Investigating the micromechanical evolutions within inherently anisotropic granular materials using discrete element method

This paper investigates the mechanical behavior of inherently-anisotropic granular materials from macroscopic and microscopic points of view. The study is achieved by simulating biaxial compression tests performed on granular assemblies by using numerical discrete element method. In the same category of numerical studies found in the literature, the simulations were performed by considering elliptical/oval particles. In the present study, however, the shape of particles is considered as convex polygons, which mostly resembles real sand grains. Particle assemblies with four different bedding angles were tested. Similar to what observed in experiment, inherent anisotropy has a significant effect on macroscopic mechanical behavior of granular materials. The shear strength and dilative behavior of assemblies were found to decrease as the bedding angle increases. Evolution of the microstructure of all samples and the influence of bedding angle on the fabric and force anisotropy during loading process were investigated. It is seen that the microscopic evolutions in the fabric can justify well the macroscopic behavior of granular assemblies. It is found that the long axis of particles tend to be inclined perpendicular to the loading axis, which results in generating more stable column-like microstructures in order to transfer the applied load. Moreover, the number of contacts as well as the magnitude of forces among particles varies in different directions during the loading process and the initial anisotropy condition totally evolves due to the induced anisotropy within samples.     

A flow rule incorporating the fabric and non-coaxiality in granular materials

Deficiencies of constitutive models in prediction of dilatancy are often attributed to simplifications associated with flow rules such as assumptions of isotropy and coaxiality. It is thus proposed here to develop a comprehensive flow rule for granular materials by including the effect of fabric and without the assumption of coaxiality. A second-order tensor is introduced as a fabric for the distribution of contact normals and contact forces. By using the energy principle in micro-mechanical scale and a suitable dissipation mechanism in granular materials, a stress-dilatancy relation is obtained. Fabric plays a “bridge-like” role in the dilatancy and non-coaxiality. Non-coaxialities between stress-strain-fabric are attributed to the non-coaxiality between stress-fabric and strain-fabric. In this formulation the constants for modeling fabric depend on non-coaxiality of the system rather than the history that determines such a state. Ability of this stress-fabric-dilatancy for modeling the non-coaxiality shows that this relation can predict the behavior of granular materials in the presence of the rotation of principal stress axes.     

On micromechanical characteristics of the critical state of two-dimensional granular materials

In micromechanics of quasi-static deformation of granular materials, relationships are investigated between the macro-scale, continuum-mechanical characteristics, and the micro-scale characteristics at the particle and interparticle contact level. An important micromechanical quantity is the fabric tensor that reflects the distribution of contact orientations. It also contains information on the coordination number, i.e. the average number of contacts per particle. Here, the focus is on characteristics of the critical state in the two-dimensional case. Critical state soil mechanics is reviewed from the micromechanical viewpoint. Two-dimensional discrete element method (DEM) simulations have been performed with discs from a fairly narrow particle-size distribution. Various values for the interparticle friction coefficient and for the confining pressure have been considered to investigate the effect of these quantities on critical state characteristics (shear strength, packing fraction, coordination number and fabric anisotropy). Results from these DEM simulations show that a limiting fabric state exists at the critical state, which is geometrical in origin. The contact network tessellates the assembly into loops that are formed by contacts. For each loop, a symmetrical loop tensor is defined, based on its contact normals. This loop tensor reflects the shape of the loop. An orientation is associated with each loop, based on its loop tensor. At the critical state, the frequencies with which loops with different number of sides occur depend on the coordination number. At the critical state, these loops have, on average, the following universal characteristics, i.e. independent of the coordination number: (1) loops with the same number of sides and orientation have identical anisotropy of the loop tensor, (2) the anisotropy of the loop tensor depends linearly on the number of sides of the loop, (3) the distribution of loop orientations is identical, (4) Lewis’s law for the loop areas, which is a linear relation between the number of sides of loops and their area, is satisfied (not exclusively at the critical state) and (5) the areas of the loops do not depend on their orientation.     

