The simulation of particulate materials packing using a particle suspension model

The behavior of particulate composite materials, such as portland cement concrete, depends to a large extent on the properties of their main constituent—the aggregates. Among the most important parameters affecting the performance of concrete are the packing density and corresponding particle size distribution (PSD) of aggregates. Better packing of aggregates improves the main engineering properties of composite materials: strength, modulus of elasticity, creep and shrinkage. Further, it brings major savings due to a reduction in the volume of binder. A simulation algorithm was developed for the modeling of packing of large assemblies of particulate materials (of the order of millions). These assemblies can represent the real aggregate systems composing portland cement concrete. The implementation of the developed algorithm allows the generation and visualization of the densest possible and loose-packing arrangements of aggregates. The influence of geometrical parameters and model variables on the degree of packing and the corresponding distribution of particles was analyzed. Based on the simulation results, different PSDs of particulate materials are correlated to their packing degree.     

Computer simulation of morphology and packing behaviour of irregular particles, for predicting apparent powder densities

This paper describes a methodology for prediction of powder packing densities which employs a new approach, designated as random sphere construction (RSC), for modelling the shape of irregular particles such as those produced by water atomization of iron. The approach involves modelling an irregular particle as a sphere which incorporates smaller corner spheres located randomly at its surface. The RSC modelling technique has been combined with a previously developed particle packing algorithm (the random build algorithm), to provide a computer simulation of irregular particle packings. Analysis of the simulation output data has allowed relationships to be established between the particle modelling parameters employed by the RSC algorithm, and the density of the simulated packings. One such parameter is η, which is the number of corner spheres per particle. A relationship was established between η (which was found to have a profound influence on packing density), and the fractional density of the packing, fd. Vision system techniques were used to measure the irregularity of the simulated particles, and this was also related to η. These two relationships were then combined to provide a plot of fractional density for a simulated packing against irregularity of the simulated particles. A comparison was made of these simulated packing densities and observed particle packing densities for irregular particles, and a correlation coefficient of 0.96 was obtained. This relatively good correlation indicates that the models developed are able to realistically simulate packing densities for irregular particles. There are a considerable number of potential applications for such a model in powder metallurgy (PM), process control. In combination with on-line particle image analysis, the model could be used to automatically predict powder densities from particle morphology.     

Link between single-particle properties and macroscopic properties in particulate assemblies: role of structures within structures

The prevalence of particulate materials in modern industrial processes and products provides a significant motivation to achieve fundamental understanding of the bulk behaviour of particulate media. The rapid progress being made with atomic force microscopy and related particle characterization techniques pushes the limits of micro- and nanotechnologies such that interparticle interactions can be engineered to fabricate particulate assemblies to deliver specific functionalities. In this paper, primarily based on discrete element method simulations that we performed over the past 10 years, we summarize the key findings on the role of force transmission networks in dense particulate systems subjected to shearing. In general, the macroscopic strength characteristics in particulate systems is dictated by the distribution of heavily loaded contacts, also referred to as 'strong' force chains. Surprisingly, they constitute only a limited proportion of all contacts in particulate systems. They act like a 'granular brain' (memory networks) at particle scale. We show that the structural arrangement of the force chains and their evolution during loading depends on the single-particle properties and the initial packing condition in particulate assemblies. Further, the 'nature' of force chains in sheared granular media induces larger 'solid' grains to behave like 'fluid' particles, retarding their breakage. Later, we probe for ways by which we can control the signature of memory networks in packed beds, for example by applying an external electrical field in a densely packed particulate bed subjected to shearing (combined electromechanical loading). Though further research is required to account for more realistic conditions and preferably to allow particles to self-organize to strength specifications, understanding the hidden memory networks in particulate materials could be exploited to optimize their collective strength.     

