Simulation Of Granular Materials Research Articles

Finite width of quasistatic shear bands

I study the average deformation rate of an amorphous material submitted to an external uniform shear strain rate, in the geometry known as the split-bottom configuration. The material is described using a stochastic model of plasticity at a mesoscopic scale. A shear band is observed to start at the split point at the bottom, and widen progressively towards the surface. In a two-dimensional geometry the average statistical properties of the shear band look similar to those of the directed polymer model. In particular, the surface width of the shear band is found to scale with the system height H as H;{alpha} with alpha=0.68+/-0.02 . In more realistic three-dimensional simulations the exponent changes to alpha=0.60+/-0.02 and the bulk profile of the width of the shear band is closer to a quarter of a circle, as it was observed to be the case in recent simulations of granular materials.

Contact dynamics method

ABSTRACT In this paper, the basics of the CD (contact dynamics) method for discrete element simulation of granular materials are presented. This method is the algorithmic formulation of nonsmooth granular dynamics at the scale of particle rearrangements where small elastic response times and displacements are neglected. Two major issues arising in this context are treated: 1) the contact laws expressed as complementarity relations between the contact forces and velocities and 2) the nonsmooth motion involving velocity jumps with impulsive unresolved forces as well as smooth motion with well-resolved static forces. A consistent description of the dynamics at the velocity level leads naturally to an implicit time-stepping scheme together with an explicit treatment of the evolution of the contact network. We discuss the role of the coarsegraining time, the precision issues and the relevance of the coefficients of restitution and their interpretation in a multicontact system.

Introduction to discrete element methods

ABSTRACT One challenge of today's research is the realistic simulation of granular materials, like sand or powders, consisting of millions of particles. In this article, the discrete element method (DEM), as based on molecular dynamics methods, is introduced. Contact models are at the physical basis of DEM. A set of the most basic force models is presented involving either elasto-plasticity, adhesion, viscosity, static and dynamic friction as well as rolling- and torsion-resistance. The examples given concern clustering in granular gases, and bi-axial as well as cylindrical shearing of dense packings in order to illustrate the micro-macro transition towards continuum theory.

Contact of granular particles and the simulation of rapid flows using event-driven molecular dynamics

ABSTRACT We discuss several models for granular particles commonly used in Molecular Dynamics simulations of granular materials, including spheres with linear dashpot force, vis-coelastic spheres and adhesive viscoelastic spheres. Starting from the vectorial interaction forces we derive the coefficients of normal and tangential restitution as functions of the vectorial impact velocity and of the material constants. We review the methods of measurements of the coefficients of restitution and characterize the coefficient of normal restitution as a fluctuating quantity. Moreover, the scaling behavior and the influence of different force laws on the dynamical system behavior are discussed. The powerful method of event-driven Molecular Dynamics is described and the algorithmic simulation technique is explained in detail. Finally we discuss the limitations of event-driven MD.

Simulation of the packing of granular mixtures of non-convex particles and voids characterization

The simulation of granular materials has considerably developed in the last decades essentially with simple geometry particles. The purpose of this paper is to study granular systems of non-convex particles which are present in many industrial processes. Two shapes of large and two shapes of small non-convex particles resulting from the cutting of a hollow cylinder are modelled, and binary mixtures containing varying proportions of small and large particles are generated with a Monte Carlo simulation. Two different states of the granular systems are studied: suspensions and packings obtained after sedimentation. No contact force model is used and only steric repulsion is taken into account. The density, the pore size distribution and the tortuosity of the granular systems are studied. The results are compared to those obtained with granular systems of convex particles.

Numerical simulation of granular materials in a rotating tumbler

Using a simple numerical model based on particle dynamics, we perform two-dimensional simulations of granular materials inside a rotating tumbler. Still in its first stages of development, the model can quantitatively reproduce some of the general behavior observed in these systems, both for monodisperse and binary mixtures. Work is currently being done to validate the model and obtain results that are directly comparable to experiments.

