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.     

Fluid–solid transition in unsteady,homogeneous, granular shear flows

Discrete element numerical simulations of unsteady, homogeneous flows have been performed by shearing a fixed volume of identical, soft, frictional spheres. A constant, global, shear rate was instantly applied to particles that are initially at rest, non interacting, and randomly distributed. The granular material exhibits either large or small fluctuations in the evolving pressure, depending whether the average number of contacts per particle (coordination number) is less or larger than a critical value. When the coordination number is less than the critical value, the amplitude of the pressure fluctuations is dependent on the shear rate, whereas, it is rate-independent in the opposite case, signatures, according to the case, of fluid-like and solid-like behaviour. The same critical coordination number has been previously found to represent the minimum value at which rate-independent components of stresses develop in steady, simple shearing and the jamming transition in isotropic random packings. The observed complex behaviour of the measured pressure in the fluid–solid transition can be predicted with a constitutive model involving the coordination number, the particle stiffness and the intensity of particle agitation.     

Ein kombiniertes Teilchen-/Rohr-Modell für den Druckabfall in gasdurchströmten geschütteten oder geordneten Kugel-Packungen

In packed columns one finds relativ great packing elements. Therefore one must consider for the pressure drop the column wall and the packing element. The new model for sphere packings uses the number of resistance of a sphere, the number of pressure drop of a pipe and the law of resistance of a sphere collective. With the tortuosity factor the pressure drop of sphere packings can later be extended to any dumped or structured packings.     

Dense granular flow at the critical state: maximum entropy and topological disorder

After extensive quasi-static shearing, dense dry granular flows attain a steady-state condition of porosity and deviatoric stress, even as particles are continually rearranged. The paper considers two-dimensional flow and derives the probability distributions of two topological measures of particle arrangement—coordination number and void valence—that maximize topological entropy. By only considering topological dispersion, the method closely predicts the distribution of void valences, as measured in discrete element (DEM) simulations. Distributions of coordination number are also derived by considering packings that are geometrically and kinetically consistent with the particle sizes and friction coefficient. A cross-entropy principle results in a distribution of coordination numbers that closely fits DEM simulations.     

Quantitative structural analysis of simulated granular packings of non-spherical particles

A set of computationally generated granular packings of frictionless grains is statistically analyzed using tools from stochastic geometry. We consider both the graph of the solid phase (formed using the particle mid-points) and the pore-phase. Structural characteristics rooted in the analysis of random point processes are seen to yield valuable insights into the underlying structure of granular systems. The graph of the solid phase is analyzed using traditional measures such as edge length and coordination number, as well as more instructive measures of the overall transport properties such as geometric tortuosity, where significant differences are observed in the windedness of paths through the different particle graphs considered. In contrast, the distributions of pore-phase characteristics have a similar shape for all considered granular packings. Interestingly, it is found that prolate and oblate ellipsoid packings show a striking similarity between their solid-phase graphs as well as between their pore-phase graphs.     

Effect of size distribution on the relation between coordination number and void fraction of spheres in a randomly packed bed

Counting the number of contact points on a particle in a packed bed is a time-consuming processes. So, usually, the number of contact points on a particle surface in a randomly packed bed, i.e. the coordination number, is estimated from the void fraction. Many researchers have proposed equations to estimate the coordination number from the void fraction in a packed bed, but all of these equations are obtained for a uniform sized sphere bed. Most actual powder has a wide size distribution, and the effect of the size distribution on the relation between the coordination number and the void fraction is very important. We investigated the effect of the size distribution on the relation using our model and a computer simulation. Based on our results, the coordination number is not changed with the size distribution of the particle bed; however, the void fraction becomes smaller for wider particle size distributions. It means that the value of the coordination number, as estimated by conventional equation for equal spheres from void fraction data, is overestimated for a randomly packed bed of multi-component spheres with wider particle size distributions.     

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.     

An anisotropic continuum model for the sintering and compaction of powder packings

A continuum model is presented for the early to intermediate stages of densification of powder packings during sintering and compaction. Material points in a particle far from a contact are assumed to deform according to the macroscopic deformation rate. This permits the force transmitted through an inter-particle contact to be related to the macroscopic deformation rate using results of unit problems for the compaction and sintering of pairs of spheres. A macroscopic stress is defined in terms of the contact force and the orientational distribution of contacts. Because the forces are related to the deformation rate, this yields a viscosity tensor that, in general, is anisotropic. Sintering tractions on the particle surfaces appear as a sintering stress that, in the isotropic case, reduces to a sintering pressure. The state is described by the orientational distribution of contacts, the contact area and length of a link connecting particle centroids (which are functions of orientation), and the coordination number and volume fraction of particles. The model is completed by specifying evolution equations for the state. Model predictions are presented for isotropic and axisymmetric cases. The predictions are compared with experiments conducted with aluminum powder compacts.     

