Algorithm for particle packing based on Aristotelian mechanics

A dynamic algorithm is proposed for three-dimensional packing of spherical solid particles. The particles are deposited within a specified region with a fixed rigid boundary. The velocity of each particle is proportional to its weight and forces due to contact of the particle with the boundary and neighbor particles. Dimensional analysis of the equations of particle motion is performed. The average density and coordination number distribution for an equilibrium packing are calculated. The dependence of these characteristics on viscosity, granulometric composition, and representation of initial conditions (numerical analogue of material pouring into a specified volume) is studied.     

Short-time dynamics of a packing of polyhedral grains under horizontal vibrations

We analyze the dynamics of a 3D granular packing composed of particles of irregular polyhedral shape confined inside a rectangular box with a retaining wall subjected to horizontal harmonic forcing. The simulations are performed by means of the contact dynamics method for a broad set of loading parameters. We explore the vibrational dynamics of the packing, the evolution of solid fraction and the scaling of dynamics with the loading parameters. We show that the motion of the retaining wall is strongly anharmonic as a result of jamming and grain rearrangements. It is found that the mean particle displacement scales with inverse square of frequency, the inverse of the force amplitude and the square of gravity. The short-time compaction rate grows in proportion to frequency up to a characteristic frequency, corresponding to collective particle rearrangements between equilibrium states, and then it declines in inverse proportion to frequency.     

An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows

A three-dimensional, incompressible, multiphase particle-in-cell method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions. A subgrid particle, normal stress model for discrete particles which is robust and eliminates the need for an implicit calculation of the particle normal stress on the grid is presented. Interpolation operators and their properties are defined which provide compact support, are conservative, and provide fast solution for a large particle population. The solution scheme allows for distributions of types, sizes, and density of particles, with no numerical diffusion from the Lagrangian particle calculations. Particles are implicitly coupled to the fluid phase, and the fluid momentum and pressure equations are implicitly solved, which gives a robust solution.     

Interfacial Stress,Interfacial Energy,and Phase Equilibria in Binary Alloys

A simple model is presented in order to explore the influence of interfacial stress, interfacial energy, and surface stress on the characteristics of phase equilibria in stressed, two-phase binary alloys. Two different system geometries are employed: concentric spheres and thin plates. The conditions for thermodynamic equilibrium are solved and equations of state for each geometry are obtained in terms of the phase fraction, alloy composition, system dimension, and several dimensionless materials parameters. Elastic stress introduces new equilibrium states that are further modified by the interfacial quantities. Those conditions for which interfacial quantities can induce significant changes in the equilibrium phase fraction and phase compositions are identified.     

The effect of surface contact conditions on grain boundary interdiffusion in a semi-infinite bicrystal

The Kirkendall effect is conditioned by active diffusion as well as by active sources and sinks for vacancies. In the case of grain boundaries under the condition of negligible bulk diffusion, the Kirkendall effect is highly localized and responsible for the formation of an extra material wedge in the grain boundary, which may lead to high stress concentrations. The Kirkendall effect in grain boundaries of a binary system is described by a set of partial differential equations for the mole fraction of one of the diffusing components and for the stress component normal to the grain boundary completed with the respective initial and boundary conditions. The contact conditions of the grain boundary with the surface layer acting as source of one of the diffusing components can be considered as equilibrium ones ensuring the continuity of generalized chemical potentials of both diffusing components. Thus, the boundary conditions are determined by the difference in chemistry (i.e. how the thermodynamic parameters depend on chemical composition) of the grain boundaries and of the surface layer. The simulations based on the present model indicate a drastic influence of the chemistry on the grain boundary interdiffusion and Kirkendall effect.     

Two remarks on extremal equilibrium states

First it is shown that each extremal equilibrium state is representable as limit of Gibbs states in finite volumes, and that an analogous statement holds for extremal invariant equilibrium states. Secondly we prove that for negative pair interactions only one equilibrium state exists which minimizes (resp. maximizes) the particle density, but that in general there are more than two extremal invariant equilibrium states with the same particle density. In this context, periodic interactions are studied.     