Three‐dimensional modelling of discrete particles by superellipsoids

AbstractDiscrete Elements are used for the simulation of granular materials (sand, ballast) as well as for molecular assemblies. Circles (2D) and spheres (3D) are often used in literature on the Discrete Element Method (DEM) however they represent a strong idealisation of the real geometry. Superellipsoids provide the opportunity to generate a wide variety of three‐dimensional geometrical shapes (e.g. sphere, cube, cylinder).The motion of each particle is described by means of rigid body dynamics. Suitable numerical integration methods are necessary which are able to conserve the essential physical quantities like momentum energy etc.. Possible choices are e.g. the explicit Verlet‐Leapfrog method for the translation and the explicit fourth order Runge‐Kutta method for the rotation. The implemented contact formulation takes damping as well as friction into account. Efficient implementation of the contact search is the main aim of this part of the work. It is subdivided into the neighbourhood search and the local search. A bisection algorithm is used to calculate the gap between two superellipsoids within the search. For the neighbourhood search two binning algorithms were implemented and compared for several packages of particles. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Investigation of performance of programming approaches and languages used for numerical simulation of granular material by the discrete element method

Performance of programming approaches and languages used for the development of software codes for numerical simulation of granular material dynamics by the discrete element method (DEM) is investigated. The granular material considered represents a space filled with discrete spherical visco-elastic particles, and the behaviour of material under imposed conditions is simulated using the DEM. The object-oriented programming approach (implemented via C++) was compared with the procedural approach (using FORTRAN 90 and OBJECT PASCAL) in order to test their efficiency. The identical neighbour-searching algorithm, contact forces model and time integration method were implemented in all versions of codes. Two identical representative examples of the dynamic behaviour of granular material on a personal computer (compatible with IBM PC) were solved. The results show that software based on procedural approach runs faster in compare with software based on OOP, and software developed by FORTRAN 90 runs faster in compare with software developed by OBJECT PASCAL.

Numerical tests of constitutive laws for dense granular flows

We numerically and theoretically study the macroscopic properties of dense, sheared granular materials. In this process we first consider an invariance in Newton's equations, explain how it leads to Bagnold's scaling, and discuss how it relates to the dynamics of granular temperature. Next we implement numerical simulations of granular materials in two different geometries--simple shear and flow down an incline--and show that measurements can be extrapolated from one geometry to the other. Then we observe nonaffine rearrangements of clusters of grains in response to shear strain and show that fundamental observations, which served as a basis for the shear transformation zone (STZ) theory of amorphous solids [M. L. Falk and J. S. Langer, Phys. Rev. E. 57, 7192 (1998); M.R.S. Bull 25, 40 (2000)], can be reproduced in granular materials. Finally we present constitutive equations for granular materials as proposed by Lemaître [Phys. Rev. Lett. 89, 064303 (2002)], based on the dynamics of granular temperature and STZ theory, and show that they match remarkably well with our numerical data from both geometries.

Numerical simulation of granular materials by an improved discrete element method

AbstractIn this paper, we present an improved discrete element method based on the non‐smooth contact dynamics and the bi‐potential concept. The energy dissipated during the collisions is taken into account by means of restitution coefficients. The interaction between particles is modelled by Coulomb unilateral contact law with dry friction which is typically non‐associated: during the contact, the sliding vector is not normal to the friction cone. The main feature of our algorithm is to overcome this difficulty by means of the bi‐potential theory. It leads to an easy implement predictor–corrector scheme involving just an orthogonal projection onto the friction cone. Moreover the convergence test is based on an error estimator in constitutive law using the corner stone inequality of the bipotential. Then we present numerical simulations which show the robustness of our algorithm and the various possibilities of the software ‘MULTICOR’ developed with this approach. Copyright © 2004 John Wiley & Sons, Ltd.

