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 On computer simulation of dry particle systems using discrete element method and the development of DEM contact forcedisplacement models
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 Zhang, Xiang, 1966
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 1998
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 Coefficient of restitution ( jstor )
Contact loads ( jstor ) Copyrights ( jstor ) Mindlin Reissner plate theory ( jstor ) Modeling ( jstor ) Plasticity ( jstor ) Simulations ( jstor ) Soybeans ( jstor ) Stiffness ( jstor ) Velocity ( jstor ) City of Gainesville ( local )
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ON COMPUTER SIMULATION OF DRY PARTICLE SYSTEMS USING DISCRETE ELEMENT METHOD AND THE DEVELOPMENT OF DEM CONTACT FORCEDISPLACEMENT MODELS BY XIANG ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 \
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To my wife and my parents.
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ACKNOWLEDGEMENTS I am grateful to my advisor, professor Loc VuQuoc for his support and guidance throughout my Ph.D. education at the University of Florida. I have greatly benefited by his stimulating approach to research and his relentless pursuit of perfection. Many thanks are extended to him for his invaluable help in preparation of this document. I am greatly indebted to Professor Otis R. Walton for his numerous advices and nonselfish help. Without his help, I could not have accomplished the achievement I have made. I thank Professor Daniel Hanes and Professor Ray Bucklin for their advice and help in improving my research. I thank also Professors N. Cristescu, I. Ebcioglu, C. Hsu, and S. Sahni for serving on my committee and for reviewing this thesis. I also benefited a lot from their graduate courses and from their help in many other aspects. I must thank Lee Lesbury for countless technical and philosophical discussions without which I could not have completed this work. Lee made significant contributions to the finite element analyses of the contact problems and the development of the contact forcedisplacement models. To the friends and colleagues, Hui Deng, Xiangguang Tan, Brian Fuller, KilSoo Mok, Vinay Srinivas, Shajeel Jaitapker, JiYao Yang, and many others, who have made my stay at Gainesville one of the most memorable periods of my life, I offer my appreciation for the uncountable enjoyable discussions as well as for the encouragement and support I received from them. Special thanks go to my wife, Yanlin Guo, for the happiness we shared and for being with me to get through the hardship together. I am forever indebted to my in
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parents and my brother, without their endless love and sacrifice, I could not have accomplished so much. The support of the Engineering Research Center for Particle Science and Technology at the University of Florida is gratefully acknowledged. IV
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii ABSTRACT viii CHAPTERS 1 INTRODUCTION 1 1.1 Why Dry Particle System? 1 1.2 Discrete Element Method 2 1.3 Contribution and Chapter Overview 4 2 DISCRETE ELEMENT METHOD 7 2.1 Algorithm of DEM 7 2.2 Geometric Modeling of Nonspherical Particles and Contact Detection 19 3 THEORIES OF CONTACT MECHANICS 29 3.1 Hertz Theory 29 3.2 Mindlin and Deresiewicz Theory 35 3.2.1 Under Constant Normal Force 35 3.2.2 Under Varying Normal Force 37 4 EXISTING FORCEDISPLACEMENT MODELS 55 4.1 SpringDashpot Models 55 4.2 Walton and Braun [1986] ForceDisplacement Models 56 4.2.1 Normal FD Model 56 4.2.2 Tangential FD Model 58 4.2.3 Implementation of the Walton Braun [1986] TFD Model 62 4.3 Thornton [1997] NFD Model 66 5 AN IMPROVED TFD MODEL FOR ELASTICFRICTIONAL CONTACT 69 5.1 Formulation of the Model 69 5.2 Accounting for Rolling Effect 71 5.3 Comparison with Other Models 73 5.4 Finite Element Validation 77
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5.4.1 FEA with SimpleLoading History 78 5.4.2 FEA for NonSimple Loading History 81 5.5 Benchmark Test: HardSphere Collisions 83 5.5.1 Simulation: HardSphere Collision of Special Material 84 5.5.2 Simulation: HardSphere Collision of Ordinary Material 89 6 ELASTOPLASTIC NFD MODEL 93 6.1 FEA Model and Loading Paths 93 6.2 Formulation of the NFD Model 95 6.2.1 Additive Decomposition of the Contact Radius 95 6.2.2 Normal Pressure Distribution 98 6.2.3 Parabola Law: Normal Displacement vs. Radius of Contact Area 102 6.2.4 Normal Force Reloading 107 6.2.5 Contact Between Two Spheres of Different Materials . 110 6.3 Algorithm: DisplacementDriven Ill 6.4 FEA Validation and Comparison to Thornton [1997] 115 6.5 Simulation of Single Sphere Collisions 120 6.5.1 Simulation Algorithm: LeapFrog Scheme 121 6.5.2 Simulation Results and Comparison with Thornton [1997] 123 7 DYNAMIC FEA OF ELASTOPLASTIC SPHERE COLLISIONS 132 7.1 Finite Element Model for Dynamic Analysis 132 7.2 FEA of Elastic Sphere Collisions 135 7.2.1 FEA Results 135 7.2.2 FEA Results of the Elastic Collisions of Soft Spheres . 142 7.3 FEA of ElastoPlastic Sphere Collisions 144 8 ELASTOPLASTIC TFD MODEL 155 8.1 Accounting for the Effect of Plastic Deformation 155 8.2 Algorithm and Implementation of the ElastoPlastic TFD Model 162 8.3 FEA Validation 167 8.4 Benchmark Tests 175 8.4.1 Single Soybean Dropping Tests 175 8.4.2 Sphere Collisions with Rotation 179 9 DEM SIMULATION OF GRANULAR FLOWS OF NONSPHERICAL PARTICLES 187 9.1 Granular Flow Experiments 187 9.1.1 GranularFlow Experimental Apparatus 187 9.1.2 Soybean Flow Experiments: Different Flow Regimes . 189 9.2 Simulations Using the ElastoPlastic FD Models 191 9.2.1 Soybean Properties and Input Parameters for Simulation 193 VI
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9.2.2 Velocity Development and Velocity Profiles 194 9.2.3 Force Statistics 201 10 CLOSURE 207 APPENDICES A IMPLEMENTATION OF LINKEDLISTS USING FORTRAN ... 209 A.l Nomenclature of Particle Types 209 A. 2 Data Structure: Linkedlists 212 A. 3 Initialization 216 A. 4 Updating the Neighbor Lists 220 B SUPERBALL AND SUPERBEAN 227 B.l Theory 227 B.2 Superball 230 B.3 Superbean 231 C HARDSPHERE COLLISIONS: THEORETICAL PREDICTION . 233 D DERIVATION OF IMPORTANT HERTZ FORMULAE 241 E EXTRACTION OF MECHANICAL PROPERTIES OF GRANULAR MATERIALS 244 E.l Experimental Measurements 246 E.2 Optimization Algorithm 248 E.3 Numerical Example: Application to Soybeans 255 E.3.1 Extraction of Mechanical Properties 255 E.3.2 Simulation Results 259 REFERENCES 263 BIOGRAPHICAL SKETCH 267 VII
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON COMPUTER SIMULATION OF DRY PARTICLE SYSTEMS USING DISCRETE ELEMENT METHOD AND THE DEVELOPMENT OF DEM CONTACT FORCEDISPLACEMENT MODELS By Xiang Zhang December, 1998 Chairman: Loc VuQuoc Major Department: Aerospace Engineering, Mechanics, and Engineering Science Particle systems are involved in a lot of engineering processes. Particle science and technology deal with the production, characterization, modification, handling, and utilization of a wide variety of particles, in both dry and wet conditions. In investigations of the behavior of dry particle systems, considerations of cost, time, and equipment may prohibit experimental works. Even in the absence of these constraints, successful experiments often yield only limited information. Computer simulation using the discrete element method (DEM) can provide more information and can be performed more quickly and at a lower cost than that of experiments. A DEM simulation requires that the interactive forces between simulated particles must be evaluated accurately as the driving factor of the motion behavior of particle systems. Therefore, to obtain reliable simulation results efficiently, it is necessary to develop simple and accurate particleparticle interactive forcedisplacement (FD) models. Most existing FD models, however, implement oversimplified versions of Vlll
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contact mechanics and do not correctly account for the plastic deformation near the contact point between particles, thus result in inaccurate simulation results. We present here the DEM simulation algorithms, the geometric modelling of nonspherical particles, and the development of a set of realistic and consistent particleparticle contact FD models that can correctly account for the effect of both the elastic and plastic deformation. The DEM simulations, implemented with the present elastoplastic FD models, can correctly predict the motion behavior of particles. In the development of the present elastoplastic FD models, we carried out a series of finite element analyses (FEA) of nonlinear elastoplastic contact problems. The modelling and results of these FEA, including the dynamic FEA of elastoplastic sphere collisions, are also presented in this dissertation. We implemented the present FD models into DEM software to carry out the simulations of experimental granular flows of nonspherical particles. Close agreement is observed between the experimental results and the results from DEM simulation on the boundary of the flow domain. Besides, additional information of the granular flow inside the flow domain, which cannot be obtained from traditional experiments, are obtained from the DEM simulation results. IX
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CHAPTER 1 INTRODUCTION 1.1. Why Dry Particle System? According to published reports, in just the southeastern region of the US, particulate systems make a more than $100 billion impact on output in industries such as wood and paper products, minerals, metals, chemicals and cement. Particles are everywhere. For example, sands on beaches, cold medicine in capsules, grains, and most mineral products are all particles. The sweetener in our soft drinks, the advanced materials that allow the space shuttle to fly around the earth and the pharmaceutical products that help to keep us healthy all are made up of particles processed and modified through manufacturing processes. Particle science and technology deal with the production, characterization, modification, handling, and utilization of a wide variety of particles, in both dry and wet conditions. An estimated minimum of 40%, or $61 billion of the value added by the chemical industry in the US is linked to particle technology. Particle science and technology are vital to many other industries, including advanced materials, food processing, mineral processing, pharmaceuticals, munitions, as well as energy, and the environment (Ennis et al. [1994]). Despite its importance, we currently lack the fundamental knowledge and engineering design tools to make reliable equipment to process particulate materials. Up to a 20% loss in advanced computer chip manufacturing can be traced to particulate contamination, while more than $15 billion in mineral value is lost due to the inadequacy of particulate material handling technology.
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In the past, particle science and technology were largely neglected by US industries and the academic community, in both research and education. In the meantime, Japan and Europe made much progress in this area and the US was left behind. For example, Japan made particle technology research a priority for its universities following World War II. Its Association of Powder Process Industry and Engineering has over 300 companies dedicated to particle research. The development of particle science and technology in Japan allows a lot of particle process related products from Japan to be more competitive than those from the USA. The deficiency in particle science and technology has harmed the overall competitive position of the US. In the 21st century, particle science and technology will become a paramount core competency to many sectors of the US economy. According to a recent assessment of the chemical industries in the US (Ennis et al. [1994]), the US needs to pay much more attention to particles in order to improve its competitiveness in the global economy. A dry particle system is a singlephase collection of particles. That is, the system contains only solid particles, without a liquid phase. Dry particle systems exist in many industrial processes. For example, equipment such as chutes, hoppers, and belt conveyers are widely used in industry to transport and to handle dry particulate or bulk materials. Thus, the flow behaviors of dry particulate materials must be understood to design transport, handling, and storage equipment properly. 1.2. Discrete Element Method The study of granular materials and particle properties involves theory, experimental measurement, and computer simulations. Because the availability of large capacity computing is a fairly recent phenomenon, computer simulations are the least developed, but potentially one of the most fruitful ways, of studying particle
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properties and their effects on bulk granular material behavior. Computer simulation of large assemblies of particles are one of the only ways of determining how individual particle properties affect bulk granular material behavior. Development of accurate, efficient computer simulation algorithms for large assemblies of particles is an essential part of improving our ability to predict the behavior of granular solids. There are two methods that can be used to develop a computer simulation of the motion of dry particles. One method is to use a continuum model to describe the motion of the particles (e.g., Bishara et al. [1981]). Computer simulation of a particulate system using a continuum model is efficient but requires the particle system to be nearly homogeneous; i.e., particles in the system should be almost evenly distributed and have the same bulk properties everywhere in the system (see Lu et al. [1995]). The other method is the discrete element method (DEM), which uses a discrete model that considers the motion of single particles individually (Cundall and Strack [1979]). DEM is regarded as a relatively reliable method to study the behavior of dry granular materials, especially when there is no uniformity in the bulk density, velocity, and other flow properties, or when the flow regime is not certain. In DEM, the ordinary differential equations describing the motion of particles are integrated numerically using a stepbystep integration procedure. Assume that the position and velocity of all particles are known at time t n ^\. The task is to compute the forces and moments that act on each particle at t n then to compute the new position and velocity of each particle. Therefore, it is crucial to correctly evaluate the contact forces between the particles in collision by employing accurate and efficient forcedisplacement models in the simulations. A detailed DEM simulation algorithm will be presented shortly.
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Both experimental methods and computational methods have their own advantages and disadvantages when applied to the study of the behavior of dry particle systems. By combining these methods together, we can expect more reliable and useful results. For example, most of the conventional experimental methods can supply direct and reliable information on the surface of granular flows; obtaining the information inside the flow domain is, however, much more difficult and expensive. For example, nonintrusive optical techniques such as the highspeed camera can only supply limited information on granular flows, the MRI method is expensive, while its measurement range is also restrained. On the other hand, insertion of fiber optical probes inevitably modifies the local behavior of the flow. Therefore, if simulation methods can reliably reproduce the bulk and surface measurements obtained in conventional experiments, then the information inside the flow domain obtained from simulations will also be reliable and useful for predicting various important quantities, such as the rate of attrition among the colliding particles. 1.3. Contribution and Chapter Overview In this dissertation, we proposed a series of forcedisplacement (FD) models for DEM simulation of dry particle systems, especially a set of elastoplastic FD models including a normal FD (NFD) model and a tangential FD (TFD) model. The first of its kind in literature, this set of elastoplastic FD models accounts for both the elastic and the plastic deformation that inevitably occurs in most impact problems. A cardinal feature of the elastoplastic NFD model and the elastoplastic TFD model is the additive decomposition of the contact radius into an elastic part and a plastic correction part. Since they both employ the same formalism to account for the effect of plastic deformation, the elastoplastic NFD model and the elastoplastic TFD are consistent with each other. Compared with previously existing FD models, our FD models are more accurate in evaluating the contact forces among particles
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during a particleparticle collision and are relatively efficient without increasing the computational complexity of DEM simulation. With the DEM implementation of our FD models, we carried out DEM simulation of granular flows and obtained satisfactory results. In Chapter 2, we present the DEM simulation algorithm and the geometric modelling of nonspherical particles. The particleparticle contact detection techniques used in our study are also discussed in this chapter. Chapter 3 contains an overview of contact mechanics theories, on which our contact FD models for DEM simulation of dry particle systems are based. In Chapter 4, we review and discuss the advantages and disadvantages of some of the FD models most widely used in DEM simulations. The latest development on attempting to account for the effect of plastic deformation on FD relation, such as the Thornton [1997] NFD model, is also introduced. Chapter 5 documents the development and verification of the improved TFD model for the FD relation in the tangential direction of an elastic frictional contact between two spheres subjected to oblique contact forces. In Chapter 6, we present an elastoplastic NFD model that accounts for the effect of plastic deformation. Finite element analyses (FEA) validated that the model can correctly represent the NFD relation when plastic deformation is involved in the contact between two spheres. The collisions between particles are all simulated by quasistatic procedures in DEM simulation. In Chapter 7, we present some dynamic FEA simulations of collisions between an elastoplastic sphere and a rigid surface. The FEA results are compared with the results of DEM simulations using the elastoplastic NFD model presented in Chapter 6.
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6 Our development and verification of an elastoplastic TFD model that accounts for the effect of plastic deformation on the TFD relation is presented in Chapter 8. This elastoplastic TFD model accounts for the plastic deformation by applying a formalism that is consistent with that used by the elastoplastic NFD model presented in Chapter 6. We implemented the FD models we present in Chapters 5, 6, and 8. We carried out DEM simulations of granular flows of ellipsoidal particles. The DEM simulation results are presented in Chapter 9 and are compared with the experiment data. In Chapter 10, we conclude our work and discuss future work.
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CHAPTER 2 DISCRETE ELEMENT METHOD In this chapter, we present the general algorithm of the computer simulations of dry particle systems using the discrete element method (DEM). In addition, the geometric modeling of nonspherical particles using clusters is discussed in details. Some of the fast contact detection methods employed in our DEM simulation are also discussed in this chapter. 2.1. Algorithm of DEM There are in general two main parts in a DEM simulation algorithm: (i) the freeflight part, where particles are only subjected to gravity, and (ii) the collision part, where particles are subjected to contact forces that arise in collisions with other particles, in addition to gravitational forces. The dynamics of motion of the particles is modeled as that of rigid bodies. For collisions, the softparticle approach is used, i.e., the contact forces resulting from a collision of a pair of particles are computed based on the amount of overlapping (penetration) of these two particles. The contact forcedisplacement relation is crucial for obtaining an accurate prediction of the contact force level and the frequency of collision, which are the two essential ingredients for predicting the rate of attrition among the colliding particles. The numerical solution of the ordinary differential equations of motion of flowing particles is based on a stepbystep integration procedure. Assume that the position and velocity of all particles are known at time t n The task is to compute the forces and moments that act on each particle at Â£ n then to compute the new position and velocity of each particle. By position of a particle, we mean both the
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8 coordinates of the center of mass and the orientation of that particle. Newton's second law governs the motion of the particle centers of mass. Euler's equation governs the orientation (rotation) of the particles. In updating the forces and moments acting on each particle at time t n there are two main problems to be addressed: (i) the detection of contact among the particles and the evaluation of the amount of overlapping in contacting particles, and (ii) the computation of the contact forces and contact moments based on this amount of overlapping. For a granular flow code to be efficient, care should be taken in designing both the contact detection algorithm and the forcedisplacement (FD) model. The evaluation of the contact forces and contact moments is in general more time consuming than the detection of contact, chiefly when spheres are employed in clusters to model nonspherical particles. Of course, an improvement in the efficiency of both tasks (contact detection and evaluation of contact forces/moments) always provides an overall improvement of the code. Remark 2.1. In discussing issues related to contact detection, we clearly distinguish a particle from a sphere. By a particle, we mean a cluster of spheres that represents an ellipsoidal particle. There can be, of course, particles that are made up of single spheres only, called singlesphere particles. Thus a sphere can be either (i) a constituent sphere in a spherecluster particle or (ii) a singlesphere particle. Since spheres are used in contact detection, we refer only to constituent spheres or singlesphere particles in discussions related to contact detection. I The stepbystep integration algorithm employed in our DEM simulations is the leapfrog scheme (see Hockney and Eastwood [1988]) implemented by Walton and Braun [1993]. We present shortly the overall algorithm in some detail. At timestep t n let's focus our attention on sphere (i) and scan the list of all spheres
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in the neighborhood of sphere (i). These spheres are said to be in the neighbor list of sphere (z) (see Section 2.2 and Appendix A). Consider next a sphere (j) in the neighbor list of sphere (i). Let (j)X n and (j)X n be the known positions of the centers of mass of sphere (i) and of sphere (j), respectively, at time t n Based on ^x n and (j)X n the amount of relative displacement (overlap) ^a in the normal direction is computed as follows: {ij) a = [{ {i) R + {j) R)  (i) xÂ„ (i) x n ] (2.1) where ^)R and yji? designate the radii of sphere (i) and of sphere (j), respectively. The calculation for the amount of relative displacement ^5 in the tangential direction is conceptually similar, but more complicated, since 3D rotation is involved. Actually, we use an incremental model in the tangential direction to evaluate the tangential force; this incremental model requires as input the relative incremental tangential displacement A( ij )5 for the evaluation of the incremental tangential force. The computation of A^j)5 involves the velocities of the spheres (i) and (j). We refer the reader to Section 4.2.3 for more details about the calculation of ^5 and of A ^S In the normal direction, the relative normal displacement ^a together with the information on the loading and unloading status, allows the computation of the normal contact (repulsive) force acting between spheres (i) and (j) (see Figure 2.1). In the tangential direction, the relative incremental tangential displacement A ^j)8 and the information on the incremental normal force (increasing or decreasing) allow the computation of the contact tangential frictional force acting between spheres (i) and (j). More details about the computation of contact force using different models are discussed in later chapters.
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10 (H) T (ij)Ot (j) x n (i^n Figure 2.1. Relative displacement in normal direction.
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11 Let (jj)F n be the resulting contact force acting on sphere (i) due to a collision with sphere (j), and let uj\t be the vector pointing from the center of mass of sphere (i) toward the center of mass of sphere (j), having a magnitude of ( ^R Â— (^a), where ^)R is the radius of sphere (i) and (ij)a the relative normal displacement as shown in Figure 2.1. At the same time, there is a reaction force from sphere (i) to sphere (j) denoted as (jj)F B such that (y)FÂ„ + (jt)F B = (Newton's third law). The resulting moment of the contact force (ij)F n with respect to the center of mass of sphere (i) is then (ij)Mn = (i)R j. T7 x (jj)FÂ„ (2.2) II (ij) r II Let (j)F n be the total contact force acting on sphere (t) from all spheres in contact with sphere (i), and (i)M n the moment with respect to the center of mass of sphere (i), then (i) F = Y. (U') F ( 2 3 ) jec(i) and (i) M n = J2 (M Â• (2.4) jec(i) where, C{i) is the set of spheres in the neighborhood of, and in contact with, sphere (z). Remark 2.2. The discussion so far has centered on singlesphere particles. For spherecluster particles, the resultant force [a]FÂ„ and the resulting moment uiMn acting at the center of mass of particle [A] have to be computed by summing the contributions from the constituent spheres that make up particle [A]. In this case, [A]X n is the position vector at time t n of the center of mass of the spherecluster particle [A]. I
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12 Remark 2.3. We refer the reader to Section A. 3 of Appendix A for the classification of various types of particles. The label [A] is mnemonic for Aparticles, which are sphere clusters. The labels for constituent spheres are in lower case italics and enclosed in parenthesis, whereas the labels for particles (sphere clusters) are in upper case italics and enclosed in square brackets. These labels appear as left subscripts to various mathematical quantities. I Let S[A] to be the set of spheres making up the spherecluster particle [A] and [Ai]P the position vector of sphere (i) in the spherecluster particle [A], with i G S[A]. Then for a singlesphere particle, S[A] = {A} and [^p = 0. Therefore, the resultant force [A]&n an d the resultant moment [4]M n acting at the center of mass of the spherecluster particle [A] are as follows [A]FÂ„ = Yl F > ( 2 5 ) ieS[A] [A] M n = J2 W M + E [Ai[P x (i)Fn Â• (2.6) ies[A] ies\A] Â• Â• The translational acceleration ^Xn of particle [A] is given by Newton's second law as follows 1 uixÂ„ = L4]F n (2.7) [A\m where ^m is the mass of sphere (z), then [ A ]m Â— ^ ^m is the mass of sphereieS[A] cluster particle [A]. Using the leapfrog scheme, the velocity of particle [A] can be updated as [A)K+$ = [Ai +(g+ [^]XÂ„) At (2.8) where [^]X n+ i is the velocity of particle [A] at the new half timestep t n+ i, [^]X n _i the velocity at previous half time step, which is known from the previous computation, g is the gravity acceleration, and At the time increment, such that
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13 t n+ i = t n + At. The new position of particle [A] can then be calculated as [A]Xn+l = [A]Xn + [A]K+ At ( 2 9 ) The rotational dynamics of these nonspherical particles (clusters of spheres in the simulation) can be integrated following the same conceptual steps as for the translational dynamics of the centers of mass of these particles. The governing equations for 3D rotations are more complex, since we no longer deal with the linear space R 3 of displacements, but the special orthogonal group of 3D rotations. Assume that from a previous calculation, we know the angular velocity [ A \uj n _i of a particle [A] at middle point of the previous timestep t n _\, and the resultant moment [^]M n about the mass center of particle [A] at time t n The rotational dynamics of particle [A] is governed by the Euler's equation fo({A} 1 [A]V) = [A]l [A]" + [A]" X {[A]l [A]" ) = [A]M (2.10) where, [ A ]I = Diag[[A]I x [A\I y [A]L] is the inertia dyadic in the principal axes { [/i]bi [A]b 2 [a]^3 } of particle [A], [ A ]U) = [ A ]U k [ A ]b k the angular velocity of particle [A], [ A ]W = [ A \w k [A]b k is the objective rate of [ A ]io In other words, [A] u> = [ir'[ M M [A] u x ( [A] I [A] u>)] (2.11) or in component form, (2.11) is written as follows: [A]W X = y[[A]M s + [ A ]U y [A]U z ([ A ]I y [A]Iz)] [A]*x \A\Zy = Â— j[[A]M y + [A \uj z [ A] UJ X ( [A] I Z [A ]I X )] (2.12) [AVy [Apz = Â—\[ A ]M Z + [A] U) X [ A ]Uy([ A ]I x [ A ]I y )] [A]*z
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14 Expressions for [ A ]I X [ A ]I y [A)h for an ellipsoidal particle are as follows TT[A]m [A]Ix 20 \ [A] W>+ [A] H% j n [A] m . t 2 .Â„2> 20 ( M L'+ M J5T a ), (2.13) with [ A ]L x [^]iy x [ A ]H being the length, width, and height of the ellipsoidal particle [A] along the three principal axes x, y, and z, respectively, as shown in Figure 2.2, and [ A ]m is the mass of particle [^4]. [A)W Figure 2.2. Ellipsoidal particle [A] and its dimensions. To integrate (2.12) with the leapfrog scheme, which is used to integrate the translational motion as described above, we employ the following predictorcorrector approach, since the state variables ( uj x u y uj z ) appear on the righthand side of (2.12) (unlike the case of the translational motion). First, the predicted angular velocity \ A ]U)' n of particle [^4] at time t n is obtained by assuming that the angular acceleration was constant in the current time step. Thus, the increment of angular velocity from time t n _\ to time t n is assumed to be half of the incremental angular
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15 velocity A[ A ]Uni from time i n 2 to time t n i, i.e., [yi]W'Â„ = [A]W n _I +A[A]W n _i (2.14) The above predicted angular velocity at time t n then yields a predicted increment of angular velocity A ^w n at time t n by virtue of (2.12) as follows A^w'n = At [A] Il [ [A] M n [A] u>'Â„ x ( [yl] I [A] u>' n )] (2.15) or in component form Aui^'n,x = p[M]Ml,a: + [>l] w 'n,y [A]w'Â„, z ( [ A ]I y [A]Iz)] [A)*x A[A]V'n,y = j[[A]M nt y + [A]^'n,z [A]^' n,x ( [A]h ~ [A]h)] [A]ly (2.16) Au]W' n ,z = ^[[A]Mri,2 + [4)^'n,x [A]u'n,y {[A]Ix ~ [A]Iy)] Â• [yi]i* The corrected angular velocity [A)<*>n of particle [A] at time t n is then obtained using A [A]w'n to yield [A] u> n = [A]W n _i +A [A )U>' n (2.17) Next, the corrected increment of angular velocity A [A\u n a t time t n is then computed using (2.12) as follows [ A ]df' n = At [A] I" 1 [ [/1] M n [A] u> n x {[ A ]l[ A ]U n )] (2.18) or in component form At_ [A]Ix A[A]Un,x Â— j[[A]M ntX + [A}^n,y [A]^n,z ( [A]Iy ~ [A]L)] '1J.T A[A]W n> j, = j[[A)M n ,y + [A}U n ,z [A]^n,x{[A]h ~ [A]I X )] \A)h A[A]V n ,z = J[[A]M ntZ + [A]OJ n ,x [A]^n,y ( [A]Ix ~ [A}Iy)] [A]*z (2.19)
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16 Then the results from (2.19) can be used to correct the angular velocity at time t n+ i as follows [A]V n+ l = [A)U n l +A[A]Wn (2.20) Remark 2.4. In the above, a relatively rudimentary integration algorithm had been employed for the rotational part of the motion, instead of more recently developed algorithms by engineers and mathematicians (e.g., Simo and Wong [1991]), that focus on the preservation of energy and momentum over long periods of simulation. There are at least two reasons for this choice: First, there are frequent collisions among the particles in granular flows. At each collision, energy is lost due to plastic deformation at impact, as reflected through the coefficient of restitution. Only during free flight that energy is to be conserved; the duration of free flight is, however, very short due to frequent collisions in the flow regimes that we have been studying. Second, the timestep size chosen for the simulation is very small, typically 1/40 times the duration of collision time, which is less than 10 3 seconds in the granular flows that we considered. In the future, it would be interesting to see whether energymomentum preserving algorithms produce any significant differences in the global results characterizing the granular flows that we are simulating. I The orientation of the principal axes { [a]^>i > [a)^>2 > [A]bz } of particle [A] are updated using an approach proposed in Evans and Murad [1977] based on singularityfree quaternions. For Euler's equations of motion and the integration of the quaternions, the torques are specified in the body or principal coordinate system for each nonspherical particle, i.e., the spherecluster particle. The contact detection and force calculations are performed in a space or global coordinate system. Let V to be a vector in the
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17 global system and [^jv a vector in the principal system for particle [A] at timestep t n and then [A? TV [4]JÂ„ (2.21) The rotation matrix [a]TÂ„ for transforming from the global system to the principal coordinate system of particle [A] is given by [A)Tl,l [A]T\,2 [A]T\,3 \ [4] 72,1 [4] 72,2 [4] 72,3 V [A]Tz,i [A]T 3 ,2 [A}T 3 ,3 J T = [A]n (2.22) where A}T h l = [Atfln + [A]Qln ~ [A\ql, n + [A]f&, n A]Ti,2 = 2( [A\qi,n [A]Q2,n ~ [4]93,n [4]94,n ) A]Ti,3 = 2( [A]q2,n [4]93,n + [4]9l,n [4]94,n ) 1 A]T 2 ,i Â— ~2( [A]Ql,n [4]92,ra + [4]93,n [4]94,n ) .4)72,2 = [Atfln ~ [A]92,Â„ ~ [A]fcn + [A] 1]^3,1 = 2( [^92,71 [4193,71 [.4)91,7! [4]94,n ) i A]T3,2 ~ 2( [4]9l,n [4)93,71 + [4]92,n [4]94,n ) i 4)73,3 = [A]9i, n [4]9l,n + [4)93,71 + [4]9f,n Â• and [A\qi, n i [4]92,n [4]93,n ) an d [4]94,7i are the quaternions for particle [A] at time t n defined as (see Evans and Murad [1977]) [A]0 n [A\^n [4]07i x [4]9i,n = szn^Â— szn(LJ Â— * Â— ) [4]0n / [4]^7i [A] 071 s [4)92,71 = sin 1 jjÂ—cos( J Â—** Â— ) [4]07i / [4] "071 + [4]07i [4)93,71 = cos^Â— sm{^ Â—^ Â— ) [4]0n / [A]1pn + [4]07i s [ A ] 94,n = COS ^Â— COS [& Â—^ ) (2.23) [4]0n i [4]VVi > an d [A\n are Euler's angles representing successive rotations (see Goldstein [1981]) of the principal axes from the global axes.
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18 The time derivatives of the quaternions can be expressed in terms of the quaternions themselves and the angular velocities of the particle (see Evans and Murad [1977]) as [A]Qi ^([A]l]9l,n+l + [A]s + ([A]g4,n+l + [A]g4,n ) [A]WÂ„+I, 2 ] ( [A]g4,n+1 ~ [A]g4, W ) x r ^ ~ 4l~U^]92,n+l + [A]g2,n j [Aj^n+I,* + ([A]gl,n+l + [A)gl, ) [/tpn+i* ~ ([4]93,n+l + [A]
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19 the quaternions that are independent, the closure of this system of equations must be obtained by the normalization relation Â£W,,n + i) 2 = I(2.28) i=l The quaternions for particle [A] at time i n +i can be put in (2.22) to get the new coordinate transforming matrix ^]T n+1 2.2. Geometric Modeling of Nonspherical Particles and Contact Detection The detection of contacts between nonspherical particles is computationally much more complex than for spherical particles. Recently, there have been several research works on efficient contact detection algorithms (Lin and Ng [1994], Cohen, Lin, Manocha and Ponamgi [1994] and Williams and O'Connor [1995], etc.). To simplify the problem of detecting contacts between ellipsoids, we approximate the geometry of an ellipsoid by a cluster of four identical spheres. The idea of simulating nonspherical particles using clusters of spheres is first employed in Walton and Braun [1993]. In out implementation, the distance between the centers of the four spheres in a cluster is adjusted to match the length L, width W, and height H (= 2R with the R to be the radius of a constituent sphere) of an ellipsoidal particle as shown in Figures 2.3(a) and 2.3(b). Even though the contact detection between spheres can easily be done by checking the distance between the centers of these spheres, and by comparing this distance with the sphere radii, one can further reduce the computational effort in the contact detection by employing, e.g., the Verlet (or neighbor) list (see, e.g., Allen and Tildesley [1987]), which we currently used in our DEM simulations. Please note that not only ellipsoidal particles can be approximated by using a cluster of spheres. Figure 2.4(a) and Figure 2.4(b) show the geometric modelling of a doughnut shaped particle using a cluster of six spheres.
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20 (a) 3D view of a cluster of four spheres. Constituent Spheres \ \ 1 l 1 / / AY R AX A^' w 1 R .... Â• A* R L (b) The position of the four spheres in the cluster. Figure 2.3. Geometric modelling of an ellipsoidal particle using a cluster of four spheres.
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21 (a) A doughnut shaped plastic pellet. (b) Cluster of six spheres. Figure 2.4. Geometric modelling of an doughnut shaped particle using a cluster of six spheres. A Verlet list has a linkedlist data structure. Here, we use a particular type of linkedlist data structure constructed based on the distance between tow constituent spheres; we call such a linkedlist a Verlet list or a neighbor list. Each constituent sphere (of all active particles and of all active singlesphere particles) has its own neighbor list. We sometimes refer to the collection of all neighbor (Verlet) lists as the linkedlists (see Appendix A for more details on the definition of different types of particles, their classification, and the construction and data structure of the neighbor (Verlet) lists). Each such sphere has a neighbor list that contains labels of the spheres in its neighborhood. The neighborhood of sphere (i) is a space region centered at the center of sphere (i), and any other spheres centered inside this region either are colliding or have a high potential of colliding with sphere (i). A typical neighborhood for sphere (i) is a sphere centered at the center of sphere (i) having a radius of 3 ^R with ^R to be the radius of sphere (i). Let C(i) be the set of spheres in the neighborhood of sphere (i), and N nb (i) its cardinal (i.e., the number of it elements). Spheres in the set C{i) either are colliding, or have high potential to collide, with sphere (i). The neighbor list of sphere (i) thus contains N nb (i)
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22 entries (or rows of data), corresponding to N nb (i) spheres in C(i). As an example, the neighbor list for sphere (1) in Figure 2.5 (here i = 1) contains entries (rows) of data for spheres (2), (4) and (5), and the neighbor list for sphere(2) (here i = 2) contains entry (rows) of data for spheres (7) but not for sphere (1) (see Appendix A Section A.l for more details about what spheres should be in the neighbor list). In each entry of data for a neighboring sphere in the neighbor list for sphere (z), we store the contact information such as contact force, displacement between this sphere and sphere (i), and most importantly the information for locating the information for the next sphere in the neighborhood of sphere (i). We refer the reader to Appendix A for more details about the implementation of the linked lists. Figure 2.5. Neighbor list: Set C(i) contains the spheres in the neighborhood of sphere (i), where i Â— 1 in this example. The use of such neighbor lists allows for a more efficient precise contact detection of sphere (i) since only spheres in the set C(i), i.e., the neighborhood of sphere (i) need to be verified. In the best case, when the size of all spheres are roughly the same, the number of spheres that could be in the neighborhood of a given sphere can be limited to be less or equal to a constant C nb To perform precise contact detection without the neighbor lists, we have to verify each sphere against all other spheres (see Appendix A. 3). The computational complexity for this verification is
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23 0(N 2 ), where TV is the total number of spheres in the simulation. With the neighbor lists, for each active sphere, we only need to check at most the number of spheres that are in the neighborhood of that sphere. The computational complexity for precise contact detection of N spheres is thus G(C n b N) when using the neighbor lists. Therefore, the complexity of precise contact detection is decreased from 0(N 2 ) to 0(C nb N). To keep up with the motion of the particles inside the simulation domain, the neighbor lists are updated periodically. The interval for updating the neighbor lists is determined by the size of the neighborhood, accounting for the highest possible velocity of the particles in the system. In our implementation, let drÂ™ ai be the maximum moving distance of all particles for integration time step t n Assume U is the timestep for the last updating of the neighbor lists. At timestep tj, when I>r x >^r, (2.29) n=i Zl where, r nb is the radius of our neighborhood sphere, then the neighbor lists need to be updated again. For the algorithm described in Appendix A, the computational complexity for updating the neighbor lists is still 0(N 2 ). For example, in Figure 2.5, the neighbor list for sphere (i) (here i Â— 1) will have rows of data for spheres (4), (5) and (6) after the updating. More details on the updating and the implementation of the linkedlists in the simulation code can be found in Appendix A. The use of the bounding box algorithm to perform the preliminary collision detection has also been implemented in our simulation code (see Cao [1996]). The bounding box algorithm is proposed by Cohen et al. [1994] and Baraff [1995] to determine approximate collisions between arbitrarily shaped objects. The bounding box for a particle [A] is a parallelepiped with its edges aligned with the axes of the Cartesian coordinate system x, y, z, and contains particle [A] tightly inside. The following three intervals characterize the bounding box of particle [A]: ( [ A ]X t [ A ]X U ),
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24 ([A]Vi, [A]Vu ) and ( [ A )Zi [ A ]Z U ), where [^rr, [ajI/j and [^i are the coordinates of the lowerbound corner of the bounding box, [a]X u [a]Vu > and \a]Z u the coordinates of the upperbound corner. Therefore, for a system of N particles, there are iV intervals in x, y, and z directions, respectively, to represent the bounding boxes of these N particles. If any two of the particles collide, or possibly going to collide, their bounding boxes must be overlapped in three directions simultaneously. Remark 2.5. Even though we use the bounding box algorithm in our simulations for the preliminary contact detection for spherical particles (nonspherical particle are represented by clusters of spheres), the bounding box algorithm can be applied equally well to nonspherical particles. Thus, in the discussion that follows, we think in terms of general nonspherical particles, unless specifically indicated otherwise. In our code, the bounding box algorithm is applied to spheres only. I The bounding box algorithm for the preliminary contact detection begins as follows. First, we keep the N lower bounds and the N upper bounds of the bounding boxes along one direction (x, or y, or z) in one array. Using heap sort, we rearrange these 2iV numbers in one array from small to large. There are three such sorted arrays, one for each direction. The computational effort for the sorting is 0(N log(N)). Since in any one of the three sorted arrays, the lowerbound point of a bounding box interval is always with a coordinate smaller than that of the upperbound point, the lowerbound point is always placed before the upperbound point in a sorted array. The following algorithm is used to construct the neighbor sets of all particles by looking at the overlapping of the bounding boxes. Designate a sorted array contains the coordinates of lowerbound points and upperbound points of bounding boxes in one direction (either x, or y, or z) as the array B. Let S be a stack used to construct the neighbor set of a given particle. The following algorithm is use to construct the neighbor sets of all particles in a given direction.