Stress propagation through frictionless granular material

We examine the network of forces to be expected in a static assembly of hard, frictionless spherical beads of random sizes, such as a colloidal glass. Such an assembly is minimally connected: the ratio of constraint equations to contact forces approaches unity for a large assembly. However, the bead positions in a finite subregion of the assembly are underdetermined. Thus to maintain equilibrium, half of the exterior contact forces are determined by the other half. We argue that the transmission of force may be regarded as unidirectional, in contrast to the transmission of force in an elastic material. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involving more recently added beads. We derive equations for the transmission of stress averaged over scales much larger than a single bead. This derivation requires the ansatz that statistical fluctuations of the forces are independent of fluctuations of the contact geometry. Under this ansatz, the d(d+1)/2-component stress field can be expressed in terms of a d-component vector field. The procedure may be generalized to nonsequential packings. In two dimensions, the stress propagates according to a wave equation, as postulated in recent work elsewhere. We demonstrate similar wave-like propagation in higher dimensions, assuming that the packing geometry has uniaxial symmetry. In macroscopic granular materials we argue that our approach may be useful even though grains have friction and are not packed sequentially.     

Propagation of waves in nonlocal porous multi-phase nanocrystalline nanobeams under longitudinal magnetic field

ABSTRACT

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality.     

Computing Particle Surface Areas and Contact Areas from Three‐Dimensional Tomography Data of Particulate Materials

Microtomography is an emerging technique for particle and particulate‐materials characterization. To use this technology effectively, robust and accurate computational algorithms are needed to compute relevant particle properties, including particle surface area and particle‐particle contact area. However, the most accurate algorithms that have been developed for computing the exposed (void/solid) surface area in a microtomography image cannot be used directly for computing surface areas or particle‐particle contact areas for individual particles in a dense packing. This paper presents an algorithm for extracting particle contact areas from a digitized, segmented image of a packed granular material, which in turn can be used to find individual particle surface areas (even if the complete surfaces are not exposed because of contacts in the packing). Results show that small errors in the binary surface‐area computations are magnified in the course of determining particle contact areas; the total error in the computation depends mainly on the size of the contact area in voxel units.     

Compactivity and transmission of stress in granular materials

We outline a statistical-mechanical theory of granular materials. Stress propagation and force fluctuations in static granular media are still poorly understood. We develop the statistical-mechanical theory that delivers the fundamental equations of stress equilibrium. The formalism is based on the assumptions that grains are rigid, cohesionless, and that friction is perfect. Since grains are assumed perfectly rigid, no strain or displacement field can enter the equations for static equilibrium of the stress field. The complete system of equations for the stress tensor is derived from the equations of intergranular force and torque balance, given the geometric specification of the material. These new constitutive equations are indeed fundamental and are based on relations between various components of the stress tensor within the material, and depend on the topology of the granular packing. The problem of incorporating into the formalism the "no tensile forces" constraint is considered. The compactivity concept is reviewed. We discuss the relation between the concept of compactivity and the problem of stress transmission. (c) 1999 American Institute of Physics.     

气粒两相流传热问题的光滑离散颗粒流体动力学方法数值模拟

在气粒两相流动问题中,颗粒间以及气体与颗粒间的传热问题不可忽略.光滑离散颗粒流体动力学(SDPH)模型作为一种新的求解气粒两相流动问题的方法,已经成功应用于模拟风沙运动等问题.在此基础上,提出了SDPH方法的热传导模型,模拟了气粒两相流动问题中的热传导过程以及颗粒蒸发过程.首先引入各相的能量方程,利用有限差分与光滑粒子流体动力学一阶导数相结合的方法,处理各相内部热传导项中的二阶导数问题,基于气粒两相间温度差及对流换热系数计算颗粒与气体间的热传导量,推导得到了含热传导模型的气粒两相流SDPH计算方程组,模拟计算了圆盘形颗粒团算例及鼓泡流化床内部热传导算例,并与双流体模型计算结果进行对比,结果基本符合;其次利用离散液滴模型中的颗粒蒸发传质传热定律计算颗粒的蒸发过程,数值模拟了颗粒射流蒸发过程,并与离散颗粒模型结果进行对比,两者符合得较好,验证了该方法的准确性及实用性.     

