Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

A coupled sliding and rolling friction model for DEM calibration

The accuracy of dense Discrete Element Method (DEM) simulations is sensitive to initial density, contact orientation, particle size and shape, and interparticle interaction parameters including contact stiffness, cohesion, coefficients of friction, and coefficients of restitution. Although studies have characterized the effects of individual particle interaction parameters on mechanical responses of loaded granular material, research combining DEM parameters for calibration is scarce. Robust DEM calibration methodology combining sliding and rolling friction coefficients was developed and validated to predict bulk residual soil strength of initially dense DEM particle assemblies.     

Micro-mechanical analysis of deformation characteristics of three-dimensional granular materials

The deformation characteristics of idealized granular materials have been studied from the micro-mechanical viewpoint, using Bagi’s three-dimensional micro-mechanical formulation for the strain tensor [Bagi, K., 1996. Mechanics of Materials 22, 165–177]. This formulation is based on the Delaunay tessellation of space into tetrahedra. The set of edges of the tetrahedra can be divided into physical contacts and virtual contacts between particles. Bagi’s formulation expresses the continuum, macro-scale strain as an average over all edges, of their relative displacements (between two successive states) and the complementary-area vectors. This latter vector is a geometrical quantity determined from the set of edges, i.e. from the structure of the particle packing.Results from Discrete Element Method simulations of isotropic and triaxial loading of a three-dimensional polydisperse packing of spheres have been used to investigate statistics of the branch vectors and complementary-area vectors of edges (subdivided into physical and virtual contacts) and of the relative displacements of edges. The investigated statistics are probability density functions and averages over groups of edges with the same orientation. It is shown that these averages can be represented by second-order Fourier series in edge orientation.Edge orientations are distributed isotropically, contrary to contact orientations. The average lengths of the branch vectors and the normal component of the complementary-area vectors are distributed isotropically (with respect to the edge orientation) and their average values are related to each other and to the volume fraction of the assembly. The other two components of the complementary-area vector are zero on average.The total deformation of the assembly, as given by the average of the relative displacements of the edges of the Delaunay tessellation follows the uniform-strain prediction. However, neither the deformation of the physical contact network nor of the virtual contact network has this property. The average relative displacement of physical edges in the normal direction (determined by the branch vector) is smaller than that according to the uniform-strain assumption, while that of virtual contacts is larger. This is caused by the high interparticle stiffness that hinders compression. The reverse observation holds for the tangential component of the relative displacement vector. The contribution of the deformation of the empty space between physical contacts to the continuum, macro-scale strain tensor is therefore very important for the understanding and the prediction of the macro-scale deformation of granular materials.     

Micromechanical study of plasticity of granular materials

Plastic deformation of granular materials is investigated from the micromechanical viewpoint, in which the assembly of particles and interparticle contacts is considered as a mechanical structure. This is done in three ways. Firstly, by investigating the degree of redundancy of the system by comparing the number of force degrees of freedom at contacts with the number of governing equilibrium equations; Secondly, by determining the spectrum of eigenvalues of the stiffness matrix for the structure that is represented by the particles and their contacts; Thirdly, by investigating the evolution with imposed strain of the continuum elastic stiffness tensor of the system. It is found that, with increasing imposed strain, the degree of redundancy rapidly evolves towards a state with small redundancy, i.e. the system becomes nearly statically determinate. The spectrum of the system shows many singular and near-singular modes at peak shear strength and at large strain. The continuum elastic stiffness tensor becomes strongly anisotropic with increasing imposed strain and shows strong non-affinity of deformation. The assumption of a constant and isotropic elastic stiffness tensor in elasto-plastic continuum constitutive relations for granular materials is generally incorrect. Overall, the plastic continuum behaviour of granular materials originates from the plastic frictional behaviour at contacts and from damage in the form of changes in the contact network.     

ROTATIONAL RESISTANCE AND SHEAR-INDUCED ANISOTROPY IN GRANULAR MEDIA

This paper presents a micromechanical study on the behavior of granular materials under confined shear using a three-dimensional Discrete Element Method （DEM）. We consider rotational resistance among spherical particles in the DEM code as an approximate way to account for the effect of particle shape. Under undrained shear, it is found rotational resistance may help to increase the shear strength of a granular system and to enhance its resistance to liquefaction. The evolution of internal structure and anisotropy in granular systems with different initial conditions depict a clear bimodal character which distinguishes two contact subnetworks. In the presence of rotational resistance, a good correlation is found between an analytical stress-force-fabric relation and the DEM results, in which the normal force anisotropy plays a dominant role. The unique properties of critical state and liquefaction state in relation to granular anisotropy are also explored and discussed.     

