘visco-inertial’ dimensionless parameter combining the inertial and Stokes numbers. Remarkably, we also find that for all combinations of system parameters, including dry granular flows, the friction coefficient is a linear function of the ratio of shear-induced porosity to the overall porosity. This relation leads to a simple expression of the friction coefficient as a function of the visco-inertial number. We also map the frictional behavior under constant confining pressure into effective viscous behavior under volume-controlled conditions. This mapping predicts the divergence of effective normal and shear viscosities in inverse square of the distance to the critical packing fraction. Our results are in excellent agreement with recent available experimental data.

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Binary mixtures of disks and elongated particles: texture and mechanical properties

Emilien Azema, Itthichai Preechawuttipong, Farhang Radjai

We analyze the shear strength and microstructure of binary granular mixtures consisting of disks and elongated particles by varying systematically both the mixture ratio and degree of homogeneity (from homogeneous to fully segregated). The contact dynamics method is used for numerical simulations with rigid particles interacting by frictional contacts. A counterintuitive finding of this work is that, the shear strength, packing fraction and, at the microscopic scale, the fabric, force and friction anisotropies of the contact network are all independent of the degree of homogeneity. In other words, homogeneous mixtures behave as segregated packings of the two particle shapes. In contrast, the shear strength increases with the proportion of elongated particles correlatively with the increase of the corresponding force and fabric anisotropies. By a detailed analysis of the contact network topology, we show that various contacts types contribute differently to force transmission and friction mobilization.

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Three-dimensional bonded-cell model for grain fragmentation

David Cantor, Emilien Azema, Philippe Sornay, and Farhang Radjai

We present a three-dimensional numerical method for the simulation of particle crushing in 3D. This model is capable of producing irregular angular fragments upon particle fragmentation while conserving the total volume. The particle is modeled as a cluster of rigid polyhedral cells generated by a Voronoi tes- sellation. The cells are bonded along their faces by a cohesive Tresca law with independent tensile and shear strengths and simulated by the contact dynamics method. Using this model, we analyze the mechanical response of a single particle subjected to diametral compres- sion for varying number of cells, their degree of disorder, and intercell tensile and shear strength. In par- ticular, we identify the functional dependence of par- ticle strength on the intercell strengths. We find that two different regimes can be distinguished depending on whether intercell shear strength is below or above its tensile strength. In both regimes, we observe a power- law dependence of particle strength on both intercell strengths but with different exponents. The strong ef- fect of intercell shear strength on the particle strength reflects an interlocking effect between cells. In fact, even at low tensile strength, the particle global strength can still considerably increase with intercell shear strength. We finally show that the Weibull statistics describes well the particle strength variability.

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Rheology of granular materials composed of crushable particles

Duc-Hanh Nguyen, Emilien Azema, Philippe Sornay, and Farhang Radjai

We investigate sheared granular materials composed of crushable particles by means of contact dynamics simulations and the bonded-cell model for particle fragmentation. 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 is kept constant and the packing is subjected to biaxial com- pression. 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 fragmentation process is highly inhomogenious both in the fragment sizes and their locations inside the packing. In particu- lar, a number of large particles never break whereas a large number of particles break up into small fragments. 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 resulting from consecutive fragmentation events whereby the memory of the initial state is lost. Due to growing polydispersity, dense shear bands are formed and the usual dilatant behavior is reduced or cancelled. Hence, the stress-strain curve no longer passes through a peak stress, and a progressive evolution towards a pseudo-steady state is observed instead. We also show that the shear strength of the packing is well expressed in terms of contact anisotropics and force anisotropics. The force anisotropy increases while the contact orientation anisotropy decreases for increasing internal cohesion of the particles. These two effects compensate each other such that the shear strength is nearly independent of internal cohesion.

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Bonded-cell model for particle fracture

Duc-Hanh Nguyen, Emilien Azema, Philippe Sornay, and Farhang Radjai

Particle degradation and fracture plays an important role in natural granular flows and in many applications of granular materials. We analyze the fracture properties of 2D disk-like 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 escribing cell size 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 size 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 as inverse square root of particle diameter, in excellent agreement with our simulation data.

