Examination of the Response of Regularly Packed Specimens of Spherical Particles Using Physical Tests and Discrete Element Simulations

Significant insight into the response of granular materials can be gained by coupling accurately controlled physical tests with complementary discrete element simulations. This paper discusses a series of triaxial and plane strain laboratory compression tests on steel spheres with face-centered-cubic and rhombic packings, as well as discrete element simulations of these tests. The tests were performed on specimens of uniform-sized steel balls and on specimens of steel balls with specified distributions of ball diameters. The packing configurations are ideal and differ considerably from real sand specimens, however, studies of such idealized granular materials can yield considerable insight into the response of granular materials and the capability of discrete element simulations to capture the response. The differences in response for the two packing configurations considered illustrate the importance of fabric. The numerical simulations captured the observed laboratory response well if the particle configurations, particle sizes, and boundary conditions were accurately represented. However, the postpeak response is more difficult to capture, and it is shown to be sensitive to the coefficient of friction assumed along the specimen boundaries. The simulations of the tests on the nonuniform-sized specimens demonstrated a clear correlation between strength and coordination number.     

Capturing Nonspherical Shape of Granular Media with Disk Clusters

In discrete numerical modeling of granular materials, idealization of individual particles is required, as it is not practical to model a large number of particles, each with its actual shape and size. To minimize computation times, researchers often use two-dimensional, circular elements. However, biaxial and direct shear tests on such specimens result in low strengths compared to granular materials, due, in part, to excessive rolling of the perfectly circular particles. In this paper, a new particle type, disk clusters, is presented. A disk cluster is a group of circular disks permanently connected to form an irregularly shaped particle that more closely represents the shape of granular materials and has less tendency to rotate. Development and implementation of disk cluster particles into a discontinuous deformation analysis program is presented. Validations of the mechanics of a single disk cluster, biaxial shear, and anchor pullout simulations illustrate the usefulness of this new particle type.     

Influence of Particle Shape and Surface Friction Variability on Response of Rod-Shaped Particulate Media

Discrete element methods are important tools for the investigation of the mechanics of granular materials. In two dimensions, the reliability of these numerical approaches can be explored using physical tests on rod assemblies. This work highlights the importance of representing the actual distribution of rod shapes and surface friction in numerical simulations. The sensitivity of the response of hexagonally packed rods to minor changes in particle geometry and friction is investigated using a combination of laboratory tests and discrete element simulations. Laboratory test results highlight the influence of small variations in rod geometry on the global response, with the peak friction angle decreasing significantly as the standard deviation of the rod size distribution increased. Small changes in rod shape are also seen to be important. The numerical simulations indicate that the peak friction angle decreases as the standard deviation of the distribution of particle surface friction increases. This paper illustrates the way in which laboratory tests and numerical simulations can be used in a complementary manner to better understand the micromechanics of the response of granular materials.     

3D Shape Characterization and Image-Based DEM Simulation of the Lunar Soil Simulant FJS-1

This paper describes a procedure used to characterize the three-dimensional (3D) grain shape of lunar soil and undertake simulations of lunar soil by image-based discrete element method (DEM). Given that detailed 3D grain-shape information is unavailable for real lunar soil, a simulant material, FJS-1, is used in this study. We use the high-resolution micro X-ray CT system at SPring-8, a synchrotron radiation facility in Japan, to visualize precise 3D images of the granular assembly of FJS-1. A newly developed image-analysis procedure is then applied to identify individual grains. Using the obtained grain-shape data, a sufficient number of FJS-1 grains are directly modeled for DEM simulation using an efficient modeling scheme. A series of particle flow simulations are then performed with the modeled grains. The resulting slope angles are in good agreement with experimental results. We discuss the effect on the slope angle of grain parameters such as contact stiffness, restitution coefficient, and interparticle friction.     