A 3D semi-digital model for the placing of granular materials

A 3D semi-digital model has been developed in order to simulate the gravitational placing of granular mixtures composed of spherical particles of different sizes allowing the optimization of mixtures proportions. The aim is to control concrete placing in formworks particularly in the presence of reinforcement bars. In this case, the representation of a large number of particles is necessary and segregation or wall effects have to be taken into account. Moreover, an easy representation of complex geometries and obstacles of arbitrary shapes should be possible. In order to respect these conditions, the used 3D semi-digital model is based on the digitization of a calculation volume and on the representation of spherical particles with real numbers. The model takes into account only steric repulsion between particles and simulates their collisions during placing by random displacements. The simplicity of the model allows the simulation of up to 2 million spherical particles without spending too much computational time. The model can therefore be used for the simulation of granular materials having a spread granulometry-like concrete. The packing density of binary mixtures of spherical particles with different size ratios and varying proportions of small and big particles is first studied. The results are compared successfully with theoretical packing density of binary mixtures without interactions. It is therefore demonstrated that the semi-digital model can be used for the simulation of binary mixtures having low size ratios (0.1), which corresponds approximately to the size ratio of fine and coarse aggregates in concrete. Secondly, digitized cylindrical obstacles are added in the systems with the aim to represent reinforcement bars in a formwork. The influence of small and big particles proportioning on the local packing density and the homogeneity of the mixture is studied. The results show that in the presence of obstacles, the optimum placing is obtained when the proportion of small particles in the mixture is higher than that without obstacles. This is due principally to the looser packing of big particles around obstacles.

A new approach for calculating strain for particulate media

AbstractDiscrete element modelling is a viable alternative to conventional continuum‐based analysis for analysing problems involving localized deformations of particulate media. However, to aid in the interpretation of the results, it is useful to express the results of discrete element analyses in terms of the continuum parameters of stress and strain. A number of homogenization methods have been proposed to calculate strain in discrete systems; however, two significant limitations of these methods remain. First, none of these methods incorporate particle rotation effects satisfactorily, although significant particle rotation occurs in shear bands in both physical tests and numerical simulations of granular materials. Additionally, observations of the particle displacement fields in shear bands in granular materials indicate that the displacements within the localizations are erratic. Consequently, existing linear, local interpolation approaches produce substantial variations in the strain values calculated in adjacent elements in the region of localization, hindering clear visualization of the strain localization as it evolves. A new method of domain discretization for calculating strain is proposed. This method is capable of capturing particle rotation and employs a non‐local meshfree interpolation procedure capable of smoothing the erratic displacements in strain localizations, which better defines their evolution. The proposed method is validated for problems involving both two and three dimensions. A number of methods are compared with the proposed method and pertinent insights are made. Copyright © 2003 John Wiley & Sons, Ltd.

Modeling of Granular Damper Using Cellular Automata

The present paper deals with flow simulations of granular materials moving inside container using Cellular Automata (CA). CA is a class of computer modeling techniques, which consists of discrete unit elements arranged uniformly on spaces, each of which can vary within a finite set of values to express the physical state of the components of the system. The time of evolution of the element state is performed synchronously according to local neighbor rules, instead of governing equations, taking into account the state of the element itself and its nearby elements. Compared with the conventional method such as DEM, it is addressed that the modeling techniques with CA provide advantages on the point of computation efficiency and numerical stability due to the discrete treatment of time and space. In this study, the damping characteristics of a granular damper is investigated numerically by Cellular Automata model and also by DEM, where a container which incorporates granular materials is attached to the mass of one-DOF vibrating system. The container is treated two-dimensionally with thickness identical to a diameter of particle. The simulated particle motions and the damping effects obtained by CA model is compared with DEM solution as well as experimental results, and the present CA model is evaluated in qualitative aspects.

Dynamic rearrangements and packing regimes in randomly deposited two-dimensional granular beds.

We study the structural properties of two-dimensional granular packings prepared by random deposition from a source line. We consider a class of random ballistic deposition models based on single-particle relaxation rules controlled by a critical angle, and we show that these local rules can be formulated as rolling friction in the framework of dynamic methods for the simulation of granular materials. We find that a packing prepared by random deposition models is generically unstable, and undergoes dynamic rearrangements. As a result, the dynamic method leads systematically to a higher solid fraction than the geometrical model for the same critical angle. We characterize the structure of the packings generated by both methods in terms of solid fraction, contact connectivity, and anisotropy. Our analysis provides evidence for four packing regimes as a function of solid fraction, the mechanisms of packing growth being different in each regime.

Shear profiles and localization in simulations of granular materials.