Tagged particle fluctuations in uniform shear flow

The nonlinear Boltzmann and Boltzmann-Lorentz equations are used to describe the dynamics of a tagged particle in a nonequilibrium gas. For the special case of Maxwell molecules with uniform shear flow, an exact set of equations for the average position and velocity, and their fluctuations, is obtained. The results apply for arbitrary magnitude of the shear rate and include the effects of viscous heating. A generalization of Onsager's assumption of the regression of fluctuations is found to apply for the relationship between the equations for the average dynamics and those for the time correlation functions. The connection between fluctuations and dissipation is described by the equations for the equal-time correlation function. The source term in these equations indicates that the “noise” in this nonequilibrium state is qualitatively different from that in equilibrium, or even local equilibrium. These equations are solved to determine the velocity autocorrelation function as a function of the shear rate.     

Grain-boundary anelastic relaxation and non-equilibrium dilution induced by compressive stress and its kinetic simulation

Grain boundaries are known to be sources and sinks for bulk vacancies, but the exchange that occurs between the grain boundary and the bulk under a low stress is still obscure. In the present paper, it is shown that grain boundaries may act as sources to emit vacancies when an anelastic deformation occurs under a compressive stress. These emitted supersaturated vacancies are combined with solute atoms to form complexes. Solute non-equilibrium grain-boundary dilution may be induced by the diffusion of complexes away from the boundary. An equation of solute concentration at grain boundary is derived under stress equilibrium during its anelastic relaxation. Furthermore, kinetic equations are also established to describe the non-equilibrium grain-boundary dilution. Additionally, an attempt is made to simulate experimental data to justify the present model.     

Scattering of spatially inhomogeneous sound waves by a spherical particle

The problem of monopole, dipole, and rotational scattering of a spatially inhomogeneous time-harmonic sound field by an arbitrary spherical particle is solved for the cases of the medium being a viscous compressible liquid or an isotropic elastic medium. Equations for the spherical mean fields at the particle are obtained. These equations are used to derive the formulas for the scattered fields. Different limiting cases of particle behavior are considered. In particular, it is shown that the dipole scattering is determined by two components of particle oscillations, one of which corresponds to translational oscillatory motion and the other to oscillations of two antiphase monopoles. For these types of particle oscillations, a scattering matrix, which determines the scattering of an arbitrary field by a particle, is constructed. The matrix allows the formalization of the processes of multiple sound scattering by particles and is valid for any distances between the particles down to their contact.     

Measurement of correlations between low-frequency vibrational modes and particle rearrangements in quasi-two-dimensional colloidal glasses

We investigate correlations between low-frequency vibrational modes and rearrangements in two-dimensional colloidal glasses composed of thermosensitive microgel particles, which readily permit variation of the sample packing fraction. At each packing fraction, the particle displacement covariance matrix is measured and used to extract the vibrational spectrum of the "shadow" colloidal glass (i.e., the particle network with the same geometry and interactions as the sample colloid but absent damping). Rearrangements are induced by successive, small reductions in the packing fraction. The experimental results suggest that low-frequency quasilocalized phonon modes in colloidal glasses, i.e., modes that present low energy barriers for system rearrangements, are spatially correlated with rearrangements in this thermal system.     