A discrete element implementation for concrete: particle generation,contact calculations,packing under gravity and modeling material response

A three-dimensional discrete element modelling capability for concrete based on rigid particles of arbitrary shape and size has been developed. The novel particle generation algorithm allows control of particle size, angularity and flakiness. General rigid body kinematics including finite rotations is accounted for, and an explicit time integration algorithm that conserves energy and momentum is implemented. An efficient contact algorithm with several features to increase the efficiency of the contact computations has been developed. This enables the gravity packing problem for arbitrary shaped particles to be solved in reasonable run time. The proposed procedure is used to generate assemblies of concrete specimens of various sizes that are homogeneous and isotropic in the bulk, and can capture the wall effect due to the formwork. The calibrated specimens are seen to be capable of accurately capturing experimentally observed macro stress strain response and failure patterns. The influence of aggregate shape on texture formation in the packed specimen, and on macro strength and failure patterns in hardened concrete, is demonstrated and is seen to be consistent with experimental results.     

Characterization of voids in spherical particle systems by Delaunay empty spheres

The pore space of mono-sized spherical particle systems of increasing density is characterized by Delaunay empty spheres. Periodic packings of densities ranging from 0.57 up to 0.70 are generated numerically by symmetric vibration. The Voronoi diagrams of these packings are then computed with an algorithm based on the research of Delaunay empty spheres. The voids distribution and the tortuosity of packings as a function of density are studied. As the density increases, the voids distribution becomes more narrow. For partly ordered packings of high density, the voids distribution presents two peaks corresponding to the size of Delaunay empty spheres of perturbed fcc or hcp packings. The tortuosity of disordered packings decreases slowly with density. However, when the system becomes partly ordered, a large increase in tortuosity is observed.     

A method for the generation of 3D representative models of granular based materials

The morphology of many naturally occurring and man‐made materials at different length scales can be modelled using the packing of correspondingly shaped and sized particles. The mechanical behaviour of this vast category of materials – which includes granular media, particle reinforced materials and foams ‐ depends strongly upon the shape and size distribution of the particles. This paper presents a method for the generation and packing of arbitrarily shaped polyhedral particles. The algorithm for the generation of the particles is based on the Voronoi tessellation technique, whilst the packing is performed using a geometrical approach, which guarantees the non‐overlapping of the bodies without relying upon any, otherwise typically computationally expensive, contact detection and interaction algorithm. The introduction of three geometrical parameters allows to control the shape, size and spacial density of the polyhedral particles, which are used to build numerical models representative of densely packed granular assemblies, granular reinforced materials and closed‐cell foams. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysis of a triangulation based approach for specimen generation for discrete element simulations

Discrete element methods are emerging as useful numerical analysis tools for engineers concerned with granular materials such as soil, food grains, or pharmaceutical powders. Obviously, the first step in a discrete element simulation is the generation of the geometry of the system of interest. The system geometry is defined by the boundary conditions as well as the shape characteristics (including size) and initial coordinates of the particles in the system. While a variety of specimen generation methods for particulate materials have been developed, there is no uniform agreement on the optimum specimen generation approach. This paper proposes a new triangulation based approach that can easily be implemented in two or three dimensions. The concept of this approach (in two dimensions) is to triangulate a system of points within the domain of interest, creating a mesh of triangles. Then the particles are inserted as the incircles of these triangles. Extension to three dimensions using a mesh of tetrahedra and inserting the inspheres is relatively trivial. The major advantages of this approach include the relative simplicity of the algorithm and the small computational cost associated with the preparation of an initial particle assembly. The sensitivity of the characteristics of the particulate material that is generated to the topology of the triangular mesh used is explored. The approach is compared with other currently used methods in both two and three dimensions. These comparisons indicate that while this approach can successfully generate relatively dense two-dimensional particle assemblies, the three- dimensional implementation is less effective at generating dense systems than other available approaches. The research presented in this paper made use of software developed by other researchers. For the two-dimensional study the program Triangle developed by Jonathan Shewchuk was used. The three-dimensional analysis used the Geompack++ program developed by Barry Joe as well as an implementation of the Jodrey and Tory (1985) algorithm by Monika Bargiel and Jacek Moscinski called NSCP3D.     

