A meso-scale model of clay matrix: the role of hydration transitions in geomechanical behavior.

While much progress has been made on the modeling of swelling clays at the molecular scale in recent decades, up-scaling to the macroscopic scale remains a challenge, in particular because the mesoscopic scale (between a few nanometers and a few hundreds of nanometers) is still poorly understood. In this article, we propose a new 2D granular model of clay at the mesoscale. This model is adapted to the modeling of a dense clay matrix representing geomechanical conditions (up to pressures of 10-100 MPa). Some salient features of this model with respect to the existing literature are: (1) its ability to capture hydration transitions occurring at small basal spacings (essential to model complex hydro-mechanical behaviors such as drying shrinkage), (2) the flexibility of the clay layers that becomes important at pressures exceeding 1 MPa, and (3) the control of the inter-layer shear strength critical to model plasticity. The model calibration is purely bottom-up, based on molecular modeling results only. The case of Na-montmorillonite (Na-Mnt) is investigated in detail, regarding isotropic compression (elasticity and plasticity), yield surface and desiccation. The behavior of the granular model appears well consistent with what is known experimentally for pure Na-Mnt, and offers valuable insight into meso-scale processes that could not be reached so far (role of hydration transition, layer flexibility, and impact of loading history). This granular model is a first step toward quantitative up-scaling of molecular modeling of swelling clay for geomechanical applications.

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

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

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.

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.

Numerical study of one-dimensional compression of granular materials. II. Elastic moduli, stresses, and microstructure.

The elastic moduli of a transversely isotropic model granular material, made of slightly polydisperse elastic-frictional spherical beads, in equilibrium along a one-dimensional (oedometric) compression path, as described in the companion paper [M. H. Khalili et al., Phys. Rev. E 95, 032907 (2017)]10.1103/PhysRevE.95.032907, are investigated by numerical simulations. The relations of the five independent moduli to stresses, density, coordination number, fabric and force anisotropies are studied for different internal material states along the oedometric loading path. It is observed that elastic moduli, as in isotropic packs, are primarily determined by the coordination number, with anomalously small shear moduli in poorly coordinated systems, whatever their density. Such states also exhibit faster increasing moduli in compression, and larger off-diagonal moduli and Poisson ratios. Anisotropy affects the longitudinal moduli C_{11} in the axial direction and C_{22} in the transverse directions, and the shear modulus in the transverse plane C_{44}, more than the shear modulus in a plane containing the axial direction C_{55}. The results are compared to available experiments on anisotropic bead packs, revealing, despite likely differences in internal states, a very similar range of stiffness level (linked to coordination), and semiquantitative agreement as regards the influence of anisotropy. Effective medium theory (the Voigt approach) provides quite inaccurate predictions of the moduli. It also significantly underestimates ratios C_{11}/C_{22} (varying between 1 and 2.2) and C_{55}/C_{44} (varying from 1 to 1.6), which characterize elastic anisotropy, except in relatively weakly anisotropic states. The bulk modulus for isotropic compression and the compliance corresponding to stress increments proportional to the previous stress values are the only elastic coefficients to be correctly estimated by available predictive relations. We discuss the influences of fabric and force anisotropies onto elastic anisotropy, showing in particular that the former dominates in sample series that are directly assembled in anisotropic configurations and keep a roughly constant lateral to axial stress ratio under compression.

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

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

Flow of wet granular materials: A numerical study.

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

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.

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

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

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

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

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

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

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

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

Solidlike behavior and anisotropy in rigid frictionless bead assemblies.

We investigate the structure and mechanical behavior of assemblies of frictionless, nearly rigid equal-sized beads, in the quasistatic limit, by numerical simulation. Three different loading paths are explored: triaxial compression, triaxial extension, and simple shear. Generalizing a recent result, we show that the material, despite rather strong finite sample size effects, is able to sustain a finite deviator stress in the macroscopic limit, along all three paths, without dilatancy. The shape of the yield surface in principal stress space differs somewhat from the Mohr-Coulomb prediction, and is more adequately described by the Lade-Duncan or Matsuoka-Nakai criteria. We study geometric characteristics and force networks under varying stress levels within the supported range. Although the scalar state variables stay equal to the values observed in systems under isotropic pressure, the material, once subjected to a deviator stress, possesses some fabric and force distribution anisotropies. Each kind of anisotropy can be described, in good approximation, by a single parameter. Within the supported stress range, along each one of the three investigated stress paths, among those three quantities: deviator stress to mean stress ratio, fabric anisotropy parameter, force anisotropy parameter, any one determines the values of the two others. The pair correlation function also exhibits short range anisotropy, up to a distance between bead surfaces of the order of 10% of the diameter. The tensor of elastic moduli is shown to possess a nearly singular, uniaxial structure related to stress anisotropy. Possible stress-strain relations in monotonic loading paths are also discussed.

Frictionless bead packs have macroscopic friction, but no dilatancy.