Bonded particle modeling of grain size effect on tensile and compressive strengths of rock under static and dynamic loading

The effect of grain (particle) size on the strength is an interesting subject in the rock engineering. Some investigations about the impact of particle size on static strength of rock have been conducted and reported in the literature. However, this issue has not received enough attention when high loading rates are involved. In this work, by utilizing the CA3 bonded particle - finite element computer program, the combined influence of loading rate and particle size on the compressive and tensile strengths of rock is examined. The bonded particle model is used to simulate the crack initiation and failure of the rock specimen and the finite element is utilized to model the elastic bars in the Split Hopkinson Pressure Bar (SHPB) apparatus employed for the dynamic testing. Specimens with four different particle sizes were prepared. The results suggest that the particle size does not affect the rock strength under static and dynamic loading. However, the particle size modifies the nominal tensile strength of the notched Brazilian specimens. For the intact Brazilian specimens under high stress rates, the particle size contributes to the tensile strength and this contribution can be justified based on the principles of fracture mechanics. The theoretical reason for these observations is derived for a 3D bonded particle system and discussed.     

A New Algorithm for Simulating the Random Packing of Monosized Powder in CIP Processes

A new model for monosized particle packing is developed in this study in simulating CIP powder compaction. The model uses a central growth method by which particles are placed from a central particle outwards until a specified container is filled. The simulation program is able to generate a particle packing of 5000 particles with characteristics that are similar to experimental results with significantly short computational time. A particle packing with a high occurrence of contacts among particles is effectively simulated using this model. It is shown that this work is an improvement in predicting the coordination number of the compact compared with the predictions of previous models. The average coordination number predicted by this model is 7·0 compared to 6·0-6·5 obtained by other simulations.     

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.     

Generation of porous particle structures using the void expansion method

The newly developed “void expansion method” allows for an efficient generation of porous packings of spherical particles over a wide range of volume fractions using the discrete element method. Particles are randomly placed under addition of much smaller “void-particles”. Then, the void-particle radius is increased repeatedly, thereby rearranging the structural particles until formation of a dense particle packing. The structural particles’ mean coordination number was used to characterize the evolving microstructures. At some void radius, a transition from an initially low to a higher mean coordination number is found, which was used to characterize the influence of the various simulation parameters. For structural and void-particle stiffnesses of the same order of magnitude, the transition is found at constant total volume fraction slightly below the random close packing limit. For decreasing void-particle stiffness the transition is shifted towards a smaller void-particle radius and becomes smoother.     

层状岩石细观构造表征及劈拉受载各向异性行为研究

层状岩石结构内部矿物之间的定向排列与胶结作用形成不同构造方向的细观结构,使岩石变形破裂及其力学性能存在明显的各向异性。该文以板岩为研究对象,通过构建空间相关函数,建立可表征不同片理方向的岩石细观颗粒离散元模型。基于细观力学参数反演,针对不同片理角度（θ）的巴西劈裂试验开展数值仿真分析,对板岩的各向异性行为进行了研究。结果表明,由于岩石受内部片理构造的影响,在劈拉荷载作用下,呈现三种破坏模式,当θ ≤ 30°时主要发生矿物颗粒之间的拉伸破坏,当θ=45°~75°时为剪切与拉伸共同作用产生的破坏,当θ >75°时为沿着片理面的拉伸破坏;岩样破坏所耗能量及劈拉强度随片理角度的增大而逐渐降低。该文提出的方法能较好地模拟层状岩石的各向异性力学特征及变形破裂规律,与试验结果表现出良好的一致性。     

Cracking process of a granite specimen that contains multiple pre‐existing holes under uniaxial compression

The pre‐existence of openings, which play an important role in the mechanical properties and cracking behaviours of rock, is prevalent in rock mass. The interaction among pre‐existing openings (or holes) complicates the instability problems when rock contains multiple holes. Studying the strength failure behaviour of rock that contains multiple pre‐existing holes contributes to the fundamental knowledge of the excavation and stability of underground rock engineering. In this study, first, a series of uniaxial compression tests were performed on granite specimens that contain multiple small holes to investigate the effect of the geometry of pre‐existing holes on the strength and fracture behaviours of rock. The crack initiation, propagation and coalescence process, and acoustic emission (AE) characteristics were investigated using photographic and AE monitoring. Three failure modes were identified, ie, splitting failure, stepped path failure, and planar failure modes. Second, a set of micromechanical parameters in the PFC3D model were calibrated by comparison with the experimental results of an intact granite specimen. The numerically simulated peak strength, peak strain, and failure mode of preholed specimens were consistent with the experimental results. In accordance with the numerical results, the failure modes of the preholed specimens were dependent on the bridge angle and number of holes. Last, the internal fracture characteristics of numerical specimens were revealed by analyzing the horizontal and vertical cross sections at different positions.     