Micromechanical behavior of granular materials with inherent anisotropy under cyclic loading using 2D DEM

This paper presents the micromechanical behavior of granular materials due to different initial inherent anisotropic conditions during cyclic loading using the discrete element method (DEM). Oval particles were used to model the samples. Three samples, with three different inherent anisotropic conditions based on the particle’s bedding direction, were prepared and subjected to biaxial cyclic loading. The differences in the inherent anisotropic conditions of the samples affect the stress–strain-dilative behavior of granular materials. The width of the stress–strain cyclic loops decreases as the preferred bedding angle changes from vertical to horizontal. Contact fabric evolution is found to be dependent on the inherent anisotropic fabric of the sample during loading and unloading. The fabric anisotropy is dominant for horizontal particle bedding at the end of loading and for vertical particle bedding at the end of unloading. A change in fabric anisotropy is observed only for the first few loading–unloading cycles for the given conditions depicted in the present study.     

Fabric,force and strength anisotropies in granular materials: a micromechanical insight

In micromechanics, the stress–force–fabric (SFF) relationship is referred to as an analytical expression linking the stress state of a granular material with microparameters on contact forces and material fabric. This paper employs the SFF relationship and discrete element modelling to investigate the micromechanics of fabric, force and strength anisotropies in two-dimensional granular materials. The development of the SFF relationship is briefly summarized while more attention is placed on the strength anisotropy and deformation non-coaxiality. Due to the presence of initial anisotropy, a granular material demonstrates a different behaviour when the loading direction relative to the direction of the material fabric varies. Specimens may go through various paths to reach the same critical state at which the fabric and force anisotropies are coaxial with the loading direction. The critical state of anisotropic granular material has been found to be independent of the initial fabric. The fabric anisotropy and the force anisotropy approach their critical magnitudes at the critical state. The particle-scale data obtained from discrete element simulations of anisotropic materials show that in monotonic loading, the principal force direction quickly becomes coaxial with the loading direction (i.e. the strain increment direction as applied). However, material fabric directions differ from the loading direction and they only tend to be coaxial at a very large shear strain. The degree of force anisotropy is in general larger than that of fabric anisotropy. In comparison with the limited variation in the degree of force anisotropy with varying loading directions, the fabric anisotropy adapts in a much slower pace and demonstrates wider disparity in the evolution in the magnitude of fabric anisotropy. The difference in the fabric anisotropy evolution has a more significant contribution to strength anisotropy than that of force anisotropy. There are two key parameters that control the degree of deformation non-coaxiality in granular materials subjected to monotonic shearing: the ratio between the degrees of fabric anisotropy and that of force anisotropy and the angle between the principal fabric direction and the applied loading direction.     

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.     

Stress-induced anisotropy in sand under cyclic loading

The anisotropy of a granular material’s structure will influence its response to applied loads and deformations. Anisotropy can be either inherent (e.g. due to depositional process) or induced as a consequence of the applied stresses or strains. Discrete element simulations allow the interactions between individual particles to be explicitly simulated and the fabric can be quantified using a fabric tensor. The eigenvalues of this fabric tensor then give a measure of the anisotropy of the fabric. The coordination number is a particle scale scalar measure of the packing density of the material. The current study examines the evolution of the fabric of a granular material subject to cyclic loading, using two-dimensional discrete element method (DEM) simulations. Isotropic consolidation modifies and reduces the inherent anisotropy, but anisotropic consolidation can accentuate anisotropy. The ratio of the normal to shear spring stiffness at the particle contacts in the DEM model affects the evolution of anisotropy. Higher ratios reduce the degree of anisotropy induced by anisotropic consolidation. The anisotropy induced by cyclic loading depends on the amplitude of the loading cycles and the initial anisotropy.     