The statement of the title is shown by numerical simulation of homogeneously sheared assemblies of frictionless, nearly rigid beads in the quasistatic limit. Results coincide for steady flows at constant shear rate gamma[over ] in the limit of small gamma[over ] and static approaches, in which packings are equilibrated under growing deviator stresses. The internal friction angle phi , equal to 5.76 degrees +/-0.22 degrees in simple shear, is independent of average pressure P in the rigid limit and stems from the ability of stable frictionless contact networks to form stress-induced anisotropic fabrics. No enduring strain localization is observed. Dissipation at the macroscopic level results from repeated network rearrangements, similar to the effective friction of a frictionless slider on a bumpy surface. Solid fraction Phi remains equal to the random close packing value approximately 0.64 in slowly or statically sheared systems. Fluctuations of stresses and volume are observed to regress in the large system limit. Defining the inertial number as I=gamma radical m/(aP), with m the grain mass and a its diameter, both internal friction coefficient mu*=tan phi and volume 1/Phi increase as powers of I in the quasistatic limit of vanishing I , in which all mechanical properties are determined by contact network geometry. The microstructure of the sheared material is characterized with a suitable parametrization of the fabric tensor and measurements of coordination numbers.

Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks.

This is the first paper of a series of three, in which we report on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. The influence of various assembling processes on packing microstructures is investigated. It is accurately checked that frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, higher solid fractions corresponding to more ordered configurations with traces of crystallization. Specific properties directly related to isostaticity of the force-carrying structure in the rigid limit are discussed. With frictional grains, different preparation procedures result in quite different inner structures that cannot be classified by the sole density. If partly or completely lubricated they will assemble like frictionless ones, approaching the RCP solid fraction Phi_{RCP} approximately 0.639 with a high coordination number: z* approximately =6 on the force-carrying backbone. If compressed with a realistic coefficient of friction mu=0.3 packings stabilize in a loose state with Phi approximately 0.593 and z* approximately =4.5 . And, more surprisingly, an idealized "vibration" procedure, which maintains an agitated, collisional regime up to high densities results in equally small values of z* while Phi is close to the maximum value Phi_{RCP}. Low coordination packings have a large proportion (>10%) of rattlers--grains carrying no force--the effect of which should be accounted for on studying position correlations, and also contain a small proportion of localized "floppy modes" associated with divalent grains. Low-pressure states of frictional packings retain a finite level of force indeterminacy even when assembled with the slowest compression rates simulated, except in the case when the friction coefficient tends to infinity. Different microstructures are characterized in terms of near neighbor correlations on various scales, and some comparisons with available laboratory data are reported, although values of contact coordination numbers apparently remain experimentally inaccessible.

Internal states of model isotropic granular packings. II. Compression and pressure cycles.

This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [I. Agnolin and J.-N. Roux, Phys. Rev. E 76, 061302 (2007)]. As functions of P , we monitor changes in solid fraction Phi, coordination number z, proportion of rattlers (grains carrying no force) x_(0) , the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Phi is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle and end up with values of z and x_(0) close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.

Internal states of model isotropic granular packings. III. Elastic properties.

In this third and final paper of a series, elastic properties of numerically simulated isotropic packings of spherical beads assembled by different procedures, as described in the first companion paper, and then subjected to a varying confining pressure, as reported in the second companion paper, are investigated. In addition to the pressure, which determines the stiffness of contacts because of Hertz's law, elastic moduli are chiefly sensitive to the coordination number z , which should not be regarded as a function of the packing density. Comparisons of numerical and experimental results for glass beads in the 10 kPa-10 MPa pressure range reveal similar differences between dry samples prepared in a dense state by vibrations and lubricated packings, so that the greater stiffness of the latter, in spite of their lower density, can be attributed to a larger coordination number. Effective medium type approaches, or Voigt and Reuss bounds, provide good estimates of bulk modulus B , which can be accurately bracketed, but badly fail for shear modulus G , especially in low z configurations under low pressure. This is due to the different response of tenuous, fragile networks to changes in load direction, as compared to load intensity. In poorly coordinated packings, the shear modulus, normalized by the average contact stiffness, tends to vary proportionally to the degree of force indeterminacy per unit volume, even though this quantity does not vanish in the rigid limit. The elastic range extends to small strain intervals and compares well with experimental observations on sands. The origins of nonelastic response are discussed. We conclude that elastic moduli provide access to mechanically important information about coordination numbers, which escape direct measurement techniques, and indicate further perspectives.

Delayed fracture in porous media.

The fracture of porous media subjected to a constant load is studied. Contrary to homogeneous solids in which fracture usually happens instantaneously at a well-defined breaking strength, the fracture of a porous medium can occur with a delay, allowing us to quantify the average lifetime of the unbroken material. We show that the average fracture probability, a key property for risk analysis in civil engineering, is given by the probability of crack nucleation. The nucleation process can be understood qualitatively by calculating the activation energy for crack nucleation, taking into account the porosity of the medium.

Elastic wave propagation in confined granular systems.

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wave front followed by random oscillations made of multiply scattered waves. We find that the coherent wave front is insensitive to details of the packing: force chains do not play an important role in determining this wave front. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wave front scales with pressure as p1/6; we compare this result with experimental data on various granular systems where deviations from the p1/6 behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

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

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