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Rheology of cohesive granular flows

Nicolas Berger, Emilien Azéma, Jean-Francois Douce and Farhang Radjai

The shear strength of dense granular flows is generally described by an effective friction coefficient, defined as the ratio of shear and normal stresses, as a function of the inertial number. However, this ratio depends on the normal stress when the particles interact via both friction and adhesion forces, and in this sense it does not properly represent a Coulomb-like friction. For the same reason, it is not a unique function of the inertial number. We use contact dynamics simulations to isolate the cohesive strength from the purely frictional strength in dense inertial flows for a broad range of shear rates and adhesion forces between particles. We find that the cohesive strength is a decreasing function of the inertial number. More generally, we show that a single dimensionless parameter, combining the inertial number and adhesion, controls not only the cohesive strength but also the normalized solid fraction and granular texture described in terms of the contact network connectivity and anisotropy.

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Shear strength and structures of highly polydisperse packings composed of pentagons

Duc-hanh Nguyen, E. Azéma, F. Radjai and P. Sornay

By means of extensive contact dynamic simulations, we present a detailed analysis regarding the combined effect of particle size and shape polydispersities (defined as size span and as the degree of irregularity of the particles) on the shear strength and structure of sheared granular media composed of pentagons. We find that the shear strength is independent of size span, but unexpectedly, it declines as shape polydispersity is increased. In contrast, the solid fraction is an increasing function of both size span and shape polydispersity revealing that the densest packings have the same shear strength as the loosest. At the scale of the particles and contacts, we analyze the connectivity, force transmission, friction mobilization as well as the associated anisotropies. We show that stronger forces are carried out by the largest particles which are propped by an increasing number of small particles. As a result, the non-variation of the shear strength with size span is shown to be due to the falloff of the geometrical anisotropy compensated by an increase of force and branch anisotropies. On the other hand, the increase of shape polydispersity induces that the sharp corners of irregular particles allow for deep contacts between neighbors that are unreachable for more regular particles. Thus, the geometrical anisotropy declines at large values of shape polydispersity, which explain the decreases of the shear strength.

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Shear flow of dense granular suspensions by computer simulations

L. Amarsid, P. Mutabaruka, J.-Y. Delenne, Y. Monerie, F. Radjai

We analyze the shear flow of dense granular materials composed of circular particles immersed in a viscous fluid by means of Molecular Dynamics simulations interfaced with the Lattice Boltzmann Method. A homogeneous flow of the suspension is obtained through periodic boundary conditions and by directly applying a confining pressure on the granular phase and shearing the fluid phase. The stead-state rheology can be described in terms of effective friction coefficient and packing fraction of the suspension as a function of the ratio of viscous shear stress to confining pressure (frictional description), on one hand, and in terms of normal and shear viscosities of the suspension as a function of the packing fraction (viscous description), on the other hand. We show that the simulation data are consistent with both descriptions and in close agreement with the corresponding scaling laws observed in recent experiments.

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Modelling soft-particle materials

S Nezamabadi, F Radjai, J Averseng and J-Y Delenne

Soft-particle materials include colloidal pastes, vesicles, many powders, microgels and suspensions. They share the common feature of being composed of particles that can undergo large deformations without rupture. For the simulation of such materials, we present a modelling approach based on an implicit formulation of the Material Point Method (MPM) interfaced with the Contact Dynamics (CD) method for the treatment of frictional contacts between particles. Each particle is discretized as a collection of material points. The information carried by the material points is projected onto a background mesh, where equations of motion are solved. The mesh solution is then used to update the material points. The implicit formulation of MPM allows for unconditional numerical stability and efficient coupling with implicit treatment of unilateral contacts and friction between the particles by the CD method. We use this model to analyse the compaction process of 2D soft-particle packings. The packing can reach high solid fractions by particle shape change and still flow plastically. The compaction is a nonlinear process in which new contacts are formed between particles and the contact areas increase. We find that the evolution of the packing fraction is a slow logarithmic function of the driving stress as a consequence of increasing contact area. We also evidence the effect of friction, which favours strong stress chains and thus the elongation of particles, leading to a larger packing fraction at a given level of compressive stress as compared to a frictionless particle packing.

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Effect of size polydispersity vs. particle shape in dense granular media

D-H Nguyen, E Azéma, F Radjai and P Sornay

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 the size and shape polydispersities of the system, respectively. The packing fraction increases with both shape and size parameters. But we find that the effect of shape polydispersity for all the structural properties investigated in this paper is significant only at low size polydispersity where crystalline structures characterized by positional and/or orientational ordering of the particles can be identified. We show that the proportion of side/side contacts is nearly independent of the polydispersity parameters. They side/side contacts do not percolate but 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 mainly captured by side/side contacts. Finally, the force transmission is intrinsically multiscale with a mean force increasing linearly with particle size. The probability density of forces is increasingly broader as size polydispersity increases with a well-defined exponential fall-off of the number of forces.