Microstructure models for cellular materials

Laguerre tessellations generated by random sphere packings are promising models for the microstructure of cellular or polycrystalline materials. In this paper, the case of hard sphere packings with lognormal or gamma distributed volumes is investigated. The dependence of the geometric characteristics of the Laguerre cells on the volume fraction of the sphere packing and the coefficient of variation of the volume distribution is studied in detail. The moments of certain cell characteristics are described by polynomials, which allows to fit tessellation models to real materials without further simulations. The procedure is demonstrated by the examples of open polymer and aluminium foams.     

A comparison of data structures for the simulation of polydisperse particle packings

Simulation of particle packings is an important tool in material science. Polydisperse mixtures require huge sample sizes to be representative. Simulation, in particular with iterative packing algorithms, therefore requires highly efficient data structures to keep track of particles during the packing procedure. We introduce a new hybrid data structure for spherical particles consisting of a so‐called loose octree for the global spatial indexing and Verlet lists for the local neighbourhood relations. It is particularly suited for relocation of spheres and contact searches. We compare it to classical data structures based on grids and (strict) octrees. It is shown both analytically and empirically that our data structure is highly superior for packing of large polydisperse samples. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient dense and stable particular elements generation method based on geometry

In this paper, a new discrete elements generation method based on geometry is proposed to fill geometric domains with particles (disks or spheres). By generating particles each one with a random radius or with a radius calculated from the iteration to ensure no overlaps exist between particles and identifying unstable particles and changing them to stable ones, a dense and stable packing can be created. A partitioning particle radius interval method and a particle stability inspection and improvement method are introduced to guarantee the algorithm's success and the stability of the particles. Some packings were created to evaluate the performances of the new method. The results showed that the algorithm was very efficient and was able to create isotropic packings of low porosities and large coordinate numbers. The partitioning particle radius interval method improved the generation efficiency significantly and increased the packing densities. Through the comparisons with several existing methods proposed recently, the method proposed in this work is found to be more efficient and can fill geometric domains with the lowest porosities. In addition, the stability of the particles is guaranteed and no complex triangular or tetrahedral mesh is required in particle generation, thereby making the new method simpler. Copyright © 2016 John Wiley & Sons, Ltd.     

Micromechanical modelling of oval particulates subjected to bi-axial compression

One of the questions that still remain unanswered among researchers dealing with granular materials is how far the particle shape affects the micro-macroscopic features of granular assemblies under mechanical loading. The latest advances made with particle instrumentation allow us to capture realistic particle shapes and size distribution of powders to a fair degree of accuracy at different length scales. Industrial applications often require information on the micromechanical behaviour of granular assemblies having different particle shapes and varying surface characteristics, which still remains largely unanswered. Traditionally, simulations based on discrete element method (DEM) idealise the shape of individual particles as either circular or spherical. In the present investigation, we analyse the influence of particle shape on the shear deformation characteristics of two dimensional granular assemblies using DEM. We prepared the assemblies having nearly an identical initial packing fraction (dense), but with different basic shapes of the individual particles: (a) oval and (b) circular for comparison purposes. The granular assemblies were subjected to bi-axial compression test. We present the evolution of macroscopic strength parameters and microscopic structural/topological parameters during mechanical loading. We show that the micromechanical properties of granular systems are significantly influenced by the shape of the individual particles constituting the granular assemblies.     

A geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing

In the discrete element method, the packing generation of polydisperse spheres with a high packing density value is a major concern. Among the methods already developed, few algorithms can generate sphere packing with a high density value. The aim of this paper is to present a new geometric algorithm based on tetrahedral meshes to generate dense isotropic arrangements of non-overlapping spheres. The method consists of first filling in every tetrahedron with spheres in contact (i.e., hard-sphere clusters). Then, the algorithm increases the packing density value by detecting the large empty spaces and filling them with new spheres. This new geometric algorithm can also generate a complex shape structure.     

