TENSILE STRENGTH FOR SPLITTING FAILURE OF BRITTLE PARTICLES WITH CONSIDERATION OF POISSON''''S RATIO

1. Introduction The core mechanism of comminution could be reduced to the breakage of individual particles that occurs through contact with other particles or with the grinding media, or with the solid walls of the mill (Potapov & Campbell, 1994). When a particle is subjected to a load, the nature of the stress field around and within the brittle particle, its material properties, and the size and distribution of micro-flaws within the particle govern the size and shape distribution of the fra…     

Numerical simulation study on particle breakage behavior of granular materials in confined compression tests

In order to study the fragmentation law, the confined compression experiment of granular assemblies has been conducted to explore the particle breakage characteristic by DEM approach in this work. It is shown that contact and contact force during the loading process gradually transform from anisotropy to isotropy. Meanwhile, two particle failure modes caused by different contact force states are analyzed, which are single-through-crack failure and multi-short-crack failure. Considering the vertical distribution of the number of cracks and the four characteristic stress distributions (the stress related to the maximum contact force, the major principal stress, the deviatoric stress and the mean stress), it is pointed out that the stress based on the maximum contact force and the major principal stress can reflect the distribution of cracks accurately. In addition, the size effect of particle crushing indicates that small size particles are prone to break. The lateral pressure coefficient of four size particles during the loading process is analyzed to explain the reason for the size effect of particle breakage.     

The motion of suspended particles almost in contact

The forces and torques on two moving solid particles suspended in a fluid and almost in contact with each other (or on a particle almost in contact with a wall) are found in terms of their relative motion by using a type of lubrication theory, the results so obtained being asymptotically valid for small gap widths. It is assumed that the surfaces of the particles involved if brought together, are such that contact would occur at a single point at which surface curvatures are finite.     

Statistics of particle interactions in dense granular material under uniaxial compression

Stress evolution in a dense granular material is closely related to interactions of contacting particles. We investigate statistics related to particle interactions and the relationship between the averaged local relative motion and the macroscopic motion. The validity of the Voigt and Reuss assumptions is examined, and extensions to these assumptions are proposed. Effects of history in the dense granular material are investigated. Statistical samples used in this paper are obtained using three-dimensional numerical simulations of dense granular media under uniaxial cyclical compression. The results show that stresses arise mostly from normal forces between particles, and direct contributions from frictional tangential forces between particles are small. Tangential friction, however, significantly increases the particle contact time, and thus reduces the rate of contact breakage. The contact breakage rate is demonstrated to be a stress relaxation rate. Therefore, stress increases significantly with friction between particles as a result of prolonged relaxation time.     

Discrete element modelling of pebble beds: With application to uniaxial compression tests of ceramic breeder pebble beds

In this paper, a discrete element simulation scheme for pebble beds in fusion blankets is presented. Each individual pebble is considered as one element obeying equilibrium conditions under contact forces. We study not only the rearrangement of particles but also the overall behaviour of an assembly under the action of macroscopic compressive stresses. Using random close packing as initial configurations, the discrete element simulation of the uniaxial compression test has been quantitatively compared to experiments. This method yields the distribution of the inter-particle contact forces. Moreover, the micro-macro relations have been investigated to relate the microscopic information, such as the maximum contact force and the coordination number inside the assembly, to the macroscopic stress variables.     

Quantifying the extent of crushing in granular materials: A probability-based predictive method

Grain crushing is one of the micromechanisms that governs the stress-strain behaviour of a granular material, and also its permeability by altering the grain size distribution. It is therefore advantageous to be able to predict the point of onset of crushing and to quantify the subsequent evolution of crushing. This paper uses the data of Discrete Element Method (DEM) simulations to inform a statistical model of granular crushing. Distributions of normalised contact forces are first obtained. If the statistical distribution of the crushing strength of the grains is then known, the onset of crushing within an assembly of grains should be predictable. Two different cases, one in which grain strength was statistically independent of grain size and one showing an arbitrary trend, were used to compare with DEM results and so confirm the validity of the statistical method.     

