Variational model of granular flow in a three-dimensional rotating container

We considered a flow of a cohesionless granular material in a partially filled three-dimensional rotating container. A variational continuum mechanics model is suggested to describe the profile of the free surface of the granular material. A combination of a level set method and a variational form of energy conservation equation allows us to describe flow of granular material in arbitrary-shaped containers.     

颗粒-颗粒接触力的热力学模型

以颗粒二体接触力模型为基础和出发点的软球离散元模拟是当前颗粒物理和力学领域广泛应用的研究手段.但文献上经常使用的、包括著名的Hertz-Mindlin和Luding在内的力模型并没有完全明确弹性势能或耗散热的计算方法,故从热力学层面看它们还需要完善.考虑到机械能的耗散行为是这类材料的重要物理内容,本文借鉴近年来提出的颗粒固体流体动力学(GSH)思路,提出一种具有明确势能和热功率的接触力建模方法.该理论除明确给出了机械能和热能的计算公式外,还能具体描述能量守恒、热力学平衡态和熵增加等基本原理,解决了传统接触力模型在这些方面的欠缺问题.初步计算显示本文模型的恢复系数可以随碰撞速率的增加而减弱,这比现有的其他模型更符合实验观测.虽然为简单起见这些公式仅局限于二维和忽略颗粒转动运动情况,文中讨论了如何推广到三维含转动情形,以及所涉及的滚动和扭转接触力的热力学处理问题.鉴于是否在Onsager非平衡热力学基础上建模是本文给出的接触力公式有别于当前其他模型的关键所在,文中强调了这里的主要建模对象应该是热力学特征函数和Onsager迁移系数,而接触力是它们的推导结果.这是一个与目前直接针对接触力进行建模的不同思路.文中对颗粒物质特有的、反映样品几何变形与弹应变之间联络的一个非对角迁移系数做了详细介绍,并且认为它与打滑等复杂力学现象关系密切,无论宏观GSH尺度上,还是细观接触力尺度上都不可忽略.     

Density fluctuations in a model for vibrated granular media

This paper presents the study of density fluctuations in a model for vibrated granular media. Their microscopic origin is shown to be linked to the microscopic disorder in grains packing. Varying vibrations amplitude and duration, several regimes are found for density relaxation. Its power spectrum is well described by power laws.     

Low-energy excitations in the three-dimensional random-field Ising model

General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method even though granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.     

From elasticity to hypoplasticity: dynamics of granular solids

"Granular elasticity," useful for calculating static stress distributions in granular media, is generalized by including the effects of slowly moving, deformed grains. The result is a hydrodynamic theory for granular solids that agrees well with models from soil mechanics.     

A new statistical mechanics approach to dense granular media

A new statistical mechanics approach to dense granular media is presented. The thermodynamic formalism is set out directly in terms of elastic potential energy, such that the configurational temperature (the intensive property which defines the steady state) relates to a quadratic function of the stresses (rather than other linear functions used in recent developments). Dense granular media are considered as canonical ensembles of noninteracting clusters, which can be identified with repeatable equilibrium configurations. Then, particles can be located in a new proposed phase space (conceived to separate the elastic potential energy levels). Although the importance of this paper lies in the method itself, it has been illustratively applied to the simple case of two-dimensional (2D) dense granular media (an arrangement of frictionless monodisperse elastic disks under isotropic horizontal stress compression). In this case, the temperature is directly replaced by the squared external pressure, and the packing ratio of the most probable microstate is close to the reported value of random close packing. Moreover, some interesting general conclusions arise.     

Shear instabilities in granular mixtures

Dynamical instabilities in fluid mechanics are responsible for a variety of important common phenomena, such as waves on the sea surface or Taylor vortices in Couette flow. In granular media dynamical instabilities have just begun to be discovered. Here we show by means of molecular dynamics simulation the existence of a new dynamical instability of a granular mixture under oscillating horizontal shear, which leads to the formation of a striped pattern where the components are segregated. We investigate the properties of such a Kelvin-Helmholtz-like instability and show how it is connected to pattern formation in granular flow and segregation.     

Effective temperature and fluctuation-dissipation theorem in athermal granular systems:A review

The definition and the previous measurements of a dynamics-relevant temperature-like quantity in granular media are reviewed for slow and fast particle systems. Especially, the validity of the fluctuation-dissipation theorem in such an athermal system is explored. Experimental evidences for the fluctuation-dissipation theorem relevant effect temperature support the athermal statistical mechanics, which has been widely explored in recent years by physicists. Difficulties encountered in defining temperature or establishing thermodynamics or statistical mechanics in non-equilibrium situations are discussed.     

颗粒介质尺度效应的抗剪试验及物理机理分析

针对颗粒介质力学特性的颗粒尺度效应研究,选用土矿物颗粒制备不同颗粒尺度的抗剪试样,进行一系列直剪快剪和三轴抗剪试验,测得了不同颗粒粒径和体分比试样的变形曲线及剪应力强度;基于颗粒间微观作用力与重力比值和胞元体模型,首次从微观和细观角度解释颗粒尺度效应的物理机理.结果表明,随着介质中粗颗粒的比例增加和粒径减小,介质变形特性增强,剪应力强度也随之提高;体分比对变形和强度特性的影响比粒径的影响更加显著.基于介质特性尺度效应物理机理分析,提出衡量介质颗粒聚集和摩擦效应的微重比判别参数以及应变梯度和变形协调微裂纹引起颗粒尺度效应的细观机理解释;文中提出的胞元体模型大大减少了颗粒物质体系的计算自由度,为工业和工程设计的计算建模提供一种可行途径.     

