颗粒流动力学及其离散模型评述

颗粒流是由众多颗粒组成的具有内在相互作用的非经典介质流动. 自然界常见颗粒流都是密集流, 颗粒间接触形成力链, 诸多力链相互交接构成支撑整个颗粒流重量和外载荷的网络, 其局部构型及强度在外载荷下演化, 是颗粒流摩擦特性和接触应力的来源.本文介绍球形颗粒间无粘连作用时的Hertz法向接触理论和Mindlin-Deresiewicz切向接触理论. Campbell依据是否生成较为稳定的力链把颗粒流分为弹性流和惯性流两大类, 其中弹性-准静态流和惯性-碰撞流分别对应准静态流和快速流, 作为两种极端流动情况通常处理成连续体, 分别采用摩擦塑性模型和动理论予以描述, 但是表征接触力链的颗粒弹性参数并不出现这两个模型和理论框架中, 如何进一步考虑颗粒弹性参数将非常困难. 目前离散动力学方法逐渐成为复现其复杂颗粒流动现象、提取实验不可能获得的内部流动信息进而综合起来探索颗粒流问题的一种有效工具, 其真实性强于连续介质理论的描述. 软球模型对颗粒间接触力简化处理, 忽略了切向接触力对法向接触力及其加载历史的依赖, 带来了法向和切向刚度系数如何标度等更艰难的物理问题, 但由于计算强度小而广泛应用于工程问题中. 硬球模型不考虑颗粒接触变形, 因而不能描述颗粒流内在接触应变等物理机理, 仅适用于快速颗粒流, 这不仅仅是由于两体碰撞的限制. 因此基于颗粒接触力学的离散颗粒动力学模型是崭新的模型,适用于准静态流到快速流整个颗粒流态的模拟, 可以细致考虑接触形变及接触力的细节,建立更为合理的颗粒流本构关系, 进而有力的促进颗粒流这一非经典介质流动的研究.      

Edge Contact Forces and Quasi-Balanced Power

We consider continuous media in which contact edge forces are present. Introducing thenotion of quasi-balanced contact force distribution, we are able to prove the conjectures by Noll andVirga [1] concerning the representation of contact edge forces. We generalize the Hamel--Nolltheorem on the Cauchy postulate. Then we adapt the celebrated tetrahedron construction of Cauchy in orderto obtain a representation theorem for stress states. In fact, we show that two stress tensors of order twoand three are necessary for such a representation. Moreover we find the relationship between the notionof interstitial working introduced by Dunn and Serrin [2] and the notion of contact edge force.     

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.     

MEASUREMENT OF ADHESION FORCES IN AIR WITH THE VIBRATION METHOD

The vibration method represents a practical method for the measurement of adhesion forces and adhesion force distributions. This method causes sinusoidally altemating stresses and yields detachment and contact forces between particles and substrate of the same order of magnitude. Alternating contact forces of the vibration method can cause an adhesion force intensification through flattening of asperities. The measuring principle of the vibration method and the analysis of experimental results are described in the article. Normal adhesion forces (pull-off forces) are measured using the vibration method and the colloidal probe technique. The results of both methods show good agreement for small particle sizes. The influence of the detachment force direction is shown by comparing tangential and normal adhesion forces measured using particle reentrainment in a turbulent air flow and the vibration method, respectively. The surface roughness of the substrate and the relative humidity are shown to significantly influence the measured adhesion forces. For the calculation of the adhesion forces, an approach by Rabinovich was combined with approximations of plastic micro asperity flattening. The Rabinovich approach accounts for roughness effects on the van der Waals force by incorporating the rms roughness of the interacting surfaces. rms-values of the particles and substrates were measured with atomic force microscopy at different scanning areas.     

A nonlocal contact formulation for confined granular systems

We present a nonlocal formulation of contact mechanics that accounts for the interplay of deformations due to multiple contact forces acting on a single particle. The analytical formulation considers the effects of nonlocal mesoscopic deformations characteristic of confined granular systems and, therefore, removes the classical restriction of independent contacts. This is in sharp contrast to traditional contact mechanics theories, which are strictly local and assume that contacts are independent regardless the confinement of the particles. For definiteness, we restrict attention to elastic spheres in the absence of gravitational forces, adhesion or friction. Hence, a notable feature of the nonlocal formulation is that, when nonlocal effects are neglected, it reduces to Hertz theory. Furthermore, we show that, under the preceding assumptions and up to moderate macroscopic deformations, the predictions of the nonlocal contact formulation are in remarkable agreement with detailed finite-element simulations and experimental observations, and in large disagreement with Hertz theory predictions—supporting that the assumption of independent contacts only holds for small deformations. The discrepancy between the extended theory presented in this work and Hertz theory is borne out by studying periodic homogeneous systems and disordered heterogeneous systems.     

Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches

The main objective of this work is to present a computational and experimental study on the contact forces developed in revolute clearance joints. For this purpose, a well-known slider-crank mechanism with a revolute clearance joint between the connecting rod and slider is utilized. The intra-joint contact forces that are generated at these clearance joints are computed by considering several different elastic and dissipative approaches, namely those based on the Hertz contact theory and the ESDU tribology-based cylindrical contacts, along with a hysteresis-type dissipative damping. The normal contact force is augmented with the dry Coulomb’s friction force. In addition, an experimental apparatus is used to obtained some experimental data in order to verify and validate the computational models. From the outcomes reported in this paper, it is concluded that the selection of the appropriate contact force model with proper dissipative damping plays a significant role in the dynamic response of mechanical systems involving contact events at low or moderate impact velocities.     

Modeling slow deformation of polygonal particles using DEM

We introduce two improvements in the numerical scheme to simulate collision and slow shearing of irregular particles. First, we propose an alternative approach based on simple relations to compute the frictional contact forces. The approach improves efficiency and accuracy of the Discrete Element Method (DEM) when modeling the dynamics of the granular packing. We determine the proper upper limit for the integration step in the standard numerical scheme using a wide range of material parameters. To this end, we study the kinetic energy decay in a stress controlled test between two particles. Second, we show that the usual way of defining the contact plane between two polygonal particles is, in general, not unique which leads to discontinuities in the direction of the contact plane while particles move. To solve this drawback, we introduce an accurate definition for the contact plane based on the shape of the overlap area between touching particles, which evolves continuously in time.     

Modeling slow deformation of polygonal particles using DEM

We introduce two improvements in the numerical scheme to simulate collision and slow shearing of irregular particles. First, we propose an alternative approach based on simple relations to compute the frictional contact forces. The approach improves efficiency and accuracy of the Discrete Element Method （DEM） when modeling the dynamics of the granular packing. We determine the proper upper limit for the integration step in the standard numerical scheme using a wide range of material parameters. To this end, we study the kinetic energy decay in a stress controlled test between two particles. Second, we show that the usual way of defining the contact plane between two polygonal particles is, in general, not unique which leads to discontinuities in the direction of the contact plane while particles move. To solve this drawback, we introduce an accurate definition for the contact plane based on the shape of the overlap area between touching particles, which evolves continuously in time.     

Experimental Studies of the Flow of Ferrofluid in Porous Media

This paper presents laboratory-scale experimental results of the behavior of ferrofluids in porous media consisting of sands and sediments. Ferrofluids are colloidal suspensions of magnetic particles stabilized in various carrier liquids. In the presence of an external magnetic field, a ferrofluid becomes magnetized as the particles align with the magnetic field. We investigate the potential for controlling fluid emplacement in porous media using magnetic fields. These experiments show that in laboratory-scale porous media experiments (up to 0.25m), with both vertical gravitational forces and lateral magnetic forces acting simultaneously, the magnetic field produces strong attractive forces on the ferrofluid, particularly in the vicinity of the magnet. These holding forces result in a predictable configuration of the fluid in the porous medium which is dependent on the magnetic field and independent of flow pathway or heterogeneity of the porous medium. No significant retention effects due to flow through variably saturated sands are observed. While the proposed field engineering applications of ferrofluids are promising, the observations to date are particularly relevant at the laboratory scale where the decrease in magnetic field strength with distance from a magnet is less of a limitation than in larger scale applications. Ferrofluids may find immediate application in any situation where it is desirable to control the motion or final configuration of fluid in an experimental flow apparatus without direct physical contact.     

Non-slipping adhesive contact between mismatched elastic spheres: A model of adhesion mediated deformation sensor

A generalized JKR model is established for non-slipping adhesive contact between two dissimilar elastic spheres subjected to a pair of pulling forces and a mismatch strain. We discuss the full elastic solution to the problem as well as the so-called non-oscillatory solution in which tension and shear tractions along the contact interface is decoupled from each other. The model indicates that the mismatch strain has significant effect on the contact area and the pull-off process. Under a finite pulling force, a pair of adhering spheres is predicted to break apart spontaneously at a critical mismatch strain. This study suggests an adhesion mediated deformation sensing mechanism by which cells and molecules can detect mechanical signals in the environment via adhesive interactions.     