Granular Temperature

Granular materials consist of a large number of small particles arranged in a random way. However, while the geometrical organization of an individual particle assembly is complex, these materials can nevertheless be well characterized by a small number of parameters. Systems of this type are the subject matter of thermodynamics, and so it seems reasonable to suppose that the methods employed in thermodynamics may be able to be applied to characterizing the behavior of particle assemblies. This paper describes an extension of classic thermodynamics to granular assemblies subjected to an energy flux through the granular assembly. By means of a Carnot engine, the granular temperature and entropy of a particle assembly is defined. The principal value of introducing the notion of granular temperature is that it defines a physically realistic internal variable for the granular assembly, which controls the evolution of the system after it is has been perturbed from its equilibrium state. The concept of granular thermodynamic equilibrium is then extended to non‐equilibrium granular thermodynamics. Chemical kinetic theory is employed to describe the relaxation of a granular assembly after it is perturbed by a sudden change in granular temperature. A particle vibration theory has been introduced to explain the behavior of soil subject to a white noise energy flux. While the theory described provides quantitative information about soil behavior, just as importantly, the theory provides a new qualitative insight into soil behavior.     

Effect of Particle Shape on Interface Behavior of DEM‐Simulated Granular Materials

This paper presents a detailed computational investigation of the effect of particle shape on the interface shear behavior of granular materials. The discrete element method (DEM) using clusters to model rough particles is used, expanding the procedure introduced in an earlier paper by Jensen et al. [1]. Seven new cluster shapes (i.e., particle configurations) of varying degrees of roughness are presented herein, and numerical experiments simulating ring shear tests are made using these clusters. From these simulations, the effect of particle shape on void ratio (e) and interface angle of friction between soil and structure surface (δ) is reported. Particle shape characteristics include roundness, angularity, and surface roughness. The results of numerical simulations using the newly formed cluster shapes are in very good qualitative agreement with laboratory tests. Simulation results showed that the void ratio of a particle mass increased as the angularity or roughness of the particles increased. They also showed an increase in interface shear strength between perfectly round DEM particles and the more angular cluster shapes, but no systematic correlations with the various definitions of particle shape parameters was found. It may be necessary to use greater accuracy in modeling the size and shape distributions of a natural medium to further investigate the influence of particle shape on interface friction. The simulations also successfully reflected the relationship between interface friction angle and structure surface roughness as demonstrated in recent physical experiments. The simulations comparing initially “dense” media to initially “loose” media demonstrated behavior that is similar to the behavior of a natural sandy soil observed in experiments.     

Input Parameters of Discrete Element Methods

This paper presents a sensitivity analysis of the input parameters of a program based on the discrete element method (DEM). Triaxial compression simulations were conducted on an assembly of ellipsoids with two particle shapes. We examine four input parameters including shear modulus of particles, density of particles, time step, and damping. Generally, these parameters are chosen by calibrating the result with certain known behavior of granular materials. In dynamic simulations, these input parameters are bounded by their physical attributions that should not be altered. However, in static simulations, they do not have the same physical implication. Validity of results may be questionable when input parameters are used without justification. A sensitivity analysis of the input parameters should shed light on this issue. In this paper, we will study the effect of the input values within the range of (1/10)–10 times the benchmark value. The benchmark values are commonly used by the writer. The results are presented against the benchmark simulation. The unbalanced forces in the simulations are kept below a prescribed value to enforce equilibrium. The result shows that the effect of all input parameters used in this paper is negligible as long as the small unbalanced forces in the system can be achieved. The runtimes are different. However, there are two simulations (one with low damping and the other with a large time step size) that cannot maintain the required small unbalanced force. In other words, equilibrium cannot be achieved for these simulations.     

Ice Boom Simulations and Experiments

A three-dimensional discrete element model (DEM) was developed to simulate ice boom operation in a rectangular channel. The model simulates the motion of each individual ice floe, the interaction between adjacent floes, the interaction of the floes with the walls and boom, and the water drag applied to the floes on the underside of the ice accumulation. The DEM simulations were compared with a parallel set of physical model tests using natural ice. The DEM successfully reproduced the observed magnitude and distribution of the forces on the boom and the channel sides as the boom retained a surge of drifting ice. Variations in channel side roughness produced similar changes in the division of forces between the boom and sidewalls in the simulations and model tests. Finally, the load distribution between the boom and the channel sides and the effect of channel side roughness in the context of granular ice-jam theory were analyzed.     