The rheology of strongly-flocculated suspensions

The rheology of strongly-flocculated dispersions of colloidal particles has been investigated at particle concentrations where a continuous network is formed rather than a collection of discrete flocs. Such networks are shown to possess a true yield stress in both shear and in uniaxial compression (as realised in a centrifuge). Properties measured as a function of particle concentration and particle size include the yield stresses in shear (σ_y) and compression (P_y); the limiting and strain-dependent, instantaneous shear moduli G_O and G(γ); the elastic recovery at finite strains, and the rate of centrifugally-driven compaction. The yield stresses and moduli appear to show a power-law dependence on particle concentration with G_O and P_y, having the same power-law index and σ_y a somewhat lower one. The data are in part consistent with predictions based on the idea that the networks have a heterogeneous structure comprising a collection of interconnected fractal aggregates. The behaviour as a function of particle size and concentration is however not completely scaleable as might be expected on this basis. Thus, whereas the shear yield stress could be scaled to remove its dependence on particle radius a and volume fraction φ (over the measured range 0.25 μm ⩽ a ⩽ 3.4 μm; 0.05 ⩽ φ ⩽ 0.25) as could the strain dependent modulus (0.25 ⩽ a ⩽ 1.3 μm; 0.08 ⩽ 0.25), the particle-size and concentration dependence of P_y and G_O could only be scaled for particles with radii between 0.16 and 0.5 μm, smaller and larger particles having different and much higher power-law index in respect of their concentration dependencies. In the case of the smaller particles the failure of the scaling is thought to be due to an anomaly since these particles distort significantly under the influence of the strong van der Waals forces and this causes the aggregates to be more compact then they otherwise would be. The reasons for the failure at larger sizes is not clear.     

Lattice–Boltzmann simulations for analysing the detachment of micron-sized spherical particles from surfaces with large-scale roughness structures

Fully resolved numerical simulations of a micron-sized spherical particle residing on a surface with large-scale roughness are performed by using the Lattice–Boltzmann method. The aim is to investigate the influence of surface roughness on the detachment of fine drug particles from larger carrier particles for transporting fine drug particles in a DPI (dry powder inhaler). Often the carrier surface is modified by mechanical treatments for modifying the surface roughness in order to reduce the adhesion force of drug particles. Therefore, drug particle removal from the carrier surface is equivalent to the detachment of a sphere from a rough plane surface. Here a sphere with a diameter of 5 μm at a particle Reynolds number of 1.0, 3.5 and 10 are considered. The surface roughness is described as regularly spaced semi-cylindrical asperities (with the axes oriented normal to the flow direction) on a smooth surface. The influence of asperity distance and size ratio (i.e. the radius of the semi-cylinder to the particle radius, R_c/R_d) on particle adhesion and detachment are studied. The asperity distance is varied in the range 1.2 < L/R_d < 2 and the semi-cylinder radius between 0.5 < R_c/R_d < 0.75. The required particle resolution and domain size are appropriately selected based on numerical studies, and a parametric analysis is performed to investigate the relationship between the contact distance (i.e. half the distance between the particle contact points on two neighbouring semi-cylinders), the asperity distance, the size ratio, and the height of the particle centroid from the plane wall. The drag, lift and torque acting on the spherical particle are measured for different particle Reynolds numbers, asperity distances and sizes or diameters. The detachment of particles from rough surfaces can occur through lift-off, sliding and rolling, and the corresponding detachment models are constructed for the case of rough surfaces. These studies will be the basis for developing Lagrangian detachment models that eventually should allow the optimisation of dry powder inhaler performance through computational fluid dynamics.     

An analysis of the chaotic motion of particles of different sizes in a gas fluidized bed

The dynamic behavior of individual particles during the mixing/segregation process of particle mixtures in a gas fluidized bed is analyzed. The analysis is based on the results generated from discrete particle simulation, with the focus on the trajectory of and forces acting on individual particles.Typical particles are selected representing three kinds of particle motion:a flotsam particle which is initially at the bottom part of the bed and finally fluidized at the top part of the bed; a jetsam particle which is initially at the top part of the bed and finally stays in the bottom de-fluidized layer of the bed; and a jetsam particle which is intermittently joining the top fluidized and bottom de-fluidized layers. The results show that the motion of a particle is chaotic at macroscopic or global scale, but can be well explained at a microscopic scale in terms of its interaction forces and contact conditions with other particles, particle-fluid interaction force, and local flow structure. They also highlight the need for establishing a suitable method to link the information generated and modeled at different time and length scales.     

Constitutive model for predicting dynamic interactions between soil ejecta and structural panels