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25 Algorithm 2.1: Construction of neighbor sets Initialize stack S to be empty, i.e., S = {} ; for (k=l, k<=2N, k++) { if (B[k] is a lowerbound point) { 1. Find the corresponding particle [A] (with label A) ; 2. Copy all particles listed in the stack S into the neighbor set for particle [A] ; 3. Add particle [A] to the stack S ; } else /* B[k] is an upperbound point */ { 1. Find the corresponding particle [A] ; 2. Remove particle [A] from the stack S ; } } Remark 2.6. In words, the construction of the neighbor sets of the particles for one direction can be described with the aid of Figure 2.6 (which actually represent a worst case). The sorted array B of lower/upperbound (LUB) points can be visualized in Figure 2.6. Loop over these LUB points from small to large. If an LUB point is a lowerbound point, then the neighbor set of the particle corresponding to that point contains all particles having the lowerbound points smaller than the current lowerbound point, except for those particles having their upperbound points that are also smaller than the current lowerbound point. If an LUB point is an upperbound point, we no longer consider the corresponding particle in the construction of the neighbor sets of subsequent parti
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26 cles. Thus for the example in Figure 2.6 (a worst case), the neighbor sets are C[3] = {}, C[4] = {3}, C[5] = {3,4}, C[2] = {3,4,5}, C[l] = {2,3,4,5} I In the end, each particle has three temporary neighbor sets, one set for each direction. Then, by scanning the three neighbor sets of a given particle, say, particle [A], any particle that belongs to three sets is identified as a neighbor particle to particle [A]. From the algorithm above, we can see that label of every particle will be inside the stack S before it is removed from the stack, since the lowerbound point for a particle appears before its upperbound point in a sorted array. The particles in the stack S are those that overlap the current particle [A] for which B[k] is the lowerbound point of its bounding box in the given direction. Therefore, the computational complexity of the search for neighbor particles can be evaluated as follows: 1. The basic operation, i.e., the loop over all 2N elements in a sorted array, and this for three arrays, requires O(N). 2. Additional computational efforts for putting all the particles listed in the stack into the temporary neighbor sets, for each particle, require C(C X + C y + C z ) with C x C y and C z being the numbers of total overlappings in x, y, and z direction, respectively. The computational complexity of the search for neighbor particles using the bounding box algorithm is 0(N + C x + C y + C z ). In the worst case when the particles are aligned in one direction and every particle overlaps with all other particles, such as the case shown in Figure 2.6, the total number of overlappings is C x + C y + C z w iV 2 The worst case computational complexity for the bounding box algorithm is then 0(N 2 ). In a sparse particle system, the total number of
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27 overlappings could be less than JV, the best case computational complexity for the search for neighbor particles using the bounding box algorithm is O(N). Certainly, we need 0(N log(N)) computational effort to initialize the sorted arrays for the bounding boxes. In some cases of our simulations, the sizes of all particles are S P5 small large [3]Xl [4]X t [ 5 ]X t [ 2 ]2f [i]X[ [ 3 ]X U [ 4 ]X U [i]X u [ 5 ]X U [ 2 }X U Array B Figure 2.6. Worst case for bounding box algorithm roughly the same, and the particles are relatively densely packed in a 3D parallelepiped simulation domain, whose edges have dimensions of similar order. The average number of overlappings of one particles to all other particles in one direction is then 0(N*) (see Figure 2.7), and therefore the total number of overlappings for all will be C x + C y + C z 0(3 N JVi) = O(Nl). Hence for these cases, the computational complexity for the search for neighbor particles using the bounding box algorithm is 0(N3).
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28 y^\ / y< fx w X >*Â•* ) ( anlcpny^ i \ Figure 2.7. For a cube densely packed with particles, the number of particles in each direction is roughly N* When we need to update the neighbor lists of the particles, it is a necessary to update the sorted arrays for the bounding boxes first. Assume that the displacements of the particles are small between the updates; in this case, at each update, the three arrays for the bounding boxes are almost in the state of being sorted. Therefore, these arrays can be sorted by using the bubble sort algorithm with a computational effort of O(N).
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CHAPTER 3 THEORIES OF CONTACT MECHANICS The theories of contact mechanics, including Hertz theory (Hertz [1882] and Johnson [1985]) and Mindlin and Deresiewicz theory (Mindlin and Deresiewicz [1953] and Mindlin [1949]), are introduced in this chapter. Hertz theory of contact mechanics deals with the contact between two elastic objects subject to contact forces in the normal direction of the contact surface; Mindlin and Deresiewicz theory gives theoretical solution for limited loading cases of the elastic frictional contact between two spheres subject to contact forces in a tangential direction. These theories of contact mechanics are the theoretical foundation of lots of the existing FD models used in DEM simulations and they are also very important to the construction of the FD models we developed. 3.1. Hertz Theory Figure 3.1 depicts the contact between two spheres subjected to normal load P. Let's define the equivalent elastic modulus E* and the equivalent contact curval_ ~R* ture Â— as the following: and r := (IzjjiL+lzJipf ( 3.i) 1 ( 1 1 \ + 715Â• ( 3 2 ) R* \ (i)R u)R where ^)R is the radius of sphere i, ^v and ^E are the Poisson's ratio and Young's modulus of the material of sphere i, respectively. Similarly ^R, (j)U
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30 sphere i sphere j Figure 3.1. Two spheres in contact, subjected to normal load P. and (j)E are those of sphere j. The contact area is circular, with a radius of a (see Figure 3.2(a)). Hertz proposed that on the contact surface, the distribution of normal pressure p is axisymmetric and shaped as half of an ellipse. At a point A in the contact area, with a distance of r from the center O of the contact area s, the normal pressure p(r) can be expressed as p(r) = p T{ r v3 a) 1/2 (3.3) where p m is the maximum normal pressure at r = 0. By integrating p over the contact area, we obtain P = IpmÂ™ 2 y (3.4)
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31 x (a) Circular contact area, Section BB is a contact area diameter. Point A 6 s. (b) Hertz normal pressure p at Section BB. Figure 3.2. Contact area and Hertz pressure. By rearranging (3.4), we arrive at Prn = 3P 2na 2 (3.5) Figure 3.2(b) depicts the distribution of Hertz normal pressure at a diameter cross section of the contact area. Hertz also found that the radius a of the contact area can be expressed as (Johnson [1985, Eq.(4.22)]) {3PR*\ 1/3 a = tlFJ (3.6) and the approach of distant points on the two spheres can be expressed as (Johnson [1985, Eq.(4.23)], see also Appendix D) (ij) a = (i) a + U) a = 9P 2 \l/3 R* VioT^*) 2 Introducing (3.6) to (3.5), we obtain 3P [6P(E*) 2 V /3 (37) Pm = 2ira 2 \ 7r 3 (i?*) 2 (3.8)
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32 For the special case of two identical spheres in contact, with ^R = (j)R = R, (i)E = (j)E Â— E, and (j)V = ^v = v. The equivalent Young's modulus by (3.1)is E R E* Â— r~rr and the radius of relative contact curvature by (3.2) is R* = Â— 2(1 v 2 ) J v 2 Therefore, the radius a of the circular contact area (by (3.6)) can be expressed as [3PR(lv 2 Y t/3 { \e ) Â• < 3 9 > and the normal displacement ^a as 1 2 a 2 n /9P 2 (1^) 2 \ 1/3 s Note that the contact between a deformable sphere and a frictionless rigid surface is equivalent to the contact between two identical spheres by symmetry. But the normal displacement are different. The normal displacement of a contact between a deformable sphere and a rigid surface is only half of that of an equivalent contact between two identical spheres. Therefore, in application to the contact between a sphere and a rigid surface, the equation (3.10) need to be modified. Hertz theory assumes that the contact area is much smaller than the size of the spheres, i.e., a < ^R and a < ^R. Therefore, the stress distribution inside the sphere can be obtained by considering concentrated forces applied to a elastic half space. The stress along the zaxis (the axis that passes through centers of the spheres and the center of the contact area, as shown in Figure 3.1) thus can be expressed as a r = o Q = p m { (1 + v) and z fa 1 tan' 1 a \z z 2 (3.11) Pm\l + ~ 2 \ (3.12) 1 When the two spheres in contact are identical, there are ^R = (j) .ft = R, and we have R* = R/2, and thus formula (3.10) has the factor 2 as in the right side.
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33 Plastic deformation occurs when the normal contact force P exceeds the incipient yield force P Y According to the von Mises criterion, yield occurs at points in the material at which the second invariant of the stress deviator satisfies ^ (313) where ay is the material's yield stress under uniaxial tension. The invariant J 2 can be expressed as h = g[(a! o 2 ) 2 + (ct 2 a 3 ) 2 + (a 3
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34 1.5 ^ I e 0.5 \ Â— = 0.48086 a A[y) = 1.613 0.2 0.1 0.1 0.2 0.3 0.4 J2 Jo (Pm) 2 (Pm) 2 Figure 3.3. Second invariant of stress deviator J 2 along the zaxis in normal contact, only for v = 0.3. Substituting (3.13) and (3.18) into (3.15), we obtain the maximum normal pressure p m< Y on the contact surface at the incipient yield Pm,Y = A Y {u)a Y where A Y (v) is a constant determined by the Poisson's ratio v A Y (u) = (3.19) (Pm (3.20) 3 {J2)z=z Y For v = 0.3, A y (0.3) it 1.613; for v = 0.4, Ay (0.4) ~ 1.738. Figure 3.3 shows the value of invariant J 2 along the zaxis for v Â— 0.3. Considering that at the incipient yield, the maximum normal stress on the contact surface still follows Hertz theory, we substitute p m y in (3.19) using (3.8) to obtain 7T 3 {R*f (1 !/) 3 Py ~ 6^ Ay y (3.21)
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35 We note that the first yield does not occur at the contact surface, but at a point on the zaxis that is about onehalf the contact area radius above the contact surface. The elastoplastic FEA results by VuQuoc and Lesburg [1998] did show that the yield starts not at a point on the contact surface but inside the sphere. In an elastic normal collision between a sphere and a rigid surface, the forcedisplacement relation during the collision can be described using Hertz theory as if there is a nonlinear spring acting between two objects; the duration of the collision is given by (Timoshenko and Goodier [1970], and also Walton [1993]) where r is the contact time during the collision, p the density of the sphere material, and v in the incoming velocity. Relation (3.22) between the contact duration time r and the incoming velocity v in for elastic collisions is validated by Walton [1993] using dynamic FEA. 3.2. Mindlin and Deresiewicz Theory 3.2.1. Under Constant Normal Force The problem of two elastic spheres in contact subjected to a constant normal force and a varying tangential force was first considered by Cattaneo [1938] and later independently by Mindlin [1949]. Consider two different spheres initially subjected to only a normal force P. While holding the normal force constant, we add a tangential force Q to the spheres, as shown in Figure 3.4. The expressions for the radius a of contact area, normal displacement a, etc., all follow Hertz theory (Section 3.1), i.e., (3.6) and (3.7).
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36 Figure 3.4. Two spheres in contact and subjected to normal and tangential forces. Depending on the normal force P, we have two kinds of loading cases to consider: (i) normal contact force P constant, and (ii) normal contact force P varying. Let Q and Q' be the tangential contact forces before and after an increment of tangential displacement AS, respectively. The relationship between Q and Q' is given by the following incremental formula Q' = Q + K T AS (3.23) where K T denotes the tangential stiffness coefficient at the current increment step,
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37 which is computed according to Mindlin and Deresiewicz [1953] as follows (see Figure 3.5) K T = I ( O0*\ 1/3 K T Jl ~ 2 p J for Q increasing^) and \Q\ < \Q*\ ( qV /3 K t ,q I 1 ^p J for Q increasing^) and \Q\ > \Q*  / Q*Q\ 1/3 K T ,o ( 1 2 p J for Q decreasing(\) and \Q\ < Q*<)3.24) K T ,o 1H Â— p ) for Q decreasing(\) and \Q\ > \Q* In (3.24), Q* is the tangential force at the last turning point in the loading history (see Figure 3.5), /x the coefficient of friction, and K Tfi the initial tangential stiffness determined by where, G\ and G 2 are the shear moduli of the two spheres, respectively. Clearly, the initial tangential stiffness K Tfi is dependent on the normal loading P, since the contact radius a is a function of P according to (3.6). Thus energy is dissipated due to friction even for elastic contact. This dissipation of energy is caused by the micro slips on the contact surface as discussed in Mindlin and Deresiewicz [1953]. In the present paper, we focus on the Q vs. 8 relationship for granular flow simulations. 3.2.2. Under Varying Normal Force In contact problems, the normal force does not in general remain constant. Mindlin and Deresiewicz [1953] proposed an incremental solution for the frictional elastic contact of two identical spheres that are subjected to varying normal and tangential forces. Mindlin and Deresiewicz [1953] theory for elastic frictional contact is limited to the socalled "simple loading" histories and includes eleven loading cases.
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38 OnE > F : K T *(W On O > A : *Â™ (' ^ Â£) 1/3 On A * L> : K T OnD / ON 1 / 3 if, = jfÂ„ (i J) Figure 3.5. Q vs. S curve and Tangential stiffness K T for constant normal load P. Remark 3.1. The "simple loading" histories (to be explained shortly) are based on the sixth assumption (rule no.6) in Mindlin and Deresiewicz [1953]. Any step of a "simple loading" history begins with an equilibrium position, for which the distribution of traction is equivalent to a state of P constant and Q varying case. The effects of a change in the state of loading are obtained by advancing to the desired state through a sequence of equilibrium positions, and the final distribution of traction is equivalent to another state of P constant and Q varying case. In the case where P is increasing and Q is increasing, for the loading to remain simple,
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39 it is required that \AQ\ > /xAP. Similarly for the case where both P and Q are decreasing, "simple loading" requires that AQ > //AP. I We briefly review below four of the eleven loading cases discussed in Mindlin and Deresiewicz [1953]. 1. P increasing, Q increasing. Figure 3.6. TFD curves for P increasing, Q increasing. Let AP and AQ be increments of the normal force P and of the tangential force Q, respectively. The state before the change of loading is {P Q Q 6 }, i.e., state shown in Figure 3.6. Assume that the loading history before state is a simple loading history, we first increase the normal force from P Q to P x = P + AP, then
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40 increase the tangential force from Q by an increment AQ. There are two subcases: AQ > nAP and AQ < fiAP. For AQ > fiAP, the tangential force increment AQ is decomposed into two parts: AQ = AQ i + AQ 12 with AQ i = (iAP. When both AP and AQ are very small (AP > 0, AQ  0), the tangential stiffness for the first part of the increment is the same as the initial tangential stiffness for P constant loading case, i.e., (Kt)oi = (Kt,o)p=p Â• (3.26) The state O reached at the end of increment AQ i (first part) is equivalent to a state of an initial loading case for constant P = P l Then the tangential stiffness for the second part of the increment follows the constant P case, i.e., by (3.24) 2 we have (#r)i2 = (tfr,o)/>=* (l ^) (3.27) where Qi = Q + AQ i = Q + fiAP as shown in Figure 3.6. Therefore, at state Q 2 = Qo + vAP + (K T ) l2 A5 l2 = Qo + ^AP + (K t ) u (a5} (K Tfi ) P=Po. and 8 2 = S + A5 = S + ASoi + A8 U = Sp I ^ AP  AQ ~ //AP ( 3 29 ) (K T ,o)p=p (K T ) 12 For AQ < /iAF, the tangential force increment AQ is not large enough to complete the transition from state to state O shown in Figure 3.6. We have the following FD relationship for this subcase Q' = Q + (K Tfi )p=p A5 (3.30) where A5 < A6 01 = Â— Â— This state is not equivalent to a virgin state of {^Tfi)P=P tangential loading under P constant; the loading history after this loading increment is no longer a simple loading history.
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41 2. P decreasing, Q increasing. Now consider the case where AP < and AQ > 0. The state before the change of loading is {P Q 5 }, i.e., state which is assumed to be one state of a simple loading history; see Figure 3.7. / Kt,o 11 AQ 23 = AQ AQoi = AQ12 = /*AP, (AF < 0) and AP > 0, the final state after the loading increment (i.e., state in Figure 3.7) is attained from the initial state after a sequence of intermediate states: Apply the increment P > P x = P + AP (AP < 0) and Qo >Â• Q\ = Qo + AQ i = Q + /j,AP to go from state to state O as shown in Figure 3.7. The tangential stiffness for this transition is the same as that in the
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42 initial loading under constant normal force P = P Therefore,
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43 and fiAP fiAP AQ Note that after the transition from state to state O subsequent transitions from state O to state and to state were executed under the the constant normal load P Pi. Therefore, if the loading history before the loading increment, i.e., P decreasing and Q increasing, is a simple loading history, then the loading history after the loading increment remains a simple loading history. 3. P increasing, Q decreasing. Consider now the case where AP < and AQ > 0. The state before the change of loading is, i.e., state {P Qo, M> as shown in Figure 3.8. Assume that the state is a state of a simple loading history. In this case, the state is equivalent to a state on a TFD curve with P = P constant, and with the turning point Q* Q and with an unloading tangential force evaluated at Q = Q ; see the solid line in Figure 3.8. The value of turning point tangential force Q* Q is determined by the loading history before the system reaches state For initial (virgin) loading, Q* is set to be zero. When Q changes direction, i.e., either from increasing to decreasing or from decreasing to increasing, Q* is set to be the value of the tangential force Q of previous time step. For all other cases, Q* of the current time step is set to be the same as Q* of previous time step. We first increment the normal force to Pj = P + AP, AP > 0. Next, we decrement the tangential force by the amount AQ. Depending on the magnitude of AQ, we observe two subcases: AQ < //AP and AQ > //AP. (Equivalently, the two cases may be stated as AQ > /jAP and AQ < //AP, the first of which is the criterion for a simple loading history.)
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44 Q Q{ Q*o Qo Qi Q2 Pi=P + AP O &2 Sq Figure 3.8. TFD curves for P increasing, Q decreasing. For AQ < //AP, we decompose the tangential force increment AQ into two parts: AQ = AQ 01 + AQ 12 with AQ i = /iAP. After the application of the first part of AQ, .e., AQ i, the system goes from state to state O with the same tangential stiffness K T ,o as that for a virgin tangential loading under P Â— P constant; see Figure 3.8. Therefore, the state O represented by {P x Q 1? S t } is related to state as follows P = P + AP (3.38) Qi = Qo + AQ 01 = Qo/iAP Si = S Q + ASqi = S + /iAP {Kt,o)p=p (3.39) (3.40)
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45 State O is a state on a simple loading history that corresponds to a TFD curve with P = Pi constant, and with the turning point Q* = QJ+//AP, and with an unloading tangential force evaluated at Q = Q x When the second part of the tangential load increment (i.e., AQ 12 ) is applied, the system goes through a transition from state O to stat following the unloading path of the TFD curve with P = P x constant; see the dotted line in Figure 3.8. The tangential stiffness for this transition is obtained from (3.24) 3 as follows (K T )n = (K T ) P=Pl (l %^y 3 (3.41) The TFD relationship is then expressed as (AP > 0, AQ < 0) r r a r c AQ12 AQ + uAP S 2 = 6,+ A8 l2 = //AP, i.e., AQ < //AP, the tangential force decrement is not large enough to complete the transition from state to state O shown in Figure 3.8. The final state of this case of loading increment lies somewhere between state and state O of Figure 3.8. Therefore, we have the following FD relationship Q' = Qo + (K Tfi ) P=Po AS (3.45) where \AS\ < \A5 01 \ = Â— Â— Since this state is not equivalent to a state in \ftTfl)P=P which P is constant and Q is unloading, the loading history after this step of loading increment is no longer a simple loading history.
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46 After this step of loading increment, the value of turning point tangential force is updated as Q\ = Q*+ //AP. 4. P decreasing, Q decreasing Consider now the case in which AP < and AQ < 0. Before the loading increment, the system is at state represented by {P Qo> <$o}> an d assumed to be a state on a simple loading history in which P = P constant and the unloading tangential force is evaluated at Q = Q after the turning point QI (see the solid line in Figure 3.9). The turning point Q* is determined as described above. To proceed, we first decrease the normal force from P to Pi = P + AP (AP < 0). Depending on the magnitudes of AQ and AP, there are three subcases: (a) Q < Q* Q + 2/iAP (this subcase is shown in Figure 3.9). When Q > Q* + 2//AP, we have the following two subcases: (b) Ql + 2//AP
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47 Q Ql Ql Pi=P + AP AQ l2 + AQ 2 3 ^3 <$2 $0 $1 Figure 3.9. TFD curves for P decreasing, Q decreasing loading increment: Situation (a) in whichQ < Ql + 2/iAP. on the unloading path of the TFD curve with P = P x constant (dotted line in Figure 3.9). Thus the tangential stiffness for this transition is then obtained from (3.24)3, to yield 1/3 (K T )i2 = {K T fi)p=p 1 1 and therefore the tangential displacement QlQi 2/xPi (3.47) S 2 = S l + A6 12 = 61 + //AP (K T ) 12 (3.48)
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48 Continue on the unloading path of the TFD curve corresponding to P = P x constant to go from state to the final state by decreasing the tangential load from Q = Q 2 = Q Q to Q = Q 3 = Q + AQ (AQ < 0). Therefore, 6 3 = 5 2 + A8 23 = 6 2 + Q (3.49) (firJ23 where the tangential stiffness {K T ) 23 is determined by (3.24) 3 as follows (Krh* = (K Tfi ) P=Pl (l ^^V 3 Â• (3.50) By combining (3.46), (3.48), and (3.49), the TFD relationship is expressed as follows Q 3 = <2o + AQ01 + AQ l2 + AQ 23 = Qo + (K T ) 23 AS 23 HAP 11AP \ ^bl > Qo + (K T ) 23 A5 (Kt,o)p=p (K T )l2, and 3 = 8 + A5 01 + AS l2 + A5 23 = 6 + .^^ P + J^+ ^ {X T ,o)p=p [K T )i2 {Kt)23 ^3 52 ) It is noted that in subcase (a), if the loading history up to state is a simple loading history, then this loading history continues to be simple at of state Remark 3.2. In the loading cases in which both P and Q decreases, for subcase (a), when Q < Q* + 2/iAP, we observe an interesting TFD behavior that when the normal force P decreases at first, the tangential force Q increases for the first transition. The reason is that when the tangential force is decreasing, there is a region of reverse (i.e., opposite to the current tangential force) tangential traction on the rim of the circular contact area. The larger the amount of tangential force decreases, the larger the region of reverse tangential traction. When the normal force decreases by AP, the contact area shrinks consequently, and thus some of the tangential traction is released. Since the outer region of the contact area is bearing
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49 reverse tangential traction, the tangential traction released by decreasing of normal force P is mainly the reverse traction. Therefore, the integration of tangential traction on the contact area, i.e., the total tangential force Q increases. Actually, the same thing happens in the two subcases (b) and (c) described below; however, in subcase (b), when Qq+/j,AP < Q the amount Q decreased previously is small, and there is not a large enough reverse traction region formed on the rim of the contact area yet. Therefore, the decrement of AP caused both the release of reverse traction and the traction in the current tangential force direction. The overall behavior of Q is still decreasing. I For the subcase (b) in which Q > Q* + 2/iAP(AP < 0), with Q* + 2/xAP < Qo < Qo + A*AP as shown in Figure 3.10. The final state is reached through the following sequence of transitions. Decrease the normal load from P to P 1 = P + AP, (AP < 0). The system goes from state to state O as shown in Figure 3.10. State O represented by {Pi,Qi,5i}, corresponds to a state on a virgin TFD curve, with P = P x constant, and with a loading tangential force evaluated at Q = Qi = Ql + //AP. The reason for this is at state the amount of decreased tangential force from the turning point Qo is not large enough and thus the decreasing normal load of AP released all the reverse tangential traction. Therefore, at state O the tangential traction on the contact surface is the same as a initial loading traction. In calculation, the stiffness for the this transition is (K T>0 )p =Po Thus the tangential displacement Si at state O is expressed as = S + AS 01 =5 + Ql ^\ P ~ Qo (3.53) Holding P = Pi constant, decrease the tangential force from Q = Q x to Q = Q2 = Qo + AQ, (AQ < 0), thus make the system goes from state O to This transition is equivalent to a first step unloading with P = P x constant. Since state
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50 Q t Ql Qo Q2 o \ = P + AP (AP < 0) LiAP $2 #0 <*1 Figure 3.10. TFD curves for P decreasing, Q decreasing: subcase (b) in which Q* + 2/iAP < Q < Q* Q + fxAP. O is equivalent to a virgin loading state, and in this transition the tangential force is unloading (decreasing), the turning point tangential force Q* need to be updated to Q\ = Q\ = Qo + {J.AP. Therefore, the tangential stiffness is determined by (3.24) 3 to yield (K\ (K \ fi Qo + ^PQo Y /3 ,,^ (A r J 12 [K Tfi )p =Pl I 1 Â— I (3.54) The tangential displacement S 2 of state therefore, can be expressed as &2 = #1 + A5i2 jr AQ 12 jg+jiAP Qq Q 2 Q*fiAP (3.55) (Kt,o)p=p (K T )l2
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51 Also, the tangential force can be expressed in terms of displacement by Q 2 = Qi + AQ 12 = Q x + (K T ) l2 AS X2 = QS + / xAP + (/r r ) 12 U 5 Q*o + ^ p Qo \ (356) \ {ftT,0)P=P o ) For subcase (c) in which Q Q > Q* + 2//AP, with Q* + jjlAP < Q as shown in Figures 3.11 and 3.12. The final TFD state of the system is reached by carrying out following sequence of transitions. Decrease normal load from P = P to P = P x = P + AP, (AP < 0). The system goes from state to state O as shown in Figures 3.11 and 3.12. At state the amount of tangential force decreased from the turning point Q* Q is very small, i.e., Ql Q < /iAP, (AP < 0), thus the shrink of contact area release all of the reverse tangential traction and a part of tangential traction in the direction of current tangential force. The overall behavior is that the tangential force decreased from Q to Q x Q^ + fiAP. State O represented by {P u Q 1; Si}, corresponds to a state of virgin TFD curve, with P = P x constant, and with a loading tangential force evaluated at Q = Q x = Q* + /iAP. The decrement of tangential force from state to state O is AQ i = Q\ Qo = Q* + V&P Qo < 0. When AP is small, the stiffness for the transition can be computed with P = P Q constant and decreasing Q at Qoi = Qo + 0.5AQoi. By (3.24) 3 we have 1/3 (tfr)oi = (K t ,o)p =Po (l ^yj^J Â• (3.57) Therefore the tangential displacement of state O is expressed as <$i = So + AÂ£ 01 = do + K T )oi (3.58) J Q + liAPQo ~ 5 + (K^ x Â• Dependent on whether  AQ\ < \AQ 0X  or  AQ\ > \AQ 0X \, the transition from state Oto state is performed differently. For AQ < AQ i shown in Figure 3.11,
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52 Q Qo Qo Q2 o 6\ 5q S 2 Figure 3.11. TFD curves for P decreasing, Q decreasing: subcase (c) in which Q* + fxAP < Q with AQ < AQ iholding P = P\ constant, increase tangential force from Q = Qj to Q Â— Q 2 Qo + AQ, (AQ < 0). The increment of tangential force AQ 12 = AQ AQ \ > 0. Therefore, the tangential stiffness for this transition is determined by (3.24) 2 to yield 1/3 {K T )u {K Tfi ) P=Pl 1 Then the tangential displacement 8 2 of state @ is expressed as S 2 = Si + A8i2 r AQ 12 = dl + = s + ^VfiAPQp Q 2 Q*nAP (K T )oi (K T )n (3.59) (3.60)
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53 To express the tangential force in terms of the increment of the tangential displacement A5, we have Q 2 = Qi + AQ l2 = Qi + {K T )n AS 12 = Q* + AP + (K T ) 12 (AS Ql + ^APQ (3.61) \ (K T ) 01 If the followed force incremental step is a Q decreasing step, the turning around Q Ql Qo Q\ Q 2 S 2 ^1 <^0 Figure 3.12. TFD curves for P decreasing, Q decreasing: subcase (c) in which Q* + pAP < Q with AQ > AQÂ„itangential force Q* should be updated as Qn+i = Q2 (3.62) else, set Q* n+l = 0. For AQ > AQoi of transition (D, there is a decreasing of tangential force from
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54 Q = Q\ to Q = Q 2 Â— Qo + AQ as shown in Figure 3.12. Since state O is equivalent to a TFD state of loading, with P = Pi constant, and with Q = Q u when tangential force decreases, the turning point tangential force need to be updated to Q\ = Qi = Qq + fiAP. Therefore, the TFD state can be expressed as $2 Â— ^1 + A(5i2 x AQ 12 and (K t ,o)p=p a (3.63) 5  Q* + fiAPQ Q  Q 2 Q* fiAP (Kt)oi (Kt,o)p=Pi Q 2 = Q. + AQu = Qi + (K T ,o)p= Pl A6i2 = Ql + fiAP + (K Tt0 ) P=Pl (AS Qo + l^PQo \ (3 64)
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CHAPTER 4 EXISTING FORCEDISPLACEMENT MODELS In this chapter, we briefly introduce some of the existing FD models that have been used in DEM simulations and discuss their advantages and disadvantages. 4.1. SpringDashpot Models The springdashpot contact FD model (Figure 4.1) is probably the most widely used FD model in DEM simulation of the motion of a multibody system with a large number of objects. We believe that one of the reasons for its popularity lies in that it is one of the simplest model used in DEM simulations. As shown in Figure 4.1, the contact FD relation is modeled using a spring and a dashpot assembled in parallel. Therefore, when two objects come into contact with each other, the NFD relation is described as follows P = K N a + C N a (4.1) where both K N and C N are constant parameters of the springdashpot model. The TFD relation has the same form of (4.1) and is limited by the friction, i.e., Q = min(K T 6 + C T S, n P) (4.2) where both K N and Cn are constant parameters of the springdashpot model, fiP is the friction limit. Despite of its highly simplicity, the springdashpot FD model has been employed in a lot of simulations and produced fairly good results (e.g., Tsuji et al. [1992], Ting et al. [1993], and Mishra [1995]). The advantages of using the spring55
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56 Figure 4.1. SpringDashpot contact forcedisplacement model. dashpot FD model are based on that the model is simple, direct, and easy to implement. The disadvantage of using this model is that the model constants K and C can not be easily chosen when applied in DEM simulation. The lack of a solid basis for obtaining the K and C is always a problem for using this model in simulation., Additionally, the dashpot in this model is designed to account for the energy dissipation. The dependence of the model on the collision velocity also prevents the use of quasistatic simulation to correctly evaluated the energy dissipation caused by plastic deformation. For the aove reasons, the springdashpot FD model is regarded as a primitive FD model for DEM simulations. 4.2. Walton and Braun [1986] ForceDisplacement Models 4.2.1. Normal FD Model The forcedisplacement (FD) law based on Hertz theory gives a nonlinear elastic relationship between the normal displacement {ij) a and the normal contact force P. Consequently, when simulating a sphere colliding with a rigid half space, the ratio of rebounding velocity to the incoming velocity of the sphere, i.e., the coefficient of restitution, obtained using the Hertz FD law is e = 1.0. For most of collision problems, plastic deformation occurs, causing a dissipation of energy, and
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57 1.0 (A) 9.5 4 8 12 16 20 24 28 32 38 Displacement > (a) Axisymmetric mesh for FEA. (b) Resulting FD curve from FEA. Figure 4.2. The FEA mesh and results that the Walton and Braun [1986] NFD model based on (originally from Walton [1993]). thus making the coefficient of restitution e less than one, i.e., e < 1.0. To account for the effect of plastic deformation, Walton and Braun [1986] proposed a bilinear FD law for normal contact of spheres based on the results of the finite element analysis of a sphere, with elastoperfectlyplastic material, in contact with a rigid surface. Figure 4.2(a) depicts the FEA mesh in Walton [1993] 2 ; Figure 4.2(b) shows the resulting FD curve. Based on the FEA result shown in Figure 4.2(b), the normal forcedisplacement (NFD) model proposed by Walton and Braun [1986] is a bilinear function of the form (Figure 4.4) P = K\a for loading, K 2 (a Â— a ) for unloading (4.3) where P is the normal contact force between two particles, a the normal displacement (relative displacement of the centers of the two spheres), K x and K 2 the slopes The FEM mesh and the result were originally presented in Walton et al. [1984]
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58 of the straight lines representing the loading and unloading stiffness coefficients, ato the residual displacement after complete unloading. The working of this NFD model can be described using the simple concept of a partiallylatching spring system shown in Figure 4.3 and Figure 4.4: During loading, only the spring with stiffness K x functions, since the latch device allows free sliding; during unloading, the latch device locks, thus making both springs with stiffness K\ and stiffness (K 2 K{) work simultaneously, yielding the resulting stiffness of K 2 The square root of the ratio of the areas underneath the loading curve and the unloading curve corresponds to the energy dissipated in the process, and is related to the coefficient of restitution e as follows 'ABC e = ^AOC^ It can be shown easily, from the above definition, that V a 2 In simulation, when the normal loading stiffness K\ and the coefficients of restitution are given, the normal unloading stiffness K 2 can be calculated using (4.5). 4.2.2. Tangential FD Model In the tangential direction, an incremental model based on a highly simplification of the Mindlin and Deresiewicz [1953] contact mechanics theory is employed. Let Q n and Q n+1 be the tangential force at time t n and tÂ„ +1 respectively. The relationship between Q n and Q n+1 is given by the following incremental formula Qn+l = Qn + K T ,Â„ A6 n (4.6) where K T n denotes the tangential stiffness coefficient at time t n and AS n the incremental of tangential displacement at time t n computed based on the motion of
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59 Friction Surface Particle A Particle B Figure 4.3. Partiallylatching spring force model p K x 1 y/ B J 1 A K 2 C .: o a a Figure 4.4. Normal contact forcedisplacement model
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60 particles in the previous time step (see (4.12)). We refer the readers to Section 4.2.3 for more details for the calculation of AS n The tangential stiffness K TtTl is a function of P n Q n and Q*, which is the value of tangential force Q at the last turning point, as follows (Walton and Braun [1986]) A' T,n Krfii 1 ~ p _qA for Q increasing^), KT ^~ %n + Q* ) for Q decreasing^), (47) where K T ,o is the initial tangential stiffness computed following (4.8) below, and /i the friction coefficient. For initial loading, we set Q* to zero. The value of Q* will be subsequently reset to the value of Q at the turning points, i.e., where the magnitude of tangential force Q changes from increasing to decreasing, or vice versa. By using partiallylatching spring system to model the contact forcedisplacement behavior in the normal direction and connecting the initial stiffness of contact in the tangential direction {Kt$) with the stiffness in the normal direction (Ki) as described in Johnson [1985, p. 220], and Mindlin and Deresiewicz [1953, p.327], the initial tangential stiffness Kt,o can be calculated as: K T ,o = Ki 2{ l~ U \ (4.8) 2 Â— v where the K\ is the loading normal stiffness in the NFD model (4.3) and v the Poisson's ratio of simulated particles, our model can reduce the stiffness in the tangential direction when plastic flow occurs in the contact region. Remark 4.1. Note that the update of Q* only depends on the turning points of tangential force Q as described above; regardless of the change in P. This approach is in contrast with the Mindlin and Deresiewicz [1953] theory in certain cases. For example, in the case where both P and Q are decreasing, Q* is updated at every
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61 time step where P is decreasing, even though there are no turning points in Q, since Q is continuously decreasing. We refer the readers to VuQuoc and Zhang [1998a] and Section 3.2 for more details. I Comparing to Mindlin and Deresiewicz [1953], the Walton and Braun [1986] TFD model uses only two expressions for the tangential stiffness K TtK for all tangential force Q loading and unloading cases, while Mindlin and Deresiewicz [1953] has eleven different cases with much more complicated conditions and expressions for the stiffness. For example, for the cases where the normal force P is constant, Mindlin and Deresiewicz [1953] calculated the tangential stiffness K T pP), In the above two cases, (4.7) this simplified TFD model (4.7) agrees with Mindlin and Deresiewicz [1953] in the computation of the tangential stiffness K Tjn Furthermore, both expression (4.7) and (3.24) ensure that the tangential stiffness K T n goes to zero as Q goes to the frictional limit fiP, beyond which sliding will occur. It should be noted that even for the cases where normal force P is hold constant, (4.7) does differ significantly from (3.24). For example, when the tangential force Q is oscillating and the magnitude of the turning point tangential force Q* is much less than the frictional limit, i.e.,  Q* < M P, (4.7) and (3.24) yield sharply different tangential stiffnesses. Another difference between (4.7) and (3.24) is the case where Q* Â£ (i.e., not the virgin loading case), and where Q reverses its direction and reaches a magnitude beyond that of Q* (i.e.,  Q \>\ Q* ) : Here,
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62 the Mindlin and Deresiewicz [1953] theory employs a formula different from the one used when  Q \<\ Q*  (see (3.24) 2 and (3.24) 4 ). Despite the difference in (4.7) and (3.24), both expressions produce similar results for the cases where P is constant. On the other hand, (4.7) can not reproduce the results obtained with (3.24) in the cases where P varies. In Mindlin and Deresiewicz [1953], there are nine cases in which P varies; for each of these cases, Mindlin and Deresiewicz [1953] provided different expressions for the tangent stiffness using derivations "of considerable complexity" (Johnson [1985, p.221]). In VuQuoc and Zhang [1998a], we presented a model that is much more accurate and robust than (4.7), and that agrees closely with the Mindlin and Deresiewicz [1953] theory of elastic frictional contact for a wide range of time histories of P and Q. 4.2.3. Implementation of the Walton Braun [1986] TFD Model The implementation of this frictional TFD model into our particle system simulation code involves some algebraic and vector manipulations, since the direction of the normal to the tangential surface at contact changes continuously during a typical contact. In the simulations, due to the small timestep size employed (typically 1/40 of the duration of contact between two spheres; this duration is far less than 10 3 second in our simulations), the displacements from one time station to the next are relatively small. Consider two spheres (i) and (j) in contact (each of these spheres could be either a singlesphere particle or a constituent sphere in a spherecluster particle). Let (j)*n (O^n /. ^\ (ij) n n = TT 2 ~ jf (4.9) II (i) r n (0 r II be the current unit vector pointing from the center of sphere (i) to the center of sphere (j), with (i )r n and ^r n being the position vectors of the centers of spheres (0 and (j), respectively, all at time t n Since in general the direction of the normal
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63 (ij)fin varies continuously, the tangential force vector at time t n has to be adjusted as follows. Let Q n d be the tangentialforce vector at the end of the previous time step. The current tangential force vector Q n at time t n is computed by carrying the vector Q n ,oid to the current tangent plane as shown in Figure 4.5. Such operation is accomplished as follows. First, Q n id is projected into the tangent plane to sphere (i), having normal (ij)n n by (see, e.g., Gurtin [1981]) Qn,0 := [1 (ij)^n (i,)nÂ„ ] Â• Qn old (4.10) Next, the current tangential force Q n is given by Q., old Figure 4.5. Direction change of tangential force *4n Â—  ^4n,old  (ij)^n i (ij)^n Qn,0 Qn,0 (4.11) Thus, the magnitude of Q n is the same as that of Q n /d whereas the direction of QÂ„ is that of the projection of QÂ„ )0 w into the tangent plane with normal Ui)^n Â• The relative displacement vector A<5Â„ is computed as follows A<5Â„ = [ 1 {ij) n n 8 (ij) nÂ„ ] Â• ( 0) x n _i w x n _i) At + (U) R 0) W ni (i)R W^ni ) X W) fi n At (4.12)
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64 where { (j)X n _i, (i)X n _} are the velocity vectors, { (j)U n _i {i)U n _i } the angular velocity vectors, of spheres (j) and (z), respectively, all at time t n _i, and At the timestep size (see Figure 4.6). We make the following approximation ( tf)*n4 W^ni) A* ~ (ii)'n ~ (#)*! in expression (4.12) for A<5 n in our implementation for the TFD model. (4.13) W ni Z II 0) X n (i) W ni Tangential plane .;Â• Figure 4.6. Quantities involved in the computation of Ad n Recall that (ij) t n is the direction of the projection of QÂ„, oW on the tangent plane having normal {ij) n n (see (4.11)). We consider the direction {ij) i n as the direction of continuing application of the tangential force Q and thus the loading history in the TFD model is to be applied in this direction. On the other hand, the direction that lies in the same tangent plane, and is perpendicular to (ii) tÂ„
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65 is considered to correspond to a new virgin loading of tangential force. We thus decompose the incremental tangential displacement Ad n into two components, one along (ij)i n and the other in the tangent plane and perpendicular to (ij)i n : AS n = A<$Â„, + AS n (4.14) as shown in Figure 4.7. With AS n known, we can compute A<5 n>  and A6 nt as A Tangent plane at contact point with normal (y)fin Figure 4.7. Decomposition of the incremental tangential displacement Ad n at time t n follows A5 n ,n = [ (ij)t n S (ij)tÂ„ ] Â• A(5 n (4.15) and A<5 n>1 = A8 n A6 n , (4.16) As mentioned above, the component of the tangential force along the direction (ij)t n is incremented form the projected tangential force Q n in the same direction as (4.11) Qn+l, = Qn + K Tt nAd n ^ (4.17)
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66 where the current tangential stiffness K T n in the direction of (ij) t n is determined by (4.7) and scaled with the change of normal contact force using Q q ip^t (4 18) The component of the tangential force perpendicular to ^j)i n and in the tangent plane with normal (i:7 )nÂ„ is computed as an initial increment (virgin loading) as Qn+i.1 = K Tfi A<$Â„ > (4.19) Finally, the intermediate updated tangential force at time t n+l is Q'n+l = Qn+l, + Qn+1,1 (4.20) which is the counterpart of (4.14). Since it is not assured that  Q' n+1  < /z P n+1 , we set the final updated tangential force at time t n+x to be Q n+1 = rmn(\\Q' n+1 \l \\^P n+1 \\)^\ (4.21) 4.3. Thornton [1997] NFD Model Ning and Thornton [1993] and Thornton [1997] proposed a simplified theoretical model for the normal contact interaction between two elasticperfectlyplastic spheres for DEM simulation. The Thornton [1997] NFD model assumes that quasistatic contact mechanics theories are valid during a sphere impact. During elastic loading, the normal traction (i.e., the distribution of normal pressure on the contact area) and NFD relationship follow Hertz theory; when plastic deformation occurs, the normal traction is less than or equal to a contact yield stress, denoted by (a Y )Th everywhere inside the contact area, as shown in Figure 4.8. Thornton [1997] believes that there is a linear relationship between the normal displacement a and the normal contact force P after the incipient plastic deformation. For unloading after
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67 Hertz Figure 4.8. Normal traction in the Thornton [1997] NFD model. plastic deformation occured, Thornton [1997] follows that normal FD law proposed by Hertz using a larger radius of relative contact curvature Rp resulted from irreversible plastic deformation. The Thornton [1997] NFD model yields the normal FD curve(s) shown in Figure 4.9. The coefficient of restitution e from the Thornton [1997] NFD model is a function of incoming velocity v in expressed as follows '6V3' 1 1 (VY_\ nV2 6 \v in J Â— 1 /1.2 0.2 \VinJ n 1/4 R22) where vy is defined as the yield velocity, i.e., the relative incoming velocity when incipient plastic deformation develops (below this velocity, no plastic deformation occurs), and is given by ?^3\ 1/2 vy = 3.194 ( <7>y> 3  where m* is the equivalent mass for collision that defined asm* = f 1 (4.23) the contact yield stress (a Y )Th is the maximum normal pressure on the contact area (Pm) when yield begins. Hertz theory together with the von Mises criterion are used
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68 \*max)H [PmaxJTh Figure 4.9. Normal forcedisplacement curve(s) of Thornton [1997] model. to obtain, (<7y)r/i = 1.61 a Y (see Section 3.1, also see VuQuoc and Lesburg [1998], and Johnson [1985, p. 155]), where oy is the yield stress of the the sphere material The radius R* of the relative contact curvature and the equivalent Young's modulus E* are given by (3.2) and (3.1), respectively, according to Hertz theory. We will show later (Chapter 6) that the Thornton [1997] NFD model produce FD curves that are too soft compared to FEA results. The contact yield stress, i.e., the maximum normal traction in the contact area, obtained from FEA results is larger than (ay) T h = 1.61 oy.