We present results from two-dimensional computer simulations of shearing granular layers, using a discrete element code, and applying a wide range of boundary conditions. We specifically investigate the distribution of shear within the granular layer and find two different modes of localization depending on the applied shear velocity, pressure, and layer thickness: (1) granular layers that develop a persistent shearing boundary region ("fluidlike" behavior) and (2) layers that switch between diffuse deformation and randomly positioned internal shear bands ("solidlike" behavior). The two end-member deformation modes can be found in laboratory experiments performed under low and high confining pressure, respectively. Micromechanical investigation reveals two different statistical distributions of the grain contacts correlating with the two different shearing modes. These results imply that rehological transitions in granular flow modes are linked to quantifiable microtstructural organization.

Dynamic triangulations for efficient detection of collisions between spheres with applications in granular media simulations

The distinct element method (DEM) is an approach to modelling and simulation well suited to the study of discontinuous phenomena. By tracking each element individually and dealing explicitly with every interaction among the elements, the DEM allows one to deduce statistical behaviour of the assembly. Realistic simulation of granular materials has largely benefited from this relatively recent approach. In particular, it is now possible to address specific phenomena such as arching effects, flows and packings in three dimensions and to make them observable, which is impossible or at least extremely complicated to achieve with physical experiments. Two major issues arise in the context of DEM: efficient detection of the interactions between the grains and numerical modelling of the physical contacts. This paper focuses on the former by presenting an efficient collision detection scheme for spherical grains of various sizes based on a dynamic three- dimensional Delaunay triangulation. Aside from the theoretical and practical discussion of this scheme, the remaining requirements for a complete DEM simulation are also addressed. Finally, examples are given to illustrate the potential of this approach.

A discontinuous future for numerical modelling in geomechanics?

Finite-element or finite-difference methods are commonly used in geomechanics to model the response of soil or rock at an engineering site. These methods assume that the material is a continuum (although known discontinuities can be included explicitly). There are two drawbacks with continuum methods. Firstly, an appropriate stress-strain law for the material may not exist, or the law may be excessively complicated with many obscure parameters. Secondly, the natural development of cracks and rupture surfaces is not well-handled by continuum approaches. It is suggested that the future trend for numerical modelling in soil and rock may consist of the replacement of continuum methods by particle methods. Assemblies of discrete particles (bonded together to represent rock, and unbonded to represent soil) capture the complicated behaviour of actual material with simple assumptions and few parameters at the micro level. Complex overall behaviour arises as an emergent property of the assembly. The application of particle methods to large-scale problems is currently difficult or impossible because of high computational demands. It is shown that such applications should be feasible within ten years, and certainly within 20 years. Examples are given of the simulation of granular material and rock by particle methods.

A discontinuous future for numerical modelling in geomechanics?

Finite-element or finite-difference methods are commonly used in geomechanics to model the response of soil or rock at an engineering site. These methods assume that the material is a continuum (although known discontinuities can be included explicitly). There are two drawbacks with continuum methods. Firstly, an appropriate stress-strain law for the material may not exist, or the law may be excessively complicated with many obscure parameters. Secondly, the natural development of cracks and rupture surfaces is not well-handled by continuum approaches. It is suggested that the future trend for numerical modelling in soil and rock may consist of the replacement of continuum methods by particle methods. Assemblies of discrete particles (bonded together to represent rock, and unbonded to represent soil) capture the complicated behaviour of actual material with simple assumptions and few parameters at the micro level. Complex overall behaviour arises as an emergent property of the assembly. The application of particle methods to large-scale problems is currently difficult or impossible because of high computational demands. It is shown that such applications should be feasible within ten years, and certainly within 20 years. Examples are given of the simulation of granular material and rock by particle methods.