A constitutive model is developed for the high-rate deformation of an aggregate comprising of mono-sized spherical particles with a view to developing an understanding of dynamic soil-structure interactions in landmine explosions. The constitutive model accounts for two regimes of behaviour. When the particle assembly is widely dispersed (regime I), the contacts between particles are treated as collisions, analogous to those between molecules in a gas or liquid. At high packing densities (regime II) the contacts are semi-permanent and consolidation is dominated by particle deformation and inter-particle friction. Regime I is modelled by extending an approach proposed by Bagnold (1954. Experiments on a gravity-free dispersion of large solid particles in a Newtonian fluid under shear. Proceedings of the Royal Society of London A 225, 49-63) to a general strain history comprising volumetric and deviatoric deformation. For regime II, classical soil mechanics models (such as Drucker-Prager) are employed. The overall model is employed to investigate the one-dimensional impact of sand against a rigid stationary target. The calculations illustrate that, unlike single-particle impact, the momentum transmitted to a rigid target is insensitive to the particle co-efficient of restitution, but strongly dependent on initial density. The constitutive model is also used to examine the spherical expansion of a shell of sand (both dry and water saturated). In line with initial experimental observations, the wet sand is predicted to form clumps while the dry sand fully disperses. The model shows that this clumping of explosively loaded wet sand exerts higher pressures on nearby targets compared to equivalent dry sand explosions.     

Discrete element simulation of effects of multicontact loading on single particle crushing

Particle crushing commonly occurs in granular materials and affects their structures and mechanical properties. Unlike idealized particles in experimental single particle crushing tests with two loading points, natural particles are crushed under multicontact loading. To date, the criteria and patterns of particle crushing under multicontact conditions are not fully understood. By using the three-dimensional discrete element method, this report explores the effect of multicontact loading on the crushing criterion of a single particle, the crushing pattern, and the relationship between the particle crushing strength and loading distribution. The particles are modelled as aggregates of glued Voronoi polyhedra. The numerical results indicate that the logarithm of the mean principal stress has a good linear correlation with the coordination number. For a specific coordination number, the number of child particles presents a significant normal distribution. For a specific number of child particles, the volumes of child particles can be statistically described as normal or gamma distribution. Three typical models are proposed to qualitatively analyse the relationship between the loading distribution and crushing strength. The relevant conclusions can be helpful in engineering practice and in further studies on crushable granular materials via the discrete element method.     

岩石单颗粒压缩破碎试验研究

岩石颗粒结构复杂、形态不规则,很难用现有经典理论进行研究,分形理论可以从统计的角度研究颗粒破碎的强度。根据600颗大理岩单颗粒破碎试验,测量单颗粒破碎的应力与竖向应变关系曲线和破碎后的颗粒分布曲线,将颗粒破碎曲线按形态分为三种类型;根据颗粒破碎后的颗粒分布特性,建立了岩石颗粒破碎的分形模型,测得了大理岩颗粒破碎的分维D为2.48;根据颗粒破碎的分形模型,导出颗粒破碎强度的理论公式;将大理岩颗粒破碎强度的试验数据与理论公式预测结果进行比较,两者十分吻合,从而验证了颗粒破碎强度的分形理论。     

An analysis of the chaotic motion of particles of different sizes in a gas fluidized bed

The dynamic behavior of individual particles during the mixing/segregation process of particle mixtures in a gas fluidized bed is analyzed. The analysis is based on the results generated from discrete particle simulation, with the focus on the trajectory of and forces acting on individual particles. Typical particles are selected representing three kinds of particle motion： a flotsam particle which is initially at the bottom part of the bed and finally fluidized at the top part of the bed; a jetsam particle which is initially at the top part of the bed and finally stays in the bottom de-fluidized layer of the bed; and a jetsam particle which is intermittently joining the top fluidized and bottom de-fluidized layers. The results show that the motion of a particle is chaotic at macroscopic or global scale, but can be well explained at a microscopic scale in terms of its interaction forces and contact conditions with other particles, particle-fluid interaction force, and local flow structure. They also highlight the need for establishing a suitable method to link the information generated and modeled at different time and length scales.     

砂土地基锚板基础抗拔承载力PFC数值分析

在模型试验的基础上,采用PFC离散单元法对条形锚板基础在中密砂土地基中的抗拔性能进行了数值分析。数值模拟采用簇颗粒单元来模拟砂粒的不规则形状,颗粒级配根据模型试验福建标准砂的级配按照相似级配法生成,细观参数根据数值双轴试验确定。水平锚板数值模拟结果与模型试验结果基本一致。与模型试验结果相比,颗粒流数值模拟能得到颗粒间接触力链的分布及其演化规律,能从细观角度来探明宏观抗拔承载力特性的演化机理。在此基础上对倾斜锚板上拔过程进行了模拟,分析了锚板前后砂粒的运动趋势以及接触力链的演化规律,并与已有承载力结果进行了对比分析。     

Correlation of powder flow properties to interparticle interactions at ambient and high temperatures