Micromechanical study of elastic moduli of three-dimensional granular assemblies

In micromechanics of granular materials, relationships are investigated between micro-scale characteristics of particles and contacts and macro-scale continuum characteristics. For three-dimensional isotropic assemblies the macro-scale elastic characteristics are described by the bulk and the shear modulus, which depend on the micro-scale characteristics of the coordination number (i.e. the average number of contacts per particle) and the interparticle contact stiffnesses in directions normal and tangential to the contact.     

Micromechanical study of elastic moduli of loose granular materials

In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the average number of contacts per particle) and interparticle contact stiffnesses in normal and tangential directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions of the elastic moduli, especially for loose systems with low coordination number. Improved predictions of the moduli are obtained here for loose two-dimensional, isotropic assemblies. This is achieved by determining approximate displacement and rotation fields from the force and moment equilibrium conditions for small sub-assemblies of various sizes. It is assumed that the outer particles of these sub-assemblies move according to the mean field. From the particle displacement and rotation fields thus obtained, approximate elastic moduli are determined. The resulting predictions are compared with the true moduli, as determined from the discrete element method simulations for low coordination numbers and for various values of the tangential stiffness (at fixed value of the normal stiffness). Using this approach, accurate predictions of the moduli are obtained, especially when larger sub-assemblies are considered. As a step towards an analytical formulation of the present approach, it is investigated whether it is possible to replace the local contact stiffness matrices by a suitable average stiffness matrix. It is found that this generally leads to a deterioration of the accuracy of the predictions. Many micromechanical studies predict that the macroscopic bulk modulus is hardly influenced by the value of the tangential stiffness. It is shown here from the discrete element method simulations of hydrostatic compression that for loose systems, the bulk modulus strongly depends on the stiffness ratio for small stiffness ratios.     

基于细观拓扑结构演化的颗粒材料剪胀性分析

剪胀性是包括岩土材料在内的摩擦性颗粒材料的重要特征之一,其形成机制与颗粒体系内部拓扑结构的演化有关.基于颗粒体系细观数据,可对颗粒体系内部的拓扑结构特征及演化进行分析,进而建立拓扑演化与宏观剪胀变形之间的联系.采用离散单元法,根据密实、中密和松散摩擦性颗粒材料双轴试验的宏微观数据,从拓扑参量演化及接触网络拓扑变化所引起...     

Micro-scale behavior of granular materials during cyclic loading

This study presents the micro-scale behavior of granular materials under biaxial cyclic loading for different confining pressures using the two-dimensional (2D) discrete element method (DEM). Initially, 8450 ovals were generated in a rectangular frame without any overlap. Four dense samples having confining pressures of 15, 25, 50, and 100 kPa were prepared from the initially generated sparse sample. Numerical simulations were performed under biaxial cyclic loading using these isotropically compressed dense samples. The numerical results depict stress–strain–dilatancy behavior that was similar to that observed in experimental studies. The relationship between the stress ratio and dilatancy rate is almost independent of confining pressures during loading but significantly dependent on the confining pressures during unloading. The evolution of the coordination number, effective coordination number and slip coordination number depends on both the confining pressures and cyclic loading. The cyclic loading significantly affects the microtopology of the granular assembly. The contact fabric and the fabric-related anisotropy are reported, as well. A strong correlation between the stress ratio and the fabric related to contact normals is observed during cyclic loading, irrespective of confining pressures.     

Micro-scale behavior of granular materials during cyclic loading

This study presents the micro-scale behavior of granular materials under biaxial cyclic loading for differ- ent confining pressures using the two-dimensional （2D） discrete element method （DEM）. Initially, 8450 ovals were generated in a rectangular frame without any overlap. Four dense samples having confining pressures of 15, 25, 50, and 100 kPa were prepared from the initially generated sparse sample. Numeri- cal simulations were performed under biaxial cyclic loading using these isotropically compressed dense samples. The numerical results depict stress-strain-dilatancy behavior that was similar to that observed in experimental studies. The relationship between the stress ratio and dilatancy rate is almost indepen- dent of confining pressures during loading but significantly dependent on the confining pressures during unloading. The evolution of the coordination number, effective coordination number and slip coordina- tion number depends on both the confining pressures and cyclic loading. The cyclic loading significantly affects the microtopology of the granular assembly. The contact fabric and the fabric-related anisotropy are reported, as well. A strong correlation between the stress ratio and the fabric related to contact normals is observed during cyclic loading, irrespective of confining pressures.     

