Micro-mechanical behaviour of granular materials under gravity-induced stress gradient

Gravity-induced stress gradient plays an important role both in terrestrial and extra-terrestrial geotechnical engineering. Its effect on the behaviour of granular materials was studied by DEM simulations for the representative volume element under triaxial conditions. The shear strength and volumetric dilatancy decrease at first and then stabilizes with increasing stress gradient; the critical gravity shows a stress level-dependency. Corresponding micromechanics was further explored. The stress gradient restrains the original decrease in valence and the increase in elongation degree of void cells. The number of void cells oriented parallel to the loading direction is also decreased. All of the original increase tendencies of those micro-parameters are also significantly restrained by the stress gradient during triaxial loading. A stress level-dependent critical gravity was observed beyond which those micro-parameters stabilize. The stress gradient leads to significantly more particles participating in load-bearing both during isotropic compression and triaxial loading, and leads to a relative stable fraction of strong force chain in the latter tests. The stress gradient appears to have no significant influence on the spatial distribution of micro slip bands, but results in a stable area fraction. Meanwhile, their overall number is reduced, whilst the average length and width are generally increased. The effect of gravity on void cell fabrics physically originates from those gravity-sensitive particles which are poorly supported and have an acceleration mainly caused by the gravity. The gravity-sensitive evolution patterns of void cells were presented finally, which gives a good explanation for both the macroscopic and microscopic observations.     

A gradient flow theory of plasticity for granular materials

Summary A flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.     

A saturated discrete particle model and characteristic-based SPH method in granular materials

Based on the discrete particle model for solid-phase deformation of granular materials consisting of dry particulate assemblages, a discrete particle–continuum model for modelling the coupled hydro-mechanical behaviour in saturated granular materials is developed. The motion of the interstitial fluid is described by two parallel continuum schemes governed by the averaged incompressible N–S equations and Darcy's law, respectively, where the latter one can be regarded as a degraded case of the former. Owing to the merits in both Lagrangian and mesh-free characters, the characteristic-based smoothed particle hydrodynamics (SPH) method is proposed in this paper for modelling pore fluid flows relative to the deformed solid phase that is modelled as packed assemblages of interacting discrete particles. It is assumed that the formulation is Lagrangian with the co-ordinate system transferring with the movement of the solid particles. The assumed continuous fluid field is discretized into a finite set of Lagrangian (material) points with their number equal to that of solid particles situated in the computational domain. An explicit meshless scheme for granular materials with interstitial water is formulated. Numerical results illustrate the capability and performance of the present model in modelling the fluid–solid interaction and deformation in granular materials saturated with water. Copyright © 2007 John Wiley & Sons, Ltd.     

Experimental micromechanical evaluation of strength of granular materials: Effects of particle rolling

Biaxial compression tests have been performed on assemblies of oval cross-sectional rods, in an effort to evaluate the effects of interparticle friction, particle shape, and initial fabric on the overall strength of granular materials. The variation in the spatial arrangement of the particles (fabric) and particle rolling and sliding are monitored by taking photoelastic pictures at various stages during the course of deformation. Based on this, the following conclusions are obtained. (1) Particle rolling appears to be a major microscopic deformation mechanism, especially when interparticle friction is large. (2) There are relatively few contacts at which relative sliding is dominant, and this seems to be true even when the assembly reaches the overall failure state; this observation is in contradiction to the common assumption that particle sliding is the major microscopic deformation mode. (3) During the course of deformation and up to the peak stress, new contacts are continually formed in such a manner that the contact unit normals tend to concentrate more in a direction parallel to the maximum principal compression. This concentration of unit normals seems to be closely related to the formation of new column-like load paths which carry the increasing axial stress under constant lateral force. After the peak stress, such a column-like microstructure disappears and considerable rearrangement of the load paths takes place, leading to a more diffused (homogeneous) microstructure in the critical state. (4) If a fabric tensor F_ij, i, J = 1,2,3, is defined to be proportional to the volume average of the quantity m_im_j, where m_i are the rectangular Cartesian components of a unit vector along a vector that connects the centroids of two typical contacting granules, then it appears that the overall stress with components σ_ij tends to become coaxial with the fabric tensor F_ij, as the overall deformation continues. For two-dimensional granules the result σ_ij = _OF_ij + β_OF_jkF_kj (k summed) obtained by Mebrabadi, Nemat-Nasser and Oda (1980) by microchemical modeling is confirmed experimentally; _O and β_O are material parameters.     