Temporal variation of contact networks in granular materials

For analogue granular materials simulated using DEM, contact networks are often drawn by joining the centroids of contacting particles. Although a limited amount of research has been done to characterise the temporal aspects of such contact networks at a micro-scale, many simple questions regarding the duration of contacts in evolving granular systems remain unanswered. This paper addresses this gap in the existing knowledge by using the open-source code LAMMPS to run and analyse nine (3D) triaxial simulations of 20,164 polydisperse spherical particles. Contacts which exist for a long duration are preferentially oriented in the direction of the major principal stress while the fabric anisotropy for contacts with a given duration increases with duration. Regardless of simulation conditions, there is a linear relationship between the percentage of contacts present for a given duration and contact duration, despite the non-linear overall behaviour of the material. The forces transmitted by contacts increase, on average, with increasing duration. The total number of unique contacts which appear during triaxial compression increases linearly after 10 % axial strain, although the number of contacts, both total and subdivided into strong and weak force subnetworks, remains quite stable. The majority of contacts in these evolving granular systems, even those participating in strong force chains, are formed and lost repeatedly.     

Fabric rates of elliptical particle assembly in monotonic and cyclic simple shear tests: a numerical study

This paper aims to investigate the evolutions of microscopic structures of elliptical particle assemblies in both monotonic and cyclic constant volume simple shear tests using the discrete element method. Microscopic structures, such as particle orientations, contact normals and contact forces, were obtained from the simulations. Elliptical particles with the same aspect ratio (1.4 and 1.7 respectively for the two specimens) were generated with random particle directions, compacted in layers, and then precompressed to a low pressure one-dimensionally to produce an inherently anisotropic specimen. The specimens were sheared in two perpendicular directions (shear mode I and II) in a strain-rate controlled way so that the effects of inherent anisotropy can be examined. The anisotropy of particle orientation increases and the principal direction of particle orientation rotates with the shearing of the specimen in the monotonic tests. The shear mode can affect the way fabric anisotropy rate of particle orientation responds to shear strain as a result of the initial anisotropy. The particle aspect ratio exhibits quantitative influence on some fabric rates, including particle orientation, contact normal and sliding contact normal. The fabric rates of contact normal, sliding contact normal, contact force, strong and weak contact forces fluctuate dramatically around zero after the shear strain exceeds 4 % in the monotonic tests and throughout the cyclic tests. Fabric rates of contact normals and forces are much larger than that of particle orientation. The particle orientation based fabric tensor is harder to evolve than the contact normal or contact force based because the reorientation of particles is more difficult than that of contacts.     

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.     

Influence of particle shape on sheared dense granular media

We study by means of molecular dynamics simulations of periodic shear cells, the influence of particle shape on the global mechanical behavior of dense granular media. At large shear deformation samples with elongated particles, independent of their initial orientation, reach the same stationary value for both shear force and void ratio. At the micro-mechanical level the stress, the fabric and the inertia tensors of the particles are used to study the evolution of the media. In the case of isotropic particles the direction of the principal axis of the fabric tensor is aligned with the one of the principal stress, while for elongated particles the fabric orientation is strongly dependent on the orientation of the particles. The shear band width is shown to depend on the particle shape due to the tendency of elongated particles to preferential orientations and less rotation.     

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.     

Anisotropy in particle contacts associated with shearing in granular media

Summary Work on directional properties in granular media is briefly reviewed with an emphasis on the angular frequency of normals to particle contact tangents. Data from a two dimensional model material is compared with that of sand sheared in plane strain with a similar stress path. This stress path included a chosen sudden change in major principal stress direction.Directional stress-strain behaviour in granular media is related to the changing angular frequency of particle contacts. Measurements of this changing frequency are related to new model predictions and actual experimental data for sand sheared in plane strain.With 5 Figures     

Discrete element method investigation on thermally-induced shakedown of granular materials

Temperature change, as a common kind of internal perturbation performed on granular materials, has a significant effect on the bulk properties of granular materials. However, few studies on thermally-induced shakedown under a long-term thermal cycling were reported. In this work, the discrete element method was used to give insight into the thermally-induced shakedown on the fabric and stress states within non-cohesive, frictional granular assemblies. Assemblies were submitted to thermal cycling at a stationary boundary condition after experiencing a one-dimensional compression. Evolution of coordination number, entropy and anisotropy was investigated as well as boundary forces and contact forces. At the same time, effects of the heating rate, the initial vertical load and the magnitude of temperature change were examined. It demonstrates that thermal cycling induces a significant force relaxation within granular materials, while the corresponding granular fabric has a small change. In addition, the entropy and anisotropy decreases with thermal cycling. Moreover, the initial vertical load can constrain the development of thermally-induced fabric change, thereby limiting force relaxation to some degree. Both high heating rate and larger magnitudes of temperature change contribute to more significant force relaxation. However, they cause smaller fabric changes even though they provide larger perturbations.     