Effect of the particle size ratio on the structural properties of granular mixtures with discrete particle size distribution

In this study, the discrete element method was used to examine the structural properties and geometric anisotropy of polydisperse granular packings with discrete uniform particle size distributions. Confined uniaxial compression was applied to granular mixtures with different particle size fractions. The particle size fraction (class) was defined as the fraction of the sample composed of particles with a certain size. The threshold value of number of particle size fractions (i.e., the value above which structural properties of assemblies remain constant) was determined. The effect of heterogeneity in particle size on the critical value of number of particle size fractions was investigated for packings with different ratios between diameters of the largest and smallest grains. The threshold number of particle size classes decreased from five to three as the diameter ratio between the largest and smallest grains increased. Regardless of the diameter ratio, the critical number of particle size fractions (above which the packing density and coordination number of the granular mixtures remained constant) was determined to be five. The study has also shown an increase in packing density of binary mixtures with particle size ratio increasing up to 2.5, which was followed by decrease in density of mixtures with larger particle size ratios, which has not so far been reported in the literature.     

Experimental study on the packing densification of mixtures of spherical and cylindrical particles subjected to 3D vibrations

To identify the dense packing of cylinder–sphere binary mixtures (spheres as filling objects), the densification process of such binary mixtures subjected to three-dimensional (3D) mechanical vibrations was experimentally studied. Various influential factors including vibration parameters (such as vibration time t, vibration amplitude A, frequency ω, vibration acceleration Γ) as well as particle size ratio r (small sphere vs. large cylinder), composition of the binary mixtures X_L (volume fraction of cylinders), and container size D (container diameter) on the packing density ρ were systematically investigated. The results show that the optimal vibration parameters for different binary cylinder–sphere mixtures are different. The smaller the size ratio, the less vibration acceleration is needed to form a stable dense packing. For each binary mixture, high packing density can be obtained when the volume fraction of large cylindrical particles is dominant. Meanwhile, increasing the container size can decrease the container wall effect and get higher packing density. The proposed analytical model has been proved to be valid in predicting the packing densification of current cylinder–sphere binary mixtures.     

FREQUENCY DISTRIBUTION OF CONTACT ANGLES IN RANDOM PACKINGS OF GRANULAR MATERIALS

For a certain spectrum of stable grain configurations in randomly packed granular aggregates it is possible to determine the frequency distribution of relative contact angles among neighboring grains. This possibility is explored in the present paper for both two-dimensional and three-dimensional random packings of granular materials. Two relationships are first derived for the local void ratios of any stable “Voronoi Cell“ within the spectrum of stable configurations for the two and the three dimensional random packings, respectively. These relationships depend on the local distribution of relative contact angles, i.e., directions of contact normals. These local void ratios are then related to the gross void ratios of the random 2-D and 3-D assemblies through two integral equations of the Fredholm type of the first kind, their arguments being the frequency distribution functions. The first integral equation corresponding to a two-dimensional random disk packing is solved, exactly by a set of exponential functional transformations. These exact distributions are shown to be generally Maxwellian, with tails favorably biased towards the population of denser “Voronoi Cells” that are statistically more stable. A discussion on the uniform solutions of the integral equation for three-dimensional random packing of spheres is also presented. However, its general solution is left for a future work.     