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CHAPTER 5 AN IMPROVED TFD MODEL FOR ELASTICFRICTIONAL CONTACT In this chapter, we present an improved TFD model for elasticfrictional contact. This model is constructed by incorporating the four basic cases of varying P and Q from the Mindlin and Deresiewicz [1953] theory (see Section 3.2) to account for the effect of changing normal force P. 5.1. Formulation of the Model Considering that when normal force P is constant, the Walton and Braun [1986] TFD model (see Section 4.2.2) gives relatively accurate results for the TFD relationship, we use Walton and Braun [1986] TFD model for loading cases with P constant. When Hertz theory is chosen as the NFD model, the initial tangential stiffness K T ,o is determined by (3.25); when the Walton and Braun [1986] NFD model is chosen as the NFD model, the initial tangential stiffness K Tfi is determined by (4.8). For the cases with both P and Q varying, our numerical experiments show that (4.6) and (4.7) cannot correctly model the TFD relationship. The reason is that there is not an appropriate change in stiffness and displacement to account for the transitions caused by the change in normal force P. Therefore, we improve the highly simplified model of Walton and Braun [1986] by implementing the TFD curve transitions of Mindlin and Deresiewicz [1953]. That is, we account for the effects on the TFD curve due to varying normal load P by transiting to a TFD curve that exhibits the tangential stiffness that is appropriate for the current normal load. We incorporate the transitions caused by varying normal force P for the following four 69
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70 cases discussed in Section 3.2.2: 1. P increasing, Q increasing 2. P decreasing, Q increasing 3. P increasing, Q decreasing 4. P decreasing, Q decreasing For example, for the case with P increasing and Q increasing, the loading state changes from {P Q 5 } to {P u Q 2 S 2 }, where P x = P + AP (AP > 0), and Q 2 = Qo + AQ (AQ > 0), as shown in Figure 3.6. There are two subcases: AQ > fj,AP and AQ < /zAP. For AQ > fiAP, the tangential force is given by (3.28), i.e., Q 2 = Qo + AP + (K T ) 12 (as ,/ A P ) (3.28) \ {^Tfi)P=PoJ For AQ < //AP, the tangential force is given by Qi = Qo + (K T ,o)p=p AS (3.30) The modification for the rest of the three cases is similar. We refer the readers to Section 3.2.2 for more detailed formulae. Remark 5.1. Both Hertz theory for normal elastic contact and Walton and Braun [1986]'s NFD model can be used to carry out DEM simulations of particle processes. When Hertz theory is used as the NFD model, the tangential stiffness for initial loading K Tfl is determined by (3.25). Since both the NFD model and the TFD model are based on the elastic frictional contact problems, these FD models are consistent with each other. When the Walton and Braun [1986] NFD model is used for the normal contact, the tangential stiffness for initial loading K T<0 is determined
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71 by (4.8). For the Walton and Braun [1986] NFD model, when the coefficient of restitution e is less than 1.0, there is plastic deformation in the normal contact that induces energy dissipation. The Walton and Braun [1986] NFD and TFD models are thus inconsistent with each other, since while their NFD model accounts for plastic deformation, their TFD model does not. Plastic deformation occurs in real contact problems, and thus must be accounted for in both NFD and TFD models (see VuQuoc and Zhang [19986]). I 5.2. Accounting for Rolling Effect We also implemented a correction for the rolling effect in our improved TFD model. A common disadvantage for most of existing TFD models is the lack of consideration for rolling effects. Consider two spheres in contact subjected to both normal force P and tangential force Q. The tangential traction is distributed on a circular contact area (the tangential force can be obtained by integrating the tangential traction over the contact area). When rolling happens between two particles in contact, as occurred frequently in particle flows, part of the contact area is separated as particles roll off from each others, and the tangential traction distributed on that area will be released. At the same time, some area in the rolling direction not previously in contact is 'rolled' into additional contact area, without any distributed tangential traction. Because of this reason, correction must be taken to account the rolling effect on the tangential force Q. For simplicity, consider the case of a sphere moving over a planar surface (itself a sphere with infinity radius). Assume that the plane is not moving and the sphere is moving with a translation velocity v parallel to the planar surface and with an angular velocity uj as shown in Figure 5.1. In a time At, the center of sphere O is translated by a distance Ax = vAt to the right, and rotates through an angle A6 = u)At in the clockwise direction (Figure 5.1). With R being the radius of the
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72 rolledoff area rolledon area Figure 5.1. A sphere moving on a planar surface Figure 5.2. Change of contact area because of rolling rolling sphere, the angular velocity lo intend to move the contact point C by the amount of RAO = RuAt. Therefore, the rolling distance of the contact point can be expressed by As R = min{Ru)At vAt) (5.1) The rest of the motion of the contact point C is due to sliding, which is the tangential displacement 6 for the calculation of tangential forces (as discussed in Section 4.2.3, also see VuQuoc et al. [1997]). In the case where the angular velocity is in the counterclockwise direction, thus making the contact point C moving in the opposite direction, all the motion of the contact point C is sliding motion, and could be counted as tangential displacement, the rolling distance is zero, i.e., As R = 0. The change of contact area is shown in Figure 5.2. Let a be the radius of the contact area. Because of the rolling distance As R the left crescentshaped area is rolled off, and is thus released of the tangential traction that was applied on it. At the same time, the right crescentshaped area is rolled on, but carry no tangential traction; see Figure 5.2. The rolledoff area is the same as the rolledon area; we have A = 7ra 2 and AA 2aAs R (5.2) where A is the contact area, and A.4 the rolledoff area or rolledon area.
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73 The effect of rolling on the contact tangential force is complex (see Kalker [1990] and Yu et al. [1995]). An approximate way to account for the effect of rolling on the TFD relationship would be to scale down the existing tangential displacement S associated with the tangential force Q by adding following adjustment due to rolling to the tangential displacement By (5.2), the adjusted tangential displacement S' with rolling effect is In our implementation of the proposed improved TFD model, the rolling effect is accounted by adding the adjustment of tangential displacement AS R to the increment of tangential displacement AS AS' = AS + AS R = AS^S (5.5) where A<5' is the adjusted increment of tangential displacement, accounting for the rolling effect, and is used to compute the increment AQ for the tangential force. 5.3. Comparison with Other Models We developed a MATLAB code for the Mindlin and Deresiewicz [1953] theory, Walton and Braun [1986] TFD model, and the proposed improved TFD model for elastic frictional contact. This code allows us to compare the proposed improved TFD model with the Walton and Braun [1986] TFD model and the Mindlin and Deresiewicz [1953] theory. The following numerical experiments are carried out on two identical aluminum alloy spheres with radius R = 0.1m, and with the following mechanical properties: Young's modulus E = 2.5 x 10 8 N/m 2 Poisson's ratio v =
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74 0.3, coefficient of friction \x = 0.5. For consistency, we use Hertz theory as the NFD model for all TFD models applied in the numerical experiments. In the first experiment, we hold the normal force P constant, while imposing a tangential force Q that varies sinusoidally, such that Q increases until it is close to the frictional limit //P, then decreases down to zero, as shown in Figure 5.3. As a result, the Q Â— 8 curve forms a closed hysteresis loop. It can be seen from Figure 5.3 that our improved TFD model produces a Q 5 curve that is close to that obtained from the Mindlin and Deresiewicz [1953] theory. Please note that in the case of constant normal force, the improved TFD model is exactly the same as the Walton and Braun [1986] TFD model. In the second experiment, we make both P and Q vary sinusoidally as shown in Figure 5.4. Note that at time t = 0, P(0) > and Q(0) = 0; the shifting of the sine curve in the time history of P upward so that P(t = 0) > is to ensure that we are dealing with the simple loading cases, which require that AQ > fi\AP\ for each increment of P and Q (see Mindlin and Deresiewicz [1953]). It can be seen from Figure 5.4 that the Walton and Braun [1986] TFD model differs sharply from the Mindlin and Deresiewicz [1953] theory. The reason lies in the fact that the Walton and Braun [1986] TFD model does not properly deal with the effect of the change of normal force P on the Q 5 curve. Our improved TFD model produces a Q6 curve that is almost on top of the curve produced by Mindlin and Deresiewicz [1953] theory. Our improved TFD model produces more accurate results for elastic frictional contact in the tangential direction than the Walton and Braun [1986] TFD model. The implementation of the improved TFD model is similar to the implementation of the Walton and Braun [1986] TFD model, but with the modification to
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75 200 150 S00 Â— a 50 ft, Experiment 1: P Constant, Q Varying 50 1 [ 1 Â— Q (tangential load) Â— P (normal load) Â— mu*P (friction limit) _.... ; "^M^ Â— i Â— ^ 1 0.2 0.4 0.6 time 0.8 Experiment 1: Q Â— 8 curves x10 Figure 5.3. Comparison of TFD models: P constant, Q sinusoidal. The TFD model by Walton and Braun [1986] (WB) produces results that are close to those of Mindlin and Deresiewicz [1953] (MD) theory.
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76 200 150.100 a 3 50 Experiment 2: P, Q Varying (simple loading) i i Q (tangential load) Â— P (normal load) mu*P (friction limit) y s s ' v. i i 0.2 0.4 0.6 time 0.8 Experiment 2: Q Â— S curves WB [1986] Present TFD MD [1953] 0.2 0.4 0.6 0.8 t(m] 1.4 1.6 1.8 x10" 4 Figure 5.4. Comparison of TFD models: P sinusoidal, Q sinusoidal (simple loading history). The TFD model by Walton and [1986] ( WB) produces results that are far from those of Mindlin and Deresiewicz [1953] (MD) theory, while our improved TFD model produces accurate results compared to the Mindlin and Deresiewicz[1953] theory.
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77 discretized volume two spheres discretized volume Figure 5.5. Discretized volumes in two contacting spheres for FEA. account for the transitions caused by the change of normal force (see also VuQuoc et al. [1997] for more details). 5.4. Finite Element Validation In this section, we compare the finite element analysis (FEA) results with the results obtained using Mindlin and Deresiewicz [1953] theory and our improved TFD model for simple loading histories. In realistic particleflow problems, nonsimple loading histories are to be expected. Comparison of the results produced by finite element analysis with the results produced by our improved TFD model applied to nonsimple loading histories are also presented. Using the commercial finiteelement software ABAQUS (version5.4), we carried out a series of FEA of different contact problems. To analyze the quasistatic
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78 2000 AQ = nAP ^ = 0.2 Figure 5.6. A Simple loading history applied for FEA validation. problems of two aluminum alloy spheres in elastic frictional contact, we constructed FE discretization of the volumes shown in Figure 5.5. The main mechanical properties of the aluminum alloy spheres are: radius R = 0.1 m, Young's modulus E = 7.00 x 10 4 MPa, Poisson's ratio v = 0.30, and coefficient of friction // = 0.20. We refer the interested readers to VuQuoc and Lesburg [1998] for more details on our 3D FEA model of static contact between two spheres. 5.4.1. FEA with SimpleLoading History The first problem for our FEA is based on the loading history shown in Figure 5.6. Since we have \AQ\ > \yAP\ in the loading history in Figure 5.6, we are having here an example of a simpleloading history (see Section 3.2.2 Remark 3.1). Please note that we are considering static contact in the FEA, so the time t is just used to show the loading sequence and without any other meaning.
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400 Tangential force vs. tangential displacement Figure 5.7. Comparison of FEA results to results produced by Mindlin and Deresiewicz (MD) [1953] theory (VuQuoc and Lesburg [1998]). As mentioned, for consistency, Hertz theory is employed as the normal forcedisplacement (NFD) model together with our improved TFD model, since we restrict our comparison to elasticfrictional contact only. Figure 5.7 shows the Q 8 curves produced by FEA and by the Mindlin and Deresiewicz [1953] theory for the simple loading history shown in Figure 5.6. The FEA results agree with the results of the Mindlin and Deresiewicz [1953] theory for this simple loading history. Figure 5.8 shows a comparison of the Q5 curves produced by our improved TFD model and by Walton and Braun [1986] TFD model with the Q 5 curve produced by the Mindlin and Deresiewicz [1953] theory for the same loading history shown in Figure 5.6. The results produced by our improved TFD model also agree with the results produced by the Mindlin and Deresiewicz [1953] theory, and thus agree with the FEA results; the Q 5 curve produced by the Walton and Braun [1986] TFD model is, however, sharply different from those by the Mindlin and Deresiewicz [1953] theory and by
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80 4000 3500 3000 Â§2500 o T3 2000 a a ^ 1500 1000 500 Simple loading history: P, Q varying 1 1 Â— Q (tangential load) Â— P (normal load) Â— mu*P (friction limit) ^ ^ s** ** rÂ— Â— 1 z^^Â— 1 i ~i 0.5 1.5 time (sec) Q Â— 5 curves WB[1986] TFI). Improved TFE' MD[1953j TFI) 2 i (nf 5 3.5 x10 4.5 6 Figure 5.8. Comparison of Q 6 curves: The loading history shown in upper part is the same as the one shown in Figure 5.6. The Q 6 curve produced by our improved TFD model agree with the result produced by the Mindlin and Deresiewicz [1953] (MD) theory, while the Q 5 curve produced by the Walton and Braun [1986] (WB) TFD model does not by a large amount.
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81 FEA. As we mentioned, the reason for the inaccuracy of Walton and Braun [1986] TFD model is that it does not properly deal with the effect of varying normal force P. 5.4.2. FEA for NonSimple Loading History Figure 5.9 displays the Q 5 curve produced from FEA results and the curve produced by our improved TFD model for the loading history shown in the upper part of Figure 5.10. This loading history can be described as follows: from time t = 0.0 sec to time t = 1.0 sec, normal force P increases from P = 20 N to Pmax Â— 2000 N at a constant rate of increasing, tangential force Q increases from Q = 0.0 N to Q m ax = 380 N at another constant rate of increasing. From time t Â— 1.0 sec to t = 2.0 sec, both P and Q decrease from their maximum values to their respective initial values at constant rates of decreasing. Since the coefficient of friction is /i = 0.2, in above loading history, the rate of increase AQ/At is always less than the rate of increase of the friction limit fiAP/At. By Remark 3.1, this loading history is a nonsimple loading history. It can be seen from Figures 5.9 and 5.10 that although the Mindlin and Deresiewicz [1953] theory is not valid for nonsimple loading histories, our improved TFD model, which is based on the Mindlin and Deresiewicz [1953] theory, can still produce results close to FEA results for the nonsimple loading history shown in the upper part of Figure 5.10. The results produced by the Walton and Braun [1986] TFD model, as shown in Figure 5.10, shows, however, a much larger energy dissipation and maximum displacement for the same loading history. Remark 5.2. We observe from the FEA results shown in Figure 5.9 that the Q 8 unloading curve of FEA results is on top of loading curve. It is observed that there is no energy dissipation for the nonsimple loading history of our example. The reason for this situation to happen is that during the loading, the rate of increase of
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SQ curves i 1 350 Â— Improved TFD FEA Q loading FEA Q unloadin 7 jj? 1 S? Q? 50 0< *t2 0.5 1.5 2 *(m) 2.5 3.5 82 x10 Figure 5.9. Comparison of FEA results with results produced by our improved TFD model for a nonsimple loading history. the normal force P is so high that the rate of increase of tangential force Q is lower than the rate of increase of the friction limit fj,AP. That AQ < p\AP\ prevented the formation of micro slips in the contact area, which are the cause for energy dissipation as mentioned in Section 3.2. During the unloading, the ratio between the rate of decrease of the tangential force Q and the rate of decrease of the normal force P are the same as the ones for the loading stage. Such an unloading path results in the tangential traction to be developed on the contact surface in exactly the same manner as the development of tangential traction for the loading but in the opposite direction. Again, this ratio prevented micro slips from being developed in the contact area. Therefore, the Q5 relationship behaves like a nonlinear elastic system. We refer the interested readers to Mindlin and Deresiewicz [1953] and VuQuoc and Lesburg [1998] for more details on the micro slips, the slip, and the stick regions. I
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83 4000 3500 3000 Â§2500 q 200015001000 500 Nonsimple loading history: P, Q varying 1 i Q (tangential load) Â— P (normal load) Â— mu*P (friction limit) n : ^ n ^ n. : .Â• v.. '' : \ ^ s : ^ ^llÂ— r^^^^^^^~rÂ— Â— 111. 0.5 1.5 time (sec) 400 Q> Q Â— S curves I i i i i: / '1 / ! y / // ','' // / '/" :'">'"': //. ; : / : // i^' i :Â•/ ; // y /y s ^ ; ;Â•;>' : / > Improved TFD \^Â— 1 *^= Â— i 1 WB[1986] i i TFD 4 5 *(m) x10 6 Figure 5.10. Comparison of Q 8 curves: The loading history shown in upper part is the same as for FEA results shown in Figure 5.9. The Q S curve produced by the Walton and Braun [1986] (WB) TFD model is sharply different with the curve produced by our improved TFD model.
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81 5.5. Benchmark Test: HardSphere Collisions We use the DEM simulation code implemented with the improved TFD model presented in this chapter to carry out the simulation of hardsphere collisions (see Appendix C). Please note that in our simulation, we use "soft" particle modeling to simulate the collision of hard spheres. 5.5.1. Simulation: HardSphere Collision of Special Material With the improved TFD model implemented in our DEM simulation code, we simulated 100 hardsphere collisions with a rigid planar surface. The spheres are of special materials so that the vibrational period of the springmass system in the TFD model is the same as the vibrational period of the springmass system in the NFD model for elastic contacts. These 100 hard spheres were given the same initial translational velocity perpendicular to the rigid surface, and subjected to different angular velocities so that the spheres have different incident angles, i.e., different v n /v Sjt at the contact points. Figure 5.11 shows such a sphere moving down to collide with a rigid planar surface. Remark 5.3. It should be noticed that in our simulation code, a planar surface is represented by a single sphere with the same radius of the impacting sphere, but of infinite mass and zero velocity. By symmetry, the behavior of two identical spheres in contact is equivalent to a single sphere in contact with a rigid planar surface. I Figure 5.12 shows the results of simulation of hardsphere collisions for the special material described in Appendix C. For this material, the period of the normal direction of contact is equal to that of the tangential direction of the contact. The Walton and Braun [1986] NFD model is employed for normal contact; this model can only accommodate a constant normal coefficient of restitution e, i.e., e
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85 Figure 5.11. Collision of a sphere with a planar surface is independent of the incoming velocity. In this simulation, except the properties for special materials, we set e = 0.5 for all spheres, and the coefficient of friction to H = 0.4. It can be seen from Figure 5.12 that the results obtained our improved TFD model show that, for sliding collision, the tangential coefficient of restitution is a linear function of ( Â—2I ; and that the tangent of the effective recoil angle f ^) \ V ^J \v'n) is also a linear function of the tangent of the corresponding effective incident angle ( Â— Â— j Those relationships are exactly as predicted theoretically in Section ?? by (C.25) and (C.26). The little circles 'o' on Figure 5.12 divide the collision regimes sliding collision ar = 1.389, and thus into sliding collision and sticky collision. At this critical incident angle, we have v s ,t *H^(Hi)ilU The above result also agrees with the inequality (C.31). From Figure 5.12, we also observe that there are two important parameters that control the collision regimes. The larger the coefficient of friction /x is, the more likely that the collision is sticky;
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86 Tangential Coefficient of Restitution (3 so. 0.5Effective Recoil Angle vs. Incident Angle 4 X 5 5,< 1 1.389 2 v st 3 Figure 5.12. Simulation results for spheres in collisions with a plane, special material: There is a large difference between the results from our improved TFD model and those results from Walton and Braun [1986] TFD model around the point of changing mode of contact. Our results agree with the theoretical results, whereas those from the WB TFD model do not.
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87 of course, the smaller the /j, is, the more likely the collision is sliding, and similarly for the incident angle ip, or the ratio f Â— Â— ) \ v s,t J The simulation results obtained by using the Walton and Braun [1986] TFD model are also presented in Figure 5.12. They agree with the theoretical results when the incident angle (ip = tan,1 Â— )) is far away from the critical angle ^o at which *= 1.389 and = (3 When the incident angle is close to the critical incident angle Vo, there is, however, a clear difference between the results from the Walton and Braun [1986] TFD model and the theoretical predictions. Figure 5.13 presents the loading histories and the Q S curves for the sphere with the critical incident angle Vo, i.e., the sphere with incoming velocity ratio Vst J= 1.389. There are five curves in the upper part of Figure 5.13 for the loading histories of the collision: The dashed line is the normal force P obtained from using the Walton and Braun [1986] NFD model used for this simulation; the line with the '+' symbols is the friction limit //P; the solid line is the tangential force Q from simulation using our improved model with 40 integration timesteps for one contact (ntcol = 40); the line with the 'o' symbols is the Q obtained from our improved TFD model with larger integration timestep size {ntcol = 20); the 'dashdot line' is the tangential force Q obtained from the Walton and Braun [1986] TFD model. As predicted in Appendix C, the sphere with the tangent of the incident angle v s ,t /v n = 1.389 is the one at the boundary of sliding collision and sticky collision; in this case, during collision, tangential force Q should be at the friction limit //P. The loading histories clearly show that the tangential force Q from our improved TFD model lies exactly on top of the friction limit /xP, as expected from theoretical considerations. On the other hand, the tangential force Q from the Walton and Braun [1986] TFD model is below the frictional limit /iP for more that two third of the collision time. The lower part of the figure shows three Q 6 curves: The
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88 0.35 0.3 ,.0.25 5 02 a 5 0.15 CO a, 0.1 0.05 Loading Histories: v s ,t / v n =1389 P:WBNFD Q: Present TFD(ntc6j=40) Q: WB TFD(ntcol=40) J Q: Present TFD{ntcol=20) +Â• mu*P: Friction Limit *^MA Time (sec) 5.05 5.1 5.15 5.2 5.25 5.3 5.35 5.4 x10" 0.14 0.12 0.1 0.08 Q Â— 5 curves Present TFD(ntcol=40) WB TFD(ntcol=40) o Present TFD(ntcol=20) Figure 5.13. Loading histories (upper) and Q 5 curves (lower), special material: At v s t / v n = 1.389, Q from our improved TFD model is at the frictional limit pP (upper figure); Q obtained from the Walton and Braun [1986] (WB) TFD model is below //P for most of the collision time.
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89 solid line is the result from our improved TFD model with small time steps (i.e., ntcol = 40, the same number of time steps as in for the simulation with the Walton and Braun [1986] TFD model); the line with the 'o' symbols is the result from our improved TFD model, but with time steps that are twice as large (or half the number of time steps, i.e., ntcol Â— 20); the dashed line is the result from the Walton and Braun [1986] TFD model. There is thus a clear difference between the Q Â— 5 curves from our TFD model and the one from the Walton and Braun [1986] TFD model. Another important feature of our improved TFD model is that the curves obtained from simulations using our improved TFD model with different time step sizes for integration are close to each other. That is, our improved TFD model is stable for different time step sizes of integration. 5.5.2. Simulation: HardSphere Collision of Ordinary Material Maw et al. [1976] presented analytical results of oblique impact of homogeneous elastic spheres on a halfspace. Their calculation was based on Hertz theory for normal contact and Mindlin and Deresiewicz [1953] theory for tangential elastic frictional contact. The Poisson's ratio of the material in Maw et al. [1976] 's calculation was v Â— 0.3. We carried out DEM simulation for the collisions discussed in Maw et al. [1976]. We used the same Poisson's ratio, v = 0.3, as in Maw et al. [1976] in our simulation. Since Hertz theory is employed for the normal contact forcedisplacement law, the coefficient of restitution for normal collision is unity, i.e., e Â— 1.0. Except for the fact that we assume the deformation involved in the collisions is limited to the elastic range, and that the friction is the only source of energy dissipation, all parameters set in our simulation are values of ordinary materials (such as aluminum). We simulate 100 spheres colliding with a rigid surface in the same way as described in Section 5.5.1. Figure 5.14 shows the simulation results.
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90 0.6 0.4 0.2 ~~ CK>0.20.4 Tangential Coefficient of Restitution /3 Improved TFD WB [1986] TFD 3 1 X 4 (1 + e) ( 1 + F)Â£ 5> I p Effective Recoil Angle vs. Incident Angle Vsjt Figure 5.14. Simulation results for spheres (ordinary material) colliding with a rigid surface: Tangential coefficient of restitution (upper), recoil angle vs. incident angle (lower).
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91 It can be seen from Figure 5.14 that there is no critical incident angle for the ordinary material considered. The upper part of Figure 5.14 shows that that the tangential coefficient of restitution in this case still follows the qualitative theoretical prediction given by (C.25) and (C.26). The improved TFD model and the Walton and Braun [1986] TFD model agree in the sliding collision regime, but differ from each other in the sticky collision regime and in the "boundary" between sliding collision and sticky collision. Even though the normal coefficient of restitution is unity, i.e., there is no energy dissipation caused by normal contact, the upper part of Figure 5.14 shows that the energy dissipation caused by the tangential force is not uniform in the sticky collision regime, as manifested through the steady decrease of the tangential coefficient of restitution /?. Such nonuniform energy dissipation in the sticky regime is different from the uniform energy dissipation found for special material, as manifested by the plateauing of the coefficient (3 in the upper part of Figure 5.12. The reason for the nonuniformity of the tangential coefficient of restitution could be due to the combined effect of friction and the mismatch in the vibrational periods of the equivalent springmass system in the NFD model and in the TFD model. When the normal contact force P decreases to zero at the end of a collision (see upper part of Figure 5.13), the tangential force Q also has to decrease to zero with P. If the vibrational period of the equivalent springmass system in the tangential direction is not the same as that in the normal direction, when P reaches zero, Q is still far from zero (there is still some stored elastic energy in the tangential direction). As a result, when P drops to zero, so must Q, and thus the remaining stored elastic energy in the tangential direction is lost, resulting in higher dissipation of energy, and thus a decrease in the coefficient /3 as we move to the right on the horizontal axis in the upper part of Figure 5.14. The curves describing the tangent of the recoil angle vs. the tangent of incident angle in the lower part of Figure 5.14 agree with the Maw et al. [1976] analytical
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92 v' results. When the tangent of the recoil angle has the value Â— Â— = Â— 1, the tangent of the incident angle is Â— Â— = 4.682, which is accurate compared to the result of v n Maw et al. [1976] of Â— Â— = 4.765, considering that we employed a highly simplified v n TFD model. The difference between our improved TFD model and the Walton and Braun [1986] TFD model in the upper part of Figure 5.14 indicates that not only different force magnitude is obtained using different TFD models, but also different behaviors of particles in collision. It is therefore important to use an accurate TFD model for an accurate prediction.