The influence of particle geometry and the intermediate stress ratio on the shear behavior of granular materials

The behavior of granular materials is very complex in nature and depends on particle shape, stress path, fabric, density, particle size distribution, amongst others. This paper presents a study of the effect of particle geometry (aspect ratio) on the mechanical behaviour of granular materials using the discrete element method (DEM). This study discusses 3D DEM simulations of conventional triaxial and true triaxial tests. The numerical experiments employ samples with different particle aspect ratios and a unique particle size distribution (PSD). Test results show that both particle aspect ratio (AR) and intermediate stress ratio \((b=({\upsigma }_{2}'-{\upsigma }_{3}')/({\upsigma }_{1}'-{\upsigma }_{3}'))\) affect the macro- and micro-scale responses. At the macro-scale, the shear strength decreases with an increase in both aspect ratio and intermediate stress ratio b values. At the micro-scale level, the fabric evolution is also affected by both AR and b. The results from DEM analyses qualitatively agree with available experimental data. The critical state behaviour and failure states are also discussed. It is observed that the position of the critical state loci in the compression \((e-p')\) space is only slightly affected by aspect ratio (AR) while the critical stress ratio is dependent on both AR and b. It is also demonstrated that the influence of the aspect ratio and the intermediate stress can be captured by micro-scale fabric evolutions that can be well understood within the framework of existing critical state theories. It is also found that for a given stress path, a unique critical state fabric norm is dependent on the particle shape but is independent of critical state void ratio.     

A hyperbolic well-posed model for the flow of granular materials

A plasticity model for the flow of granular materials is presented which is derived from a physically based kinematic rule and which is closely related to the double-shearing model, the double-sliding free-rotating model and also to the plastic-potential model. All of these models incorporate various notions of the concept of rotation-rate and the crucial idea behind the model presented here is that it identifies this rotation-rate with a property associated with a Cosserat continuum, namely, the intrinsic spin. As a consequence of this identification, the stress tensor may become asymmetric. For simplicity, in the analysis presented here, the material parameters are assumed to be constant. The central results of the paper are that (a) the model is hyperbolic for two-dimen-Specifically, sional steady-state flows in the inertial regime and (b) the model possesses a domain of linear well-posedness. it is proved that incompressible flows are well-posed.     

A contact model for the yielding of caked granular materials

We present a visco-elastic coupling model between caked spheres, suitable for Distinct Element Method simulations, which incorporates the different loading mechanisms (tension, shear, bending, torsion) in a combined manner and allows for a derivation of elastic and failure properties on a common basis. In pull, shear, and torsion failure tests with agglomerates of up to 10.000 particles, we compare the failure criterion to different approximative variants of it, with respect to accuracy and computational cost. The failure of the agglomerates, which behave according to elastic parameters derived from the contact elasticity, gives also insight into the relative relevance of the different load modes.     

A hyperbolic well-posed model for the flow of granular materials

A plasticity model for the flow of granular materials is presented which is derived from a physically based kinematic rule and which is closely related to the double-shearing model, the double-sliding free-rotating model and also to the plastic-potential model. All of these models incorporate various notions of the concept of rotation-rate and the crucial idea behind the model presented here is that it identifies this rotation-rate with a property associated with a Cosserat continuum, namely, the intrinsic spin. As a consequence of this identification, the stress tensor may become asymmetric. For simplicity, in the analysis presented here, the material parameters are assumed to be constant. The central results of the paper are that (a) the model is hyperbolic for two-dimensional steady-state flows in the inertial regime and (b) the model possesses a domain of linear well-posedness. Specifically, it is proved that incompressible flows are well-posed.     

The influence of rolling resistance on the stress-dilatancy and fabric anisotropy of granular materials

The effects of rolling resistance on the stress-dilatancy behavior and fabric anisotropy of granular materials were investigated through a three-dimensional discrete element method (DEM). A rolling resistance model was incorporated into the DEM code PFC3D and triaxial DEM simulations under simulated drained and undrained conditions were carried out. The results show that there existed a threshold value of the rolling friction. When the rolling friction was smaller than this value, the mechanical behavior of granular materials under both drained and undrained conditions were substantially influenced by the rolling friction, but the influence diminished when it was larger than the threshold value. A linear relationship has been observed between the dilatancy coefficient and the natural logarithm of the rolling-friction coefficient when it was smaller than the threshold value. An increase in the rolling friction led to an increase in the fabric anisotropy of all strong contacts under both testing conditions until the threshold value was attained. The investigation on the effect of rolling friction on the microstructure of granular materials revealed that the rolling friction enhanced the stability of force chains, which resulted in the difference in the stress-dilatancy behavior. Finally, the relationship between the stress ratio q/p\(^{\prime }\) and the fabric measure at strong contacts \(\hbox {H}_{\mathrm{d}}^{\mathrm{s}} /\hbox {H}_{\mathrm{m}}^{\mathrm{s}}\) was found independent of the inter-particle friction, rolling friction and testing conditions.     