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Initiation of immersed granular avalanches

P Mutabaruka, J-Y Delenne, K Soga & F Radjai

Depending on its packing fraction, a granular bed immersed in a viscous fluid and inclined above its angle of repose is either unstable or stabilized by a negative overpressure induced by slow creep and expansion of the bed due to dilatancy. In this paper, we use a 3D coupled DEM/LBM algorithm with appropriate boundary conditions to investigate the spatiotemporal process of slope failure in this configuration. Our findings are in quantitative agreement with the available experimental results. We analyze the evolution of shear strain, packing fraction and pore overpressures for different values of the initial packing fraction and slope angle. We show that the time evolutions of shear strain and packing fraction scale excellently with a characteristic time extracted from a model based on the balance of granular stresses in the presence of a pore overpressure and the relation of the latter with dilatancy due to darcian drag forces. The cumulative shear strain at failure is found to be approximatively 0.2, as in experiments, irrespective of the initial packing fraction and slope angle. The triggering time and packing fraction at failure are correctly predicted by using this shear strain as failure criterion. We also analyze the evolution of the contact network during creep. Remarkably, the network deforms by distortion at a nearly constant connectivity, and slope failure is triggered when the anisotropy saturates. This work clearly demonstrates the feasibility of realistic numerical simulations for immersed granular materials and opens in this respect quite far-reaching perspectives in this field.

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Archi-texture of inertial granular flows

E Azéma & F Radjai

By means of extensive 3D numerical simulations, we analyze the microstructure of dense granular flows and its relation with shear strength as a function of the inertial number I representing the ratio of particle relaxation time to shear time. We find that the shear strength increases with I only as a consequence of increasing anisotropy of the contact network whereas the anisotropy of force chains remains nearly constant. The contact network undergoes topological transitions, and beyond I ≃ 0.25 the force chains break into clusters immersed in a background “soup” of floating particles. We show that this transition coincides with the divergence of the size of fluidized zones.

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Rheology of 3D packings of aggregates: microstructure and effects of nonconvexity

E Azéma, F Radjaï, B Saint-cyr, J-Y Delenne and P Sornay


We use 3D contact dynamics simulations to analyze the rheological properties of granular materials composed of rigid aggregates. The aggergates 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 nonconvexity. In contrast, the packing fraction increases first to a maximum value but declines as nonconvexity further increases. At 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 friction mobilization are enhanced by multiple contacts between aggregates and interlocking, revealing thus the mechanical and geometrical origins of shear strength.

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

E Azéma, F Radjaï and F Dubois


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 (octahedron-like shape) to 596. The shear strength increases with angularity up to a maximum value and saturates as the particles become more angular (below 46 faces). This finding extends the results of a previous study of regular polygons in two dimensions to irregular polyhedra in three dimensions. We also find that the packing fraction increases with angularity to a peak value but declines for more angular particles. We analyze the connectivity and anisotropy of the contact network and show that the increase of the shear strength with angularity is due to a net increase of fabric and force anisotropies but at higher particle angularity a rapid fall-off of the fabric anisotropy is compensated by an increase of force anisotropy, leading thus to the saturation of shear strength.

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Effect of the platy shape of the particles

M Boton, E Azéma, N Estrada, F Radjaï and A Lizcano


This is the first paper of a series devoted to the study of clayey soils from a micromechanical perspective. We specifically focus on the effect of the platy shape of particles, typical of clays, in the mechanical behavior and microstructure of dry assemblies by means of discrete element simulations. The particles are three-dimensional square plates, approximated as spheropolyhedra. Several samples composed of particles of different levels of platyness (ratio of length to thickness) were numerically prepared and sheared up to large deformations. We analyzed the shear strength, solid fraction, orientation of the particles, connectivity, fabric of the interactions network, and interaction forces as functions of the particles' platyness. We found that both the mechanical behavior and the microstructure were strongly dependent on the particles' platyness. In particular, we found that the principal phenomenon underlying these dependences is the alignment of the particles' faces along a particular direction. This ordering phenomenon, which emerges even for shapes that deviate only slightly from that of a sphere, enhances the ability of the packing to develop an anisotropic structure, developing large shear strengths, especially as a consequence of the interactions fabric and the mobilization of friction forces. Additionally, the connectivity of the packings and their solid fraction also evolve with the particles' platyness. In particular, the solid fraction evolves in a non-monotonic fashion, as is usual for granular materials made up of non spherical particles.