A combination of a continuum approach and a particle–particle approach to describe the multi-scale nature of the mechanical properties of bulk solids may be beneficial to scientific and engineering applications. In this paper, a procedure is proposed to estimate the interparticle forces beginning with the bulk flow properties as measured with standardized techniques. In particular, the relationship between interparticle forces and bulk solid tensile strength is adopted based on the microscale approaches of Rumpf (1970) and Molerus (1975). The flow properties of fluid cracking catalyst (FCC), corundum and glass bead powders were all characterized with a modified Schulze ring shear cell capable of operating at temperatures up to 500 °C. The powder test conditions were selected such that the van der Waals forces were the most significant particle–particle interactions. The model equations describe two cases, in which either elastic or plastic deformation of the contact points is assumed. The results indicate that the model provides the correct order of magnitude for the values of the tensile strength when proper values for the mean curvature radius at the contact points are taken into account. A sensitivity analysis for the main parameters in the model was performed. This analysis indicated that the assumption of plastic deformation at contact surfaces coupled with a decrease in porosity justified an increase of the tensile strength with consolidation stress. Furthermore, the effect of temperature on the measured flow behavior can be explained as a change in the strength of the material.     

The elastic response of a cohesive aggregate—a discrete element model with coupled particle interaction

A model is presented for the deformation of a cohesive aggregate of elastic particles that incorporates two important effects of large-sized inter-particle junctions. A finite element model is used to derive a particle response rule, for both normal and tangential relative deformations between pairs of particles. This model agrees with the Hertzian contact theory for small junctions, and is valid for junctions as large as half the nominal particle size. Further, the aggregate model uses elastic superposition to account for the coupled force–displacement response due to the simultaneous displacement of all of the neighbors of each particle in the aggregate. A particle stiffness matrix is developed, relating the forces at each junction to the three displacement degrees of freedom at all of the neighboring-particle junctions. The particle response satisfies force and moment equilibrium, so that the model is properly posed to allow for rigid rotation of the particle without introducing rotational degrees of freedom. A computer-simulated sintering algorithm is used to generate a random particle packing, and the stiffness matrix is derived for each particle. The effective elastic response is then estimated using a mean field or affine displacement calculation, and is also found exactly by a discrete element model, solving for the equilibrium response of the aggregate to uniform-strain boundary conditions. Both the estimate and the exact solution compare favorably with experimental data for the bulk modulus of sintered alumina, whereas Hertzian contact-based models underestimate the modulus significantly. Poisson's ratio is, however, accurately determined only by the full equilibrium discrete element solution, and shown to depend significantly on whether or not rigid particle rotation is permitted in the model. Moreover, this discrete element model is sufficiently robust, so it can be applied to problems involving non-homogeneous deformations in such cohesive aggregates.     

On the definition of the stress tensor in granular media

This paper investigates the definition of the stress tensor within a granular assembly, when inertial effects are likely to occur. It is shown that the stress tensor can be expressed as a sum of two terms. A first term corresponds to the standard definition of the stress, according to the Love–Weber formula; this term is related to the contact forces existing within adjoining particles. A second term accounts for dynamic effects related to rotation velocities and accelerations of the particles. These results are checked from discrete numerical simulations in order to examine in which context the contribution of inertial effects should not be omitted. With this aim, the simulation of a granular specimen collapse and then a silo discharge is considered.     

基于分形统计强度理论的煤颗粒冲击破碎概率研究

基于分形统计强度理论对煤颗粒的冲击破碎概率进行研究。以Hertz接触假设为基础,得到煤颗粒冲击破碎概率与最大接触压应力之间的函数关系,并结合碰撞动力学理论,建立了冲击破碎概率的分形模型。对不同矿区工作面的煤颗粒进行冲击破碎试验,统计分析表明：分形模型可以对煤颗粒的冲击破碎概率进行很好的描述,冲击破碎概率与冲击速度在对数坐标中为线性关系。通过试验确定分形模型的分形维数和破碎常数,可以得到不同冲击速度下煤颗粒的冲击破碎概率以及煤颗粒全部破碎需要的冲击速度,为冲击破碎效果评价以及冲击速度的确定提供了理论指导。     

Dynamic stress chain formation in a two-dimensional particle bed

Cure cast plastic bonded explosives (PBXs) consist of relatively hard particles in a soft binder. Under compressive loading, the explosive cyrstals come into contact that causes high stress concentrations. The lines along which the crystals are loaded are called stress chains. Damage done to these particle beds during compressive loading can lead to reaction. The photoelastic effect of PMMA is exploited to examine the stress state within a two-dimensional particle bed. Stress chain development within the bed is recorded and is shown to increase the stress state within some particles while leaving others unloaded. These concentrations form early in the loading process, leading to fracture along the stress bridges and generating likely reaction initiation sites. Through material point method simulations, contact friction is shown to have a large effect on the stress distribution within the particle bed.