Modeling of scale effects of damage in cemented granular rocks

We propose a probabilistic model of cemented granular material consisting of elastic grains and of weak brittle cement which may be present in contacts between grains. The central assumption of the model is that the damage of such a material under the effect of external mechanical loading is principally controlled by the two following factors: 1) heterogeneous distribution of cement in the sample; and 2) heterogeneous (typically exponential) distribution of inter-granular forces and the associated elastic energy produced by the macroscopic strain. These assumptions, which can be justified by direct observations in granular rock (Menéndez et al., 1996) and by experimental and numerical studies of force distributions in granular media (Mueth et al., 1998; Makse et al., 2000), combined with a simple fracture criterion for a single cemented contact, lead to a percolation-type damage behavior. An important feature of the proposed model is an explicit scale dependence of certain macroscopic damage parameters for a material with fixed microscopic properties and given distributions of heterogeneities. This scale dependence may be important for interpretation of numerical simulations of granular materials using a moderate number of grains, as well as in attempts to extrapolate damage criteria in rock from laboratory to larger scales. The predictions of the probabilistic damage model are qualitatively confirmed by 3D numerical simulations using a Discrete Element Model of cemented granular material.

Similarity ‘hot-spot’ solutions for a hypoplastic granular material

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Hill J. M. 2000Similarity ‘hot-spot’ solutions for a hypoplastic granular materialProc. R. Soc. Lond. A.4562653–2671http://doi.org/10.1098/rspa.2000.0631SectionRestricted accessResearch articleSimilarity ‘hot-spot’ solutions for a hypoplastic granular material J. M. Hill J. M. Hill School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author J. M. Hill J. M. Hill School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 November 2000https://doi.org/10.1098/rspa.2000.0631AbstractHypoplasticity is a theory which has been developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe to accurately model the behaviour of granular materials such as sand, soil and certain powders. The earlier hypoplasticity theories were formulated on the basis of the two assumptions of rate independence and homogeneous dependence of the stress rate on the stress. These two assumptions mean that the governing partial differential equations remain invariant under a general family of stretching similarity transformations, which in turn imply the existence of similarity solutions for which the partial differential equations can be reduced to ordinary differential equations. In two recent papers involving uniaxial compaction and cylindrical and spherical cavity expansion problems, these similarity solutions have been identified and the appearance of uniaxial ‘hot–spot” solutions has been noted. In this context, ‘hot–spot” refers to the material undergoing an infinite stress in a finite time. Here, for planar, cylindrical and spherical geometries, we summarize the general picture relating to these similarity ‘hot–spot” solutions and we give a number of simple analytical expressions for them not included in the recent papers. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Yuu S, Umekage T, Mitsuiki Y and Koga F (2008) Constitutive Relations Based on Distinct Element Method Results for Granular Materials and Simulation of Granular Collapse and Heap by Smoothed Particle Hydrodynamics, and Experimental Verification, Advanced Powder Technology, 10.1163/156855208X293808, 19:2, (153-182), . Yuu S and Umekage T (2008) Constitutive Relations and Computer Simulation of Granular Material, Advanced Powder Technology, 10.1163/156855208X294609, 19:3, (203-230), . Wu W (2006) On High-order Hypoplastic Models for Granular Materials, Journal of Engineering Mathematics, 10.1007/s10665-006-9040-7, 56:1, (23-34), Online publication date: 1-Sep-2006. Hill J and Selvadurai A Mathematics and mechanics of granular materials Mathematics and Mechanics of Granular Materials, 10.1007/1-4020-4183-7_1, (1-9) Hill J and Selvadurai A (2005) Mathematics and mechanics of granular materials, Journal of Engineering Mathematics, 10.1007/s10665-005-2729-1, 52:1, (1-9), Online publication date: 1-Jul-2005. Hill J and Selvadurai A (2005) Mathematics and mechanics of granular materials, Journal of Engineering Mathematics, 10.1007/BF02694027, 52:1-3, (1-9), Online publication date: 1-Jul-2005. Cox G and Hill J (2005) Hypoplasticity theory for granular materials—I: Two-dimensional plane strain exact solutions, International Journal of Non-Linear Mechanics, 10.1016/j.ijnonlinmec.2005.05.007, 40:9, (1171-1188), Online publication date: 1-Nov-2005. Cox G and Hill J (2002) Non-dilatant double-shearing theory applied to dynamical granular chute flow, Acta Mechanica, 10.1007/BF01171451, 159:1-4, (125-142), Online publication date: 1-Mar-2002. This Issue08 November 2000Volume 456Issue 2003 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0631Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/11/2000Published in print08/11/2000 License: Citations and impact Keywordsgranular materialsuniaxial compactionhypoplasticityhot spotssimilarity solutionscavity expansion