A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion

Summary In this paper we will explore the consequences of the Mohr-Coulomb criterion on the constitutive equation proposed by Rajagopal and Massoudi [1]. This contunuum model which is based on the earlier works of Cowin [2] has also the ability to predict the dilatancy effect which is related to the normal stress effects. At the same time, if a proper representation is given to some of the material parameters, this model would also comply with the Mohr-Coulomb criterion. We also present, as a special case, an exact solution for the case of simple shear flows.     

The influence of particle rolling and imperfections on the formation of shear bands in granular material

This paper examines the development and evolution of shear bands in granular assemblies when particle rolling and imperfections are taken into account. Simulated biaxial tests in two-dimension are conducted using the discrete element method. The progressive development of rotational angles and effective strain are presented to describe the emergence and evolution of shear bands in biaxial tests. The simulated results reveal that when rolling resistance is taken into account in DEM, the development of shear bands is more distinct as the evolution of the minor shear bands is limited while the major shear bands are preferably promoted in granular materials, and that the local rotating bearings not only influence the onset of shear bands and the width of the shear bands, but also decrease the resistance and reduce the strength of the granular material. Also, it is demonstrated that the primary shear bands initiate from the imperfect areas and develop preferentially along the direction of imperfections. Therefore, the emergence and development of shear bands, which will result in a decline in strength and eventually lead to instability and destruction of the material, can be effectively simulated when rolling resistance is incorporated in DEM and the initial distribution of imperfections in the granular material is defined.     

A heterarchical multiscale model for granular materials with evolving grainsize distribution

Naturally occurring granular flows, such as landslides, debris flows and avalanches typically have size ratios of up to \(10^{6}\) between the smallest and largest constituent particles. For the purposes of modelling, however, it is generally assumed that a single representative size can adequately describe the grains. Polydisperse flows are not described more completely primarily because of two reasons: The first is a lack of understanding of the physical mechanisms which affect polydisperse flows. The second is a lack of models with which to describe such systems. Here, we present a heterarchical multiscale model which accounts for both the microstructural evolution within representative elementary volumes, and also the associated changes in bulk flow properties. Three key mechanisms are addressed; segregation, comminution and mixing. Granular segregation is an important mechanism for industrial processes aiming at mixing grains. Additionally, it plays a pivotal role in determining the kinematics of geophysical flows. Because of segregation, the grainsize distribution in a granular medium varies in space and time during flow. Additional complications arise from the presence of comminution, where new particles are created, potentially enhancing segregation. This has a feedback on the comminution process, as particles change their local neighbourhood. Simultaneously, particles are generally undergoing remixing, further complicating the segregation and comminution processes. The interaction between these mechanisms is explored using a stochastic lattice model with three rules: one for each of segregation, comminution and mixing. The interplay between these rules creates complex patterns, as seen in segregating systems, and depth dependent log-normal grading curves, which have been observed in avalanche runout.     

Influence of particle breakage on the dynamic compression responses of brittle granular materials

The dynamic compression responses of dry quartz sand are tested with a modified spilt Hopkinson pressure bar (MSHPB), and the quasi-static compression responses are tested for comparison with a material testing system. In the experiments, the axial stress–strain responses and the confining pressure of the jacket are both measured. Comparison of the dynamic and the quasi-static axial stress–strain curves indicate that dry quartz sand exhibits obvious strain-rate effects. The grain size distributions of the samples after dynamic and quasi-static loading are obtained with the laser diffractometry technique to interpret the rate effects. Quantitative analyses of the grain size distributions show that at the same stress level, the particle breakage extent under quasi-static loading is larger than that under dynamic loading. Moreover, the experimental and the theoretical relationships of the particle breakage extent versus the plastic work show that the energy efficiency in particle breakage is higher under quasi-static loading, which is the intrinsic cause of the strain-rate effects of brittle granular materials. Using the discrete element method (DEM), the energy distributions in the brittle granular material under confined compression are discussed. It is observed that the input work is mainly transformed into the frictional dissipation, and the frictional dissipation under dynamic loading is higher than that under quasi-static loading corresponding to the same breakage extent. The reason is that more fragmentation debris is produced during dynamic breakage of single grains, which promotes particle rearrangement and the corresponding frictional dissipation significantly.     