考虑颗粒转矩的接触网络诱发各向异性分析

颗粒材料的宏观力学行为与接触网络的组构各向异性密切相关, 根据接触点的滑动与否、转动与否和强弱力情况, 可以将颗粒间的接触系统分为不同的子接触网络. 一般而言, 不同的子接触网络在颗粒体系中的传力机制不同, 对宏观力学响应的贡献也有不同. 采用离散单元法(discrete element method, DEM)模拟了不同抗转动系数$\mu_r$下颗粒材料三轴剪切试验, 分析了剪切过程中不同子接触网络的组构张量的演变规律, 并探究了颗粒抗转动效应对子接触网络各向异性指标演变规律的影响. 研究发现: 剪切过程中转动、非转动接触的组构张量变化不是独立的, 受到颗粒间滑动与否的影响; 非滑动、强接触网络是颗粒间的主要传力结构, 非滑动接触网络的接触法向和法向接触力各向异性均随$\mu_r$的增大而增大, 其对宏观应力的贡献程度随$\mu_r$的增大而减小;强接触网络的接触法向各向异性随$\mu_r$的增大而增大, 但法向接触力各向异性随$\mu_r$的增大无明显变化, 强接触网络对宏观应力的贡献程度在不同$\mu_r$情况下均相同.      

Analysis of three-dimensional micro-mechanical strain formulations for granular materials: Evaluation of accuracy

An important objective of recent research on micro-mechanics of granular materials is to develop macroscopic constitutive relations in terms of micro-mechanical quantities at inter-particle contacts. Although the micro-mechanical formulation of the stress tensor is well established, the corresponding formulation for the strain tensor has proven to be much more evasive, still being the subject of much discussion. In this paper, we study various micro-mechanical strain formulations for three-dimensional granular assemblies, following the work of Bagi in two dimensions (Bagi, 2006). All of these formulations are either based on an equivalent continuum approach, or follow the best-fit approach. Their accuracy is evaluated by comparing their results, using data from Discrete Element Method simulations on periodic assemblies, to the macroscopic deformation. It is found that Bagi’s formulation (Bagi, 1996), which is based on the Delaunay tessellation of space, is the most accurate. Furthermore, the best-fit formulation based on particle displacements only did unexpectedly well, in contrast to previously reported results for two-dimensional assemblies.     

Analysis of local behaviour in granular materials

A local scale, called the meso-scale, has recently been introduced to the multi-scale approach for 2D granular materials. This local scale is defined at the level of meso-domains enclosed by particles in contact. Stress and strain have been defined at this local scale, and their relation with the local structure has been studied. The purpose of this paper is to analyse the behaviour of granular materials at the meso-scale, i.e. the stress–strain–structure relationship at this scale. Analyses are performed on a 2D numerical granular sample subjected to a biaxial compression test and simulated with the Discrete Element Method (DEM). The sample is quite dense and it is loaded at a relatively low strain rate so that the state of the sample can be considered as being quasi-static. The size of sub-domains in the sample varies largely from 3 to 12 particles. It is shown that the evolution of the internal state of the sample corresponds, at the meso-scale, to a clear evolution of the quantity of meso-domains oriented in different directions. In addition, the behaviour of meso-domains is highly governed by their orientation rather than their density, especially for the strongly elongated meso-domains: the meso-domains oriented in the compression (resp. extension) direction behave like a dense (resp. loose) granular material.     