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CHAPTER 6 ELASTOPLASTIC NFD MODEL In this chapter, we present the formulation and validation of a new normal forcedisplacement (NFD) model that accounts for the effect of both the elastic and plastic deformations in DEM simulations. To construct the NFD model, we carry out a series of finite element analyses (FEA) of contact problems at first. The NFD model proposed in this chapter is based on the continuum theory of plasticity and the observation of the FEA results. 6.1. FEA Model and Loading Paths Figure 6.1 shows the FEA model employed in our FEA of the normal contact between two identical spheres. The case of two identical spheres in contact, subjected to normal force only, can be analyzed in an efficient manner by considering a single sphere contacting a frictionless rigid surface, as a result of the symmetry in the problem. Using ABAQUS, version 5.4, we constructed an axisymmetric FE model of a sphere contacting a frictionless rigid surface, as shown in Figure 6.1(a), The sphere material is elastic/elastoperfectlyplastic. Invoking SaintVenant principle, we only discretize the domain within the square in dashed line in Figure 6.1(a). Figure 6.1(b) depicts the discretization of our axisymmetric finite element model, which is a much finer mesh around the contact area than the discretization used in Walton [1993] (see Figure 4.2(a)). The sphere in our FEA has a radius of R = 0.1 m, with the material properties chosen as of an aluminum alloy: Young's modulus E = 7.0 x 10 10 N/m 2 Poisson's ratio v = 0.3. When plastic plastic property is imposed, the yield stress is set to 93
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94 sphere discretized zone //////////////////fr/////////////// frictionless rigid planar surface (a) A sphere in contact with a frictionless rigid planar surface. (b) Axisymmetric FEA mesh for normal contact problems. Figure 6.1. FEA model for contact problems. cry = 1.0 x 10 8 JV/ra 2 and the perfectly plastic model was selected as the material behavior in the plastic regime. Using the von Mises yield criterion and Hertz contact theory, the normal contact force at incipient yield is obtained to be P Y = 36.4 N (by (6.19) in Section 3.1). Figure 6.2 shows three loading paths of the normal force P used in the FEA. We refer the readers to VuQuoc and Lesburg [1998], VuQuoc and Zhang [1998a], and VuQuoc et al. [1997] for more details on the FEA models and FEA results of contact problems, including elastic, elastoplastic contact problems for both normal contacts and tangential frictional contacts. Before performing the FEA for elastoplastic contact problems, we validated the FEA model by applying the loading path AFG {P max = 1500 N, in Figure 6.2), with the elastic material properties as listed above, without considering the yield condition and plastic deformation. Then the FEA results are compared with the
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95 P(N) 1500 1000 500 ABC: P max = 5007V "ADE:P mQX =T000/V~ AFG: P max = 1500/V y F B C A/ E G Figure 6.2. Loading paths of normal force. quantities obtained from Hertz contact theory (3.10), (3.9), and (3.3). Close agreement between our FEA results and the results from Hertz theory is observed. 6.2. Formulation of the NFD Model 6.2.1. Additive Decomposition of the Contact Radius Let a ep be the radius of the contact area of an elastoplastic contact under normal contact force P, a e the radius of the contact area of elastic contact under the same normal force P. We have the following relationship 7 ep a e for P a e for P > P Y In other words, the effect of plastic deformation is to increase the size of contact area. Figure 6.3 shows the FEA result of the elastoplastic contact radius a ep versus normal contact force P for P loading from to 1500 N. The elastic contact radius a e determined by Hertz theory (i.e., by (3.6)) is also shown in Figure 6.3.
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96 1.6 x10" FEA (elastoplastic) oe> e eo Hertz (elastic) 500 1000 1500 P(N) Figure 6.3. Contact radius a ep versus normal force P for elastoplastic contact, with comparison to Hertz theory (elastic). Based on (6.1) the observation from Figure 6.3, we propose the following additive decomposition of the elastoplastic contact radius a ep a ep = a e + a p (6.2) where a 6 is the elastic part of the contact radius determined by Hertz theory's (3.6), and a? the plastic part of the contact radius. The above decomposition is motivated by the formalism of the theory of elastoplasticity and the principle that the total strain e ep is the sum of elastic strain e e and the plastic strain e p Figure 6.4 shows the plastic contact radius a p versus the normal force P for loading P to 1500 N and unloading. We observe from Figure 6.4 that, during loading, the plastic radius a p increases with P in an approximately linear fashion. During unloading, the plastic contact radius a? does not obviously decrease with the
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97 x10" 6 2 a, P Y = 36.45 N + loading o unloading 500 P(N) 1000 1500 Figure 6.4. Plastic contact radius a p versus normal contact force P. Symbols (+, o): FEA results. Solid line: model for loading. Dashed line: model for unloading. decreasing P; and there is a permanent deformation left after complete unloading. In other words, the contact radius goes to a nonzero residual value (denoted later as a res ) as the normal force P goes to zero. Further, we note that the elastic part of a ep is nonlinear with respect to P, as can be seen from (3.6). Based on these observations, the additional radius a? of contact area caused by plastic deformation (from now on, we call a p the plastic part of the radius of the contact area) in the proposed NFD model is approximated as or C a (P Â— P Y ) for loading C a (Pmax Py) for unloading (6.3) where C a is a constant that can be determined from the properties of the spheres in
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98 contact. For example, C a for the current elastoplastic contact problem, the FEA results indicate C a = 2.33 x 1(T 7 N/m. The symbol ( ) denotes the MacCauley bracket defined by v f for x < , (X) = {x fora^O Â• ^ This additive decomposition of elastoplastic contact radius is not only the foundation of the proposed NFD model, but is also crucial in the elastoplastic TFD model (see Section 8.1, also see VuQuoc, Lesburg and Zhang [1998] and VuQuoc and Zhang [1998 c]) since the initial tangential contact stiffness is closely related to the radius of the contact area. The additive decomposition of the elastoplastic contact radius a ep into an elastic part (a e ) and a plastic part (aP). We observe that the relationship between a? and the normal load P given by (6.3) parallels that between the plastic strain e p and the stress a in uniaxial problems in the classical model of elastoplasticity with isotropic hardening, in which the following relationships hold e ep = e e + e p (6.5) where e ep is the elastoplastic strain, e e the elastic strain, and e p the plastic strain, which is related to the stress a by e P  K h (aa Y ) for loading \ Kh(&max cry) for unloading ^ ' where K h is the coefficient of isotropic hardening. Figures 6.5 and 6.6 depict relations (6.6) and (6.3), respectively. The similarity between the continuum theory of elastoplasticity with hardening and the elastoplastic contact between spheres form a point of departure in the construction of our elastoplastic NFD model. 6.2.2. Normal Pressure Distribution Figure 6.7 shows for the normal pressure p FE distribution at P max = 1500 TV. This figure shows that the normal pressure on the contact surface is approximately
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99 unloading unloading Umax O *max P Figure 6.5. e p Â— a curve for linear isotropic hardening material. Figure 6.6. a? P curve for normal contact between elastoperfectlyplastic spheres. constant at a level of 2.3 x 10 8 N/m 2 or 2.3 times the material's yield stress a Y 1.0 x 10 8 N/m 2 FEA results for maximal force levels of P max = 500 N and Pmax Â— 1000 N show similar results for the maximum normal pressure p FE but show different elastoplastic contact area radii a ep (Figure 6.8). That is, when the normal contact force is much greater than the incipient yield force, i.e., P > P Y the maximum normal pressure is always roughly twice the material yield stress. For detailed results of these other loading histories, see VuQuoc and Lesburg [1998]. Figure 6.7 also shows the Hertz prediction (via (3.3)) for the distribution of normal pressure p Hz for the same normal force level P max = 1500 N. Comparing p H with p FE we see that the maximum normal pressure from Hertz theory is much larger than that from the FEA results. On the other hand, the radius of contact area from (elastic) Hertz theory is smaller that that from FEA results. Since both normal pressures shown in Figure 6.7 arise from the same normal force level P max the integrals of the normal pressures over the respective contact areas should be the same.
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100 x10 ft, Hertz (elastic), puz present model, p pm / FEA (elastoplastic), pfe V
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101 x10" PHz for {Prnax)AFG 1500 N (Pmax) ADE = 1000 N Â— Hertz o ABC x ADE AFG (Pmax)ABC = 500 A^ 0.6 0.8 1 r (m) x10 1.8 3 Figure 6.8. Distribution of normal pressure p on contact surface for three normal force levels. Broken lines with symbols (o, x, +): FEA results. Solid lines: Hertz theory p# in Figure 6.7. This normal pressure can be expressed as P P m(r) = (Pm) c]> ^1/2 a e Pj (6.7) where the maximum normal pressure (p m ) ep is determined by setting the integral of p pm of (6.7) over the elastoplastic contact area equal to the normal force P. We can see that the shape of the normal pressure distribution of our model is similar to that of Hertz theory, but the distribution is over the elastoplastic contact area. Such an approximation of normal pressure is motivated by the use of Mindlin and Deresiewicz [1953] formalism in constructing the elastoplastic TFD model. It is important to keep in mind that although this elliptic approximation of normal pressure is crucial
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102 for the construction of our TFD model (Section 8.1), but will not affect the NFD relationship in the present NFD model. 6.2.3. Parabola Law: Normal Displacement vs. Radius of Contact Area Hertz theory gives a parabolic relation between the normal displacement and the radius of the contact area. In the cases of one sphere contacting a rigid frictionless plane, the parabola law (3.7) can be simplified to K) 2 a H = R (6.8) where a H is the normal displacement between the sphere and the plane, a H the radius of contact area. Note that a# and a# are both for elastic contact and are both determined via the relations of Hertz theory. 2.5 xlO 1.5 0.5 Pmax = 1500 N Hertz (elastic) FEA (elastoplastic) 0H, o ep {pi) x10 1.5 3 Figure 6.9. Normal displacement a versus the radius of total contact area (a ep for elastoplastic contact, a H for elastic contact).
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103 Figure 6.9 shows the FEA result for the relationship between normal displacement a and elastoplastic contact radius a ep for monotonic loading form P Â— to P = 1500 N. The an versus a# relation obtained from Hertz theory (via (6.8)) is also shown. In Figure 6.9, the a Â— a ep curve from FEA results follows a roughly parabolic relation that is similar to the curve from Hertz theory. Our analyses of other elastoplastic contact cases reveal the similar trends (see VuQuoc and Lesburg [1998]). Based on the FEA results, and considering that the first half of (3.7) is largely derived from geometry relation (see Johnson [1985, p. 8489] and Appendix D), we assume that the a Â— a ep relation is parabolic for the loading portion of the force history. That is, when P is increasing, we have ( ep ) 2 , a = 5T> ( 6 9 ) where R* is the equivalent radius of the local contact curvature accounting for the effect of plastic deformation. The quantity R* is defined in a manner similar to (3.2) by ^ := f"V + ^) (610) R p \ (i)R P u)Rp J For the contact between two identical spheres or between a sphere and a rigid plane, we have R* Â— Rp/2. Therefore, the parabola law can be written as (a ep ) 2 a = i^ (6.11) tip where Rp is the radius of local contact curvature accounting for plastic deformation. Using (6.12) this radius can be computed based on the original radius R and on the level of plastic deformation. Figure 6.10 shows such a change in the local contact curvature. Therefore, we propose to compute the radius Rp of relative curvature by the following Rp = C R (P)R (6.12)
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104 Figure 6.10. Plastic deformation increases the radius of relative contact curvature. where Cr(P) is the coefficient for adjusting the contact radius to account for the plastic deformation. Considering that the plastic deformation tends to flatten the contact surface and that a larger normal force P produces a larger the radius of local contact curvature, we propose to approximate Cr(P) as 1.0 for P < P Y C R (P) = (6.13) 1.0 + K c (PPy) for P>P Y where K c is a constant determined by the sphere properties. For the sphere used in FEA in this chapter, we obtained K c = 2.69 x 10" 4 N~ l Therefore, when P < P Y C R (P) = 1.0 leads to Rp = R; and when P > P Y C R (P) > 1.0 leads to Rp > R. For the case of normal force P unloading after loading to a maximum normal force P max we assume that the relation between normal displacement and contact area still follows the parabola law. Since the plastic deformation is irreversible, there must exist a residual normal displacement a Tes after the complete unloading
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105 of normal force P. Considering that the unloading should be an elastic process, we propose the parabola law for the P unloading stage to be a are* = j^r, E ( 6 14 ) Let Qmax and (a e ) max be the normal displacement and the elastic contact radius corresponding to the maximum normal force P mai respectively. Substituting a max (a e )max, and P max into (6.14), we obtain ( e ), 2 'max (6.15) (C R ) P=Pmax R The residual normal displacement a res depends on the maximum normal force P mai Remark 6.2. Apart from the additive decomposition of the contact area, the use of (6.12) to correct the radius of the local contact curvature is another crucial component for our successful FD models. This portion also plays an important role in the TFD model in accounting for the effect of the plastic deformation. I Please note that the present NFD model is developed to apply to elastoplastic contact in combination with the TFD model presented in Chapter 8 (also see VuQuoc, Lesburg and Zhang [1998] and VuQuoc and Zhang [1998c]). The results from our 3D FEA show that the presence of a tangential force does not significantly affect either the P Â— a relationship or the plastic flow inside the sphere. In other words, the plastic deformation is mainly caused by the the normal contact force P. Consequently, the present NFD model does not need to be adjusted when a tangential force is present. The two important parameters in our model are C a and K c which are functions of the other parameters such as oy, E, u, R, etc. Similar to the idealization of elastoplastic behavior illustrated in Figure 6.11 (Lubliner [1990]), the NFD model
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106 (a) a e curve: Uniaxial tensile/compressing experiment on mild steel. (b) a e curve: Elastoplasticity models of continuum theory. Figure 6.11. Idealization of elastoplasticity: Model parameters. The yield stress o Y and the hardening coefficient K are extracted from experimental data. presented here is general, and the model parameters of course depend on the material and the geometry of the particles. To illustrate the workability of our model, the values of the model parameters that appear in this paper correspond to a specific material and geometry. One should not use these values for other examples without verifying their validity. This situation is similar to the case where one should not use a set of plasticity model parameters for a given material (e.g., mild steel) to model the behavior of another material (e.g, powder), using the same plasticity model. While the plasticity model employed is in itself general, the values of the model parameters have to be changed depending on the material under study. In exactly the same manner, the NFD model proposed in this paper is in itself general, whereas the values of the model parameters have to be measured depending on the material and the geometry of the particles. Further theoretical work can be done to relate these model parameters to other basic parameters (e.g., a Y E, v, R, etc.).
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107 c l*A \ o r / ^^ i jS\Q r A 1 1 A \ < / I ^\_ \/ ^o \ ^ / Y v o Figure 6.12. An example of a polycrystal; Nickel based Inconel 718 alloy. From experimental observations, idealized plasticity models have been proposed, with model parameters (e.g., a Y K, etc.) to be measured from experiments. On the other hand, steel itself if looked closely is not homogeneous, but is a polycrystal (Figure 6.12, Lemaitre and Chaboche [1990]). Thus a more fundamental question is how to come up with homogenized plasticity model (with parameters evaluated) from singlecrystal plasticity. Here, in our work, we follow a similar philosophy: (i) Use numerical experiments to observe the behavior of certain critical quantities (e.g., the contact radius, normal stress distribution on contact surface, etc.) so that a model can be invented, (ii) propose simple experiments to measure the model parameters (Zhang and VuQuoc [19986]), (iii) indicate fundamental future work to link the model parameters to other model parameters of the material (e.g., cr y E, v) and of the geometry of the particle (e.g., R). 6.2.4. Normal Force Reloading The simplest contact loadingunloading cycle is the one in which during the contact, the normal force increases monotonically from zero, and then decreases monotonically till the end of the contact. A normal force reloading case is more complicated than that of a simplest contact loadingunloading cycle. In such a case,
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108 the status of the normal contact force P changes from decreasing to increasing. Such reloading cases are encountered in DEM simulation of granular flows, especially for frictional (or dense) flows. When the maximum normal force is less than or equal to the incipient yield normal force P Y the loadingunloading of the contact is an elastic process, and its forcedisplacement (FD) relationship follows Hertz theory. When the maximum normal force P max exceeded the yield force P Y plastic deformation occurs, and the loadingunloading process is more complicated because of the effect of plastic deformation. Figure 6.13 shows the NFD relation of such a case. Figure 6.13. Normal force reloading when P max > P Y In Figure 6.13, the normal force P increases from state of P = to state O of Pmax = Pi at first, then P decreases from state O to state of P = P 2 > before it starts to reload. Since that P max = P x > P Y we have a nonzero residual normal displacement a res = a 4 caused by plastic deformation that can be determined by
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109 (6.15). That is, if we continue to decrease the normal displacement a from a 2 to a4, the normal force P will be decreased from P 2 to as indicated by state O Our problem is to determine the P Â— a relationship for the case where the normal force P reloads, i.e., P increases from P 2 at state and for the case of the P unloading after such reloading. As stated in Section ??, the unloading of the normal force P from state O to is an elastic process, with a nonzero residual normal displacement and with a modified effective contact curvature (recall (??)) to account for the effect of plastic deformation. The reloading NFD relationship should follow the same rule (i.e., parabola law) for the initial normal loading, and should account for the effect of previous loading and unloading history. Therefore, from state to state we let (a e + a?) 2 2{a a Â™ ) = k(r)if(6 16) In (6.16) the radius Cr(P)R* of effective contact curvature is the same as the one for initial loading to avoid increasing the complexity in the implementation and in the computation. For unloading after a reloading stage, as from state to shown in Figure 6.13, the NFD relationship follows the parabola law 2(q a res a' res ) = {a) (6.17) K^R)p=PU ax H where P^ ai = P 3 is the secondary maximum normal force, the secondary residual normal displacement a' re3 can be determined similar to (6.15) as R)P=P> nax ti where a' max = a 3 is the maximum normal displacement of reloading, and the corresponding elastic contact radius (a e ') max is determined by Hertz theory. In (6.17), a a res a' res is the effective normal displacement. Therefore, the effect of previous plastic deformation is accounted for by accumulating the residual displacements.
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110 6.2.5. Contact Between Two Spheres of Different Materials In VuQuoc, Zhang and Lesburg [1998], we derive in detail the following expression for the yield normal force P Y i.e., the value of P at which an incipient yield occurs inside the sphere, p y = fcp L Mf 4 Â• (6.19) where the coefficient A Y (u) is a function of the Poisson's ratio v of the sphere material, a Y the yield stress. Therefore, when the contacting spheres are of different materials (with different Poisson's ratio, or yield stress), the two spheres will yield at different magnitudes of the normal force. Since the value of P Y is important to judge whether there is plastic deformation, in our DEM implementation of the elastoplastic FD models, the following method is employed to compute the incipient yield forces for spheres of different materials in contact. Consider the case of that when sphere (i) is in contact with sphere (j), we have ir 3 (R*) 2 (l{i) v 2 ) 2 4 v (1) Y = 6& MW)* Wy (6.20) and mPY = tm^n M{nv?vya >. (6 21) And then the smaller value of W P Y and {j) P Y is set to be the incipient yield contact force for the two spheres in contact. When the contact force is larger than the incipient yield force, i.e., P > P Y = min{ {i) P Y {j) P Y ), the contact between the two spheres is treated as a contact with plastic deformation. In addition, when two spheres of different materials are in contact with each other, the initial tangential stiffness K^ Q is computed by using the approximation in (??) and (??) for the equivalent elastic moduli for elastoplastic contact.
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Ill 6.3. Algorithm: DisplacementDriven From Section 6.2, to apply the NFD model discussed above in a forcedriven manner is straight forward. In general the DEM simulation of granular flow is, however, based on displacementdriven version. Assume that at time t n i, the normal force P n \ and the normal displacement a n i are known. With the geometric and mechanical properties of the the moving spheres known, the normal displacement a n at time t n Â— fÂ„_x + At can be computed by integrating the equation of motion of the moving spheres. The goal is to compute the normal force P n at time t n resulting from the displacement a n For simplicity, we only present the formulae for the normal contact between two identical spheres in this section. The algorithm for two different spheres in contact is similar to the one presented here. Based on the stress analysis in Hertz theory and on the von Mises yield criterion the incipient yield normal force Py can be computed using (6.19). Consequently, at incipient yield, the contact area radius ay and the normal displacement ay can be computed using (3.9) and (3.10), i.e., ar __ (sa^aj* and ay = ^f (6.23) During loading, as long as a n is less than or equal to the incipient yield displacement ay, the NFD relationship follows Hertz theory. The normal contact force P n can be computed using (3.10), i.e., p w^) al <6 24) When a n > a Y from (6.11), we have ( a ep ) 2 a = m (6 25)
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112 Let a e n p be decomposed according to (6.2), and C R (P n ) as given in (6.13); we obtain [1.0 + K c (P n P Y )} Ra n = ( < + alf (6.26) where a e n and aP n can be expressed in terms of P n using (3.9) and (6.3). We thus have {[1.0+ c (ftiV)] Jta.}'" = (5^!))' /3 p./3 + C^Py) V iE I (6.27) With the definition 1 {3R(lu 2 )\ 1/s relation (6.27) can be rewritten as HP n ) = (PÂ„iV) + c^ 1 / 3 i{[l.o + ^^p^)]^} 1 / 2 = 0. (6.29) The nonlinear equation (6.29) can be solved for the normal force P n using the NewtonRaphson method as follows pi Â— pil F\"n ) > " p(pii ) (O.30J where the P l n designate the th iterative value of P n and the derivative of function HPir 1 ) is jTUprU = 1 C L (P^) J_ KcR V2 a l/2 3 U) "2C Q[1 + ^ ( p rl Pr)]1 / 2 (63i) The initial guess for the NewtonRaphson procedure in (6.30) and (6.31) can be obtained by extrapolating from previously computed solution: P Â— D i *l Â— Pn2 i s , r n P n i + {a n a n _! (6.32 After having computed the normal force PÂ„, the elastic contact area radius a e n can be determined by (3.9), and the plastic contact area radius < can by (6.3).
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113 A negative increment of the normal force, i.e., AP n = P n Â— PÂ„_i < 0, indicates that the normal force is unloading after it reaches the maximal normal force P ma x', the normal displacement at that turning point is then recorded as a max Let a e max and a p max be the corresponding elastic and plastic contact area radii, respectively, at the turning point. The forcedisplacement relationship for unloading is nonlinear elastic with the modified radius of relative contact curvature R* p = C R {P max )R now held fixed at the last value reached at P = P max The residual normal displacement a res therefore, can be determined using (a e ) 2 Â„ Â„ <> max / (a oq\ Ores = a max (0.33) During unloading, the plastic contact area radius remains constant, i.e., (634) The elastic contact area radius a e n is determined by Hertz theory for an equivalent contact with relative contact curvati a res Hence, from (3.10), we obtain contact with relative contact curvature Â— rrr and normal displacement a n ^R\Pmax)tl < = [C R (P max )R(a n a res )] l/2 (6.35) and p = (astrb)) ( )S (636) The detailed pseudocode for the implementation of the proposed elastoplastic NFD model is presented in Algorithm 6.1.
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114 Algorithm 6.1. Elastoplastic NFD model: Displacementdriven version. ; Data : R,E,v,ay. 2 Calculate Py via (6.19). 3 Calculate ay via (6.23). 5 Input : displacements {aÂ„_ 2 aÂ„_i, a n },and forces {P n 2, ^ni}6 Goal : compute P n a** a e n a v n 8 Calculate AqÂ„ = a n Â— aÂ„_i. io if AaÂ„ = n Update P n = PÂ„_i, < p = <%_ x < = <_j a p n = o^_i 13 elseif Aa n > (Loading) 14 set P inc = true. is if a n < ay (elastic) 16 Calculate P n via (6.24). (P n < Py) n Calculate a e n via (3.9). is < = by (6.3). 19 < = < by (11). 20 (Delseif a n > ay (elasto Â— plastic) 21 Find P n via Algorithm 6.2. 22 Calculate a e n via (3.9). 23 Calculate a p via (6.3). 24 Calculate a* p via (6.2). 25 endif 26 elseif AoÂ„ < (Unloading) 27 if P inc = true *o Set *max Â— nÂ— 1* 29 SeZ Ct m ax Â— Â™nÂ—
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115 30 set a max Â— Q> n \ 31 Calculate a res via (6.33) 32 set P inc = false. 33 (Dendif 3.{ set aP n = aP max as in (6.34). 35 Calculate aÂ„ via (6.35). 36 Calculate P n via (6.36). 31 Calculate off via (6.2). 38 endif Algorithm 6.2. Solving (6.29) for P n : NewtonRaphson method. ; Data : Tolerance Tol. 2 Compute P via (6.32). 3 while riPft > Tol. 4 set i = i + 1. 5 Calculate P^ via (6.30). 6 endwhile 7 set P n = Pi 6.4. FEA Validation and Comparison to Thornton [1997] In order to compare the proposed model with FEA results and other models, we performed nonlinear elastoplastic FEA for the problem of a sphere being pressed against a frictionless rigid surface as shown in Figure 6.1(a), applied with the loading paths ABC, ADE, and AFG that shown in Figure 6.2. The geometric and material properties are the same as listed in Section 6.1. In FEA, the NFD relationship is produced in a forcedriven manner, i.e., the normal force path is the
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116 input and the normal displacement path is the output of the computation. The normal contact displacement cxfe obtained from the FEA is then used as the input to calculate the corresponding normal force P by the MATLAB code implemented with displacementdriven versions of both the proposed elastoplastic NFD model and the Thornton [1997] NFD model. The forcedisplacement (FD) curves as well as the related coefficients of restitution generated by our elastoplastic NFD model are compared with FEA results and with the results generated using the Thornton [1997] NFD model in Figures 6.14, 6.15, and 6.16. In these Figures, the NFD curves from Hertz theory are also presented for reference. 1500 0.4 0.6 0.8 1 1.2 Normal Displacement a (m) x10 Figure 6.14. Normal force P versus normal displacement a by different models from the FEA displacement path for the loading path AFG in Figure 6.2.
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117 Figure 6.14 shows various P versus a curves produced by FEA, the proposed elastoplastic NFD model, and the Thornton [1997] NFD model. The coefficient of restitution from the results of the proposed elastoplastic NFD model is e pm = 0.7541, while the coefficient of restitution from FEA results is e FE = 0.7372; the difference is only 2.3 %. It can be seen that the P a curve produced by the proposed elastoplastic NFD model agrees with the P a curve produced by FEA. The maximum normal force {P ma x) P m from the proposed NFD model is 1495 N; the difference is only 0.3 %. The P a curve produced by the Thornton [1997] NFD model is, however, much too soft, i.e., one obtain a much smaller maximum contact force (P ma x)Th for the same displacement level, compare to FEA results. At the maximum normal displacement a max ~ 1.56 x 10~ 5 m, the normal force by the Thornton [1997] NFD model, {P max ) T h = 770.5 N is about half of the corresponding FEA force level, {P ma .x)FE = 1500 N. The corresponding coefficient of restitution from the Thornton [1997] NFD model is e Th = 0.5641, suggesting a much larger energy dissipation (i.e., the area enclosed by loading and unloading curve and the x axis) ratio, resulting in a difference with the FEA coefficient of restitution about 23.5 %. Similar to Figure 6.14 is Figure 6.15 that shows the P a curves for the FEA displacement, which is produced in a forcedriven manner for the loading path ADE (P max = 1000 N) shown in Figure 6.2. The results from the proposed NFD model agree closely with FEA results. The corresponding coefficient of restitution from the proposed elastoplastic NFD model is e pm = 0.7994, while the coefficient of restitution from FEA results is e FE = 0.7757; the difference between them is small (3.1 %). The maximum normal force (P max )pm from the proposed NFD model is 962.1 N, differs from the FEA maximum force {P max ) FE = 1000 N by 3.8 %. The coefficient of restitution from the Thornton [1997] NFD model is e Th = 0.6091, which differs from e FE and e ep by about 22 %. The results from the Thornton
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118 1000 Cl, 600 V o o 500 (ji  400 o I 1 1 1 71 + e FE = 0.7757, (P max ) FE = 1000 N / *Zf / // e pm = 0.7994, {P max ) pm = 962.1 N / **Â£/ e Th = 0.6091, (P max ) T h = 559.6 AT / ,+y / ^/ / 4/ / / *;*/ / Hertz (elastic) / y / + FEA (elastoplastic) present model /y / Thornton [1997] Ay />'' s*x / 4,y / sirs' / / y^ y/ y' *^ < i +^ i 0.2 0.4 0.6 0.8 Normal Displacement a (m) x10 1.2 5 Figure 6.15. Normal force P versus normal displacement a by different models from the FEA displacement path for the loading path ADE in Figure 6.2. [1997] NFD model display a much too soft behavior, as in the previous case, with much smaller maximum force and much larger energy dissipation ratio. Quantitatively, the maximal force (P ma x)Th obtained from the Thornton [1997] NFD model is {Pmax)Th = 559.6 AT at the maximum displacement a max ~ 1.14 x 10~ 5 m and differs from the FEA force level by 44 %. Similar to Figure 6.14 and Figure 6.15 is Figure 6.16 that shows the P a curves for the FEA displacement, which is produced in a forcedriven manner for the loading path ABC (P max = 500 N) shown in Figure 6.2. Again, it can be seen that the results from the proposed elastoplastic NFD model agree closely with FEA results. The corresponding coefficient of restitution from the proposed elastoplastic
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119 500 450 400 ^.350 g, a, 300 v o r o250 a 200 u o Z 150 e FÂ£; = 0.8407, (F mai ) FE = 500 N e pm = 0.8620, (P mai ) pm = 468.1 N e Th = 0.6929, {P max ) T h = 325.2 AT Hertz (elastic) + FEA (elastoplastic) present model Thornton [1997] 2 3 4 5 Normal Displacement a (m) x10 7 6 Figure 6.16. Normal force P versus normal displacement a by different models from the FEA displacement path for the loading path ABC in Figure 6.2. NFD model is e pm = 0.8620, which agrees well to the coefficient of restitution from FEA results, e FE = 0.8407 with a difference of 2.5 %. The maximum normal force {Pmax)pm from the proposed NFD model is 468.1 N, differs from the FEA maximum force (P ma x)FE = 500 N by 6.4 %. Similar to the previous two cases, the Thornton [1997] NFD model yield results that are much too soft, with the maximum force (PmaxWh = 325.2 N, which is much smaller than the corresponding maximum force level of FEA with the maximum displacement a max ~ 0.68 x 10 5 m; the difference is 34.9 %. Similarly, the energy dissipation ratio in the Thornton [1997] NFD model is also much larger than that in FEA results. The coefficient of restitution from the
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120 Thornton [1997] NFD model is e Th = 0.6929, which is about 17.8 % smaller than e FE In summary, the proposed elastoplastic NFD model produces not only accurate P a relationship, but also correct coefficient of restitution and energy dissipation compared with FEA results. The Thornton [1997] NFD model produces much softer P a relationship, smaller coefficient of restitution, and larger energy dissipation ratio, for the same maximum normal displacement level. 6.5. Simulation of Single Sphere Collisions Using the proposed elastoplastic NFD model, we carry out the simulation of a sphere in normal collision against a frictionless rigid planar surface with various magnitudes of the incoming velocity; see Figure 6.17, in which v in designates the incoming velocity, and v out the outgoing velocity. Due to symmetry, this collision problem is equivalent to two identical spheres colliding with each other with the relative incoming velocity of 2v in and the relative outgoing velocity 2v out Also, in our simulation, the properties of the spheres are: radius fi = 0.1 m, Young's modulus E 7.0 x 10 10 N/m 2 Poisson's ratio v = 0.3, and yield stress o Y = 1.0 x 10 8 N/m 2 The normal force at incipient yield given by (6.19) is P Y = 36.4 N. The mass m of the sphere is determined by m = p ^ 7T R 3 (6.37) where p is the density of the sphere material. For our simulation, we choose the density to be the same as that of the aluminum alloy, i.e., p = 2.699 x 10 3 kg/m 3 the mass of the sphere is computed to be m = 11.306 kg. We also simulate the above sphere collision problem using the Thornton [1997] NFD model. The results produced by both the proposed elastoplastic NFD model
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121 Figure 6.17. A sphere colliding with a frictionless rigid planar surface. and the Thornton [1997] NFD model Â— such as the outgoing velocities, coefficient of restitution, contact force histories, etc. Â— are presented and compared. 6.5.1. Simulation Algorithm: LeapFrog Scheme Since the tangential force and the rotation of the sphere are absent, the algorithm is much simpler than when these two quantities are present (see VuQuoc et al. [1997]). Let x to be the distance from the sphere center to the rigid surface. Initially, at time t = 0, set the initial position of the sphere center to x and the velocity at t to v in i.e., v\ = v in A typical time step is as follows: Assume that at time t n _i, the position x n _i and the velocity v n _i are known. The velocity is evaluated at half time steps in the leapfrog algorithm. At time t n = t n _i + At, the position x n of the sphere can be calculated by Â• / x n \ +v n _i At (6.38) where At is the integration timestep size. For the simulation results shown in Figures 7.9, 6.18, 6.19, and 6.20, the timestep size is set to At = 1.0 x 10~ 6 sec.
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122 If x n < R, the sphere is in contact with the rigid surface. In this case, the normal contact displacement a n can be evaluated by a n = Rx n (6.39) If Aa n = a n Â— a n _i > 0, the collision is in the compression stage, i.e., P increases; else if Aa n = a n Â— aÂ„_i < 0, the collision is in the rebounding stage, i.e., P decreases. The normal contact force P n at time t n can be computed using Algorithm 3.1. Then the acceleration v n of the sphere at time t n can be determined by Newton's second law as v n = Â— (6.40) m where v n is the acceleration of the mass center of the sphere, and the negative sign indicates that the normal contact force is in the opposite direction to the incoming velocity. Since the acceleration of gravity is not considered in the simulation, the sphere moves at constant velocity when there is no contact, with the plane remains fixed. Therefore, the velocity of the sphere at time t n+ i can be expressed as v n+ i = v n _+v n At. (6.41) Attention should be paid for the stage when sphere is separating from the rigid surface. Since the normal contact force should always be positive or zero, when P is decreasing and reaches zero, the sphere is considered separated from the surface. Consequently, P is then set to be zero thereafter, even though the normal displacement may not be zero (there maybe some residual normal displacement due to plastic deformation as shown in Figures 6.14 6.16). The coefficient of restitution for such a collision can be defined as v out e := f* (6.42)
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123 or equivalently based on the energy dissipation as follows 1/2 (Area under unloading curve \ Area under loading curve J loading Â£ iÂ€{ataj_i>0} 1 \ (a i a i 1 )(P i + P i+1 ) 1/2 unloading i Â£ 5 \iÂ€{a^c^_i<0} (o i Qt j 1 )(Pj + P i+ l) (6.43) In our simulation, both (E.3) and (6.43) yield the same numerical result for the coefficient of restitution. When we use the Thornton [1997] NFD model in our simulation, the coefficient of restitution computed using (6.43) agrees with expression (4.22). 6.5.2. Simulation Results and Comparison with Thornton [1997] The simulation of a sphere in normal collision with the rigid planar surface is carried out with various magnitudes of the incoming velocity. With the convergent criterion for the NewtonRaphson method (see Algorithm 3.2) set to pi piÂ— I n Â„. n < 0.005 Tit Â— (6.44) and with the timestep size for numerical integration set to At = 1.0 x 10~ 6 sec, the maximal number of NewtonRaphson iterations in our simulation was two. The simulation results are presented and discussed below. Figure 6.18 shows the normal force P versus the contact time and the NFD relationship according to two models for the incoming velocity v in = 0.02 m/s. Not only that there are differences in the values of the outgoing velocity v^t and coefficient of restitution e as obtained from the present NFD model and the Thornton [1997] model, there are also large differences in the values of the maximum normal force P max and the maximum normal displacement a max as obtained from the two
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124 1000 800Time History of P 60013 400O 200 0.6 0.8 1 Time t (sec) x10 700 600 P versus a fe; 500 g 400 300 o 200 100I I 1 Present Model e= 0.8382 Pmax = 635.77V Thornton [1 997] X Â€ 0.6308 ... ; y// Pmax = mAN//V / / Â— Present Model Thornton [1997] / / / / \*~^ 1 Â— ^^Â— 1 _jÂ£ s I 0.2 0.4 0.6 0.8 Normal Displacement a (m) x10~ Figure 6.18. Simulation of a sphere in collision with a frictionless planar surface for incoming velocity v in = 0.02 m/s. Top: Time history of P. Bottom: P versus a.