Stress triaxiality effect on void nucleation in ductile metals

The stress triaxiality effect on the strain required for void nucleation by particle‐matrix debonding has been investigated by means of micromechanical modelling. A unit‐cell model considering an elastic spherical particle embedded in an elastic‐plastic matrix was developed to the purpose. Particle‐matrix decohesion was simulated through the progressive failure of a cohesive interface. It has been shown that the parameters of matrix‐particle cohesive interface are correlated with macroscopic material properties. Here, a simple relationship for the maximum cohesive opening at interface failure as a function of material fracture toughness and yield stress has been derived. Results seem to confirm that, increasing stress triaxiality, the strain at which void nucleation is predicted to occur decreases exponentially in a similar way as for fracture strain. This result has substantial implications in modelling of ductile damage because it indicates that if the stress triaxiality is high enough, ductile fracture can occur at plastic strain lower than that necessary to nucleate damage for moderate or low stress triaxiality regime.     

Sintering models and the development of instabilities

A mathematical model is developed to describe, at least approximately, the densification and reorganization of a random stacking of particles due to internal transport of material. In this model, local stresses due to time varying coordination of particles are allowed which are found to alter the overall sintering behaviour significantly. Further, variations on stacking density and coordination on both a local and a global scale are investigated for their influence on small and large scale particle reorganization during sintering. It is found that these local variations will easily give rise to the development of a porosity of high coordination along with local densification. The overall effect is that this porosity disappears after a large sintering period when grain growth has become already substantial.Global variations in coordination are seen to be responsible for defect formation. A number of criteria will be derived to estimate under which conditions this formation of defects may be expected. The present model will be discussed with the help of own and a number of examples found in the literature.     

Controlling the onset of numerical fracture in parallelized implementations of the material point method (MPM) with convective particle domain interpolation (CPDI) domain scaling

The material point method is well suited for large‐deformation problems in solid mechanics but requires modification to avoid cell‐crossing errors as well as extension instabilities that lead to numerical (nonphysical) fracture. A promising solution is convected particle domain interpolation (CPDI), in which the integration domain used to map data between particles and the background grid deforms with the particle, based on the material deformation gradient. While eliminating the extension instability can be a benefit, it is often desirable to allow material separation to avoid nonphysical stretching. Additionally, large stretches in material points can complicate parallel implementation of CPDI if a single particle domain spans multiple computational patches. A straightforward modification to the CPDI algorithm allows a user‐specified scaling of the particle integration domain to control the numerical fracture response, which facilitates parallelization. Combined with particle splitting, the method can accommodate materials with arbitrarily large failure strains. Used with a smeared damage/softening model, this approach will prevent nonphysical numerical fracture in situations where the material should remain intact, but the effect of a single velocity field on localization may still produce errors in the post‐failure response. Details are given for both 2‐D and 3‐D implementations of the scaling algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Microcracks size growth prediction based on microdefects nucleation number

The mechanical reason for rock and concrete failure is trans-scale fracture, which can be divided into three phases: (1) microcrack evolution, (2) macrocrack nucleation, (3) macrocrack growth and run-through. Using the idea that a microcrack could be regarded as a well-organized aggregation of nucleated microdefects, the size growth model of the largest microcrack based on the accumulated number of microdefect nucleation is established. In order to test the validity of the model, trans-scale fracture of a plate made of heterogeneous material is numerically simulated to display the microcrack’s evolution. Statistical analysis of the number and sizes of the microcracks indicates that the predicted size of the largest microcrack according to the model is in close agreement with the measured crack size prior to peak stress, but not at all close to the measured values after the peak. At the end of the paper, some remaining problems are proposed for the further work.     

Numerical simulation with experimental verification of the fracture behavior in granite under confining pressures based on the tension-softening model

The use of the tension-softening model for analyzing fracture processes of rock is examined with special reference to the effect of confining pressure on the fracture extension. Tension-softening curves are measured by means of theJ-based technique from unconfined tests performed on compact tension (CT) specimens of granite. On the basis of the determined tension-softening relation, numerical analyses are executed using a boundary element method (BEM) to simulate fracture of the granite under confining pressures. Numerical results are compared to the experimental results of two series of tests for which CT specimens and thick-walled cylindrical specimens were loaded to failure under confining pressures ranging from 0 to 26.5 MPa. It is shown that the BEM analyses can predict the observed fracture behavior. Based on the results, it is demonstrated that the tension-softening relation provides a suitable model to analyze the fracture process in the rock. The source mechanism for the pressure sensitive fracture is discussed by examining the growth of the fracture process zone.