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.     

Crushing of particles in idealised granular assemblies

Four idealised assemblies of equally sized spherical particles are subjected to a range of macroscopic compressive principal stresses and the contact forces on individual particles are determined. For each set of contact forces the stress fields within individual particles are studied. A failure criterion for brittle materials is imposed and indicates that crushing (or rupture) occurs when the maximum contact force reaches a threshold particle strength value, irrespective of the presence and magnitude of other lesser contact forces acting on the particle and the material properties of the particle. Combining the crushing mechanism with an assembly instability mechanism enables failure surfaces to be drawn in the three-dimensional stress space. A simple spatial averaging technique has been applied to the failure surfaces to remove the effects of assembly anisotropies. Sections of the failure surfaces on π planes have similarities to those commonly used in sand modelling.     

High-amplitude elastic solitary wave propagation in 1-D granular chains with preconditioned beads: Experiments and theoretical analysis

Elastic solitary waves resulting from Hertzian contact in one-dimensional (1-D) granular chains have demonstrated promising properties for wave tailoring such as amplitude-dependent wave speed and acoustic band gap zones. However, as load increases, plasticity or other material nonlinearities significantly affect the contact behavior between particles and hence alter the elastic solitary wave formation. This restricts the possible exploitation of solitary wave properties to relatively low load levels (up to a few hundred Newtons). In this work, a method, which we term preconditioning, based on contact pre-yielding is implemented to increase the contact force elastic limit of metallic beads in contact and consequently enhance the ability of 1-D granular chains to sustain high-amplitude elastic solitary waves. Theoretical analyses of single particle deformation and of wave propagation in a 1-D chain under different preconditioning levels are presented, while a complementary experimental setup was developed to demonstrate such behavior in practice. The experimental results show that 1-D granular chains with preconditioned beads can sustain high amplitude (up to several kN peak force) solitary waves. The solitary wave speed is affected by both the wave amplitude and the preconditioning level, while the wave spatial wavelength is still close to 5 times the preconditioned bead size. Comparison between the theoretical and experimental results shows that the current theory can capture the effect of preconditioning level on the solitary wave speed.     

考虑接触应力非线性分布的接触力元模式及其验证分析

在作者提出的非连续变形计算力学模型中,采用接触力元模型描述多体接触界面上的接触特性.由于这种模型中假定接触应力沿接触界面为线性分布,从而得到的接触界面应力分布往往出现跳跃等非光滑性特征,该文对此进行了改进,采用具有高阶光滑性的非线性函数建立了能够考虑界面上接触应力非线性分布的接触力元模式,以期合理地揭示多体系统中界面的接触特性.对某一典型算例进行了数值计算,通过与大型通用非线性有限元结构分析软件ABASQUS的计算结果对比,验证了所建议计算模型的合理性与有效性.两种方法计算得到的界面接触对上的接触力基本相同;而由于采用的应力分布模式的假定不同,接触应力有所差别,由于在该文计算模型中接触对上的接触应力是按照未知量直接求得的,因此按照所建议的非线性接触力元模式所得到的接触应力更为合理.     

凹形多面体离散单元的水平集函数接触算法

多面体模型理论上可构造任意颗粒形态,然而受单元接触算法的限制,仅用于凸形颗粒材料的离散元模拟。对于具有凹形特征的多面体单元,单个接触点的搜索算法难以精确计算单元间的作用力。考虑多面体单元间存在单个或多个接触点的计算特性,本文发展了适用于凸形和凹形多面体颗粒材料的水平集函数接触算法。该方法通过点-三角形单元距离计算方法和奇-偶数判定方法建立多面体单元的零水平集函数和空间水平集函数,并对水平集函数进行三线性插值,可得到多面体单元间的单个或多个接触点。为检验水平集函数接触算法的可靠性,对球形和凹形多面体颗粒材料的堆积和倒塌过程进行离散元模拟,并分析颗粒形状对堆积密度和休止角的影响规律。     

Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.     

A Molecular Dynamics Model for Single Adhesive Contact

The normal adhesive contact between a pair of asperities is performed using molecular dynamics. To simplify the problem, the equivalent contact problem of sphere–plane interaction is solved. Displacements in the contact zone are very small compared to the asperity size, therefore, the computational model is focused on the neighborhood of the contact area. The adhesion between the asperity and the plane is calculated as a sum of interactions between atoms of the asperity and the plane. A computational experiment of pull-on and pull-off is carried out to study the influence of the adhesion on the formation of the contact forces and deformations. The numerical results are compared with theoretical predictions.     

Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.