Critical State of Granular Materials Based on the Sliding-Rolling Theory

By representing the assembly by a simplified column model, a constitutive theory was recently developed for a two-dimensional assembly of rods. This theory, referred to as the sliding-rolling theory, is extended in this paper to represent the triaxial stress-strain behavior of granular materials. The sliding-rolling theory provides a dilatancy rule and an expression for the slope of the line of zero dilatancy in the stress space. These rules are then combined with triaxial observations to provide a microstructural interpretation of the critical state of granular materials. According to the theory, the slope of the critical state line in the stress space depends on the interparticle friction angle and the degree of contact normal anisotropy. To verify the basic ideas of the sliding-rolling theory, numerical experiments are conducted using the discrete-element method on three-dimensional assemblies of spheres.     

Quantification and Simulation of Particle Kinematics and Local Strains in Granular Materials Using X-Ray Tomography Imaging and Discrete-Element Method

Microfeatures of granular materials have significant effects on their macrobehaviors. Unfortunately, three-dimensional (3D) quantitative measurements of microfeatures are rare in literature because of the limitations of conventional techniques in obtaining microquantities such as microdisplacements and local strains. This paper presents a new method for quantifying the particle kinematics and local strains for a soft confined compression test using X-ray computed tomography and compares the experimental measurements with the simulated results using the discrete-element method (DEM). The experimental method can identify and recognize 3D individual particles automatically, which is essential for quantifying particle kinematics and local strains. 3D DEM simulations of the soft confined compression test were performed by using spherical particles and irregular particles. The simulated global deformations and particle translations that were based on irregular particles showed better agreement with the experimental measurements than those that were based on spherical particles. The simulated movements of spherical particles were more erratic, and the material composed of spherical particles showed larger vertical contraction and radial dilation.     

Micromechanical Modeling of a Dump Material

Micromechanical modeling of a fragmented claystone—a difficult waste material produced by open‐pit coal mines in Northwestern Bohemia—is presented in this article. The PFC2D code, which accounts for the discrete nature of geomaterials and represents them as an assembly of unbonded or bonded particles, has been used. First, synthetic claystone was generated and the deformability and strength parameters were calibrated via numerical testing and comparing the results with those of available laboratory and field tests. The pre‐peak, peak, and post‐peak behavior of synthetic claystone was studied, and microscopic indicators of macroscopic behavior were selected and visualized. In order to simulate the dump material, joints were introduced that divided the claystone specimen into fragments. The appropriate microproperties of synthetic dump material were selected by means of a similar numerical testing and calibration procedure. Distribution and redistribution of particle contact forces before, during, and after failure of the dump material specimen were visualized and velocities corresponding to strain localization plotted. According to the study and some previous references, the compressive contact force chain acting in the direction of major principal stress appears as a backbone of microstructure, and compression induced tension as its basic failure mode at particle level.     

Specimen Size Effect in Discrete Element Simulations of Granular Assemblies

The paper addresses the question of whether the number of particles in a noncemented granular assembly will affect the mechanical characteristics of the assembly: its strength and stiffness. The question is answered by applying the discrete element method to assemblies of different sizes. To isolate the effect of assembly size, apart from the scatter that usually accompanies such simulations, multiple assemblies were tested. The two-dimensional assemblies had nearly identical initial porosities and fabrics, and they were all loaded in biaxial compression. Two different boundaries were tested: periodic and wall boundaries. We find that the peak compressive strength decreases with assembly size for both types of boundaries and over a range of assembly sizes that contain 256 particles to over 66,000 particles. Stiffness is only slightly reduced and only with wall boundaries. Deformation is less uniform in the larger assemblies, with deformation concentrated in a smaller fraction of the assembly area. An analysis of deformation patterning shows that at least a few thousand particles are required for realistic microband patterning.     