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125 mentioned NFD models. Using the present NFD model, we obtain (P ma x)ep = 635.7 N, which much larger than the value {P ma x)Th = 478.4 N obtained by using the Thornton [1997] NFD model. The values of the collision time are also different with r ep ~ 1.17 x 10~ 3 sec for the present model and r T ^ ~ 1.29 x 10 3 sec for the Thornton [1997] model. In Remark 6.3, we provide an explanation on the difference in the collision time. Parallel to Figure 6.18 is Figure 6.19, which shows the contact force P versus contact time and the P versus a relationship for the sphere collision problem with incoming velocity v in Â— 0.04 m/s. Similar to the case with v in = 0.02 m/s, there are large differences in the results obtained from the proposed NFD model and the Thornton [1997] NFD model. The maximum normal force obtained from the proposed elastoplastic NFD model, (P ma x)ep = 1435 N, is much larger than that obtained from the Thornton [1997] NFD model, (P ma x)Th = 957 N. It is noted that this displacementdriven simulation produces a value of (P ma x)ep that is close to the maximum applied normal force P max = 1500 N in the forcedriven FEA using the loading path AFG in Figure 6.2. It follows that the P Â— a relationship produced by the proposed NFD model, as shown at the bottom part of Figure 6.19, is close to the P Â— a relationship by the proposed NFD model shown in Figure 6.14; the same can be said for the coefficient of restitution. The coefficient of restitution er/i by the Thornton [1997] NFD model in the current simulation is e T h = 0.5331, suggesting a larger energy dissipation by the collision than that from the proposed NFD model for the same incoming velocity. We notice that the similarity between the FD relationships and coefficients of restitution depends only on the maximum value of the normal force, and not the rate of loading and unloading. The reason is that both the proposed elastoplastic NFD model and the Thornton [1997] NFD model are based on time independent plasticity. Again, it should also be noticed that the collision times from the two models are different: r ep ~ 1.00 x 10~ 3 sec
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126 2500 2000 o Time History of P timestep size: At == 1.0 x 10~ 6 sec Present Model v in = 0.04 m/s
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127 from the proposed NFD model, and r Th ~ 1.20 x 10 3 sec from the Thornton [1997] NFD model. Figure 6.20 is similar to Figures 6.18 and 6.19, but for the incoming velocity of v in = 0.10 m/s, with the same observations made on the previous figures. The maximal normal force produced by our elastoplastic NFD model, (P max )ep = 4404 N, is much larger than the maximal normal force produced by the Thornton [1997] NFD model, {P ma x)Th = 2393 N. The collision time from the proposed elastoplastic NFD model elastoplastic NFD model, r ep ~ 0.76 x 10~ 3 sec, is much shorter than the collision time from the Thornton [1997] NFD model, r Th ~ 1.10 x 10~ 3 sec. Remark 6.3. In the above simulation results, the P Â— a relationship obtained from the Thornton [1997] NFD model is much softer than that from the proposed NFD model 3 Let K T h and K ep be the tangential stiffness in the P a relationship, for the Thornton [1997] NFD model and for the proposed NFD model, respectively. We have K Th < K ep The collision time from the Thornton [1997] NFD model is larger than that from the proposed model, i.e., r T h > T ep Assume that we are working with linear models to simplify the discussion. Based on the impulse principle, we have 2mv in = P Th T Th = P ep r ep (6.45) where P Th is the average normal force from the Thornton [1997] NFD model, and P ep the average normal force from the proposed NFD model. Since r Th < r ep we have P Th < P ep This simple argument explains the reason for obtaining (P max ) T h < (Pmax)e P as shown above. A similar situation occurs when we use the same normal force model, but different tangential force models as in VuQuoc and Zhang [1998a]. The difference in the flow velocity is not as pronounced as the difference in the force statistics. I 3 See Figures 6.14, 6.15, 6.16, 6.18, 6.19, and 6.20.
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128 7000 6000 5000 4000 Time History of P timestep size: At = 1.0 x 10 6 sec Present Model Thornton [1997] Vm = 0,10 m/s .,.; %Â„. = 0...10 m./.S; Vout = 0.0617 m/sv out = 0.0426! m/s Present Model Thornton [1997] 0.4 0.6 0.8 Time t (sec) x10 4500 4000O3500 ^3000 o 2500 SH ^ 2000 B 1500u Â•z 1000P versus a Present Model Thornton [1997] e = 0.6143 Â€ = 0.4252 max = 4404JV P max = 2393 Present Model Thornton [1997] 2 3 4 Normal Displacement a (m) x10 Figure 6.20. Simulation of a sphere in collision with a frictionless planar surface for incoming velocity v in = 0.10 m/s. Top: Time history of P. Bottom: P versus a.
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129 Figure 7.9 shows the simulation results in the velocities and coefficient of restitution produced from the proposed elastoplastic NFD model and from the Thornton [1997] NFD model. The incoming velocity v in ranges from 0.02 m/s to 0.20 m/s. The top part of Figure 7.9 shows the outgoing velocity zw versus the incoming velocity v in For a given incoming velocity, the outgoing velocity from the Thornton [1997] NFD model is less than the corresponding outgoing velocity from our elastoplastic NFD model. The bottom part of Figure 7.9 shows the coefficient of restitution e versus the velocity ratio v in /v Y With the incoming velocity for incipient yield determined by (4.23) to be v Y = 1.67 x 10 _3 m/sec 4 the velocity ratio v in /v Y thus ranges from 12 to 120. From Figure 7.9, the coefficient of restitution from the proposed elastoplastic NFD model is larger than that from the Thornton [1997] NFD model for a given incoming velocity. In other words, the kinetic energy dissipation caused by plastic deformation produced by the proposed elastoplastic NFD model is less than that corresponding value produced by the Thornton [1997] NFD model. Considering that (i) both models are quasistatic models, (ii) quasistatic FEA results agree with the proposed elastoplastic NFD model, while the Thornton [1997] NFD model produces softer Pa relations in all tests, the proposed NFD model is the more accurate and reliable of the two. Remark 6.4. It is not guaranteed that a NFD model that produces acceptable coefficient of restitution e can produce accurate NFD relationship. Since the coefficient of restitution e is usually obtained from the square root of the ratio of releasing energy to the storing energy, i.e., the area under the unloading curve to the area under the loading curve on the figure of the NFD curves. For example, in Figure 6.22, two different NFD relationships are depicted, with the solid line repRecall that we are dealing here with the case of a sphere colliding against a rigid surface. For the collision of two identical spheres, the relative incipientyield incoming velocity is doubled to 2v Y ~ 3.34 x l(r 3 m/s.
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130 V ut VS. V in o Present Model + Thornton [1997] 0.1 (m/s) 0.2 0.9 u 0.8 o 0.7 55 0.6s a; g0.5 o OQ.4 0.3 e VS L Vin/vy i r I i O Present Model Thornton [1997] s 1.67 x 10~ 3 m/s + Vy k \ "~. vh. i i i 20 40 60 Vin/v Y 80 100 120 Figure 6.21. Simulation of a sphere in collision with a frictionless planar surface. Top: outgoing velocity v^t versus incoming velocity v in Bottom: coefficient of restitution e versus impact velocity ratio Vin/vy, where vy = 1.67 x 10 3 m/sec determined by (4.23).
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131 p { II A / D / 1 + * / 1 ^* / / I / 1 Â„ / / / +** / / '"' / / / '[ / S J / s ** / / /*' / / ^s / / / 4r j/ / B EC F a Figure 6.22. Sharply different NFD models may yield similar coefficients of restitution. resenting either the presented NFD model or the Thornton [1997] NFD model, and the dotted line representing the Walton and Braun [1986] NFD model. It is possible that even though the two NFD relationships are completely different, the resulting coefficients of restitution may be very close, i.e., area of BAC ^ area of EDF area of OAC ~ axea. of ODF' ^ 6 46 ^ Hence, it is not sufficient to use the coefficient of restitution to compare the accuracy of NFD models. A combination of both NFD relationship and coefficient of restitution may be necessary (see, e.g., Zhang and VuQuoc [19986]). I
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CHAPTER 7 DYNAMIC FEA OF ELASTOPLASTIC SPHERE COLLISIONS 7.1. Finite Element Model for Dynamic Analysis Figure 6.17 shows a sphere colliding against a frictionless flat plate, a situation equivalent to two identical spheres with the same velocity amplitude colliding against each other. In our nonlinear dynamic FEA, the size and the material properties of the sphere are chosen to be: Radius R = 0.1 m, Young's modulus E 7.0 x 10 10 N/m 2 Poisson's ratio v = 0.3, and density p = 2.699 x 10 3 kg/m 3 For elastoplastic collisions, elastoperfectlyplastic model with von Mises yield criterion is employed. The yield stress of the material is chosen to be a Y = 1.0 x 10 8 N/m 2 Since that there is no rotation of the particle about itself in the collision that we are studying, axisymmetric FE models are employed to carry out the analyses. All axisymmetric elements used are CAX6 elements of the nonlinear FE code ABAQUS [1995]. Figure 7.1 shows one of the meshes employed in our FEA. In this FE model, the half sphere is discretized into 1640 axisymmetric sixnode triangular elements (Figure 7.1(a)) with a total of 3288 nodes, and with three levels of mesh refinement around the contact area (Figure 7.1(b)). We designate this FE model as model B. The other two FE models, model A and model C, employed in our analyses are similar to the one shown in Figure 7.1, but with different number of levels of mesh refinement and different numbers of elements. Model A has 928 axisymmetric sixnode triangular elements and 1886 nodes, with two levels of mesh refinement around the contact area. Model C 2951 axisymmetric sixnode triangular elements and 5892 nodes, with four levels of mesh refinement around the contact area. When not specified, the FEA results presented later are obtained using model B (Figure 7.1). 132
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133 (a) Sphere discretization. (b) Zoomedin view around the contact area. Figure 7.1. Axisymmetric finite element model of the sphere.
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134 Figure 7.2. Incompatible elements connected by multipoint constraints (MPC's). For low velocity impacts, the deformation of the sphere during a collision is concentrated in a small region around the contact area (see later results, such as the contours of Mises stress presented in Sections 7.2 and 7.3). To accurately represent the overall response, we refine the FE mesh in the small region close to the contact point (Figure 7.1(b)). The mesh refinement is achieved by the use of incompatible interelement matching at the boundary of the different zones of refined mesh, as illustrated in Figure 7.2. In our FE models, these incompatible elements are connected to each other using multipoint constraints (MPC's). In Figure 7.2, the second order triangular elements O . and are connected using quadratic MPC's. Node m of element and node n of element do not have independent degrees of freedom (DOF); their displacements are determined by the quadratic functions of the displacements of nodes i, j, and k, which are common to elements , and The contact detection and contact analysis between the sphere and the rigid surface are carried out using the ID IRS21A contact elements of ABAQUS [1995]. For the FE model shown in Figure 7.1, the size of the contact elements around the
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135 contact area is about 1.02 x 10 4 m (half of the size of the triangular element), which is much less than the radius of the contact area. For example, the maximum radius of contact area is about 2.32 x 10~ 3 m for the elastoplastic collision with an incoming velocity v in = 0.10 m/s (see Section 7.3). Therefore, it can be concluded that the discretization of the sphere is fine enough to describe the collision behavior accurately. 7.2. FEA of Elastic Sphere Collisions Using the nonlinear FE code ABAQUS [1995], with the FE models described in Section 7.1, we carry out a series of dynamic FEA for elastic collisions between an elastic sphere and a frictionless rigid surface with different incoming velocities. As mentioned in Section 1, the behavior of such collisions can be solved theoretically using Hertz theory through a quasistatic procedure. In this section, we compare our dynamic FEA results of elastic collisions with the corresponding results obtained by applying Hertz theory through a quasistatic procedure. In addition, the error, which may be caused by the energy dissipation of elastic wave propagation inside the sphere, is discussed by comparing the results. 7.2.1. FEA Results An important result from our dynamic FEA of elastic collisions is the coefficient of restitution. Recall that the coefficient of restitution in a collision in the normal direction (Figure 6.17) is defined by (E.3). In our FEA, the incoming velocity Vi n of the sphere is an input parameter, whereas the outgoing velocity v out of the sphere is obtained by averaging the velocities of all nodes of the FE mesh at a time right after the sphere is separated from the rigid surface. The coefficient of restitution obtained from FEA is presented in Table 7.1. The results show that using either model A, model B, or model C, the coefficient of restitution obtained
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136 12000 100008000600040002000 1 ( 1.144 x : 4.95 x 1 1 "max Qmax = 10 4 o5 W Hertz + FEA Loading o FEA unloading
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137 Figure 7.3 shows the normal contact force versus normal displacement curve for the elastic collision with incoming velocity v in Â— 0.20 m/sec. The unloading path of the forcedisplacement (FD) curve from FEA results is almost on the top of the loading path of the FD curve, meaning that there is almost no energy dissipation. The loading curve produced using Hertz theory by (3.8) is also presented in Figure 7.3, which shows that the FEA results agree well with the Hertz theory for elastic contact. At the points with the highest normal contact force P max Â— 1.144 x 10 4 N, the maximum normal displacement obtained from FEA results is a max Â— 4.95 x 10~ 5 m; while the corresponding normal displacement produced using Hertz theory by (3.8) is {oL max ) H z = 4.99 x 10 5 m. The difference between the result from dynamic FEA and the result from Hertz theory is only 0.8 %. Table 7.2. Collision duration (in sec) vs. incoming velocity for elastic collisions Vi n (m/sec) Hertz Theory Model A Model B Model C 0.02 1.200 x 103 1.16 x 10~ 3 1.16 x 103 1.16 x 10~ 3 0.06 9.298 x 104 9.33 x 10~ 4 9.27 x 104 9.30 x 104 0.10 8.395 x 10~ 4 8.38 x 104 8.36 x 10" 4 8.38 x 10" 4 0.20 7.308 x 104 7.28 x 104 7.28 x 10~ 4 7.30 x 104 We also extracted the collision duration time r from our dynamic FEA by subtracting the time t c when the sphere comes into contact with the rigid surface, from the time t s when the sphere completely separates from the rigid surface, i.e., r = t s t c The maximum possible error committed on r is the integration timestep size, which is very small; for example, the timestep size around the separation of the sphere for the collision with incoming velocity v in = 0.10 m/sec using FE model B is 3.207 x 10~ 6 sec. The results from FEA using different models are presented in Table 7.2 and Figure 7.4, and are compared with the theoretical prediction using Hertz theory, i.e., (3.22). Compared to the timestep size, the error of the computed collision duration time for the collision with incoming velocity v in = 0.10 m/sec
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138 Vin {ml sec) Figure 7.4. Collision duration vs. incoming velocity for elastic collisions. is about 0.38%, or less than 1%. Again, our FEA results agree closely with the theoretical prediction using Hertz theory with a quasistatic procedure. Figures 7.5(a) and 7.5(b) show the distributions of the normal pressure on the contact surface obtained from our dynamic FEA results. In these figures, the Hertz normal pressure by (3.3) is represented by the solid line; the original FEA results are represented by the small circles 'o\ To remove the spurious oscillations in the original FEA results, we perform an averaging process to produce a much smoother curve, shown by the symbols 'x'. The oscillations in the original FEA results is due probably to the type of contact elements employed. The contact radius a from FEA results is obtained by using cubic spline interpolation at zero normal pressure p = 0. This cubic spline is based on the two averaged data points
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139 10' Hertz o ABAQUS Dynamic x ABAQUS Average a = 0.000877m a hz ss 0.000887zh 10' 1 1 1 Â— Hertz o ABAQUS Dynamic x ABAQUS Average a = 0.00170m a hz = 0.00169?jrc 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 X10 / \ xlC r (m) (a) Collision with vi n = 0.02 m/sec, at (b) Collision with Vi n = 0.10 m/sec, at maximum normal force P max = 716.6 N. maximum normal force P max = 4938 TV. Figure 7.5. Normal pressure profile over contact area. Original dynamic FEA data (o); averaged data (x). that are closest to the the r axis, and their two mirror symmetric points about the r axis. The distribution of the normal pressure p for the elastic collision with incoming velocity v in = 0.02 m/sec at the time when the normal contact force P reaches its maximum value P max = 716.6 N is shown in Figure 7.5(a). Even though the original FEA results are oscillating around the Hertz normal pressure, the averaged normal pressure from FEA results agrees closely with Hertz theory. In addition, the contact area radius a = 0.000877 m from FEA results agrees with that from Hertz theory, &hz Â— 0.000887 m, with a small difference of 1.1 %. In this case, from Figure 7.5(a), there are nine contact elements involved in the contact. Similarly, the distribution of the normal pressure p of the collision with incoming velocity v in Â— 0.10 m/sec at the time when the normal contact force P is at its maximum value PÂ„ 4938 N is presented in Figure 7.5(b). Again, we observe good agreements between FEA results and Hertz theory. In Figure 7.5(b), there are 17 contact elements involved in the contact.
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140 a Â• 2.72EMI8 h = 2.33E4<>8 c = 1.94E*<)K d= 1.55E+OK e=1.16E+<>8 f = 7.76E+7 h 3.84EI02 O.OOlm Mises Stress t=S.SOE~OS P99.1 m5.l7E+08 a*4.52Et<)8 h Â• 3.87E<08 t = 3.23E+OK (1 1 58] M IS c l.WE+08 b f=l.29E
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141 m Â• 6.78Erf>8 a 5.94E+OK b = 5.INEHI8 c=4.24E+< d 3..WEMW e = 2.54E+08 1= 1.7IIE+0K g 8.48E+08 h = 2.65Et<>3 0.001m M = 6.1KJE+0K a 5.33Et<>8 h = 4.57Et<)8 c = 3.81EM)8 8 g = 7.6IE*<>8 h 3.55E403 0.001m (a) Collision with t>;Â„ = 0.20 m/sec, loading (b) Collision with v in = 0.20 m/sec, unat t = 3.64 x 10 4 sec and P = 1.144 x loading at t = 5.06 x 10 4 sec and P = 10 4 N. 8279 TV. Figure 7.7. Contour of Mises stress inside the sphere during elastic collisions. 7.6(b) show that the highest Mises stress is not on the contact surface, but inside the sphere, at about half of the radius a of contact area above the contact surface. During a collision, high stress level are concentrated in a small region close to the contact surface. Figure 7.7(a) shows the contour of the Mises equivalent stress inside the sphere for the elastic collision with incoming velocity v in = 0.20 m/sec at time t = 3.64 x 10 4 sec when the normal contact force reaches its maximum value P = 1.144 x 10 4 TV. Figure 7.7(b) shows the contour of the Mises equivalent stress inside the sphere for the elastic collision with incoming velocity v in = 0.20 m/sec at time t = 5.06 x 10 4 sec when the normal contact force decreases to P = 8279 TV after it reaches its maximum value. From Figure 7.7(a) to Figure 7.7(b), even though the stress contours look similar, the value of the Mises equivalent stress on these contours decreases. The stress state inside the sphere changes gradually, i.e., without any
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142 dramatic changes, going from loading to unloading. This feature in elastic collisions is very different from that of elasto plastic collisions (see Section 7.3 below). 7.2.2. FEA Results of the Elastic Collisions of Soft Spheres To show the effect of the elastic modulus on the collision behavior, we also carry out dynamic FEA using the FE model B described in Section 7.1, with different Young's moduli for the sphere material. For the elastic collision with incoming velocity v in = 0.10 m/sec, the coefficient of restitution obtained from FEA by averaging the rebounding velocity of all nodes in the model is shown in Table 7.3. Table 7.3. Coefficient of restitution for the sphere with different Young's moduli in elastic collisions, v in Â— 0.10 m/sec. E (N/m 2 ) 7.0 x 10 10 7.0 x 10 6 7.0 x 10 3 e 0.9984 0.9890 0.7094 Figure 7.8 shows the distributions of the rebounding velocity of all the nodes in the FE model right after the sphere separates from the rigid surface. When the Young's modulus of the sphere material is E = 7.0 x 10 10 N/m 2 the one we use for most dynamic FEA in this paper, the coefficient of restitution of the elastic collision is almost equal to one, with a relatively uniform distribution of velocity, except at a small region close to the contact area, where there is some fluctuation in the magnitude of the rebounding velocity, due to wave propagation. When the Young's modulus is E = 7.0 x 10 6 N/m 2 which is ten thousandth of the one used in most of our FEA, the coefficient of restitution is a little smaller than the one with higher Young's modulus, but it is still very close to one. When the Young's modulus is set to a very small value E = 7.0 x 10 3 N/m 2 the effect of wave propagation is clearly shown in Figure 7.8(c), resulting in a coefficient of restitution much less than one, when the rebounding velocity is computed by averaging the instantaneous velocity of all nodes. In this case, because the sphere material is much softer than
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143 u u ag ."Â£ 0.08 u o > 0.06 bp ^0.04 3 9 _0.02 2 K Incoming velocity: Vin = 0.1 Avg. outgoing velocity: Vout = 0.09964 Coet. of rest, (from vel): e = 0.9984 Node Numbers 13 iNWWMuw ,"*5 0.08 o _o > 0.06 a ^0.04 3 3 ^0.02 K Incoming velocity: Vin = 0.1 Avg. outgoing velocity: Vout = 0.0989 Coel of rest, (from vet): e = 989 'tlnH1 Node Numbers (a) Sphere with E = 7.0 x 10 10 N/m 2 at (b) Sphere with E = 7.0 x 10 6 N/m 2 at time t = 9.24 x 10" 4 sec. time i = 3.27 x 10"^ sec O 0.08_o > 0.06 bO d 0.04 o S 0.02 O Q o & Incoming velocity: I Avg. outgoing velocity: Voul Coet. ol rest, (from vel): e 0.7094 1000 1500 2000 2500 3000 3500 Node Numbers (c) Sphere with E = 7.0 x 10 3 N/m 2 at time t = 5.29 x 10 _1 sec. Figure 7.8. Rebounding velocity in the sphere versus node numbers for elastic collision with incoming velocity v in = 0.10 m/sec.
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144 in the two previous cases, disturbance propagates at a slower speed and with higher amplitude. As a result the average velocity is much smaller in amplitude compared to the incoming velocity, and thus a sharp decrease in the coefficient of restitution. Clearly, from Table 7.3 and Figure 7.8, the computation of the coefficient of restitution as shown in (E.3), with v out being the average velocity of all nodes at rebounding, depends on the material properties such as Young's modulus E, mass density p, etc. In other words, when the ratio E/(pR) is smaller, the time time for the elastic wave propagating across the sphere is longer, thus the effect of elastic wave propagation on v^t is larger. For most of the elastic and elastoplastic collisions studied in this paper, the ratio E/(pR) is large, resulting in much higher elastic wave propagation speed. In the case where Young's modulus E Â— 7.0 x 10 10 N/m 2 mass density p Â— 2.699 x 10 3 kg/m 3 and sphere radius R = 0.1 m, the collision duration time r is hundreds times longer than the time for the elastic wave to propagate across the diameter of the sphere, thus make valid the use of quasistatic FD model at the contact point. 7.3. FEA of ElastoPlastic Sphere Collisions In this section, we present the dynamic FEA results for elastoplastic collisions between a sphere of elastoperfectlyplastic material and a frictionless rigid planar surface. In addition, we compare our FEA results with the results of DEM simulation using the VuQuoc and Zhang [19986] elastoplastic NFD model (discussed in Chapter 6) to show the correctness of the NFD model. Similar to the FEA of elastic collisions presented in Section 7.2, we carry out dynamic FEA of elastoplastic collisions between the sphere described in Section 7.1 and a frictionless rigid surface with various incoming velocities using the nonlinear FE code ABAQUS [1995]. Elastoperfectlyplastic material properties as described
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145 in Section 7.1 are used. The coefficient of restitution for collisions with different incoming velocities obtained from FEA results are listed in Table 7.4. The coefficient of restitution in Table 7.4 are all less than one, showing that there are energy dissipations caused by plastic deformation. The coefficient of restitution decreases with increasing incoming velocity, because the larger the incoming velocity, the higher the level of plastic deformation, and thus the higher the level of energy dissipation. Table 7.4. Coefficient of restitution for elastoplastic collisions Vin (m/sec) Model A Model B Model C 0.02 0.8076 0.8132 0.8132 0.06 0.6951 0.6858 0.6930 0.10 0.6352 0.6524 0.6229 0.20 0.5635 0.5853 0.5481 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Incoming velocity Vi n (m/sec) Figure 7.9. Coefficient of restitution from DEM simulations and FEA using different FEA models.
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146 Figure 7.9 shows coefficient of restitution versus the incoming velocity with results coming from (i) FEA, (ii) discrete element method (DEM) simulations using the VuQuoc and Zhang [19986] elastoplastic NFD model (denoted by VZ NFD model", see Chapter 6), and (iii) the Thornton [1997] NFD model. The results from FEA using FE models A, B, and C (see Section 7.1) are presented using the symbols '+', '*', and 'o', respectively. For collisions with incoming velocities v in = 0.02 m/sec, 0.06 m/sec, and 0.10 m/sec, the results from DEM simulation using the VuQuoc and Zhang [19986] NFD model agree well with the FEA results. For the collision with incoming velocity v in = 0.20 m/sec, which is about 120 times of the yield velocity 5 v Y and for which the maximum normal contact force reaches Pmax Â— 8373 N, the coefficient of restitution from FEA is e = 0.5853, while that from the VuQuoc and Zhang [19986] NFD model is e vz = 0.4773; the difference between the two results is 18.5 %. We will discuss the cause of this difference later. The coefficient of restitution from the Thornton [1997] NFD model given by (4.22) is also shown in Figure 7.9 for comparison. Note that in (4.22) and (4.23), the incoming velocity v in and yield velocity v Y are the relative incoming velocity of two spheres in collision, we need to double the incoming velocity when considering the collision between one sphere and a frictionless rigid surface which is equivalent to the collision of two identical spheres with doubled relative incoming velocity. In comparison with our dynamic FEA results, the coefficient of restitution from DEM simulation using the VuQuoc and Zhang [19986] NFD model is clearly superior to that obtained from the Thornton [1997] NFD model, for incoming velocity in the range shown in Figure 7.9, namely, when v in /v Y < 120. We refer the readers to VuQuoc and Zhang [19986] for more detailed comparison of the VuQuoc and Zhang [19986] NFD model and the Thornton [1997] NFD model. 5 Plastic deformation begin to develop inside the sphere for the incoming velocity v in = vy = 1.667 x 10~ 3 m/sec computed by (4.23).
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147 700 600 ^W\f BOO VZNFD Mod* FEA Loading o FEA unloading (X 300 200 i ...jey. i ............. Â•y y a (m) (a) Collision with Vi n Â— 0.02 m/sec. a (m) (b) Collision with V{ n Â— 0.06 m/sec. Figure 7.10. Normal contact force P vs. normal displacement a. Figure 7.10 shows the normal contact force P versus the normal displacement a for the collisions with incoming velocity v in = 0.02 m/sec (vm/vy Â— 12) and v in = 0.06 m/sec (vin/vy Â— 36). The results produced by the DEM simulations using the VuQuoc and Zhang [19986] elastoplastic NFD model agree closely with the corresponding FEA results. The maximum normal contact force P max the maximum normal displacement a max and the residual normal displacement a res caused by plastic deformation are listed in Table ??. Table 7.5. Maximum force P mQI maximum displacement a max and residual displacement a res vs. incoming velocity v in (m/sec) *max \P> ) Umax (m) a res (m) FEA VZNFD FEA VZ XFD FEA VZNFD 0.02 647.8 635.7 8.25 x 10~ 6 8.48 x l(r (1 2.78 x 10~ 6 2.22 x 10" 6 0.06 2224 2341 2.11 x 10" 5 2.12 x 10" 5 1.07 x 105 1.05 x 105 0.10 3922 4404 3.30 x 10" 5 3.21 x 10" 5 1.90 x 10~ 5 2.00 x 10" 5 0.20 8373 1.045 x 10 4 6.08 x 10~ 5 5.55 x 10~ 5 4.01 x ur 5 4.31 x 10~ 5 Similar to Figure 7.10, Figure 7.11 shows the normal contact force P versus the normal displacement a for the collisions with incoming velocities v in = 0.10 m/sec
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148 1OOO0 i 1 i 1 1 i 1 jr o 1/ : Ml VZ NFD Modtf + FEA Loading o FEA unloading /' // a (m) (a) Collision with ViÂ„ = 0.10 m/sec. 5 6 7 a (m) (b) Collision with Vj n = 0.20 m/sec. Figure 7.11. Normal contact force P vs. normal displacement a. {vin/vy = 60) and v in = 0.20 m/sec (v in /v Y = 120). In Figure 7.11(a), with Vin/vy = 60, the NFD relation produced by the DEM simulations using the VuQuoc and Zhang [19986] elastoplastic NFD model agrees reasonably well with the corresponding FEA results. By (6.19), the incipientyield contact force is Py = 36.45 N for the sphere with properties given in Section 7.1. In this case, the maximum normal contact force is P max = 3922 N from FEA results (Table 7.5), and thus the ratio Pmax/Py = 107.6. In Figure 7.11(b), with v in /v Y = 120, there is a clear departure of the results using the VuQuoc and Zhang [19986] NFD model from the FEA results. In this case, the maximum normal contact force P max Â— 8373 AT is much larger than the yield normal force Py by a ratio of P max /Py = 229.7. Such large maximum normal force is the reason for the departure of the results using the VuQuoc and Zhang [19986] NFD model from FEA results, because the parameters C a and K c employed in the VuQuoc and Zhang [19986] NFD model were extracted for maximum normal force less than 1500 N (i.e., Pm ax /Py = 41.1). Clearly, the maximum normal force P max = 8373 A^ is way outside the range of validity of the parameters employed. In other words, to make the results using the VuQuoc and
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149 Zhang [19986] NFD model work up to P max = 8373 N, we need to extend the range of Pmax to this force level in the extraction of the model parameters C a and K c Table 7.6. Comparison of collision time r vs. incoming velocity u in v in (m/sec) Hertz (elastic) FEA (elastoplastic) VZ NFD (elastoplastic) 0.02 1.16 x 103 1.12 x 10~ 3 1.18 x 103 0.06 9.30 x 104 9.01 x 104 8.99 x 10~ 4 0.10 8.40 x 10~ 4 8.22 x 10~ 4 7.64 x 10" 4 0.20 7.31 x 10~ 4 7.31 x 10~ 4 5.97 x 104 10" Hertz (elastic) FEA (elastoplastic) x VZ NFD (elastoplastic) x^ 10" 10 Vin (m/sec) Figure 7.12. Collision duration time for elastoplastic collision vs. incoming velocity. The collision duration time r for the elastoplastic collision obtained from our FEA and from the DEM simulations using the VuQuoc and Zhang [19986] NFD model is shown in Figure 7.12 and in Table 7.6. The computation method is the same as explained in Section 7.2.1. Since the integration timestep size for the nonlinear dynamic FEA is automatically chosen by ABAQUS, and the timestep
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150 a = 0.00232m a hz = 0.00156m Hertz o ABAQUS Dynamic x ABAQUS Average (a) Collision with Vi n = 0.02 m/sec, at maximum normal force P max = 647.8 N. (b) Collision with V{ n = 0.10 m/sec, at maximum normal force P max Â— 3922 N. Figure 7.13. Normal pressure over contact area for elastoplastic collisions. size for elastoplastic analysis is even smaller compared to that for elastic analysis, the error for the data presented in Figure 7.12 and in Table 7.6 is much smaller than 1%. For example, the timestep size around the separation time of the FE model B from the rigid planar surface, for Vi n = 0.10 m/sec is 7.663 x 10 7 making the error on r of about 0.09%. For the results using DEM simulation, the timestep size is fixed at 1.0 x 10~ 6 sec, thus making the error on r about 0.13%. The collision duration time for elastic collision obtained using Hertz theory as given by (3.22) is also presented for comparison. Again, when v^jvy < 60, the results from DEM simulation using the VuQuoc and Zhang [19986] NFD model agree accurately with those from FEA. It is also observed that the collision duration time from FEA for elastoplastic collision is very close to the collision time for elastic collision from Hertz theory, meaning that the effect of plastic deformation on the collision duration time is very small, at least for the cases in which 12 < v in /v Y < 120. Such observation deserves further study.
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151 Figure 7.13 shows the distribution of the normal pressure on the contact surface obtained from our dynamic FEA for elastoplastic collisions. The Hertz normal pressure as given by (3.3) (for elastic contact) is also presented; the original FEA data are presented by the symbols 'o\ Similar to what presented in Figure 7.5, we average the original oscillating FEA data to obtain a much smoother curve shown in Figure 7.5 using the symbols 'x'. The contact area radius a of area from FEA results is obtained by using cubic spline interpolation in the same manner as for results shown in Figure 7.5. The distribution of the normal pressure p for the elastoplastic collision with incoming velocity v in = 0.02 m/sec at the time when the normal contact force P reaches its maximum value P max = 647.8 N is shown in Figure 7.13(a). The contact area radius a = 1.07 x 10 3 m from the FEA results for the elastoplastic collision is larger than the contact area radius a hz = 8.58 x 10~ 4 m from Hertz theory for the elastic collision under the same normal force level. Similarly, the distribution of the normal pressure p for the elastoplastic collision with incoming velocity v in = 0.10 m/sec at the time when the normal contact force P reaches its maximum value P max = 3922 N is presented in Figure 7.5(b). Again, we observe that the contact area radius from the FEA results for the elastoplastic collision is larger than that from Hertz theory for the elastic collision under the same normal force level. In both Figures 7.13(a) and 7.13(b), the magnitude of the normal pressure on the contact surface inside the contact area is roughly constant, and is more than twice that of the material yield stress cry. The normal pressure distribution on the contact surface from the dynamic FEA results agrees with that obtained from the static FEA results for elastoplastic contact problems presented in VuQuoc and Lesburg [1998] (Figure 6.7). We refer the readers to VuQuoc and Lesburg [1998] and VuQuoc, Zhang and Lesburg [1998] for more detailed discussion, on the static FEA results of elastoplastic contact problems.
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152 a = 9.99E+07 h = 7.WE+07 t 5.yyEM)7 iii iwim C= 1 IKE+07 L 0.001m (a) Collision with v in = 0.10 m/sec, loading at t = 1.15 x 10" 4 sec and P = 999.7 N. a = 9.99E+07 h = 7y9E+07 C = S.99E+07 U UMf e= I.00EH17 f2.54E+03 (b) Collision with vt n = 0.10 m/sec, loading at t = 4.78 x 10" 4 sec and P = 3920 N. Figure 7.14. Contour of Mises stress inside the sphere during elastoplastic collisions. Figure 7.14(a) shows the contour of the Mises equivalent stress inside the sphere for the elasto plastic collision with incoming velocity v in Â— 0.10 m/sec at time t = 1.15 x 10 4 sec when the normal contact force increases to P = 999.7 N. The sphere material is modeled by the elastoperfectlyplastic model with the von Mises yield criterion, according to which the material will yield when the Mises equivalent stress q (given by (7.1)) is equal to the yield stress o Y With o Y = 1.0 x 10 8 N/m 2 the area encircled by the contour line marked a in Figure 7.14(a) is the plastic zone. The same applies to Figures 7.14(b), 7.15(a), 7.15(b), and 7.16. Figure 7.14(a) shows that the plastic deformation is first developed not on the contact surface but at a point inside the sphere, and close to the contact area. When the normal contact force increases, the plastic zone expand to reach the contact surface, beginning from the edge of the circular contact area, while the material on the contact surface and around the center of the contact area remains elastic. Compare Figure 7.14(a) to Figure 7.6(a), we see that the maximum Mises stress inside a sphere with elastic
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153 u = y,yyE+07 ti = 7.99E+07 I = 5.99E407 <1 = .1.00E+07 e=1.00EtO7 f=3.l4Et03 0.001m 1 9.99E+07 b = 7.99E<>7 t = 5.99E*(r7 d = J.OOE107 e=1.00E07 f3.03E03 0.001m (a) Collision with u in = 0.20 m/sec, loading (b) Collision with v in = 0.20 m/sec, beginat t = 4.41 x 10 4 sec and P 8373 JV. ning unloading at t = 4.74 x 10 4 sec and P = 8347 N. Figure 7.15. Contour of Mises stress inside the sphere during elastoplastic collisions. perfectlyplastic material is limited to the material yield stress a Y which is much lower than the stress reached in an elastic collision. Figure 7.15(a) shows the contour of Mises equivalent stress inside the sphere for the elastoplastic collision with incoming velocity v in = 0.20 m/sec at time t = 4.41 x 10~ 4 sec when the normal contact force increases to P = 8373 N. When the normal loading is high, the plastic zone develops to the contact surface on most part inside the contact area. Figure 7.15(b) shows the contour of Mises equivalent stress inside the sphere for the elastoplastic collision with incoming velocity v in Â— 0.20 m/sec at time t = 4.74 x 10~ 4 sec when the normal contact force decreases to P = 8347 N, right after the normal contact force reaches its highest value, and when the sphere starts to separate from the rigid surface. At this time, even though the normal contact force P = 8347 iV is much larger than the incipient yield force P Y = 36.45 N, once the normal force starts unloading, the Mises equivalent stress inside the sphere immediately decreases to a level q < a Y everywhere. With
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154 a = 9.99E+07 b = 7.99E+07 c = 5.99E+07 d = 3.00E+07 e= 1.00E+07 f=6.14E+02 Mises Stress t=6.93E04 P=1017. 0.001m element refinement profile Figure 7.16. Contour of Mises stress inside the sphere during the elastoplastic collision with v in Â— 0.20 m/sec, unloading at t = 6.93 x 10" 4 sec and P = 1017 N. the plastic deformation frozen, the material behaves elastically during the normal contact force unloading session. Figure 7.16 shows the contour of Mises equivalent stress inside the sphere for the elastoplastic collision with incoming velocity v in = 0.20 m/sec at time t = 6.93 x 10 4 sec when the normal contact force decreases to P = 1017 N after this force reaches its highest value and after the sphere is separates from the rigid surface. Comparing Figure 7.16 (P = 1017 N, unloading) and Figure 7.14(a) (P Â— 999.7 N, loading), we see that the stress distribution during unloading with plastic deformation can be very different from the stress distribution during loading, even at the same normal force level. Unlike elastic collision, the distribution of stress inside the sphere in an elastoplastic collision is dependent on the loading history.