Collision dynamics of wet particles: Comparison of literature models to new experiments

Collision dynamics of wet particles are often investigated in literature, since their knowledge is important for the design and modeling of granular process involving liquid layers or moisture. Several models were already reported predicting rebound behavior of wet particles. However, most of them are either developed for a viscous dominant regime, neglecting capillary effects all together, or capillary effects are strongly simplified. This work summarizes the various models and compares them to new experimental results for liquids at small and moderate viscosities, to check for applicability also for these materials. Several discrepancies between experiments at small liquid viscosities and models were found and reasons for these differences are discussed. Mainly, the negligence or simplification of capillary forces regarding energy dissipation during the collision leads to an overprediction of rebound velocities compared to the experiments reported in this work. Furthermore, selection of appropriate models for viscous forces during wet particle collisions has to be conducted with care.     

Macro deformation and micro structure of 3D granular assemblies subjected to rotation of principal stress axes

This paper presents a numerical investigation on the behavior of three dimensional granular materials during continuous rotation of principal stress axes using the discrete element method. A dense specimen has been prepared as a representative element using the deposition method and subjected to stress rotation at different deviatoric stress levels. Significant plastic deformation has been observed despite that the principal stresses are kept constant. This contradicts the classical plasticity theory, but is in agreement with previous laboratory observations on sand and glass beads. Typical deformation characteristics, including volume contraction, deformation non-coaxiality, have been successfully reproduced. After a larger number of rotational cycles, the sample approaches the ultimate state with constant void ratio and follows a periodic strain path. The internal structure anisotropy has been quantified in terms of the contact-based fabric tensor. Rotation of principal stress axes densifies the packing, and leads to the increase in coordination numbers. A cyclic rotation in material anisotropy has been observed. The larger the stress ratio, the structure becomes more anisotropic. A larger fabric trajectory suggests more significant structure re-organization when rotating and explains the occurrence of more significant strain rate. The trajectory of the contact-normal based fabric is not centered in the origin, due to the anisotropy in particle orientation generated during sample generation which is persistent throughout the shearing process. The sample sheared at a lower intermediate principal stress ratio \((b=0.0)\) has been observed to approach a smaller strain trajectory as compared to the case \(b=0.5\), consistent with a smaller fabric trajectory and less significant structural re-organisation. It also experiences less volume contraction with the out-of plane strain component being dilative.     

Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr-Coulombium and beyond

Fluid-particle systems, in which internal forces arise only from viscosity or intergranular friction, represent an important special case of strictly dissipative materials defined by a history-dependent 4th-rank viscosity tensor. In a recently proposed simplification, this history dependence is represented by a symmetric 2nd-rank fabric tensor with evolution determined by a given homogeneous deformation. That work suggests an essential physical link between idealized suspensions (“Stokesium”) and granular media (“Mohr-Coulombium”) along with possible models for the visco-plasticity of fluid-saturated and dry granular media. The present paper deals with the elastoplasticity of dilatant non-cohesive granular media composed of nearly rigid, frictional particles. Based on the underlying physics and past modeling by others, a continuum model based on parametric hypoplasticity is proposed, which involves a set of rate-independent ODEs in the state-space of stress, void ratio and fabric. As with the standard theory of hypoplasticity, the present model does not rely on plastic potentials but, in contrast to that theory, it is based explicitly on positive-definite elastic and plastic moduli. The present model allows for elastic loading or unloading within a dissipative yield surface and also provides a systematic treatment of Reynolds dilatancy as a kinematic constraint. Some explicit forms are proposed and comparisons are made to previous hypoplastic models of granular media.