Coupled Continuum-Discrete Model for Saturated Granular Soils

A coupled hydromechanical model was used to analyze the mesoscale pore fluid flow and microscale solid phase deformation of saturated granular soils. The fluid motion was idealized using averaged Navier–Stokes equations, and the discrete element method was employed to model the assemblage of solid particles. The fluid–particle interactions were quantified using established semiempirical relationships. Simulations were conducted to investigate the three-dimensional response of sandy deposits when subjected to critical and overcritical upward pore fluid flow. These simulations revealed complex response patterns after the onset of quicksand conditions and provided valuable insight into the associated mechanisms. The employed model provides an effective tool to assess the microscale mechanisms and characteristics of the partially drained response of saturated granular media.     

DEM Simulation of Particle Damage in Granular Media — Structure Interfaces

Using the discrete element method (DEM) with clustering, a novel means of numerically modeling damage of particles is presented. Damage, such as grain crushing, is treated by allowing clusters to break apart according to a failure criterion based upon sliding work. If the accumulated work done on an individual DEM particle of a cluster exceeds a threshold, that particle is allowed to break from the cluster. A value for the critical energy density is determined by comparing the degree of particle breakage from numerical simulations to data from laboratory tests. Numerical simulations were also conducted to determine the impact of particle damage on interface behavior. It was found that a very distinct shear zone was evident when particle damage was considered and that this occurred without significant reduction of the maximum shear strength of the medium. Also, the degree of damage was shown to be related to the angularity of the clusters.     

Particulate flow analysis in inclined pipes and transfer chutes using tomography imaging,discrete element simulations and continuum modeling approaches

In particulate material transfer systems,traditional shear test based steady state analysis can provide some insight into the strength of the bulk material and subsequent resistive frictional forces during flow.For fast flowing transfer points,dynamic flow conditions dominate and additional modelling techniques are required to improve design guidance.The research presented shows the evolution of a design solution which utilises two distinct processes;a continuum method and a discrete element method(DEM). Initially,the internal structure of dense granular flow,down vertical and inclined pipes was investigated using a twin sensor,12 electrode electrical capacitance tomography device.Subsequently,DEM simulations were conducted using the commercial software,PFC3D.Initially,two particle types and their flow behaviours were analysed:plastic pellets and sand.The pipe angle was varied between 0°and 45°to the vertical.For both the plastic pellets and the sand,good qualitative agreement was found with the spatial particle concentration analysis.Generally,the flow had a dense particle region at its core with the particle concentration reducing away from this core.As expected,at 0°, the core was centrally located within the pipe for both the plastic pellets and sand.At pipe angles 5°or greater,the dense core of particles was located on or near the pipe wall.Average flow velocity analysis was also conducted using the results of wall friction test analysis.The velocity comparisons also showed good agreement between the ECT image analysis and the DEM simulations. Subsequently,the DEM method was used to analyse a complex transfer system(or chute) with the continuum method providing comparative flow analysis with the DEM flow analysis.     

Evaluation of Two Homogenization Techniques for Modeling the Elastic Behavior of Granular Materials

This paper discusses the capabilities of two homogenization techniques to accurately represent the elastic behavior of granular materials considered as assemblies of randomly distributed particles. The stress-strain relationship for the assembly is determined by integrating the behavior of the interparticle contacts in all orientations, using two different homogenization methods, namely the kinematic method and the static method. The numerical predictions obtained by these two homogenization techniques are compared to results obtained during experimental studies on different granular materials. Relations between elastic constants of the assembly, interparticle properties, and fabric parameters are discussed, as well as the capabilities of the models to take into account inherent and stress-induced anisotropy for different stress conditions.     