Elastic mechanical grain interactions in polycrystalline materials; analysis by diffraction-line broadening

Abstract

Experimental investigations have revealed that the Neerfeld–Hill and Eshelby–Kröner models, for grain interactions in massive, bulk (in particular, macroscopically isotropic) polycrystals, and a recently proposed effective grain-interaction model for macroscopically anisotropic polycrystals, as thin films, provide good estimates for the macroscopic (mechanical and) X-ray elastic constants and stress factors of such polycrystalline aggregates. These models can also be used to calculate the strain variation among the diffracting crystallites, i.e. the diffraction-line broadening induced by elastic grain interactions can thus be predicted. This work provides an assessment of diffraction-line broadening induced by elastic loading of polycrystalline specimens according to the various grain-interaction models. It is shown that the variety of environment, and thus the heterogeneity of the stress–strain states experienced by each of the individual grains exhibiting the same crystallographic orientation in a real polycrystal, cannot be accounted for by traditional grain-interaction models, where all grains of the same crystallographic orientation in the specimen frame of reference are considered to experience the same stress–strain state. A significant degree of broadening which is induced by the heterogeneity of the environments of the individual crystallites is calculated on the basis of a finite element algorithm. The obtained results have vast implication for diffraction-line broadening analysis and modelling of the elastic behaviour of massive polycrystals.     

Molecular rototranslation in condensed phases : Single particle theory

A multidimensional expansion of the Mori equation in terms of a chain of Markov equations is used to develop a theory of molecular rototranslation in condensed phases. The stochastic equations of motion are solved for transient and equilibrium averages of the relevant dynamical variables. The single particle rototranslational Langevin equations correspond to the first equation of the Markov chain and (with a rotational constraint) are solved using Wiener matrix algebra for a possible sixteen autocorrelation functions. The Einstein result for the mean-square velocity and angular velocity is generalized. The third dimension of the Markov chain corresponds mechanically to the (constrained) rototranslation of a molecule bound to a cage of nearest neighbours by a dissipative matrix γ. The cage is itself undergoing a rototranslational Brownian motion. The problem of evaluating the formal theory with experimental measurements is discussed in terms of the number of parameters associated with each approximant (or dimensionality of the Markov chain). It is possible to avoid using a least-mean-squares fitting procedure by using a broad enough range of data and simulator results.     

Numerical simulation of Gaussian beam scattering by complex particles of arbitrary shape and structure

An efficient numerical method based on the surface integral equations is introduced to simulate the scattering of Gaussian beam by complex particles that consist of an arbitrarily shaped host particle and multiple internal inclusions of arbitrary shape. In particular, the incident focused Gaussian beam is described by the Davis fifth-order approximate expressions in combination with rotation defined by Euler angles. The established surface integral equations are discretized with the method of moments, where the unknown equivalent electric and magnetic currents induced on the surfaces of the host particle and the internal inclusions are expanded using the Rao–Wilton–Glisson (RWG) basis functions. The resultant matrix equations are solved by using the parallel conjugate gradient method. The proposed numerical method is validated and its capability illustrated in several characteristic examples.     

Lattice Kinetic Formulation for Ferrofluids

The lattice Boltzmann approach is used to solve continuum equations describing colloids of ferromagnetic particles (ferrofluids) in a regime, where the particle spins are in equilibrium with magnetic torques. This limit of rapid spin adjustment yields a symmetric total stress tensor that is essential for a kinetic formulation based on the Boltzmann equation. The magnetisation equation is solved using a vector-valued distribution function analogous to the earlier treatment (J. Comput. Phys. 179, 95) of the induction equation in magnetohydrodynamics, but the details are rather more complex because the magnetisation equation is not in conservation form except in a weakly magnetised limit.     

Equation of state of a classical anharmonic oscillator

Static properties of a particle moving in an anharmonic potential in equilibrium with a temperature bath and an external field are discussed in the framework of classical statistical mechanics. This model system represents the basic unit in current theories of structural phase transitions. Hierarchies of equations for the correlations (cumulants) and irreducible vertices are derived from the equilibrium condition. Approximate solutions are obtained from the hierarchies by truncation. Alternatively, one can write the equilibrium condition as differential equation which may be solved exactly, if appropriate initial conditions are known. Both methods have been worked out for a single- and a double-well potential. By truncation of the hierarchies one obtains as quantitatively correct result only a low-temperature expansion for the single-well potential.