New 3D geometrical deposition methods for efficient packing of spheres based on tangency

The morphology of many natural and man‐made materials at different length scales can be simulated using particle‐packing methods. This paper presents two novel 3D geometrical collective deposition algorithms for packed assemblies with prescribed distribution of radii: the ‘planar deposition’ and the ‘3D‐clew’ method. The ‘planar deposition’ method mimics an orderly granular flow through a funnel by stacking up spirally ordinated planar assemblies of spheres capable of achieving the theoretical maximum for monodisperse aggregates. The ‘3D‐clew’ method, instead, mimics the winding of a clew of yarn, thus yielding densely packed 3D polydispersed assemblies in terms of void ratio of the aggregate. The morphologies of such geometrically generated assemblies, achieved at several orders of magnitude reduced computational cost, are comparable with those consolidated uni‐directionally by means of discrete element method. In addition, significantly faster simulations of mechanical consolidation of granular media have been performed when relying upon the proposed geometrically generated assemblies as starting configurations. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal and effective technique for particle packing

In various fields ranging from manufacturing or pharmaceutical industry to agriculture or automotive industry air filters and -dryers filled with porous desiccant media are used for industrial air preparation. It is necessary to increase the mass of particulate material filled into the dryer for improving its efficiency. Practically this means decreasing porosity of the filler media. In order to achieve this, mechanical excitation (ensured by a vibration platform) is commonly used. However, applying outer excitation decreases the efficiency of filling procedure because of the high energy consumption of the vibratory device. To avoid unnecessary energy consumption it is recommended to examine the possibility of filling more particles into the canister without outer excitation. According to this in our present paper the gravitational deposition of particulate materials was investigated from viewpoint of porosity. Main idea of this research was the usage of the so called “snowstorm” filling technique. A conical hopper and under this multiple cylindrical rods were used to reach uniform deposition and high value of particle packing density. Thus efficiency of air preparation units may be increased as it is possible to fill more material into the container without additional energy usage compared to normal gravitational deposition.     

Mesoscopic particulate system assembled from three-dimensional irregular particles

Particles around us are generally in the form of irregular characteristics in shape and size. Establishing rational mesoscale models that are suitable for different types of particles is of great significance to comprehensively evaluate the mechanical properties of particulate systems assembled from irregular particles. The preponderances of previous works are mainly focused on particle simulations using regular-shaped geometries or simple polyhedrons, and little is known about quantitative characterization for the particles with complex shape characteristics. In this paper, a series of novel and efficient algorithms are presented to generate three-dimensional particulate systems assembled from particles with irregular characteristics in shape and size. According to the developed particle generation algorithm and vector-growth algorithm, the convex- and concave particles with different shape- and size configurations are obtained. Parametric analysis of corresponding algorithm parameters on the shape- and size configurations of particles are quantitatively investigated in terms of different indexes, such as sphericity, angularity and size, which provides an important guidance for the shape- and size control of irregular particles. Based on the generated irregular particles, a series of algorithms are proposed to generate the particulate systems with random spatial characteristics as well. Employing the developed compaction algorithm, mesoscopic particulate systems with different particle gradations and compactness are generated precisely. To sum up, the present particle model provides insights into capturing and studying the meso-structure characteristics and macroscopic mechanical properties of particulate systems.     

Computer Simulation of Random Sphere Packing in an Arbitrarily Shaped Container

Most simulations of random sphere packing concern a cubic or cylindric container with periodic boundary, containers of other shapes are rarely studied. In this paper, a new relaxation algorithm with pre-expanding procedure for random sphere packing in an arbitrarily shaped container is presented. Boundaries of the container are simulated by overlapping spheres which covers the boundary surface of the container. We find 0.4~0.6 of the overlap rate is a proper value for boundary spheres. The algorithm begins with a random distribution of small internal spheres. Then the expansion and relaxation procedures are performed alternately to increase the packing density. The pre-expanding procedure stops when the packing density of internal spheres reaches a preset value. Following the pre-expanding procedure, the relaxation and shrinking iterations are carried out alternately to reduce the overlaps of internal spheres. The pre-expanding procedure avoids the overflow problem and gives a uniform distribution of initial spheres. Efficiency of the algorithm is increased with the cubic cell background system and double link data structure. Examples show the packing results agree well with both computational and experimental results. Packing density about 0.63 is obtained by the algorithm for random sphere packing in containers of various shapes.