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CHAPTER 8 ELASTOPLASTIC TFD MODEL The elastoplastic TFD model proposed in this paper is consistent with the elastoplastic NFD model presented in VuQuoc and Zhang [19986] (Chapter 6). It is developed from the VuQuoc and Zhang [1998a] TFD model for elasticfrictional contact introduced in Chapter 5. The account for the effect of plastic deformation is based on the same formalism used for the elastoplastic NFD model, i.e., the additive decomposition of the radius a ep of elastoplastic contact area by (6.2), and the modification of the radius R* of local contact curvature by (6.12). In the following, we discuss the formulation of the TFD model that accounts for the plastic deformation and then present the algorithm and pseudo code of the proposed elastoplastic TFD model. 8.1. Accounting for the Effect of Plastic Deformation In the VuQuoc and Zhang [19986] NFD model, to account for the effect of plastic deformation on the normal forcedisplacement relationship, we introduce the additive decomposition of the radius a ep elastoplastic contact area into the elastic part a e and the additional part caused by plastic deformation a p according to (6.2). Following Hertz theory, the radius a e of elastic contact area is determined by (3.6) as follows where R* is the equivalent radius of relative contact curvature defined by (3.2), and E* the equivalent elastic modulus for the contact defined by (3.1). In addition to the increase of contact area, another feature of elastoplastic contact is that 155
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156 the plastic deformation is irreversible, and tend to flatten the contact surface, thus make the relative contact curvature decreasing. The increase of the radius of contact curvature, as caused by plastic deformation when a sphere contacts a rigid planar surface, is shown in Figure 6.10. In VuQuoc and Zhang [19986], we use a coefficient Cr{P) > 1.0 to modify the local radius of contact curvature. For given material properties and radii of spheres in contact, the coefficient Cr(P) is determined by the normal force P (using, e.g., (6.13)). In 3D finite element analyses (FEA) of two identical elasticperfectlyplastic spheres contacting against each other with friction, we observe the following forcedisplacement (FD) behavior of the tangential contact stiffness (see VuQuoc and Lesburg [1998]): 1. When the contact is in the elastic range, i.e., without yield and plastic deformation, the FD behavior in normal direction follows the Hertz theory; the FD behavior in tangential direction follows the Mindlin and Deresiewicz [1953] theory. 2. For the aluminum alloy, which is the material employed in the FEA in VuQuoc and Lesburg [1998], while the plastic zone under the combined loading of P and Q remains very close to the plastic zone created by the normal force P alone, plastic deformation clearly affects the TFD behavior (see Remark 8.1 for more explanation). 3. In the case where the normal force P is applied on the sphere until its magnitude becomes very large compared to the yield normal force Py (P 3> Py), and then held constant while the tangential force Q is applied with increasing magnitude, even though there is a large amount of plastic deformation involved, the TFD behavior is stiffer than that obtained for the elastic case
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157 using the Mindlin and Deresiewicz [1953] theory (see Remark 8.1 and VuQuoc and Zhang [19986] for more details). Figure 8.1 shows the TFD curves, computed by FEA and by Mindlin and Deresiewicz [1953] theory, for two identical aluminum alloy spheres of radius R = 0.1 m in contact with each other. The spheres are subjected to a constant normal load P = 2600 N and varying tangential force Q. FEA results were obtained for elastoplastic behavior of the material. 4. When both the normal force P and the tangential force Q are increasing, the FD behavior in the normal direction follows what described in VuQuoc and Zhang [19986], i.e., softer than Hertz (elastic) theory; the FD behavior in the tangential direction is also softer than that from the Mindlin and Deresiewicz [1953] (elastic) theory, as a result of the plastic deformation. 5. The TFD curves for both elastic and elastoplastic materials display a hysteresis effect when the applied tangential force Q goes through cycles of loading, unloading and reloading. The TFD curve for the elastoplastic material display a softer behavior during loading. Figure 8.2 shows the FEA results of the same sphere as in Figure 8.1, subjected to varying normal and tangential forces (P and Q), and a comparison with the Mindlin and Deresiewicz [1953] elasticfrictional contact theory. The step like behavior in the TFD curve during loading is due to the change in the normal load P throughout the loading phase. The maximum magnitude of P is P max = 1500 N. Remark 8.1. Since Q < jjlP throughout the loading and unloading stages, and since the coefficient of friction chosen was /j, = 0.2, the effect of the tangential force Q on the plastic deformation is relatively smaller than that of the normal force P.
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158 Figure 8.1. Two identical aluminum alloy spheres in contact: TFD curves from FEA of elastoplastic sphere material, and from Mindlin and Deresiewicz [1953] (MD[1953]) elastic theory. Constant normal force P = 2600 N, and varying tangential force Q. Also, it is interesting to observe that plastic deformation makes the TFD curve stiffer, compared to that of elastic material, in the case where the normal load P is constant. The reason is that when the sphere yields under the normal force P that is much larger than the yield normal force P Y the contact area is also much larger than that of the elastic case. It follows that there is more contact area to resist the tangential force Q through friction. On the other hand, when both P and Q are varying, plastic deformation affects the TFD relationship that is softer and has a steplike pattern shown in Figure 8.2. We refer the readers to VuQuoc and Lesburg [1998] for more details. I
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159 ouu 250 200 ^150 100 P Py I 1 1 LJt = 1500 [N = 36.45 N A .1 4 J 4 Â•) j: ** ^ J 1 \ 1 \ t \ + //:/ / 1 : + / / 4 M* i i i i 50 r i / 1 / i + FEA MD[1953] oi i i i 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 6{m) xio~ 6 Figure 8.2. Two identical aluminum alloy spheres in contact: TFD curves from FEA of elastoplastic sphere material, and from Mindlin and Deresiewicz [1953] (MD[1953]) elastic theory. Varying normal force P and tangential force Q.
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160 Based on the FEA results, it can be concluded that the plastic deformation affects the TFD behavior in following ways: (i) Increase the contact radius a, and thus increase the tangential stiffness, (ii) Increase the radius of local contact curvature R*, and thus increase the tangential stiffness, (iii) The plastic zone weakens the resistance to the tangential force, and thus decreases the tangential stiffness. For most of the loading cases considered, FEA yields a TFD relationship that is softer than that of the Mindlin and Deresiewicz [1953] theory for elastic frictional contact. Let us look at the TFD relationship described in Chapters 3 and 5. Only the calculation of initial tangential stiffness K T $ (see (3.25)) is connected to the effect of plastic deformation described above. Considering that the TFD curves from elastoplastic FEA results retain the same shape as those from the Mindlin and Deresiewicz [1953] theory, we retain much of the formulation described in Chapters 3 and 5 to account for the effect of changing in the normal force, and this for all loading cases. Recall that Kr> = Sa( 2 ^f+ 2 ^f.)~\ (3.25) If one inserts the increased elastoplastic contact radius a ep into (3.25), the tangential stiffness is increased to account for the effect of plastic deformation. In this case, the weakening of tangential stiffness by plastic deformation could not be properly represented. For this reason, we introduce the equivalent Young's modulus (E*) ep for the elastoplastic contact in the tangential direction. For the cases where the normal force P is increasing, we replace a by a ep R* by R* p and E by (E*) e ?, in (8.1) to obtain = (s8Â£f Â• < 8 2 >
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161 Therefore, for P increasing, { ] 4(a e P) 3 4(a<*) 3 [ j From (8.3) It can be seen that since the contact radius a ep > a e when there is a plastic deformation, this enlarged contact area tends to decrease (E*) ep if the other quantities are held constant. On the other hand, since the radius of contact curvature /?* > R* when there is plastic deformation, this increase in the radius of contact curvature tends to increase (E*) ep For most cases, the effect of a ep is stronger, and thus the equivalent elastoplastic Young's modulus (E*) ep is usually less than the original material Young's modulus E, as if the Young's modulus is weakened by the plastic deformation. It should be noticed that since (E*) ep is proportional to R* p = C R (P)R* as given in (8.3), the use of (E*) ep in the computation of the tangential stiffness in the TFD relationship therefore also accounts for the effect of the increase in the radius of contact curvature due to plastic deformation. Recalling that G = Â— , and assuming that the two spheres in contact have the same Poisson's ratios, i.e., i^v = (j) i/ = u, we can rewrite (3.25) as follows fr,.^( P"4g"> + P+'>g' r. 8 .4) V (i)k (j)E J We can express the initial tangential stiffness for the TFD relationship in terms of the Young's modulus E* by using (3.1) in (8.4). Once this step is done, we again replace E* by (E*) ep to account for plastic deformation in the computation of K Tfi and obtain A' y, = 4a ep (ETr (IrjJ) Â• (8.5) Therefore, when the normal force P is increasing, expression (8.5) is used in the computation of the tangential stiffness in the TFD relationship to account for the effect of plastic deformation.
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162 When the normal force P is decreasing, the plastic deformation is frozen, as the spheres undergo elastic unloading. Therefore, the following formula is used to modify the equivalent elastic modulus (T?*\ep 3PCR{P max )R* {E ] = Â— ijSy Â— ( } where the value of C R is frozen at is value at P = P max and the contact radius in the denominator is the elastic part a e The tangential stiffness in the TFD relationship is computed using expression (8.5) for the initial tangential stiffness K T ,o, but with {E*) ep computed from (8.6). It should be noticed that since plastic deformation is mainly attributed to the normal force P, when P remains a constant during the loading or unloading of the tangential force Q, the plastic zone is considered as unchanged. Therefore, the modification of tangential stiffness for the case where P is a constant should be similar to that of the case where P is decreasing, i.e., (8.6) is used to compute the equivalent Young's modulus (E*) ep The modification of the tangential stiffness of the TFD relationship can properly account for the effect of plastic deformation for a range of the ratio of increment of P over increment in Q. Further work is needed to extend this range, which is large enough for granular flow simulations. We refer the readers to VuQuoc, Lesburg and Zhang [1998] for more detailed discussion. 8.2. Algorithm and Implementation of the ElastoPlastic TFD Model In a DEM simulation code, the increment of tangential displacement between two spheres in contact is evaluated by the relative position and velocities of these two spheres in each timestep of numerical integration (see VuQuoc et al. [1997]). At time tn+i, the normal contact force P n+1 is computed by the elastoplastic NFD model
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163 that is consistent with the present TFD model, and whose algorithm is presented in VuQuoc and Zhang [19986]. We use this present elastoplastic TFD model to compute the current tangential contact force Q n +\ based on the increment of tangential displacement AÂ£ n+1 = 5 n+ i Â— 5 n and on the loading history previously calculated. We can see, from previous discussion, that the effect of plastic deformation on the TFD relationship is accounted for by using (8.5) to compute the initial tangential stiffness using the radius a ep of the elastoplastic contact area and the equivalent Young's modulus (E*) ep Noticing that the shape of the TFD curves obtained from elastoplastic FEA is similar to that obtained from Mindlin and Deresiewicz [1953] theory and from the VuQuoc and Zhang [1998a] TFD model, we employ the same procedure as described in Chapter 5 to account for the effect of varying normal force P. The implementation of the present elastoplastic TFD model is described algorithmically in the pseudo code below. In Algorithm 8.1, the elastoplastic NFD model is the one presented in Chapter 6 (and VuQuoc and Zhang [19986]). Algorithm 8.1. Elastoplastic TFD model: Displacementdriven version. 1 Input : (i)R, (j)R,E,v,a Y 2 Py calculated using the elasto Â— plastic NFD model. s P n ,P n+ i, and Pmax calculated using the elasto Â— plastic NFD model. 4 o^ + i,On+i) an d a n+i calculatedusing the elasto Â— plastic NFD model. 6 Displacement {S n S n+l }, forces {Q n Q* n }. 7 Goal : compute next tangential force Q n +\ 8 Initialization : Q* Q Â— 0, Qi nc Â— true. 10 Calculate AP n+ i = P n+ \ Â— P n li Calculate A6 n+i = 5 n+ i Â— 5 n
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164 13 Set Q* n+l = Ql 15 if APÂ„ +1 = (P constant) n if P n+1 < P Y {elastic) is Calculate Kt,q via (3.25). 19 elseif P n+1 > P Y (plastic) 20 Calculate (E*) ep via (8.6). Calculate Kt,o via (8.5). 21 22 endif 24 if A6 n+ i > (Q increasing) 25 Calculate Kr, n via (3.24). 26 Calculate Q n +\ via (4.6). 27 Set Q inc = true. 28 elseif A8 n+i < (Q decreasing) 29 if QiÂ„c = true so Set Q* n+l = Q n 3i endif 32 Calculate Kr, n via (3.24). 33 Calculate Q n +\ via (4.6). 34 Set Q inc = false. 35 endif 37 elseif AP n+ i > (P increasing) 39 if P n+1 < P Y (elastic) 40 Calculate Kt,o via (3.25). 4i elseif P n+1 > P Y (plastic) 42 Calculate (E*) ep via (8.3). 43 Calculate K T ,o via (8.5). 44 endif
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165 46 if A5 n+ i >0(Q increasing) 47 if A<5 n+1 > (K T ,o)p=p n 48 Calculate K T n via (3.27). 49 Calculate Q n+ \ via (3.28). so elseif AÂ£ n+1 < (K T fi)p=p n si Calculate Q n+ \ via (3.30). 52 WARNING : NOT a simple loading step. 53 endif 54 Set Q inc = true. 56 elseif AS n+ i < (Q decreasing) 57 if Q inc true 58 Set Q* n+l = Q n 59 endif fiAP if HA n+l (K T ,o)p=p n 6i Calculate K T n via (3.41). 62 Calculate Q n+ \ via (3.43). 63 elseif A(J n+1  < {Kt,o)p=p u 64 Calculate Q n +\ via (3.42). 65 WARNING : NOT a simple loading step. 66 endif 67 Update Q* n+l = Q* n + pAP. 68 Set Q inc = false. 69 endif 7i (D elseif AP n+1 < (P decreasing) 73 if P n+1 < P Y {elastic) 74 Calculate Kt,q via (3.25). 75 elseif P n+1 > P Y (plastic)
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166 76 Calculate (E*) ep via (8.6). 77 Calculate Kt,o via (8.5). 78 (Dendif so if AS n+ x >0(Q increasing) si Calculate Kr t n via (3.32). 82 Calculate K' Tn via (3.35); 83 Calculate Q n +\ via (3.36). 84 Set Q inc = true. 86 (Delseif A6 n+ i < (Q decreasing) 87 if Q inc = true s* Set Q* n+1 = Q n 89 endif 90 if Q n < Q* n+l + 2/iAP 9i Calculate K T Q* n+1 + 2//AP 97 lf Q'n+l + 2 AP n via (3.54); 99 Calculate Q n+ i via (3.56). 'oo Update Q* n+l = Q* n + /iAP ">i elseif Q* n+1 + /iAP < Q n 102 Calculate K' Tn via (3.57). 103 if AJ < \\Qn+l + V&P ~ Qn K T,n '4 Calculate K T n via (3.59).
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167 105 106 107 108 109 110 112 m 115 in (Dendif endif endif Set Qi n c = false. endif elseif AJ <  Q* n+1 +liAPQ n Set K T n = K Tfi K T,n endif Calculate Q n +\ via (3.61). Update Q* n+l =Q* n + /iAP. 8.3. FEA Validation We implemented the present elastoplastic TFD model into a MATLAB code. The TFD curves produced by using the present TFD model are compared to the corresponding TFD curves obtained from 3D FEA results for the static contact problem between two identical aluminum alloy spheres. Finite element analyses are performed for the loading history shown in Figure 8.3 and Table 8.1. Table 8.1. Force parameters for loading history shown in Figure 8.3. Loading History *max I*' J Wmax \Â™j PQbg W VPmax (N) A 1500 270 300 300 B 500 90 100 100 C 250 45 50 50 D 2600 500 2600 520 The mechanical properties of the aluminum alloy sphere are: Young's modulus E = 7.0 x 10 10 N/m 2 Poisson's ratio v = 0.3, material yield stress a Y = 1.0 x 10 8 N/m 2 and coefficient of friction between the sphere and the planar surface fi = 0.2. The radius of the aluminum alloy sphere is R = 0.1 m. For the FEA, elastic
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168 i P Qbg fA*max (AQ = /iAP), '.max I max H = 0.2 AQ = /iAP Figure 8.3. Loading histories for the comparison of TFD curves. perfectlyplastic and timeindependent plasticity are employed. For the purpose of comparison, only some of the tangential forcedisplacement curves are presented in this chapter, and we refer the interested readers to VuQuoc and Lesburg [1998] for more details on the related FEA models and results. The TFD curves shown in Figure 8.4 are produced using the following procedures: At first, we use the loading history A (shown in Figure 8.3 and Table 8.1) as the input for the FEA using ABAQUS with elasticperfectlyplastic sphere material in the contact problem to compute the time history of the normal displacement denoted by a fe and of the tangential displacement S fe Since the TFD model presented in Section 8.2 is of the displacementdriven type, the FEA displacement results a fe and Sf e are used as input into our MATLAB code, which is based on Algorithm 8.1, to compute the tangential force Q pm The normal force P is calculated using the elastoplastic NFD model described in VuQuoc and Zhang [19986]. The TFD
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169 curve from Mindlin and Deresiewicz [1953] theory is obtained by inputing the loading history A (shown in Figure 8.3 and Table 8.1) to obtain the output of tangential displacement history 5 m i n for elastic contact. The results with the loading histories B,C, and D shown in Figures 8.5, 8.6, and 8.7, respectively, are produced using a similar procedure. Remark 8.2. The TFD curves in Figures 8.4, 8.5, 8.6, and 8.7 are produced from procedures that are different in nature. That is, the FD curves from the FEA results are produced using a forcedriven procedure, while the FD curves from the proposed elastoplastic TFD model are produced using a displacementdriven procedure. Since at the beginning we do not know the time history of the tangential displacement, we use FEA to produce such time history using a forcedriven procedure, in which the time history of P and Q is given. The time history of the tangential displacement obtained from FEA is then used as input for the present TFD model to obtain the time history of the tangential force in a displacementdriven procedure. The results from Mindlin and Deresiewicz [1953] theory are produced in a forcedriven procedure for comparison. I Figure 8.4 shows the TFD curves under varying normal force P with P max = 1500 N. It can be seen that the TFD curve produced by the present elastoplastic TFD model agrees with the TFD curve from FEA results. Since P max = 1500 N > Py = 36.45 N, both the curve from FEA results and the curve from the present TFD model show much larger energy dissipation in the tangential direction than the TFD curve from the Mindlin and Deresiewicz [1953] theory for elastic contact. Clearly, Mindlin and Deresiewicz [1953] theory cannot predict the TFD relationship correctly for elastoplastic contact when there is plastic deformation. Quantitatively, the maximum tangential force employed in FEA is (Qf e ) max = 270 N, which is part of the time history of the applied forces that produce the time history of
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170 300 250200Â•150 Q? 100 Figure 8.4. TFD curves for loading history A (Figure 8.3 and Table 8.1): Comparison of the present elastoplastic TFD model, FEA, and Mindlin and Deresiewicz [1953] (MD[1953]) elastic contact theory. tangential displacement used as input for the present TFD model. In this fashion, the maximum tangential force produced from the present TFD model is (Q pm ) ma x = 295.5 TV, which differs from {Qf e ) ma x by 9.4 %. There is very good overall agreement between the TFD curve from FEA and the TFD curve from the present TFD model as shown in Figure 8.4. It is noted that even the steplike behavior during the loading stage is reproduced by the present TFD model. We emphasize that there is thus far no TFD model in the literature that exist for elastoplastic contact, while existing TFD model for elastic contact cannot achieve the overall quality shown in Figure 8.4. The coefficient of restitution /3 for tangential direction represents the energy dissipation ratio in the tangential direction of the contact, which is computed
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171 by taking the square root of the ratio between the restoring energy (area under the unloading curve) and the storing energy (area under the loading curve) in tangential direction, i.e., P := /Area under tangential unloading curve \ \ Area under tangential loading curve / 1/2 loading E ie{6i6ii>0} M8ii)(Qi + Qni) \ 1/2 unloading Â• (8.7) ViÂ€{44i<0} Figure 8.4 shows the tangential coefficient of restitution from the present TFD model is f3 pm = 0.6192, which differs from from the FEA result, f3 fe = 0.5545, by 11.7 %. Both P pm and (3 fe are much smaller than the coefficient (3 min = 0.9716 from Mindlin and Deresiewicz [1953] theory. Figure 8.5 shows the TFD curves under varying normal force P with P max = 500 AT (loading history B in Figure 8.3 and Table 8.1). The TFD curve produced by the present elastoplastic TFD model is close to the TFD curve from FEA, especially for the loading part. Since P max = 500 N > P Y = 36.45 N, both the TFD curve from FEA and the TFD curve produced by the present TFD model show much larger energy dissipations in tangential direction than that from the Mindlin and Deresiewicz [1953] theory for elastic contact. Again, Figure 8.5 shows that Mindlin and Deresiewicz [1953] theory cannot predict the TFD relationship correctly for elastoplastic contact when there is plastic deformation. On the other hand, the maximum tangential force used for the FEA is (Q /e ) mox = 90 N, and the maximum tangential force produced from the present TFD model is {Q pm ) m ax 93.1 N, i.e., a difference of only 3.4%. The tangential coefficient of restitution from the present TFD model, (3 pm = 0.8063, differs from that of FEA results, f3 fe = 0.7352, by only 9.7 %. Both p pm and fe are sharply different from p min = 0.9716 from Mindlin and
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172 100 90 80 70 60 o> Pmax = 500 N Py = 36.45 N {Qfe)max=90N {ycpmimax = "o.liV /? /e = 0.7352 /< stir/j P pm = 0.8063 FEA Present TFD model MD[1953] S(m) 1.5 x10 Figure 8.5. TFD curves for loading history B (Figure 8.3 and Table 8.1): Comparison of the present elastoplastic TFD model, FEA, and Mindlin and Deresiewicz [1953] (MD[1953]) elastic contact theory. Deresiewicz [1953] theory. The present elastoplastic TFD model correctly predicts not only the tangential force level, but also the energy dissipation of this elastoplastic contact in tangential direction. Figure 8.6 is one more example showing the comparison of TFD curves under varying normal forces, and computed using different models. In this case, the maximum normal force is P max = 250 N (the loading history C in Figure 8.3 and Table 8.1). It shows again that the TFD curve produced by the present elastoplastic TFD model is close to the TFD curve from FEA, with both of those TFD curves for elastoplastic contact being quite different from the TFD curve produced by Mindlin and Deresiewicz [1953] theory for elastic contact. The maximum tangential force
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173 O 1 50 45 40 35 30 25 20 15 10 5 Py = 36.45 N {Qfe)max = 45 N (^ipm)max ~ *M.ifi JV P Ie = 0.8309 ppm = 0.8839 And = 0.9743 i i i i /a *' /A '' / 7* / S / : / .*' 4// / Â•+ FEA Â— Present TFD model MD[1953] 1 in^Â—H Â— i b,. 1 1 i i i 4 6{m) x10~ Figure 8.6. TFD curves for loading history C (Figure 8.3 and Table 8.1): Comparison of the present elastoplastic TFD model, FEA, and Mindlin and Deresiewicz [1953] (MD[1953]) elastic contact theory. used for the FEA is (Qf e )max Â— 45 N, and the maximum tangential force produced from the present TFD model is (Q pm )max = 46.2 N, i.e., a difference of only 2.6%. The tangential coefficient of restitution from different models are: /?/ e = 0.8309, ppm = 0.8839, and AmÂ„ = 0.9743. Figure 8.7 is an example showing the comparison of TFD curves under a varying tangential force Q and a constant normal force P = 2600 N (the loading history D in Figure 8.3 and Table 8.1). As described in Section 8.1, we can see in Figure 8.7 that the TFD curve from FEA with the effect of plastic deformation (P = 2600 N P Y = 36.45 N) is stiffer in the early loading stage than that from Mindlin and Deresiewicz [1953] theory for elastic contact. Figure 8.7 also shows
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174 600 500 400 ^300 200 100 P Y = 36.45 N {Qfe)max = 500 N (Qpm)max = 519.5 N fe = 0.6887 13pm = 0.6693 (3 md = 0.8155 + : FEA Â— Present TFD model 4.5 6 Figure 8.7. TFD curves for loading history D (Figure 8.3 and Table 8.1): Comparison of the present elastoplastic TFD model, FEA, and Mindlin and Deresiewicz [1953] (MD[1953]) elastic contact theory. that the energy dissipation in the tangential direction from FEA is larger than that from the Mindlin and Deresiewicz [1953] theory, because energy dissipation caused by plastic deformation is accounted for in FEA while only the energy dissipation caused by friction is accounted for in the Mindlin and Deresiewicz [1953] theory. The TFD curve produced by the present elastoplastic TFD model (Figure 8.7) agrees qualitatively incorporate the stiffening effect that discussed in Section 8.1. After an overshoot in the stiffening effect at the beginning of the loading stage, the present model softens to an accurate value for the maximum tangential force (probably caused by the friction limit), and then follows an unloading curve with the same slope as in FEA (which is also stiffer than the TFD curve from Mindlin and Deresiewicz [1953] elastic contact theory). Thus despite a decrease in accuracy
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175 in the loading stage due to excessive normal load P, the present TFD model shows good recovery of the stiffness overshoot and accuracy in the unloading stage. In addition, Figure 8.7 shows that the tangential coefficient of restitution from the present TFD model, (3 pm = 0.6693, agrees with the FEA results, /3 fe = 0.6887, with a difference of only 2.8 %. Remark 8.3. From the results shown in Figures 8.4, 8.5, 8.6, and 8.7, we observe the following behavior of the present elastoplastic TFD model: (i) For the case where both P and Q are varying, the results produced by the present TFD model are quantitatively accurate in both the force magnitude and the dissipation of the energy in comparison with FEA results, (ii) For the case where P = constant ( ,.. = oo ) and P C Py, there is a substantially larger amount of plastic deformation and contact area, the present TFD model displays some stiffness overshoot at the beginning of the loading stage, but a good unloading curve. I 8.4. Benchmark Tests Singlesoybean dropping tests (see Appendix E Section E.l) and the sphere collisions with rotation are employed as benchmark tests to validate the DEM implementation of our elastoplastic FD models. The problem of sphere collision with rotation is based on the hardsphere collision problem that was used in Walton [1993], who described the hard sphere collision operator. In this problem, a series of spheres all are launched at the same translational velocity, and with different angular velocity, toward a rigid plane. The spheres, each having a different angular velocity, strike the rigid plane with different angle of incidence at the contact points, and thus present many different cases of tangential loading. By studying the post collision behavior of these spheres, we obtain a validation of the implementation of the proposed FD models.
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176 8.4.1. Single Soybean Dropping Tests Based on the data of soybeancompression experiments and on the results of singlesoybean drop tests (see LoCurto et al. [1997]), we develop an optimization procedure that is used together with the VuQuoc and Zhang [19986] elastoplastic NFD model to extract the mechanical properties of granular materials (see Appendix E, also see Zhang and VuQuoc [19986] for more details). To simplify the DEM simulation of the singlesoybean drop tests, a soybean involved in this simulation is represented by a sphere with radius R = 4.05 x 10~ 3 m, which is the average radius of curvature of the soybean at the contact point of the dropping tests. The mass of a soybean is the average value measured from experiments, i.e., m Â— 1.49 x 10~ 4 kg. Other parameters extracted from experimental data are: E = 1.288 x 10 8 N/m 2 v = 0.4134, o Y = 2.75 x 10 6 N/m 2 the coefficient for plastic correction contact radius used in (6.3) is C a = 2.202 x 10 5 m/N, the coefficient for the modification of the effective radius of contact curvature (as in (6.13)) is K c = 2.97 x 10~ 6 1/N, and the incipient yield normal force in (6.19) is P Y = 0.3952 N. As in singlesoybean drop tests that the soybeans are dropped from three different heights to a hard surface, we collide a single soybean (represented by a sphere here) to a rigid surface with three different initial velocities in our DEM simulations. The DEM simulation results are shown in Figures 8.8, 8.9, and 8.10. In addition, the results obtained from the optimization process for the extraction of the mechanical properties and model parameters of soybeans are also shown in Figures 8.8(a), 8.9(a), and 8.10(a) for comparison (see Zhang and VuQuoc [19986]). Since the results from DEM simulation agree with those obtained from the mechanical property extraction process, the correctness of the implementation of the elastoplastic NFD model is thus validated.
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177 q (m) (a) Result from the mechanical property extraction process. Integration time step At = 1.00 x 10~ 6 sec. a (m) (b) Result from DEM simulation with small integration time step, ntcol = 60 and At = 2.53 x 10 6 sec. a (m) (c) Result from DEM simulation with small integration time step, ntcol = 20 and At = 7.60 x 10 6 sec. Figure 8.8. Comparison of P a curves for soybean collision with incoming velocity v in = 1.705 m/s. Figure 8.8(a) is the PÂ—a curve, obtained in the mechanical property extraction process for soybean collision against a rigid surface with an impact velocity of v in = 1.705 m/s. Results from DEM simulation for this collision using different integration timestep sizes are presented in Figures 8.8(b) and 8.8(c). In Figures 8.8(b) and 8.8(c), ntcol stands for the number of time steps for a collision, for example, with ntcol = 40 the integration timestep size is determined to be about one fortieth of the collision duration. The results from the DEM simulations agree well with the results from the extraction process in the rebounding velocity, coefficient of restitution, maximum contact force level, maximum normal displacement, and residual normal displacement, despite the difference in integration timestep sizes. The comparison between results from DEM simulation and results from the extraction process for soybean collision against a rigid surface with different impact velocities are presented in Figures 8.9 and 8.10. The agreement among the results has not only validated the correctness of the implementation of the elastoplastic NFD model, but also implies that the results of DEM simulation using the Vu
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178 Vout = 1.702 e = 0.7148 a (m) (a) Result from the mechanical property extraction process. Integration time step At = 1.00 x 10 6 sec. m) (b) Result from DEM simulation with small integration time step, ntcol = 60 and At = 2.53 x 10 6 sec. m) (c) Result from DEM simulation with small integration time step, ntcol Â— 20 and At = 7.60 x 10~ 6 sec. Figure 8.9. Comparison of P Â— a curves for soybean collision with incoming velocity V{ n Â— 2.381 m/s. a (m) (a) Result from the mechanical property extraction process. Integration time step At = 1.00 x 10 6 sec. Vout = 2.096 e = 0.6641 a (m) (b) Result from DEM simulation with small integration time step, ntcol = 60 and At = 2.53 x 10 6 sec. a (m) (c) Result from DEM simulation with small integration time step, ntcol Â— 20 and At = 7.60 x 10 6 sec. Figure 8.10. Comparison of P Â— a curves for soybean collision with incoming velocity v in = 3.156 m/s.
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179 Quoc and Zhang [19986] NFD model are not sensitive of the timestep size. Such conclusion is made based on the small differences in the results of DEM simulations using very different timestep sizes, namely At = 2.53 x 10 6 sec (ntcol = 60) and At = 7.60 x 10~ 6 sec (ntcol = 20). Therefore, we can use a relatively large timestep size (such as with ntcol = 20 instead of ntcol = 40) in DEM simulations to decrease the computation time by half. 8.4.2. Sphere Collisions with Rotation As shown in Figure 5.11, when a sphere collides with a frictional rigid plane with a fixed translational velocity v n and with different angular velocity to, the incident angle at the contact point will be different. The motion behavior of the sphere after the collision can be theoretically predicted based on rigidbody dynamics (Appendix C, Walton [1993], and VuQuoc and Zhang [1998a]). In the DEM simulation of this problem, a hardsphere is represented by a deformable sphere with high elastic modulus. With various incident angles in the collisions, various loading histories of the normal/tangential forces will be encountered. If the DEM simulation produces the expected results for hardsphere collisions, then the DEM implementation of the elastoplastic TFD model is correct. Recall that the normal coefficient of restitution e of the collision is defined as e = ~f > (88) where v n is the normal incoming velocity of a sphere and v' n the normal outgoing velocity, with the negative sign '' indicating that the velocities are opposite in directions. We also define the tangential coefficient of restitution in a similar way, as V 's,t 0~j*. (8.9)
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180 where v Syt and v' st are the tangential velocities of the contact point relative to the rigid surface before and after the collision, respectively. For the case shown in Figure 6.17, we have v Stt = v t + uR. In VuQuoc and Zhang [1998a] and (Appendix C), we derive an expression for P for sliding collisions (C.25), in which sliding between the sphere and the rigid plane occurs during the entire collision 01 + L(l + e)^, (8.10) v s ,t where n is the coefficient of friction between the sphere and the plane. We also show that when there is no sliding between the sphere and the rigid surface during the collision (i.e., the case of "sticky collision"), we have (3 < 1 + l^l + e)^. (8.11) v s,t Especially when we let v = 0.83333333 (a ficticious material) so that the normal collision period is equal to the tangential collision period, we have = e (8.12) Our first example is the simulation of elastic collisions. The properties of the spheres are chosen as follows: Radius R = 4.05 x 10 3 m, mass m Â— 1.49 x 10 4 kg, Young's modulus E Â— 1.288 9 N/m 2 (ten times harder than that of soybean), and with a special Poisson's ratio u = 0.83333333 to verify (8.12). The yield normal force is set to be Py = 3000.0 N to ensure that all collisions will be elastic. We set the incoming normal velocity for all spheres to be v in = v n = 0.10 m/s, but with different angular velocities so that the incident ratio v n /v Sft varies between 0.0405 and 4.010. The results of this example in comparison with the theoretical prediction are presented in Figure 8.11. In the top figure, we plot /? versus the 7 v n quantity fi(l + e)Â—^=: x. It can be seen that for sliding collisions, the curve is 2 v aJt
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181 Tangential Coefficient of Restitution (3 2.5 2 1.5 0.5 Sticky Collision Sliding Collision Simulation Theory 20 30 40 7 /i i \ v n 2/ni + e)Â— v s,t Effective Recoil Angle vs. Incident Angle e = 0.98 Simulation Theory 50 Figure 8.11. Simulation results for spheres in elastic collisions with a frictional rigid plane: Special material with Poisson's ratio v = 0.83333333.
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182 a straight line = 1 + % as predicted by (C.25). For sticky collisions, we have (3 < 1 + x as stated in (8.11). Even though hard spheres are represented by elastic spheres in this DEM simulation, the simulation results agree very well with theoretical predictions, especially for sliding collisions. Figure 8.12 shows the DEM simulation results of spheres in elastoplastic collisions with a rigid plane. The sphere properties are chosen to be the same as those in the simulation of singlesoybean drop tests, i.e., radius R = 4.05 x 10" 1 m, mass m = 1.49 x 1(T 4 kg, Young's modulus E = 1.288 x 10 8 N/m 2 Poisson's ratio v = 0.4134, yield stress a Y = 2.75 x 10 6 N/m 2 and yield normal force P Y = 0.3952 N. Again, the normal incoming velocities for the spheres are all Vin = v n = 0.10 m/s, but with different angular velocities so that the incident ratio v n /v s
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183 0.4 0.2 Tangential Coefficient of Restitution j3 co. 0.2 0.4 0.6 0.8 r i i i 1 ..+rr\ Sticky Collision ;i :. R j^L^ Sliding Collision i i 1 V s ,t Effective Recoil Angle vs. Incident Angle "p 10 15 20 25 30 35 40 45 ^(1 + e) Â— 0.5 0.5 1 v a ,t Figure 8.12. Simulation results for spheres in elastoplastic collisions with a frictional rigid plane.