Calibration of the SMCS Criterion for Ductile Fracture in Steels: Specimen Size Dependence and Parameter Assessment

The stress modified critical strain (SMCS) criterion provides a local index for the initiation of ductile fracture in metals as a function of plastic strain and stress triaxiality. Previous research has confirmed the SMCS criterion to be an accurate index for fracture initiation in mild steels and demonstrated its application to civil/structural engineering. To facilitate practical implementation of the SMCS criterion, two key aspects of its calibration for steel materials are examined. The first pertains to the sensitivity of the measured SMCS material toughness parameter to the size of the test coupon. New results from 23 tests of cylindrically notched tension (CNT) specimens of various sizes and notch geometries indicate that the toughness parameter is relatively insensitive to calibration specimen size. This finding validates the use of miniature bar specimens to calibrate the SMCS model for thin plate steels and in-service structures, where extraction of larger coupons is impossible. The second aspect involves the development of closed-form expressions to determine directly the SMCS toughness parameter from CNT tests, thus avoiding the need for interpretation of the test data through finite-element simulations. Based on the results of 54 numerical simulations, encompassing a range of material constitutive properties, specimen geometries, and applied deformations, a semiempirical relationship (based in part on Bridgman’s solution for necked tension rods) is proposed to determine the toughness parameter directly from the CNT bar tests.     

Anisotropic Nonlinear Elastic Model for Particulate Materials

This paper presents the development of an elastic model for particulate materials based on micromechanics considerations. A particulate material is considered as an assembly of particles. The stress–strain relationship for an assembly can be determined by integrating the behavior of the interparticle contacts in all orientations and using a static hypothesis which relates the average stress of the granular assembly to a mean field of particle contact forces. Hypothesizing a Hertz–Mindlin law for the particle contacts leads to an elastic nonlinear behavior of the particulate material, we were able to determine the elastic constants of the granular assembly based on the properties of the particle contacts. The numerical predictions, compared to the results obtained during experimental studies on different granular materials, show that the model is capable of taking into account both the influence of the inherent anisotropy and the influence of the stress-induced anisotropy for different stress conditions.     

Three-dimensional reconstruction: methods of improving image registration and interpretation

(i) Image registration. The use of serial images for computerised three-dimensional reconstruction necessitates the inclusion of three separate sources of information at the stage of data input. These are (i) artificial registration points or fiducials, (ii) a calibration scale and (iii) an outline of each slab of the section to be included in the reconstruction. Most traditional methods rely on the production of drawings of the contours of the structure under investigation which also include both registration points and a calibration scale. We report on a method which considerably reduces the time involved at this labour intensive stage of reconstruction and in addition allows subsequent reconstructions of different structures to be performed without new drawings. Use is made of computerised alignment of tissue sections and the production of composite photomicrographs of both the tissue under investigation and an accurately registered stage micrometer scale. (ii) Improving image interpretation. Images derived from computerised three-dimensional reconstruction can be affected by the number of coordinates used to form the contour of each slice of a structure and by the number of slices that are used to construct the final model. Too little or too much data can considerably reduce the ability of the observer to interpret accurately the image generated by the computer. We report on a feature-based method which enables the experimenter to assess objectively the amount of data required in the two-dimensional plane, i.e. the number of data points per slice, and the three-dimensional plane, i.e. number of slices per structure, so that optimal reconstructions are generated.     

Three-Dimensional Digital Representation of Granular Material Microstructure from X-Ray Tomography Imaging

This paper presents the use of x-ray tomography imaging to reconstruct a three-dimensional (3D) digital representation of individual particles in a granular system. The granular system is represented by the mass center coordinates and the morphology representation of each particle. An automated procedure using pattern recognition to identify related particle cross sections in adjacent serial images was developed. Procedures to calculate quantities needed for subsequent simulation of particle behavior including the volume and the momentum of inertia of each particle are also presented. The developments described in the paper enable modeling and simulation of the behavior, and experimental observations of the particle kinematics of real microstructures of granular materials in a true 3D platform.