Modeling of stress distribution in granular piles: Comparison with centrifuge experiments

The classical method to compute stress and strain distributions in granular materials is recalled using continuum mechanics approach, and different rheological laws described. It is recalled that granular materials exhibit highly nonlinear response such as nonlinear elasticity, dilatancy and plastic flow. Finite element technique is used to predict the stress field distribution below a conic and a triangular pile. The dependence of the stress distribution on the rheological law, the bottom boundary condition and the building process (horizontal or inclined strata) is demonstrated. These results are compared to experimental data obtained in centrifuge. (c) 1999 American Institute of Physics.     

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input–output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.     

On accounting for the effect of particles of a condensed dispersed phase on radiant energy transfer in gaseous media

Questions concerning the formation of the optical properties of dense gaseous and plasma media in relation to the specific features of radiant energy transfer are considered. The integral equations describing the radiation trapping are investigated as a new class of generalized wave equations of Schrödinger type. Starting from the methods of quantum mechanics, original analytical and numerical approaches are suggested for solving problems of the radiative kinetics of both spatially homogeneous and inhomogeneous absorbing media containing dispersed particles. In terms of the quasi-classical approximation, two classes of reference problems for determination of phase factors are formulated. Solutions for a number of model problems are presented that demonstrate the efficiency of the methods developed.     

Physics of particulate flows: From sand avalanche to active suspensions in plants

Flows of granular media in air or in a liquid have been a research field for physicists for several decades. Sometimes solid, sometimes liquid, these particulate materials exhibit peculiar behaviors, which have motivated many studies at the frontiers between nonlinear physics, soft matter physics and fluid mechanics. This paper presents a summary of the recent advances in the field, with a focus on the development of continuous approaches, which make it possible to treat granular media as a complex fluid and to develop a granular hydrodynamics. We also discuss how the better understanding of granular flows we have today may help to address more complex materials, such as colloidal suspensions or some biological systems.     

A roe-average algorithm for a granular-gas model with non-conservative terms

A Roe-average algorithm has been derived for a granular-gas model, proposed by Goldshtein and Shapiro [Goldshtein, Shapiro, Mechanics of collisional motion of granular materials: Part 1. General hydrodynamic equations, J. Fluid Mech. 282 (1995) 75–114], which contains non-conservative terms in the Euler-like hyperbolic governing equations apart from sink terms, which arise from inelastic collision of granules and are present only in the energy equation. The non-conservative terms introduce non-isentropic effects in acoustic-wave propagation within granular media and they also contribute to the Rankine–Hugoniot relations across a discontinuity. A Roe-average algorithm, based on the same granular-gas model, was derived in the literature [V. Kamenetsky, A. Goldshtein, M. Shapiro, D. Degani, Evolution of a shock wave in a granular gas, Phys. Fluids, 12 (2000) 3036–3049] which then required the implementation of a shock-fitting technique at a discontinuity. In the present work, Roe-averaged variables have been obtained from the Rankine–Hugoniot jump relations and the non-conservative terms have been incorporated in the numerical flux formula consistent with upwind principles associated with the granular speed of sound. Results for unsteady one-dimensional granular flows, colliding with a wall, demonstrate the capability of the proposed algorithm to capture strong shocks in addition to flow features not found in molecular gases, such as a fluidized region downstream of the shock and a compacted solid-block region adjacent to the wall.     

A new empirical model for the acoustic properties of loose granular media

This paper examines physical parameters of loose granular mixes and their empirical relations to the acoustic performance of these mixes. In this work a new classification of granular media has been proposed which is related to the characteristic particle dimension and the specific density of the grain base. It has been shown that this classification is a useful characteristic for rapid evaluation of the acoustic performance of loose granular mixes. The characteristic impedance and propagation constant have been measured for a representative selection of grain mixes and used to develop a new empirical model. This model relates the above acoustic characteristics to the characteristic particle dimension, porosity, tortuosity and specific density of the grain base, which are routinely measurable parameters. A very good agreement with the experimental data is illustrated in the frequency range of 250-4000 Hz for materials with the grain base of 0.4-3.5 mm and specific densities between 200 and 1200 kg/m³. Unlike many theoretical models for the prediction of the acoustic properties of porous media, the proposed expressions do not involve any special functions of complex argument, empirical shape factors or sophisticated characteristics of porous structure. These are practical enough to be of interest to acoustic and noise control engineers and material manufacturers.     

Quantum dynamics of theSU(2)-Skyrme model in Nelson stochastic mechanics

In the framework of Nelson stochastic mechanics the SkyrmeSU(2) model is quantized. A new term is added to a classical skyrmion mass. It coincides with the term obtained by Fujiiet al. by modifying the canonical quantization. This example illustrates that stochastic mechanics as an alternative method of quantization is convenient for theories with collective coordinates and for nonlinear theories, as some problems related to operator ordering and modification of canonical formalism are naturally solved.     

Variational problems on vector bundles

A variety of problems in quantum physics and classical statistical mechanics, in particular the quantization of topological solitons and the statistical mechanics of defects in ordered media, are described. These problems can be studied within a semi-classical approximation, or with the help of low-temperature expansions, respectively. The calculation of the leading term in such expansions gives rise to variational problems for sections of vector bundles characterized by certain topological constraints. Examples of such problems are the quantization of kinks in the two-dimensional ⁴-theory and the analysis of Bloch walls in a Landau-Ginzburg model of a threedimensional anisotropic ferromagnet. We state a general existence result for variational problems of this kind and develop regularity and decay estimates for solutions of the Landau-Ginzburg model describing Bloch walls with prescribed boundaries. For certain boundary configurations stability results are established. The relation between the minimizers of the Landau-Ginzburg model in a certain strong-coupling limit and minimal surfaces is pursued in some detail. An open question is whether, asymptotically, the stability of the limit (minimal) surface will imply the stability of the minimizers of the Landau-Ginzburg model.