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184 a (m) 0.8 1 0.2 4 0.6 0.8 1 1.2 14 xio' t (sec) X10' (a) Normal force P vs. displacement a curve. (b) Normal force P vs. time t curve. Figure 8.13. Normal force P of sphere collisions with v in = 2.0 m/s. force and tangential forces are presented in Figures 8.13 and 8.14. Two spheres with the same incoming normal translational velocity provide the same characteristics in the normal force behavior. Figure 8.13(a) shows the NFD (i.e., P vs. a) curve for the collisions with v in = 2.0 m/s and Figure 8.13(b) shows the normal force loading history during a collision. On the other hand, the tangential force in these collisions are different since the initial angular velocities of the spheres are different from each other. Figure 8.14(a) shows the tangential force behavior for the sphere collision with initial angular velocity w = 50.0 rad/s. The upper part of Figure 8.14(a) is the time history of the tangential contact force, with a comparison between the results obtained from DEM simulation code ABOD and the results from the MATLAB code of out NFD and TFD models. The MATLAB code mentioned here corresponds to the displacementdriven version of the elastoplastic TFD model (see Chapter 8 and VuQuoc and Zhang [1998c]), which is implemented in the DEM code ABOD. The MATLAB results are produced by in
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185 putting the tangential displacement obtained from the DEM simulation to compute the corresponding tangential force. The reason to present the MATLAB results is to provide a verification on the correctness of the implementation of our TFD model in the DEM simulation code ABOD. For a better visualization, the MATLAB results are intentionally shifted a small amount along the time axis. The lower part of Figure 8.14(a) shows the TFD (Q vs. 6) behavior during the collision. It can be seen from this figure that our elastoplastic TFD model can handle complex loading cases. The TFD behavior of the sphere collision with initial angular velocity w = 500.0 rad/s is shown in Figure 8.14(b). Compare to Figure 8.14(a), this TFD curve is relatively simpler.
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186 co in k \ \ / J CD 5< J 1 I 1 I 1 N in ^ 1 U so \ X1 783 iÂ— i II tytnax m So s< 1 1 1 1 1 1 d to 6 d CJ d (N)& (N)& in 3 c ~ a o II 3 a o o Â•t 8 g ^d s O CM DO Â£3 c O 8 o> u 0> CJ '+J a
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CHAPTER 9 DEM SIMULATIONS OF GRANULAR FLOWS OF NONSPHERICAL PARTICLES DEM simulations of granular flows of ellipsoidal particles are presented as numerical examples for the application of the NFD, TFD models presented in this dissertation. At first, let's look at the granular flow experiments on which the DEM simulations are based. 9.1. Granular Flow Experiments A series of experiments of granular flows of soybeans have been carried out at the University of Florida (LoCurto [1997]). Mechanical properties of soybeans such as the coefficient of restitution and the normal stiffness during loading are measured from single soybean tests. In this section, we focus our discussion on the experiments of soybeans flowing down an inclined chute with a bumpy bottom. 9.1.1. GranularFlow Experimental Apparatus Figure 9.1 shows a sketch of the experimental apparatus for the granular flow experiments and the dimensions of the chute. As shown in Figure 9.1, the total length of the chute is 3.42 m, the width of the chute is 0.152 m. Glass beads of 3.0 mm in diameter are attached to the chute bottom to make it bumpy, and thus making the flow more energetic. The inclination angle of the chute can be adjusted. A highspeed video camera is attached to the chute, and is pointed to the center of section six of the chute (close to the lower end of the chute) In granular flow experiments, granular materials are put in the hopper and supplied to the higher end of the chute with the help of a vibrator. Granular 187
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Belt Conveyor Hopper 188 Particle Collector Chute Supply Control Vibrator i i 4$ \/f ^ /r 1 ''2 1 5. 2cm SlS w s s s s Â•' 5 S si S s *' 3 ^ yS\^ ^ S s Si ,''6 Camera at Section 6 V h \ Figure 9.1. The experimental apparatus: Overall set up (top), chute dimensions and camera position (bottom).
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189 materials flowing out of the lower end of the chute are collected, and sent back to the hopper by a belt conveyor. The flow velocity profile along the plexiglass side wall of section six can be obtained by processing the images obtained from the highspeed video camera. Since there are dropping gates dividing the chute into seven sections, other information about the flows, such as the overall velocity and the massholdup rate, can also be obtained using conventional measurement, such as dropping the gates and weighing the granular materials in each section. 9.1.2. Soybean Flow Experiments: Different Flow Regimes By adjusting the chute inclination angle, different flow regimes can be observed. Figure 9.2 depicts the soybean flow regimes obtained with different chute angle. It can be seen that soybean flows are sensitive to the chute angle. The different soybean flow regimes shown in Figure 9.2 are obtained by adjusting the chute to a certain inclination angle, and by running the chute for a long enough amount of time with out any interference. When the chute angle is 9 Â— 17.5, soybeans are packed up at the upper end of the chute; no flow is observed in this case. When the chute angle is 9 = 19.1, soybeans flow down the chute with increasing velocity, and with the flow thickness decreasing from the upper end to the lower end of the chute. Despite this accelerating flow regime, the particles remain in close contact with each other in a dense (or frictional) flow fashion. At the chute angle 9 = 21.1, one obtains a steady frictional flow that is fully developed, i.e., the thickness of the flow changes little along the length of the chute, from the upper end to the lower end, while the average velocities of the flow at any given spatial point reaches a steady state, and is almost uniform along the length of the chute. Figure 9.2 shows that, when the chute angle is 9 = 21.1, the thickness of the flow is about 0.038 m at the upper end of the chute, and about 0.035 m at the lower end of the chute. With the chute angle
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190 bumpy bottom No Flow 5.4 cm (sect. 2) 3.8 cm (sect. 2) dense flow dilute flow Figure 9.2. Different soybean flow regimes, which are highly dependent on the chute angle 9.
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191 increases to reach the magnitude of 9 = 21.7, the thickness of the flow becomes much thinner at the lower end of the chute, and the flow there evolves from a dense, frictional flow into a dilute, collisional flow. In a collisional flow, particles are no longer packed to the bottom of the chute, but bounce around and collide with each other, while flowing down. Another interesting behavior that can be observed from Figure 9.2 is that when there is a steady frictional flow, soybeans flow down most efficiently, i.e., the mass flow rate reaches a maximum, for the range of the chute angle as shown in Figure 9.2. We refer to LoCurto [1997] for more details on the experiments. 9.2. Simulations Using the ElastoPlastic FD Models Using DEM as described in Chapter 2, together with the VuQuoc and Zhang [19986] elastoplastic NFD model (Chapter 6) and the VuQuoc and Zhang [1998c] elastoplastic TFD model (Chapter 8), we simulate the granular flow of soybeans down an inclined chute with a bumpy bottom. Here, the simulation domain is a parallelepiped representing a small part of the chute. As shown in Figure 9.3, the two vertical sides of the parallelepiped simulation domain that are perpendicular to the flow direction are periodic boundaries. Whenever a particle flows out of the simulated domain through a periodic boundary, another particle is set to flow into the simulated domain on the opposite vertical side. The other two vertical sides of the parallelepiped simulation domain are real boundaries representing the plexiglass walls. For simplicity, we simulate the case with the chute angle 9 Â— 21.1, since at this situation we have the granular flow close to a fully developed flow, as mentioned above (see Figure 9.2). The input parameters and the simulation results are presented below.
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bl b3: Periodic Boundaries b2 b4: Real Boundaries (Plexiglass Walls) 192 Flow Directio, Chute Angle Bumpy Bottom Figure 9.3. Parallelepiped simulation domain.
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193 9.2.1. Soybean Properties and Input Parameters for Simulation The main input parameters for computer simulation of soybean flow in an inclined chute with a bumpy bottom are listed in Tables 9.1 and 9.2. Recall that in our DEM simulation, the ellipsoidal particles (soybeans) are represented by clusters of four spheres. Therefore, the contact among particles are modelled by the contact between the constituent spheres of the sphere clusters. Based on this geometric modeling, the material properties of the soybeans related to the contact FD models such as E, u, ay, C a and K c are extracted from the experimental data of singlesoybean dropping tests and singlesoybean compression tests, using an optimization process presented in Zhang and VuQuoc [19986] (see Appendix E. Please note that the geometric modelling is different from what described in Appendix E, thus lead to different parameters). Table 9.1. Input parameters for particles Number of Ellipsoidal Particles N p 1200 Mass of Single Particle m p 0.149 g Young's Modulus E 2.098 x 10 8 N/m 2 Poisson's Ratio V 0.4314 Yield Stress Oy 4.857 x 10 6 N/m 2 Yield Normal Force Py 0.3804 N Coefficient for Plastic Contact Radius C a 1.504 x 10" 5 m/N Coefficient for Contact Curvature K c 2.954 x 10" 6 1/N Coefficient of Friction (particle/particle) fi 0.267 Coefficient of Friction (particle/glass) V 0.328 Radius of a Constituent Sphere R c 0.275 cm Length of a Particle Lp 0.73 cm Width of a Particle w p 0.61 cm Height of a Particle H p 0.55 cm Most other parameters presented in Table 9.1 are measured experimentally, either from physical property measurements on single soybeans (such as the coefficient of friction), or from flow experiments with soybeans (such as the number of
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194 particles inside the simulation domain, see LoCurto et al. [1997] for more details). For example, the number of particles used in the simulation domain is determined by the measured value of the mass holdup of section six of the chute, as obtained from the granular flow experiments shown in Figure 9.2. We refer the readers to VuQuoc et al. [1997], Zhang and VuQuoc [19986], and Appendix E for more details about the extraction of parameters from experimental data. Table 9.2. Input parameters for the simulation domain of chute Width of Simulation Domain w. 15.20 cm Length of Simulation Domain L s 4.16 cm Inclination Angle e 21.1 The size of the simulation domain, as shown in Table 9.2 is made large enough to capture the essential features of the flow, while remains small enough to have a low number of particles inside the flow domain to reduce the computational time. The length of our simulation domain, i.e., the distance between two periodic boundaries perpendicular to the flow direction (see Figure 9.3), is set to be six or seven times the length of a soybean. The chute angle 6 is set to the value that yields a flow behavior closest to a fully developed flow, i.e., 9 = 21.1. 9.2.2. Velocity Development and Velocity Profiles The simulation of the chute flow of ellipsoidal particles using the input parameters described in Section 9.2.1 (Tables 9.1 and 9.2) was carried out over 7.0 sec. Initially, 1200 ellipsoidal particles were randomly distributed over the whole parallelepiped simulation domain shown in Figure 9.3. These particles were let to fall down under the action of gravity toward the chute bottom (with an angle). Recall that 4800 spheres made up these 1200 ellipsoidal particles, since each ellipsoidal particle is formed by four spheres. As the particles hit the chute bottom, they began to flow down the chute. The average velocity of the flow at an instant of time is com
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195 puted using the velocity of all particles in the flow domain. At the very beginning of the simulation, the average velocity of the flow varies rapidly (see Figure 9.4(a)). After some time, the flow settled down to a steady state as manifested by the average velocity reaching an asymptotic value. We present below results obtained from this simulation. The average velocity obtained from the simulation results are shown in Figures 9.4 and 9.5. Figure 9.4(a) shows the average velocity of all moving particles in the simulation domain from the time t = 0.0 sec to t Â— 1.0 sec. The solid line in the figure shows the average velocity the direction of the flow (x direction) shown in Figure 9.3; the dashed line depicts the average velocity in the direction perpendicular to the chute bumpy bottom (y direction). The development of the average velocity of the flow from time t = 1.0 sec to t = 7.0 sec is shown in Figures 9.4(b), 9.4(c), 9.4(d), 9.5(a), 9.5(b) and 9.5(c). It can be seen that from the very beginning of the simulation, the velocity of the particles change rapidly. From time t = 2.00 sec to t Â— 6.00 sec, the average velocity in the direction of the flow is still fluctuating but increases gradually. After time t = 6.0 sec, the flow developed into a frictional steady state flow. From time t = 6.0 sec to t = 7.0 sec, the average velocity of all particles in the direction of the flow is about 28.0 cm/sec; this result is close to the experimental measurements, i.e., 30.0 cm/sec as shown in Figure 9.2, with a difference of about 7%. Therefore, the mass flow rate from the simulation results agrees with the mass flow rate measured from experiments. The experimental velocity profile viewing from the side of the chute for the granular flow is obtained from the processing of the images from the highspeed video camera (see VuQuoc et al. [1997]). Since only the particles close to the transparent side wall of the chute can be seen from the highspeed video camera, the experimental velocity profile represents the velocity distribution of the flowing
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196 (VX) <Vy) (vy) 01 02 0.3 04 0.5 0.6 07 0.8 09 1 1 1.1 1.2 1.3 1.4 15 1.6 1.7 18 1.9 2 Tims (sac) Time (sec) (a) From t = 0.00 sec to t = 1.00 sec. (b) From t = 1.00 sec to t = 2.00 sec. 25 20 1 1 1 A y~^\ i 1 I 1,5 Â— (VX) f (Vy) >,0 1 /v v K ."Si '''V^"' V \' V ''/"'/>'' >//Â•!Â•"/'' 5 1 2.1 2.2 2.3 2. 4 2.5 2.6 2.7 2 8 2.9 3 V~";' >>! (c) From i = 2.00 sec to t = 3.00 sec. (d) From f. = 3.00 sec to i = 4.00 sec. Figure 9.4. Average velocities of all particles from t = 0.00 sec to t = 4.00 sec. v x and v y : x and y components.
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197 (Vx) (Vy) Â„/.v i( ,_4 Â— 'v./'.'\/v"\, >, 4 41 4.2 4.3 4.4 4.S 46 4.7 4.8 4.9 5 Time (sec) 25 1 Â— 1 1 1 1 r If a 15 > Â— (Vx) (Vy) I11 V/A," '' v ..' v_'\/"\ ,'V, *,, \' s '*^* ^ *;/* J _i 1 1 1 1 1 y Â— *"'"Â•' 5 8.1 5 2 5 3 5 4 5.5 5.6 5 7 U 5 9 6 Time (sec) (a) From t = 4.00 sec to t = 5.00 sec. (b) From t = 5.00 sec to t 6.00 sec. 30 25 20 i 1 r oS 15 10 I < (Vx) (Vy) ' .* .~ w N ^ fi M ft ,. ^s ^ ^ , v y^ 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 Time (sec) (c) From t = 6.00 sec to i = 7.00 sec. Figure 9.5. Average velocities of all particles from t = 0.00 sec to t = 6.00 sec. v x and v y : x and y components.
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198 I 5 5 j cm/s ^. \ 33.2 i 25.8 \ 22.7 \ 18.6 1/5.0 10.8 4.0 3.0  I 2.0 1.0 Average Velocity Profile {Side) of epsim7{6.027sec) over 99 frames cnVsec 30.41 28 43 2427 20.02 15.67 11.97 8 886 4.232 05369 cnvsec 0.3552 cnvsec 0.3637 crrVsec 0 1878 crrVsec 0.03622 cnvsec 1333 cnvsec 0.05167 cnvsec 0.07014 cnVsec (a) Experiment 15 2 25 3 Along Chute X (cm). Z[0, 1)cm (b) Simulation Figure 9.6. Comparison of velocity profiles of soybeans close to the side wall of the chute. soybeans that are close to the side wall. Figure 9.6(a) shows the velocity profile obtained from experiments, with the chute angle 9 = 21.1. Figure 9.6(b) shows the velocity profile of the soybeans close to the side wall as obtained from simulation results. 6 The velocity profile from simulation is the average velocity distribution from time t Â— 6.02 sec to t = 7.00 sec when the simulated flow is close to a steady state. Comparing Figure 9.6(b) to Figure 9.6(a), we can see that the velocity profile from simulation agrees with the velocity profile from experiment. In the simulation, we measure the origin of the ordinate along the height of the chute is taken to be the level of the centers of the glass beads that are attached to the chute bottom; in the experiment, the origin of the ordinate along the chute height is set at the top of the glass beads. It follows from this difference that there is a slight difference between Figures 9.6(a) and 9.6(b) (in particular the velocity with magnitude 4.2 cm/sec at the bottom of 9.6(b)). 6 Only the soybeans centered within 1.00 cm from the side wall are taken into the calculation for this velocity profile.
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199 Average Velocity Profile (Side) of epsim7(6.027sec) over 99 frames 50.41 44.95 39.65 33.65 27.49 21.27 15.12 7.191 cnVsec 0.05713 cm/sec 0.0842 cm/sec 0.05846 cm/sec 0.02443 cm/sec 0.02928 crrvsec 0.009127 cm/sec 0.03681 cm/sec 0.006638 cm/sec Average Velocity Profile (Side) ot epsim7{6.027sec) over 99 frames cm/sec 1.5 2 2.5 3 Along Chute X (cm), Z1 15)cm 56.12 53.27 46.01 39.34 31.7 24.43 17.36 8.086 0.03722 cm/sec 0.1901 cm/sec 0.05152 cm/sec 0.0589 cm/sec 0.1283 cm/sec 0.1094 cm/sec 0.1535 cm/sec 0.02051 crrVsec 1 1.5 2 2.5 3 Along Chute X (cm). Zf7.2, 8.2)cm (a) Velocity profile of all soybeans. (b) Velocity profile of soybeans close to the center line of the chute Figure 9.7. Velocity profile of the soybean flow in chute (side view) averaged from t = 6.02 sec to t = 7.00 sec. Figure 9.7(a) shows the velocity profile of all soybeans, as viewed from the side wall, and obtained by averaging the velocity from time t = 6.02 sec to t = 7.00 sec, when the simulated flow is steady. Figure 9.7(b) shows a similar result but for the soybeans close to the center line of the chute. 7 Since the soybeans inside the flow domain cannot be seen by the highspeed camera, the results shown in Figures 9.7(a) and 9.7(b) are not obtainable from experimental data. Figures 9.6(b), 9.7(a), and 9.7(b) show that, along the side wall, the velocity of the flowing soybeans increases from the bottom of the chute to the top of the flow. Also, the velocity of the flow increases from where close to the side walls of the chute to the center of the chute. Figure 9.8 shows the velocity profile of all soybeans, as viewed from the top of the chute, and obtained by averaging the velocity from time t = 6.02 sec to 7 Since the width of the chute is 15.2 cm, the soybeans whose centers fall within the range from 7.2 cm to 8.2 cm, on one side of the flow center, are used in the calculation of this velocity profile.
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200 15 Average Velocity Profile (Top) of epsim7(6.027sec) over 99 frames 10 5h 19.14 0.005603 cm/sec 22.97 0.07979 cm/sec 26.79 0.07499 cm/sec 30.25 0.07769 cm/sec 32.11 0.07033 cm/sec 32.41 0.1094 cm/sec 33.04 0.1792 cm/sec 32.91 0.1115 cm/sec 31.72 0.221 cm/sec 31.47 0.1936 cm/sec 29.02 0.1762 cm/sec 25.4 0.1209 cm/sec 19.74 0.05432 cm/sec 0.5 1 1.5 2 2.5 Along Chute X (cm) Y[0, 5]cm 3.6 Figure 9.8. Velocity of all soybeans (top view) average from t 6.02 sec to t = 7.00 sec. 15Average Velocity Profile (Top) of epsim7(6.027sec) over 99 frames Average Velocity Profile (Top) of epsim7(6.027sec) over 99 frames I I 30.82 35.78 42.29 47.6 51.19 51.54 52 52.35 50.95 48.53 45.85 41.27 33.76 0.3598 0.1451 0.02399 0.05814 0.04149 0.000294 0.1543 0.2168 0.215 0.2594 0.1834 0.1629 0.1841 cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec cm/sec 12.11 0.04862 cm/sec 14.75 0.1856 cm/sec 17.09 0.1102 cm/sec 18.63 0.319 cm/sec 19.85 0.2005 cm/sec 197 0.1429 cm/sec 19.44 0.1134 cm/sec 19.35 0.08682 cm/sec 19.21 0.2339 cnvsec 19.17 0.1781 cm/sec 17.64 0.1658 cnvsec 15.36 0.1271 cm/sec 11.82 0.01128 cnvsec 1 1.5 2 2.5 3 Along Chute X (cm) Y[3.5. 4.5)cm 3 5 1 1.5 2 2.5 3 Along Chute X (cm) YJ0.5. 1 .5)cm (a) Velocity profile of soybeans close to the top surface of the flow. (b) Velocity profile of soybeans close to the bottom surface of the flow. Figure 9.9. Velocity profile of soybean flow (top view) average from t = 6.02 sec to t = 7.00 sec.
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201 2.5 ^ 2 I" SH O 2 J3 0.5 + + + + + + I < ^> + + + ++ + + + +" 1 '"* '"MS?* : + + ++ ++ + + t ^*sS t (sec) Figure 9.10. Maximum normal contact force P inside the flow. t = 7.00 sec. Figure 9.9 shows similar results, but Figure 9.9(a) is for soybeans close to the top surface of the flow (top view), i.e., soybeans centered in the interval [3.5 cm 4.5 cm] from the chute bottom. Figure 9.9(b) shows the velocity profile of the soybeans close to the bottom of the chute (viewing from the top), i.e., the soybeans centered in the interval [0.5 cm 1.5 cm] from the bottom of the chute. The results shown in Figures 9.8 and 9.9 are not obtainable from experimental data. Figures 9.8 and 9.9 confirm the velocity behavior of the flowing soybeans as observed from Figures 9.6 and 9.7. 9.2.3. Force Statistics Figure 9.10 shows the maximum normal contact force P max versus time t, as obtained from the simulation results. It can be seen from Figure 9.10 that at the
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202 beginning of the simulation, the maximum normal force P max is much higher than what obtained at a later time. The reason is because at the very beginning of the simulation, the particles are distributed randomly throughout the simulation domain; incompatible overlapping among the particles exists, thus generating high contact forces. After this initial stage (e.g., t > 1.0 sec), the maximum normal contact force decreases to a normal level, because the development of the flow eliminates the incompatible overlapping in the simulation domain. Also, we observe that from time t = 2.0 sec on, the maximum normal contact force P max is mostly about 0.5 N. Comparing with the average velocity of the soybean flow shown in Figures 9.4(c), 9.4(d), to 9.5(c), it can be concluded that the level of the maximum normal contact force is compatible with the development of the average velocity of the flow. Like the average velocity, the maximum normal contact force also has a steadystate regime toward the end of the simulation time interval, i.e., t G [5.0 sec 7.0 sec]. In another word, there is few exceptions of the maximum normal force level for the time interval t G [5.0 sec 7.0 sec]. Figure 9.11 shows the force statistics results of the soybeanflow simulation at time t = 6.00 sec. The force statistics results are obtained by recording all the contact events and the contact force level inside the simulation domain for a very short period of simulation around time t = 6.00 sec. At a time close to t = 6.00 sec, the soybean flow is already close to a steady state, and therefore the force statistics around this time point are representative. It can be seen from the top part of Figure 9.11 that more than 95% of contact forces at that time are less than or equal to 0.10 N, and among them, more than 90% of the forces are less than 0.05 N. By sampling of the contact forces during ten (10) integration time steps around time t Â— 6.00 sec, we see that the average number of collisions is about 1,000 in that particular sampling period (middle part of Figure 9.11), meaning that there are
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203 100 80 a 60 c 0) 1 Q. Accumulated Percentage of Force Level => I 1 1500 CO C g CO = 1000 o O I 500 E Z 300 O200 o 6 6,100 > < 0.05 0.1 0.15 Resultant Contact Force (N) 0.2 2 4 6 8 10 Sampling Time Step Number Histogram of epsim7 af 6.000sec 1076 Avg. No. of Colls. ! n I II II lrinr,^Â„Â„r, J, 0.05 0.1 0.15 Resultant Contact Force (N) 0.2 0.25 Figure 9.11. Contact force statistics of the soybean flow simulation at time t = 6.00 sec.
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204 300 Histogram of epsim7 at 6.000sec 1 076 Avg. No. of Colls. 250 r. C o o d Z 100 > < 50 1 1 1 1 1 Â— 1 1 Â— II Â— I irrii inÂ— n Â— i i Â— ii Â— i L 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Resultant Contact Force (N) Figure 9.12. Zoomedin histogram of contact force distribution of the soybean flow simulation at time t = 6.00 sec. about 1,000 (actually 1,076) soybeansoybean or soybeanboundary contact events inside the simulation domain at any instant around time t = 6.00 sec. The bottom part of Figure 9.11 shows the histogram of the distribution of the contact forces with different force levels. Figure 9.12 is a zoomedin figure of this part. The height of a bar in this figure represents the average number of collisions for the corresponding contact force level. For example, the second bar from the left of Figure 9.12 indicates that there are about 210 collisions, among the total of 1,076 collisions, having the contact force level in the range [0.0055iV, 0.0111AT]. The sum of the heights of all bars in this histogram is equal to the average number of total collisions (1,076 in this case). Figure 9.13 is similar to Figure 9.11 but shows the force statistics of the simulation at time t = 7.00 sec. The basic features of the force statistics at time t = 7.00 sec agree to those at time t = 6.00 sec as shown in Figure 9.11. Since
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205 Accumulated Percentage of Force Level 100 80 o a 60 c CD I 40 Q. 201000 CO  800
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206 350 300 250 O200 150 100 50 Histogram of epsim7 at 7.000sec 968.1 Avg. No. of Colls. 1 1 I I I i i ni iml ll lr 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Resultant Contact Force (N) Figure 9.14. Zoomedin histogram of contact force distribution of the soybean flow simulation at time t = 7.00 sec. the number of samplings for the force statistics at one time point is limited, the difference between Figure 9.11 and Figure 9.13 are reasonable. Figure 9.14 shows the zoomedin figure of the bottom part of Figure 9.13, i.e., the histogram of the distribution of the contact forces with different force levels. Again, the basic features shown in Figure 9.14 agree with those in Figure 9.12. The results in Figures 9.12 and 9.14 show that the flow achieves a steady state in the contact force statistics, and thus agree with the steady state of the flow velocity around the same time.
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CHAPTER 10 CLOSURE We presented in this dissertation the DEM simulation algorithms and the geometric modelling of nonspherical particles. We also presented in this dissertation a set of realistic and accurate forcedisplacement models for DEM simulation that can correctly account for the effect of both elastic and plastic deformation inside the contacting particles. Close agreement is observed between the FD relation of contact elastoplastic spheres produced by these FD models and the results obtained from finite element analyses (FEA). In addition, we carried out computer simulations of granular flows of nonspherical particles using discrete element method (DEM). The velocity distribution on the boundary of the granular flow domain extracted from DEM simulations using the presented elastoplastic FD models match the corresponding results obtained from real granular flow experiments. Important informations of the granular flow, such as the collision frequency and the contact force statistics, are also obtained from the DEM simulation results. Those informations can not be measured from traditional experiments without alternating the flow condition and introducing huge cost of money. We must point out here that the present elastoplastic FD models and the formalism to account for the effect of plastic deformation are general approaches. That is, the methodology based on the additive decomposition of the contact radius and the modification of the contact curvature employed in the present FD models to account for plastic deformation is general. On the other hand, the values of the model parameters are not universal, but depend on the material properties and geometry of the particles, and are thus functions of other more basic parameters 207
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208 (oy, E, v, R, etc). When applied to DEM simulation, the model parameters (e.g., C a and K c ) are either obtained from FEA results or extracted from a set of data from simple experiments (Zhang and VuQuoc [19986]). Further theoretical work can be done to connect the model parameters to other basic parameters related to the material model (e.g., ay, K, E, u, etc.) and to the geometry (e.g., R) of the particles. We successfully validated the FD models against FEA results for the contact/collision between two identical spheres, or between a sphere and a frictionless rigid surface. Even though some suggestions are made to adjust the models when applied to the contact/collision between spheres with different properties, further validation and generalization are still to be carried out. In our FEA of contact /collision problems related to the presented FD models, we focused our attention on the FD relation. We also found some interesting behavior of contact from the FEA results, such as the plastic deformation does not affect the collision duration much and the existence of the bumps of the TFD curves from the FEA results (Figure 8.2), deserve further investigations. We carried out simulations, using DEM code implemented with the presented FD models, of the granular flow of ellipsoidal particles. The results obtained from the DEM simulation match with the results from experiment measurement on the boundary of the flow domain successfully. At the same time, simulations of particle systems of particles in different properties, and with different flow domain are still to be carried out.
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APPENDIX A IMPLEMENTATION OF LINKEDLISTS USING FORTRAN In this appendix, we document the implementation of the contact detection algorithm based on the neighbor (Verlet) lists. Recall that the contact detection strategy employed here is for spheres, and thus applicable to nonspherical particles modelled using cluster of spheres. By a particle, we always mean either a spherecluster particle, formed by a number of constituent spheres, or a singlesphere particle. In the next section, we describe the types of particles implemented in our code. A.l. Nomenclature of Particle Types The following types of particles in the code can be currently employed for simulations. All particles are numbered sequentially following the convention stated below. Â• Aparticles: spherecluster particles. Let na be the number of Aparticles. The Aparticles are labeled from 1 to na. Â• Â• Bparticles: singlesphere particles. Let nb be the number of Bparticles, the first Bparticle is labeled indl = na + 1. With nab = na + nb, the Bparticles are labeled from indl to nab. Cparticles: constituent spheres that constitute the Aparticles (spherecluster particles). In the case where the Aparticles are sphere clusters, each made up of of four spheres, the number of Cparticles is nc = 4*na. The first Cparticle is labeled ind2 = nab + 1 Let nabc = na + nb + nc to be the total number 209
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210 of all the A, Band Cparticles, then the Cparticles are labeled from ind2 to nabc. Â• Boundary particles: singlesphere particles employed to form boundary walls or bumpy boundaries (e.g. the bumpy bottom of a chute). The boundary particles are allowed to move at a certain fixed velocity but not to collide among themselves. Let the number of boundary particles be nbound, and let ind3 = nabc + 1; then the boundary particles are labeled sequentially from ind3 to (nabc + nbound). Â• Fixed particles: singlesphere particles that are fixed in the simulation domain, and that are not boundary particles. Let the number of fixed particles be nf ix; then the fixed particles are labeled from (nabc + nbound + 1) to (nabc + nbound + nfix). Â• Cylindrical particles: particles employed to form cylindrical boundaries. For example, when simulating the particles moving inside a drum, the drum boundary is simulated using a cylindrical particle. Let the number of cylindrical particles be ncyld, then they are labeled from (nabc + nbound + nfix + 1) to np, where np = (nabc + nbound + nfix + ncyld) is the number of all particles involved in the simulation. The total number of particles in a simulation is np = na + nb + nc + nbound + nfix + ncyld. The definitions of the particle types, their labels, their numbers are summarized in Figure A.l. Note that the boundary particles, the fixed particles, and the cylindrical particles are not allowed to collide among themselves and with each others. The following definition formalizes the constraint just mentioned. Definition: active and inactive particles Inactive particles are boundary particles, fixed particles, cylindrical par
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211 tides and particles in reserve (i.e., particles to be inserted into the simulation domain in a future time, but are currently not yet in the simulation domain). Active particles are spherecluster particles (Aparticles), singlesphere particles (Bparticles), and constituent spheres (Cparticles) that make up the spherecluster particles. label type number status na indl nab ind2 nabd ind3 np A B C boundary fixed cylinder na V nb nc V active V nbound nf ix ncyld V inactive formation of linked lists Figure A.l. Data structure of particles: Types, labels, numbers, status (active or inactive). In the construction of the neighbor lists of the constituent spheres of the active particles, we exclude the spherecluster particles (Aparticles), since these particles are not used for contact detection. Only singlesphere particles (Bparticles) and constituent spheres (Cparticles) that make up spherecluster particles are used in contact detection and therefore have neighbor lists. Since Bparticles and Cparticles are all spherical particles, there are only neighbor list for spherical particles. Remark A.l. A neighbor list for a given sphere does not contain that sphere itself. The reason for excluding a given sphere from its own neighbor list is because
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212 its neighbor list contains the information of the contact between that given sphere and those in its neighbor. Since no information is needed for a given sphere and itself, we do not include that given sphere in its own neighbor. I Remark A.2. The data structure of the neighbor lists to be explained below is valid for general nonspherical particles (i.e., not sphere clusters). But since we are using sphere clusters to model nonspherical particles, we present the construction of the neighborlist data structures using spheres. I A. 2. Data Structure: Linkedlists A large common block named /linklist/ is used to store all the neighbor lists of the Bparticles and the Cparticles. The Fortran code for this common block is given below: common/linklist/ ndx(l) ,next(l) ,a(l) ,aO(l) fntot(l) ,tfx(l) $ ,tfy(l) ,tfz(l) ,tm(l + nwd*mp*nnave) Table A.l. Structure of the common block /linklist/. ndx next a aO fntot tfx tfy tfz tin We can think of the data in the common block /linklist/ as being organized in matrix form as shown in Table A.l, where each row contains the following information related to a sphere: ndx, next, a, aO, fntot, tfx, tfy, tfz, tm. The common block /linklist/ in Fortran 77 can be thought of as an array with a single row formed be putting the rows in Table A.l sequentially on after another as shown in Table A.2. Let nwd be the number of (64bit) words for each row in
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213 Table A. 2. Single row of common block /linklist/. ndx I next I a I Â• Â• Â• I tfz I tm I I ndx I next I a I Â• Â• Â• I tf z I tm Table A.l. The length of the common block /linklist/ in number of 64bit words is (nwd + nwd*mp*nnave). The definitions of nwd, mp and nnave will be given later. Each row in Table A.l contains two integers ndx and next, and seven (64bit) real numbers a, aO, fntot, tfx, tfy, tfz, and tm. The data in each row represent the information on the contact between the sphere associated to that row (called, say, sphere (x)) and the sphere whose neighbor list contains sphere (x). Let sphere (i) be the sphere whose neighbor list is being considered, and let sphere (x) be a sphere in the neighbor list of sphere (i). The meaning of the nine variables in a row associated with sphere (x) is given below (see Table A.l). Â• ndx : (integer) label (x) of sphere (x) in the neighborhood of a sphere (i). Â• next : (integer) address in common block /linklist/ pointing to the beginning of the row associated with the next sphere in the neighborhood of sphere (i). Â• a : (64bit real) normal displacement a between sphere (i) and sphere (x) used in the NFD model (see Section 4.2.1). Â• aO : (64bit real) residual normal displacement ao in the NFD model (see Section 4.2.1). Â• fntot : (64bit real) total normal force in the contact between sphere (i) and sphere (x). Â• tfx, tfy, tfz : (64bit reals) components (in the global coordinates) of tangential contact force between sphere (i) and sphere (x).