DEM investigation of particle anti-rotation effects on the micromechanical response of granular materials

The importance of particle rotation to the mechanical behavior of granular materials subject to quasi-static shearing has been well recognized in the literature. Although the physical source of the resistance to particle rotation is known to lie in the particle surface topography, it has been conveniently studied using the rolling resistance model installed typically on spherical particles within the DEM community. However, there has been little effort on assessing the capability of the rolling resistance model to produce more realistic particle rotation behavior as exhibited by irregular-shaped particles. This paper aims to eliminate this deficiency by making a comprehensive comparison study on the micromechanical behavior of assemblies of irregular-shaped particles and spherical particles installed with the rolling resistance model. A variety of DEM analysis techniques have been applied to elucidate the full picture of micromechanical processes occurring in the two types of granular materials with different particle-level anti-rotation mechanisms. Simulation results show that the conventional rheology-type rolling resistance models cannot reproduce the particle rotation and strain localization behavior as displayed by irregular-shaped materials, although they demonstrate clear effects on the macroscopic strength and dilatancy behavior, as have been adequately documented in the literature. More insights into the effects of particle-level anti-rotation mechanism are gained from an in-depth inter-particle energy dissipation analysis.     

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu–Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.     

A theory for granular materials exhibiting normal stress effects based on Enskog's dense gas theory

Granular materials exhibit phenomena such as normal stress differences, which are typical of materials whose response is non-linear. For example, when a non-linearly elastic slab is sheared, its motion is not determined by the shear force but by the normal forces that manifest themselves due to the shearing (Poynting effect). Another example is a non-linear fluid which exhibits normal stress differences that lead to phenomena like “die-swell” or “rod-climbing,” which is again a manifestation of the stresses that develop orthogonal to planes of shear.

In this paper, an expression for the stress tensor of a granular material that can exhibit normal-stress effects due to a solids fraction gradient is derived from both continuum and kinetic models. The continuum model motivates and develops the form of the stress tensor, but introduces undetermined coefficients. The kinetic model evaluates those coefficients using Enskog's dense gas theory. The dependence of the granular stress tensor on the solids fraction gradient arises by requiring that the correlating factor that links the two-particle distribution function to the two single-particle distribution functions be the contact value for the radial distribution function of a non-homogeneous, hard-sphere fluid. A representation for that contact value is found by developing the generalized van der Waals theory expression for a stress tensor element of a nonhomogeneous fluid (a fluid that exhibits a density gradient) in equilibrium, and comparing it to the exact expression. That representation of the contact value is introduced into the two-particle distribution function, and its contribution to the stress tensor is found. The resulting stress tensor expression is applied to a simple shear flow problem in which a linear, solids-fraction profile is transverse to the flow. The resulting normal-stress effects increase with the solids-fraction and its gradient.     

Use of CFD-DEM to evaluate the effect of intermediate stress ratio on the undrained behaviour of granular materials

This study investigates the effect of intermediate stress ratio (b) on the mechanical behaviour of granular soil in true triaxial tests. A CFD-DEM solver with the ability to model compressible fluid and moving mesh has been developed and calibrated based on existing experimental test results on Nevada sand. The effect of b on the undrained true triaxial test, which has been neglected in the literature, was investigated using a reasonable number of models. The effects of the initial confining stress and initial void ratio also have been studied. The developed model was used to calculate the hydrodynamic forces on the particles and evaluate the ratio of the particle–fluid interaction force to the resultant force on the particles. It has been demonstrated that, in numerical studies, the effect of these forces cannot be neglected.     

An entropy model for shear deformation of granular materials

A statistical mechanical analogy for characterization of granular materials is discussed by using such notions as the state of the material, the density of states, entropy, canonical distribution and the partition function. The transition law of states during shear deformations of the material is microscopically investigated in the case of two-dimensional model granular materials. The assumption of entropy growth is shown to characterize the dilatancy of the material. A rough proof is given by assuming the measure preserving property of the transition and showing its ergodicity.     

A frictional Cosserat model for the flow of granular materials through a vertical channel

Summary A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, in contrast to experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus, localized couple stresses exist, and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as ^-1/3.