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Cohesive granular materials composed of nonconvex particles

B Saint-Cyr, F Radjai, Ph Sornay, J-Y Delenne

By means of contact dynamics simulations, we investigate the mechanical strength and the internal structure of granular materials composed of adhesive nonconvex aggregates with increasing nonconvexity. The issue is how the shape nonconvexity will affect the macroscopic Coulomb cohesion emerging from particle interaction with a given bond number. We show that the nonconvexity amplifies considerably the macroscopic cohesion that increases with the degree of nonconvexity. We also found that the Coulomb cohesion evaluated from different shear tests, strongly depends on the boundary conditions due to the heterogeneity of either the strains or stresses inside the material. The largest cohesion is obtained for nearly homogeneous deformation at the beginning of axial compression without confinement (simple compression test). In the residual state, during the biaxial shear tests, the samples characterized by a unique level of nonconvexity, develop a heterogeneous structure strongly marked by shear bands localization (movie eta 5). Because of this, they flow in coherent block larger than particles sizes. Furthermore these samples show different schemes of strain localization, which lead to fluctuating values of the apparent cohesion as a function of the nonconvexity. On the contrary, by considering uniaxial test without the lateral walls, we found that the unconfined yield stress and the resultant apparent cohesion increases monotically with the nonconvexity, but an heterogeneous stress distribution occurs in the different packings (movies eta 4 and eta 7). A local analysis of the granular structure, shows that the more nonconvex are the aggregates the more heterogeneous are the distribution of the local packing fraction.

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Tensile strength and fracture of cemented granular aggregates

R. Affes, J.-Y. Delenne, Y. Monerie, F. Radjai, V. Topin

Cemented granular aggregates include a broad class of geomaterials such as sedimentary rocks and some biomaterials such as the wheat endosperm. We present a 3D lattice element method for the simulation of such materials, modeled as a jammed assembly of particles bound together by a matrix partially filling the interstitial space. From extensive simulation data, we analyze the mechanical properties of aggregates subjected to tensile loading as a function of matrix volume fraction and particle-matrix adhesion. We observe a linear elastic behavior followed by a brutal failure along a fracture surface. The effective stiffness before failure increases almost linearly with the matrix volume fraction. We show that the tensile strength of the aggregates increases with both the increasing tensile strength at the particle-matrix interface and decreasing stress concentration as a function of matrix volume fraction. The proportion of broken bonds in the particle phase reveals a range of values of the particle-matrix adhesion and matrix volume fraction for which the cracks bypass the particles and hence no particle damage occurs. This limit is shown to depend on the relative toughness of the particle-matrix interface with respect to the particles.

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Collapse dynamics and runout of dense granular materials in a fluid

V. Topin, Y. Monerie, F. Perales, F. Radjai

We investigate the effect of an ambient fluid on the dynamics of collapse and spread of a granular column simulated by means of the contact dynamics method interfaced with computational fluid dynamics. The runout distance is found to increase as a power law with the aspect ratio of the column and, surprisingly, for a given aspect ratio and packing fraction, it may be similar in the grain-inertial and fluid-inertial regimes but with considerably longer duration in the latter case. We show that the effect of fluid in viscous and fluid-inertial regimes is to both reduce the kinetic energy during collapse and enhance the flow by lubrication during spread. Hence, the runout distance in a fluid may be below or equal to that in the absence of fluid due to compensation between those effects.

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Stability, deformation and variability of granular fills composed of polyhedral particles

J C Quezada, P Breul, G Saussine and F Radjai

By means of extensive contact dynamics simulations, we investigate the mechanical equilibrium and deformation of a granular material composed of irregular polyhedral particles confined between two horizontal frictional planes. We show that, as a consequence of mobilized wall-particle friction force at the top and bottom boundaries, the transient deformation induced by a constant vertical load is controlled by the aspect ratio (thickness over width) of the packing as well as the stress ratio. The transient deformation declines considerably for increasingly smaller aspect ratios and grows with the stress ratio. From the simulation data for a large number of independent configurations, we find that sample-to-sample fluctuations of the deformation have a broad distribution and they scale with the average deformation. We also analyze the evolution of particle connectivity during settlement and with the applied force. The face-face and edge-face contacts between polyhedral particles are shown to concentrate strong force chains with a growing proportion as a function of the applied force.

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