Study of particle rearrangement during powder compaction by the Discrete Element Method

This paper presents simulations of cold isostatic and closed die compaction of powders based on the Discrete Element Method. Due to the particulate nature of powders, densification of the compact proceeds both through the plastic deformation at the particle contact and the mutual rearrangement of particles. The relative weight of each mechanism on the macroscopic deformation process depends on the contact law, the relative density, and the type of stress exerted on the particles (shear or pressure). 3D computer simulations have been carried out to investigate the role of these parameters on the deformation mechanisms of powder compacts. The effect of rearrangement is studied by comparing simulations that use a homogeneous strain field solution for which local rearrangement is omitted and simulations that include local rearrangement. It is shown that local rearrangement has some effect on average quantities such as the average coordination number, the average contact area and the macroscopic stress. The effect on averaged quantities is much stronger for closed die compaction than for isostatic compaction. However the main effect of local rearrangement is to widen the distribution of the parameters that define the contact (contact area in particular). The results of these simulations are compared to available experimental data and to statistical models that use a homogeneous strain field assumption.     

DEM investigation on the effect of sample preparation on the shear behavior of granular soil

The effect of initial fabric anisotropy produced by sample preparation on the shear behavior of granular soil is investigated by performing discrete element method (DEM) simulations of fourteen biaxial tests in drained conditions. Numerical test specimens are prepared by three means: gravitational deposition, multi-layer compression, and isotropic compression, such that different initial inherent soil fabrics are created. The DEM simulation results show that initial fabric anisotropy exerts a considerable effect on the shear behavior of granular soil, and that the peak stress ratio and peak dilatancy increase with an increase in the fabric index a_n that is estimated from the contact orientations. The stress–dilatancy relationship is found to be independent of the initial fabric anisotropy. The anisotropy related to the contact orientation and contact normal force accounts for the main contribution to the mobilized friction angle. Also, the occurrence of contractive shear response in an initial shearing stage is accompanied by the most intense particle rearrangement and microstructural reorganization, regardless of the sample preparation method. Furthermore, the uniqueness of the critical state line in e–log p′ and q–p′ plots is observed, suggesting that the influence of initial fabric anisotropy is erased at large shear strains.     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates. Three different systems have been considered. In the first system the interaction between particles has been simulated using the JKR （Johnson, Kendall and Roberts） contact theory, while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively. In order to compare the mechanical behaviour of the three systems, the magnitude of the maximum attractive force between particles has been kept the same in all cases. However, the relationship between force and separation distance differs from case to case and thus, the range of the interparticle force. The results clearly indicate that as the range of the interparticle force increases, the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

Methods for incorporating particle rearrangement into compaction using thermodynamic approaches

This paper presents a novel, yet thermodynamically consistent, model of the isothermal compaction of loose granular material based on the principle of maximum dissipation rate. The method is first tested out on a simple version of the Bingham model and a hard particle model of rate-independent granular flow where it is seen that only the dissipation function and dilatancy rule are required in either case and the procedures are identical. This hard particle model is subsequently modified by the introduction of damage. Yield surface and flow rules are produced that are broadly in accordance with experimental findings. The key to the above modification is the concept of a dilatancy rule with two contributions. (1) A shear induced negative dilatancy, where any shear deformation has a tendency to produce densification. (2) Under many circumstances, this is countered by positive dilatancy such as at the critical state where the two mechanisms balance. This modification uses the idea that the first contribution is encouraged by microscopic damage local to the particle contacts that might permit compaction to occur under hydrostatic pressure alone. A mechanism is postulated whereby shear stresses operating at the microscopic level, while cancelling out at the macroscopic level, might occur with low levels of damage but produce no overall shear strains.     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates.Three different systems have been considered.In the first system the interaction between particles has been simulated using the JKR (Johnson,Kendall and Roberts) contact theory,while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively.In order to compare the mechanical behaviour of the three systems,the magnitude of the maximum attractive force between particles has been kept the same in all cases.However,the relationship between force and separation distance differs from case to case and thus,the range of the interparticle force.The results clearly indicate that as the range of the interparticle force increases,the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

Determination of contact parameters for discrete element method simulations of granular systems

Both linear-spring-dashpot (LSD) and non-linear Hertzian-spring-dnshpot (HSD) contact models are commonly used for the calculation of contact forces in Discrete Element Method (DEM) simulations of granular systems.Despite the popularity of these models, determination of suitable values for the contact parameters of the simulated particles such as stiffness, damping coefficient, coefficient of restitution, and simulation time step,is not altogether obvious.In this work the relationships between these contact parameters for a model system where a particle impacts on a flat base are examined.Recommendations are made concerning the determination of these contact parameters for use in DEM simulations.