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214 Â• tm : (64bit real) value of the tangential force at the last turning point, i.e., Q* in the TFD model (see Section 4.2.2). Remark A. 3. The code is designed to be run on both 32bit machines (e.g., DEC, Sun workstations) as well as 64bit machines (e.g. Cray supercomputers). An integer is represented by 32 bits on a 32bit machine, and by 64 bits on a 64bit machine. We use 64bit real numbers (real*8) on both 32bit machines and 64bit machines. The length of a row in Table A.l, in terms of 64bit words, is stored in the variable nwd. Thus, on a 32bit machine, nwd = 8, whereas on a 64bit machine, nwd = 9. In the code, nwd = 10 i vers, where i vers (integer version) designates the number of integer (s) in a 64bit word, i.e., ivers = 2 on a 32bit machine, and ivers = 1 on a 64bit machine. I Remark A.4. The length of common block /linklist/ is (nwd + nwd*mp*nnave) where mp represents the maximum number of spheres in the code, and nnave the average number of spheres in a neighborhood of any given sphere. In the code, nnave is set to 30. It does not mean that the number of actual spheres in a neighborhood of a given sphere cannot exceed nnave. I Table A. 3 shows the 'row's of data in the common block /linklist/. Table A.3. Row's in the /linklist/ ndx next a aO fntot tfx tfy tfz tm rowl row2 We now describe the indices in the arrays in the common block /linklist/ in the case of 32bite machines. In this case nwd = 8, and two 32bit integer words occupy the same length as a 64 bit real variable. In a row associated with a sphere
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215 (see e.g., Table A.4), the variables ndx and next are 32bit integers. Therefore, the first ndx is indexed as ndx(l) where as the second ndx at the head of the second row is indexed as ndx(2*nwd+l). Further, next(l) and ndx(2) point to the same word Table A.4. Indices for the arrays in the common block /linklist/ rowl 1 row2 1+nwd row3 l+2*nwd ndx next a aO fntot tfx tfy tfz tm > > Â• Â• Â• tm(l) tin (1+nwd) tm(l+2*nwd) in /linklist/. The variable next in the second row is next(2*nwd+l). Generally, the variable ndx in the i'th row is indexed as ndx(2*(il)*nwd+l), whereas the variable next in the i'th row is indexed as next(2*(il)*nwd+l). For all other 64bit real*8 variables, a, aO, fntot, ftx, tfy, tfz, tm, a variable x on the i'th row is x((il)*nwd+l). In the code, we introduce the integer variable idx, which serves as the index for the integer variables ndx and next, and the integer variable jdx as the index for the real*8 variables: jdx = (i 1) *nwd + 1 (A.l) idx = i2orl jdx ilorO (A.2) On a 32bit machine, set i2orl = 2 and ilorO = 1; we have idx = 2 (i Â— 1) nwd + 1 On a 64bit machine, set i2orl = 1 and ilorO = 0; we have idx = jdx (A.3) (A.4)
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216 The variables in the i'th row of /linklist/ are indexed as ndx(idx), next(idx), a(jdx), aO(jdx), fntot(jdx), tfx(jdx), tfy(jdx), tfz(jdx), and tm(jdx). There are two more important variables for the data structure, the array neborO, dimensioned to mp (the maximum number of spheres), and mtl. The variable nebor(i) is the "jdx" address pointing to the beginning of the neighbor list of sphere (i). When nebor(i) = 0, there is no sphere labeled with index larger than i in the neighborhood of sphere (). The variable mtl is the "jdx" address pointing to the beginning of the empty list (i.e., the space in /linklist/ not being used). Here mt is the mnemonic for "empty", while 1 means the beginning. A.3. Initialization We use the example shown in Figure A. 2 to illustrate the initialization of the /linklist/, neborO, and mtl. Â— w \ N / \ / N tÂ£\''' LÂ— A *'"\. (2) i S H' ' : vi> rÂ— \ ~ / \ / Figure A.2. An example for the initialization of the neighbor lists. Remark A. 5. In the initialization, to avoid repeating the calculation, we only put in the neighbor list of a given sphere (say, sphere (i)) those spheres in the
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217 neighborhood of sphere (i), with labels of greater than (z). Considering that the labels of the inactive particles are all larger than the labels of the active particles, and that there is no need for contact detection among the inactive particles, there is no neighbor list for the inactive spheres. I By checking the distances between the spheres and by following Remark A. 5, we find the following neighbor lists for the configuration of spheres in Figure A. 2: neighbors of sphere 1: { } neighbors of sphere 2: { } neighbors of sphere 3: { 4, 6, 8 } neighbors of sphere 4: { } neighbors of sphere 5: { 6, 8, 9 } neighbors of sphere 6: { 8 } neighbors of sphere 7: { } neighbors of sphere 8: { } neighbors of sphere 9: { } At the beginning, all numbers in the array neborQ and in the common block /linklist/ are initialized to zero. The integer variable mtl is initially set to 1 to indicate that the whole /linklist/ is initially empty. For the example of Figure A.2, the contents of some of the mentioned typical arrays and variables at the initialization are shown in Table A. 5. Let's begin with sphere (1) to fill in array nebor and common block /linklist/. and used them to represent Since both sphere (1) and (2) have empty neighborhood, the values of nebor (1) and nebor (2) remain zero. The neighborhood of sphere (3) is on the other hand nonempty. To add sphere (4) in to the neighbor list of sphere
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218 Table A. 5. Initial state of the data structure sphere (i) 1 2 3 4 5 6 7 8 9 nebor mtl \T] jdx idx 1 17 33 ndx next a aO fntot tfx tfy tfz tm 1 9 17 (3), we set jdx = mtl = 1, idx = 2*jdx 1, nebor(3) = jdx = 1, and reset mtl. The new value for mtl is set by adding nwd to point to the head position of the next empty 'row', i.e., mtl = mtl + nwd = 9. Finally, set ndx (idx) = 4, which is the label of sphere (4). See Table A. 6 for the status of the data structure at this stage. Table A. 6. Data structure after sphere (4) is put in the neighbor list of sphere (3) sphere (i) 1 2 3 4 5 6 7 8 9 nebor 1 mtl T] jdx idx 1 17 33 ndx next a aO fntot tfx tfy tfz tm 1 4 9 17 The next sphere to be put in the neighbor list of sphere (3) is sphere (6). For this reason, we set jdxold jdx = 1, idxold = idx = 1, and then set jdx = mtl = 9, idx 2*jdx 1 = 17, next (idxold) = jdx = 9 and mtl = mtl + nwd = 17. Finally, put ndx (idx) = 6, which is the label of the sphere (6) See Table A. 7 for the status of the data structure at this stage.
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219 Table A. 7. Data structure after sphere (6) is put in the neighbor list for sphere (3) sphere (i) 1 2 3 4 5 6 7 8 9 nebor 1 mt 1 17 jdx idx 1 17 33 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 The last sphere to be put in the neighbor list of sphere (3) is sphere (8). Again, we set jdxold = jdx = 9, idxold = idx = 17, and then set jdx = mtl = 17, idx = 2* jdx 1 33, next (idxold) = jdx = 17 and mtl = mtl + nwd = 25. Finally, put ndx (idx) = 8, which is the label of the sphere (8). See Table A. 8 for the status of the data structure at this stage. Table A. 8. Data structure after sphere (8) is put in the neighbor list of sphere (3) sphere (i) nebor 1 2 3 4 5 6 7 8 9 1 mt 1 25 jdx idx 1 17 33 47 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 25 Finally, the initialization of the neighbor lists is completed by adding spheres (6), (8), (9) into the neighbor list of sphere (5); and by adding sphere (8) into the neighbor list of sphere (6). See Table A.9 for the status of the data structure at the end of the initialization stage.
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220 Table A. 9. Data structure after the initialization sphere (i) 1 2 3 4 5 6 7 8 9 nebor 1 25 49 rati 57 jdx idx 1 17 33 49 65 81 97 113 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 25 6 33 33 8 41 41 9 49 8 57 A.4. Updating the Neighbor Lists Assume that after a period of simulation time, the configuration of the particle system shown in Figure A.2 is changed into that shown in Figure A.3. I I 1 'l \ / V I I \ ; V /\ I I N I a / \ Figure A. 3. New configuration of an example particle system. Consequently, the neighbor lists are changed into the following
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221 neighbors of sphere 1 : { 9 } neighbors of sphere 2: { } neighbors of sphere 3: { 4, 6, 7, 8 } neighbors of sphere 4: { } neighbors of sphere 5: {6,8} neighbors of sphere 6: { 7, 8 } neighbors of sphere 7: { } neighbors of sphere 8: { } neighbors of sphere 9: { } To update the data structure, we need to do some of insertions into, and deletions from, the data structure as discussed below. 1. Insertion into the neighbor lists. We need to insert sphere (9) into the neighbor list of sphere (1), and insert sphere (7) into the neighbor list of both sphere (3) and sphere (6). First, the following procedure is used to insert sphere (9) into the neighbor list of sphere (1). Since nebor(l) = 0, there is no entry for sphere (1). Yet, we set jdx = mtl = 57, idx = 2*jdx 1 = 113, nebor(l) = jdx, and then update mtl. To update mtl for an insertion, if next (idx) = 0, then set mtl = mtl + nwd; else set mtl = next (idx). Here, since next (113) = 0, set mtl = mtl + nwd = 65. Finally, set ndx(idx) = 9. See Table A. 10 for the status of the data structure at this stage. To insert sphere (7) into the neighbor list of sphere (3), first we verify whether sphere (7) is already in the neighbor list sphere (3) by performing the following procedure
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222 Table A. 10. Data structure after insertion of sphere (9) into the neighbor list of sphere (1). phere (i) 1 2 3 4 5 6 7 8 9 nebor 57 1 25 49 mtl 65 jdx idx 1 17 33 49 65 81 97 113 129 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 25 6 33 33 8 41 41 9 49 8 57 9 65 found = false jdx = nebor (3) idx = 2*jdx 1 while (next (idx) .ne.O) .or. (found.ne.true) if (ndx(idx) .eq.7) found = true else jdxold = jdx idxold = idx jdx = next (idxold) idx = 2*jdx 1 end if end while
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223 After running above procedure to check out that sphere (7) is not in the neighbor list of sphere (3) yet, we also have jdx = 17, idx = 33 pointing to the address for the last sphere in the neighbor list of sphere (3). To insert, we set next (idx) = mtl = 65, jdx = mtl = 65, and idx = 2*jdx 1 = 129. Again since next (129) = 0, reset mtl = mtl + ndw = 73. Finally, set ndx(idx) = 7. The status of the updated data structure is shown in Table A. 11. Table A.ll. Data structure after insert sphere (7) into the neighbor list of sphere (3) sphere (i) 1 2 3 4 5 6 7 8 9 nebor 57 1 25 49 mtl 73 jdx idx 1 17 33 49 65 81 97 113 129 145 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 65 25 6 33 33 8 41 41 9 49 8 57 9 65 7 73 Similarly, sphere (7) can also be inserted into the neighbor list of sphere (6). We obtain the data structure shown in Table A. 12. Remark A.6. The procedure used in the code for finding contact information between sphere (i) and sphere (j) for previous timestep is similar to the above procedure for verifying whether sphere (7) is in the neighbor list of sphere (3). When j > i, just replace (7) with (j) and (3) with (i). If after running the procedure, the variable found = true, means sphere (j) is already in the neighbor list of sphere
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224 (t), the information a( jdx), aO(jdx), tfx(jdx), tfy(jdx), tfz(jdx), fntot(jdx), tm(jdx). If sphere (j) is in the neighborhood of sphere (i), and after running the procedure, the variable found = false, means sphere (j) is not in the neighbor list of sphere (z); we need to insert the entry for sphere (j) into the neighbor list of sphere (i) as described in this section. I 2. Deletion from the neighbor lists. According the particle configuration of Figure A.3, we need to delete sphere (9) from the neighbor list of sphere (5). First, we find the position of sphere (9) in the neighbor list of sphere (5) by performing the following procedure found = false jdx = nebor(5) idx = 2* jdx 1 while (next(idx) .ne.O) .or. (found.ne.true) if (ndx(idx) .eq.9) found = true else jdxold = jdx idxold = idx jdx = next (idxold) idx = : !*jdx 1 endif end while After running the above procedure, we obtain found = true and jdxold = 33, idxold = 65, jdx = 41, idx = 81. To delete sphere (9) from the neighbor list of sphere (5), first, we set next(idxold) = next(idx) = 0, then set ndx(idx) =
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225 Table A. 12. Data structure after all insertions rticle i 1 2 3 4 5 6 7 8 9 nebor 57 1 25 49 mtl 81 jdx idx 1 17 33 49 65 81 97 113 129 145 161 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 65 25 6 33 33 8 41 41 9 49 8 73 57 9 65 7 73 7 81 0, and next (idx) = mtl = 81. Then, we set mtl = jdx = 41 to indicate that the first empty list is at jdx = 41. See Table A. 13 for the status of the data structure at this stage. It should be noticed that at the beginning, there is an ordering in the neighbor lists in /linklist/ in ascending order of the labels of spheres, both for the neighbor lists and for the entries in the neighbor lists. After a number of updates (insertions and deletions), such an ordering will no longer holds. For example, in Table A. 9, when the initialization is completed the neighbor lists for sphere (3), (5), and (6) are stored sequentially in /linklist/ in the ascending order of the labels of spheres. Also, in each neighbor list, the entries for spheres are also in the ascending order of the labels of the neighbor spheres, e.g., in the neighbor list of sphere (5), the entries for sphere (6), (8), and (9) in the neighborhood of sphere (5) are stored in the ascending order of related labels of spheres. In Table A.13, after some insertions and deletions the beginning position of the neighbor list of sphere (1) is after the
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226 Table A. 13. Data structure after deletion sphere (i) 1 2 3 4 5 6 7 8 9 nebor 57 1 25 49 mtl 41 jdx idx 1 17 33 49 65 81 97 113 129 145 161 ndx next a aO fntot tfx tfy tfz tm 1 4 9 9 6 17 17 8 65 25 6 33 33 8 41 81 49 8 73 57 9 65 7 73 7 81 beginning position of several neighbor lists of spheres of larger labels. Also, in the neighbor list of sphere (3), the entry for sphere (7) is behind the entry for sphere (8). Actually, after a number of insertions and deletions, the neighbor lists of spheres in /linklist/ could be in any arbitrary order of the labels, so do the entries for neighbor spheres in any neighbor list of a given sphere.
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APPENDIX B SUPERBALL AND SUPERBEAN Superball (spherical) and superbean (nonspherical) are two particles with designed properties and initial conditions. Which can be used as benchmark tests to verify the implementation of the DEM simulation and check the implementation of the forcedisplacement (FD) models. B.l. Theory Figure B.l depicts a particle in frictional collision with the horizontal plane. Let m to be the mass of the particle, and I the moment of inertia of the particle about its center of mass O. Assume that, before contact, the incoming translation velocity of the particle is {v x v y }, whereas the angular velocity is u z All other components of translational velocity and angular velocity are zero. The qualifier super is ascribed to a particle (e.g., superball in the case of a spherical particle, or a superbean in the case of an ellipsoidal particle) when the velocity vector of the center of mass O of the particle, just after contact, is equal in magnitude, but with opposite direction to the velocity of the same point before contact: v' x = v x v' y = v y ,and J z = u) z (B.l) Also, assume that during the contact, the normal contact force P passes through the contact point C and the center of mass O, then the line OC has to be perpendicular to the tangential contact force Q Let r be the distance between the center of mass O and the contact point C. To have v' y = v y it suffices to that the normal spring is elastic, with spring coefficient K N (see Figure B.l). It follows that the normal coefficient of restitution for a super particle is e = 1.0 (B.2) 227
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228 //////// x Figure B.l. A particle colliding with frictional (tangential) force The balance of linear momentum in the x direction (Figure B.l) of the super particle yields mv x = mv x j Qdt => f Qdt = 2mv x (B.3) Whereas the balance of angular momentum about the center of mass yields J ^2 = h u>, / Qrdt (B.4) Assume that the deformation of the particle at the contact C is small compared with the size of the particle, we can consider r as a constant. Therefore, from (B.4) and (B.l), 2u z I [Qdt (B.5)
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229 Combining (B.3) and (B.5), we obtain m.r (B.6) u> z mr v x h The tangential contact force Q is caused by friction. To avoid energy loss due to sliding, the frictional coefficient must be large enough to ensure that Q < fiP during the entire contact, thus preventing sliding. One way is to choose very large friction coefficient fi. When the particle detaches from the horizontal plane, the normal force P goes down to zero, thus forcing the tangential force Q to also go down to zero, if the condition < Q < jU  P \ is to hold. Since the normal force P and the tangential force Q are modelled using the springs K N and K T as shown in Figure 4.3, we want these springmass systems to have the same natural frequency (or period). The period of the normal spring K N with mass m is given by *
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230 Thus, the conditions for obtaining a super particle motion are (B.2), (B.6), (B.ll) and a very large coefficient of friction to ensure that there will be no energy dissipation caused by frictional slip. B.2. Superball A superball is a spherical particle with carefully selected properties and initial conditions, such that when a superball collides with a horizontal floor, it bounces back with exactly the same magnitude of translational and rotational velocities, but in opposite direction, without any loss of energy. Under gravity, the superball follows the same trajectory (a parabola connecting two points on the floor) back and forth indefinitely. To fully recover all kinetic energy after a collision with the floor, we prescribe the following conditions for a super particle as discussed in Section B.l: (i) Unit coefficient of restitution, i.e., e = 1, to ensure no loss of energy in the normal direction; (ii) a very large coefficient of friction, /j, = 1000; (iii) a ratio of initial tangential stiffness K Tfi to normal loading stiffness Ki as specified by (B.ll), i.e., Kf o /o 2 ~kT = ToT^ = 7 (B 12) since I = mr 2 (B.13) for a sphere, where m is the mass of the sphere, and r its radius; (iv) a ratio between the angular velocity u> z and the horizontal component v x of the translational velocity given by (B.6) as follows u), mr Â— = T" ( B 14 ) with all other velocity components set to zero. In our test problem, the superball has a radius r = 0.002m. At t = 0.0, the superball is set at the initial position having coordinates (0.1m, 0.1m, 0.1m) with
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231 initial velocities v x = OAm/s and initial angular velocity to z = bOO.Orad/s All other components of velocities are zero. The acceleration of gravity is g Â— 9.8m/ s 2 in the negative y direction. The floor is at y = 0. The trajectory are similar to the one shown in Figure B.2, but with a spherical particle. B.3. Superbean A superbean is an ellipsoidal super particle, whose initial conditions must be set carefully following the guidelines in Section B.l. A cluster of four spheres is used for contact detection and contact force calculation (see Section 2.2). The condition for a superbean include those spelled out in Section B.l, with (B.ll) and (B.6) now written as K\ I + ma 2 and (B.15) co z ma = Â— B.16 v x I respectively, with a being half of the length L of the ellipsoid (Figure 2.2), since we want the ellipsoid to impact the floor on its end, i.e., the longest axis of the ellipsoid is normal to the floor at contact. In our test problem, the ellipsoid has the following geometric dimension L = 6.330 x l(T 3 m, W = 5.250 x 10~ 3 m, H = 4.00 x 103 m (see Figure 2.1). With 7T7TL h = ^(Â£ 2 + H 2 ), we hay e stiffness ratio Kn/Ki = 0.252378665, as computed using (B.15). The initial position and initial velocities must be selected carefully as the ellipsoid rotates as it falls toward the floor, and we want it to impact the floor on one of its end, as mentioned above. To this end, we set the initial position of the ellipsoid at (0.1m, 0.097093... m, 0.1m), and the initial velocity components to v x = OAm/s, lo z = 374.38231356 rad/s; and all other components are set to zero.
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232 The numerical results we obtained using DEM simulation with Walton and Braun [1986] FD models implemented indicate good agreement with the theoretical behavior of expected from a super bean. We also noticed that the superbean test is very sensitive to the material stiffness (i.e, K\) and the initial position. The reason is that the stiffness affects the contact time and the amount of displacement at the contact point. Therefore, the softer the superbean is, the longer the contact time is, and the larger the displacement at the contact point is; the result is larger error accumulated after each collision of the ellipsoid with the floor. In out test, we set K x = 4.0 x 10 6 N/m. Both the initial position and initial velocities have to be set with appropriate number of significant digit, lest the ellipsoid would not impact the floor with its longest axis normal to the floor, thus generate error at each rebounding. Figure B.2 shows a sketch of the trajectory of the superbean obtained using our DEM simulation code. y o Figure B.2. A superbean: Rotation (indicated by small arrows), contact with floor on ends, trajectory of center of mass.
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APPENDIX C HARDSPHERE COLLISIONS: THEORETICAL PREDICTION The problem of hardsphere collisions with friction, presented by Walton [1993], is employed in our research as a benchmark test to validate our implementation of different TFD models. In this test, a number of hard spheres collide with a rigid planar surface with a fixed translational velocity, and with different angular velocities, thus making the angles of incidence of the contact points on the spheres different from each other. With known properties of the spheres, and the incoming velocities, the incident angles of the contact points, we can predict the rebounding velocities and the recoil angles of the spheres using first principles of dynamics. With the various angles of incidence, we have the combination of the normal loading with various cases of tangential loading, thus providing an excellent test for the TFD models we implemented. In our DEM simulations, we use "soft" particle modeling to simulate the collision of hard spheres as mentioned in Section 2.1. Figure C.l depicts the collision of two spheres. Let ^m and ^m to be the masses of sphere (i) and of sphere (j), (i )7 and yj/ the moments of inertia of these spheres about their centers of masses, respectively. Assume that, before contact, (j)V and yjv are the velocities of the two spheres, (i)<*> and (j)U> the angular velocities, respectively, and that the spheres are moving on a plane before and after the collision. The two spheres are colliding at the contact point C as shown in Figure C.l. The conservation of the momentum of the two spheres after collision yields (i)v' {i) m + (j) v' {j) m = (i) v (i) m + (j) v {j) m (C.l) where (i )v' and (j )v' are the velocities of sphere (i) and of sphere (j) after collision, 233
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234 n (i)R (i) \ (0 \*t i (,)V j>< 1 CL c U) R U) (i) u> or Figure C.l. Collision of two hard spheres respectively. Denoting A w v = (i) v' (i) v and A (j) v = (j) v' y) v, we have A w v w m = A 0) v 0) m (C.2) Let (,)u/ and (j)u/ to be the angular velocities of sphere (i) and of sphere (j) after collision, respectively. The conservation of angular momentum about the contact point C before and after collision yields A(,)W {i) I {i)Rkij x A (i) v (i) m = A (j) u> 0) 7 W) flky x A (j) v {j) m (C.3) where k^ is the unit vector from the center of sphere (i) to the center of sphere (j), and passes through the contact point C (see Figure C.l); A (i) u; = (i) u/ mu and ^ 0') w = 0) a,/ ~ 0) w are tne changes in angular velocities of the two spheres by the collision; ^R and
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235 angular momenta of the colliding spheres are related to the moment of the contact forces about the spheres center as follows A (i) w (i) 7 = I ftRkij (ji)F {t)dt (C.4) J ta and / (,)flkyx {ii )F(t)dt, (C.5) J t s where A ^u: ^1 and A (j)U> ^)I are the changes of angular momenta of sphere (i) and of sphere (j), respectively, t s the starting time of the collision, and t e the ending time of the collision. Since ^F (t) Â— Â— ^j)F (t), from (C.4) and (C.5), we have A (0^ (0 1 A U)" U) 1 {i) R {j) R (C.6) Combining (C.2), (C.3), and (C.6), we obtain A(i)W (i )7 = (j)%xA(i)V(j)ffl, (C.7) and A (j) w U) 1 = V) R tq x A 0) v U) m ( C 8 ) The velocities of the contact point on sphere (i) before and after collision are (i)V C = (i)V + (,Â•) x tfRkij , cg (t) v c = W v + (<)*>' x (O^i Â• For sphere (j), U) v c (i)V (,) x tfRkij (r 1f1 x (i) v c (j) v W) w x U) Kk ii Let v s = (j)V C ( j)V C and v^ = ^v^ ^)v' c be the relative velocities between the contact point ^C on sphere (i) and the contact point y)C on sphere (j) before and after the collision, respectively. Inserting (C.9) and (CIO) inAv s = v^ v s
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236 then using (C.7) and (C.8), we can express the change of the relative velocity of the contact points on the two spheres as Av, a)Tn u)R 2 (i)m d)R 2 A v ^ A (0 v + ^ Â— A 0) v (C.ll) 0)1 ~ vv U)I where A v is the change of the relative velocity of the centers of two spheres given by Av = ( (l) v' 0)V ')( w v y) v) (C.12) For homogeneous spheres, (j) m d) R2 (i) m (j)R 2 5 (Â•)/ = . Therefore, U) 1 2 Av = Av +^( A o) v A w v ) = ( 1 + ^) Av (C.13) where k 2 := . 5 Let v = (j)V y)V be the relative velocity of the two sphere centers before collision, and v' = ( i)v' (j )v' the relative velocity after collision. We decompose v and v' into components along the t and n directions (Figure C.l). With v n and v' n being the normal components of v and v' the normal coefficient of restitution is defined as (C.14) We define the tangential coefficient of restitution := u s,t V s ,t (C.15) where v' s>t and w 5jl are the tangential components of the relative velocities v^ and v s of the two spheres at the contact point C after and before the collision, respectively. Therefore, AVr, v n v n (1 + e) v n (C.16)
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237 and Av Stt = v' tti v Stt = (1 + 0) v aA (C.17) The component form of (C.13) then leads to (1 + 0) ty, = (l + ^)A t (C.18) For this hardsphere collision problem, there are two casessliding collision and sticky collisionthat may arise depending on the incident angle at the contact point, i.e., the angle V = tan 1 [ Â—2) When ib = 90, the initial relative velocity at the contact point is perpendicular to the tangent surface of the two spheres at the contact point C. We now discuss the two cases of sliding collision and sticky collision separately. (i) Sliding collision: The two spheres slide at the contact point C during the collision. By the impulse principle, the momentum change of sphere (i) caused by the collision can be expressed as A w v w m = f e {ji) F dt. (C.19) In component form, we have fte A(<)UÂ„ (j)ra = / (ji)F n dt > (C20) A (i)V t w m = / {ji) F t dt During a sliding collision, {ji) F t = H(ji)F n where // is the (constant) coefficient of friction for the two spheres in contact. It follows that A {i) v t = //A {i) v n (C.21) Similarly, for sphere (j), A {j) v t =n&(j)V n (C.22)
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238 Combining (C.21) and (C.22) together, yields Av t = (iA v n (C.23) From equations (C.16), (C.18), and (C.23), we obtain (1 + (3) v s t = (l + p) /" [(1 + e)] v n (C.24) i.e., the tangential coefficient of restitution (3 can be expressed in terms of the coefficient of friction /x, the normal coefficient of restitution e, the coefficient k 2 V defined earlier, and the angle of incidence ip (through the ratio Â— Â— ) v s ,t ,l+,
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239 Similar to the derivation of (C.24), we obtain I (1 + v,, t  < (l + 1) ft [(1 + e)} v n  (C.29) Therefore, for a sticky collision P< 1 + ^ 1 + e )( 1 + ^)^;( c 3 ) If consider only for special cases that when the vibrational period of the springmass system in the TFD model is the same as the vibrational period of the springmass system in the NFD model (similar to the cases for superball and superbean discussed in VuQuoc et al. [1997], in these cases, (5 = e), the criterion for a sticky collision to happen is l+//(l + e )(l + ^)^>A), (C31) where /3 = e, i.e., the normal coefficient of restitution of the collision. On the other hand, (C.31) does not imply that the tangential coefficient of restitution (3 for a sticky collision is less or equal to O Examples for collisions of special materials and ordinary (not special) materials are given in Sections 5.5.1 and 5.5.2. Remark C.l. As discussed in VuQuoc et al. [1997] regarding the superball and the superbean tests, when the vibrational period of the springmass system in the TFD model is the same as the vibrational period of the springmass system in the NFD model, we have the following relationship K T I K N ~ Io+mR* (C 32) where K T and K N are stiffness coefficients in the tangential and in the normal directions, respectively. I the moment of inertia of the sphere, m the sphere's
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240 mass, and R the sphere's radius. For a homogeneous sphere, Iq = m/? 2 and therefore !H Â• (c 33) We call this type of materials that have (C.33) as special materials 8 We refer the readers to VuQuoc et al. [1997] for more details on a related discussion (also see the discussion in Appendix B). I We note that (3.25) or (4.8) for elastic contact is not to be applied to (C.33), since it would yield a Poisson's ratio of v = 0.83333 > 0.5.
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APPENDIX D DERIVATION OF IMPORTANT HERTZ FORMULAE Except for the additive decomposition of the radius of elastoplastic contact, which is novel, the present elastoplastic NFD model begins with Hertz theory. Here, we present the detailed derivation of some important Hertz formulae, which is not given explicitly in Johnson [1985]. For two solids with smooth surfaces in contact to each other, the gap h between the undeformed solids can be given by (Johnson [1985, Eq.(4.3)]) h = Ax 2 + By 2 (D.l) For solids of revolution, and by (3.2), we have 1 ( 1 1 \ 11 Â„ Now consider Figure D.l that shows the contact region of the two solids in contact. Let 51 and S 2 be two points on the surface of the two contacting solids, such that these points have the same x and y coordinates. At those points, we can make the following statement: The sum of the material deformations (u zi + u z2 ) and the gap h must be equal to the sum of the displacement of the two distant points Ti and T 2 (Figure D.l), denoted by (^a + ( 2 )&), i.e., u z i +u z2 + h = ( i)a + (2)a (D.3) Substituting h in (D.3), using (D.l) and (D.2), we obtain u z i+u z2 = ( i)tt + ( 2 )Oj ^ r2 > (D.4) where r 2 = x 2 + y 2 Equation (D.4) corresponds to Johnson [1985, Eq.(4.17)]. 241
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(2) 242 (i) a T t Figure D.l. Two solids in contact, Johnson [1985, p. 88]. The displacement in ^direction caused by a Hertz pressure (3.3) is expressed by Johnson [1985, Eq. (3.41a)] as u, Â— 1 Â— v 2 np m /r, 2 2 E 4a (2a 2 r 1 ) r
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243 which is Johnson [1985, Eq.(4.18)]. Evaluating (D.7) at r = 0, we obtain irp m a (1) a + (2) a = ^, which is Johnson [1985, Eq.(4.20)]. Evaluating (D.7) at r = a, we obtain np m a a 2 ^ = (D + (2) ^ Substituting (D.8) into (D.9), we have a = nPmR* IE* which is Johnson [1985, Eq.(4.19)]. Substituting (3.5) into (D.10), we obtain (3.6), i.e., a = (3PR*\ 1/3 \~Je~* which corresponds to Johnson [1985, Eq.(4.22)]. Dividing (D.8) by (D.10), we obtain (1 )Q + (2)0 = Â— which corresponds (3.7) and Johnson [1985, Eq.(4.23)]. (D.8) (D.9) (D.10) (D.ll) The boxed equations are important for the construction of present elastoplastic FD models.
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APPENDIX E EXTRACTION OF MECHANICAL PROPERTIES OF GRANULAR MATERIALS Basic mechanical properties of some materials can be measured using traditional experiments. For example, the yield stress of steel and aluminum can be measured by uniaxial tests on steel bars and aluminum bars, respectively. The yield stress of a steel bead, however, may not be the same as the yield stress measured from uniaxial test on a steel bar. The reason is that the micro structure of steel beads may be altered by the processing that made the steel bead. For some other materials that are inherently heterogeneous, such as soybeans (Figure E.l), it is not possible to find the yield stress, although a questionable value of Young's modulus can be found in ASAE [1996] standards. The soybean has a structure which includes the embryo and seed coat. Figure E.l (a) from Aldrich and Scott [1983] shows the interior structure of a soybean. The embryo is composed of two cotyledons, and the hypocotylradicle axis which rests in a shallow depression formed by the cotyledons. The cotyledons make up the majority of the soybean by volume and weight. The seed coat is composed of four layers: the epidermis, the hypodermis, the parenchyma layer, and the endosperm remains. Figure E.l(b) from American Society of Agronomy, Inc. [1973] shows the structure of soybean seed coat. In our own experiments on soybeans (LoCurto et al. [1997]), it is difficult to determine the yield stress based solely on the departure of the measured FD relation from the Hertz contact theory. Any attempt to guess the yield stress this way would unlikely lead to a good agreement between the normal FD (NFD) model and experimental measurements (for both the NFD relation and the coefficient of restitution). Moreover, it is not possible to fabricate barshaped specimens out of soybean materials to perform uniaxial tests to determine the yield stress, because of the inherent inho244
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245 Cotyledon h *d CMC (a) The interior structure of a soybean. (b) The structure of soybean coat. Figure E.l. The structure of a soybean and soybean coat. mogeneity of soybean material. Even in the case where such barshaped specimens can be fabricated, it is not clear that the specimens and the measurements produce appropriate values for the yield stress, which when used in our NFD model would produce accurate model for granular flow simulations. Of course, the yield stress is a critical parameter in our NFD model for the computation of the amount of plastic deformation and the amount of energy dissipation (coefficient of restitution) in the collision of the flowing particles. To apply the new elastoplastic NFD model and TFD models in simulations of granular flows, we need to know that Young's modulus E, Poisson's ratio u, and yield stress oy in the granular material, and other parameters particular to these new FD models. Due to the difficulty in obtaining these mechanical properties for many granular materials (e.g., soybean) as discussed above, we propose here a very simple procedure to extract these mechanical properties using the new NFD model of VuQuoc and Zhang [19986] and based on simple experiments on the granular materials. Here, soybeans are used as an example to illustrate the proposed procedure.
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246 E.l. Experimental Measurements To obtain all five parameters (E, v, ay, C a K c ) for use in the new NFD model for a given granular material, we propose to use two simple experiments: the singleparticle drop test and the singleparticle compression test. The singleparticle drop test is designed to measure the coefficient of restitution of a particle colliding against a plate made of a certain material (LoCurto et al. [1997]). In this experiment, particles are dropped vertically from a height H to collide with a flat plate mentioned above. A hard surface is used when a coefficient of restitution of two particles made of the same material is to be measured, due to symmetry (see VuQuoc and Zhang [19986]). To compute the coefficient of restitution e, the rebounding height H' is measured so that where v is the incoming velocity of the particle before impacting the flat plate and v' the rebounding velocity of the particle right after impact. The negative sign in (E.3) indicates the velocities v and v' are in opposite directions. Several values of the drop height H are used. We performed more than 400 singlesoybean drop tests. After we selected the best test results, the coefficient of restitution at each height represents the averaged value over about seven (7) data points. These test results have been discussed and published in LoCurto et al. [1997]. The data shown in Table ?? thus represent more than 21 experimental data points. In this experiment, single soybeans are dropped to fall vertically down, and collide against an aluminum plate (Figure E.2). The height at the end of the rebounding stage are obtained from highspeed video recording. The incoming velocities at impact are computed from the initial drop heights of the soybeans. Figure E.3 depicts a singleparticle compression test. In this test, a particle is
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247 Bean Holder sD bean Plastic wall with scale Aluminum Plate A X w II 1Vacuum Pump Â— .... '; iy'l Figure E.2. Sketch of the singleparticle drop test apparatus. Table E.l. Results from more than 21 singlesoybean drop tests Dropping Height (m) Impact Velocity (m/s) Coefficient of Restitution 0.148 1.705 0.71 0.289 2.381 0.70 0.508 3.156 0.69 placed between two very hard surfaces that are parallel to each other. If the particle is symmetrically compressed, then the normal displacement at one contact point is half of the approachment of the two parallel surfaces. The relationship between the normal force P and the normal displacement a for the compressed particle can be obtained by applying different magnitudes of the normal force P and by recording the corresponding normal displacements. Figure E.4 shows the curvefitting result from the single soybean compression test.
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248 ii P Figure E.3. Singleparticle compression test. E.2. Optimization Algorithm Assume that for certain particles the radius of curvature R at the contact point can be measured. Let ml be the number of data points from the single particle compression tests, and ra2 the number of data points from the singparticle drop tests. The experimental values for the normal force P and normal displacement a coming from the compression tests are denoted by P exp ,i, P e x P ,2, Â•Â•Â• % P e x P ,mi and c*exp,i, Â£*exp,2i Â•Â• exp,miThe experimental values of the incoming velocity v in and of the coefficient of restitution e are denoted by v in ^, i>i ni 2, Â•Â•Â• fin,m2 and ei, e 2 Â•Â•Â• ,e m 2. To simulate the compression tests using the elastoplastic NFD model proposed in VuQuoc and Zhang [19986] (see Chapter 6), we need the five parameters already mentioned above (E, v, P Y C a K c ). These parameters are determined by a least square fit through the experimental data points. We first select the trial values of the five model parameters denoted as E try v try Py y C l J y and K* y Based on the VuQuoc and Zhang [19986] NFD model, we then define the first objective function to be minimized in the leastsquare sense
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249 1.5 % u 1 cd Â£ 0.5 \ U C / / /: /" / / / /' / ; .}'. i. / : /' /* / i / Â•/ / s' '. / / ; ; V ; ; y' : : Â• ': : s' s' ; **' i i i i i 0.005 0.01 0.015 0.02 0.025 0.03 Single Side Compression a (xlO 3 m) Figure E. 4. Curvefitting result of soybean compression test: Function P = 70.24 x (2 x 10 3 a) 1 318 N (a is in m). as follows ml ?* = Ek
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250 collision be v^ y ti According to a definition of the coefficient of restitution, we have try e try ._ _^out Â£ (E 3) Vin,i Therefore, the second objective function to be minimized in the leastsquare sense is defined as m2 K = E [4 y (v in ,u E*y, v*\ Pp, (Â£, K?) e$ (E.4) t=i where the trial coefficient of restitution ef y is a function of the incoming velocity Vi nt i, and (implicitly) of the trial values of the model parameters E tTy u try P Y ry 0?*, and K*T*. The objective of the optimization problem is to find the values of E, u, P Y C a and K c such that the objective function T := uj a T' a (P Y C a K c ) + u e T' e (P Y C a K e ) (E.5) is minimized. In (E.5), u a and u e are the weighting coefficients, and T' a and T' e are the objective functions related to T a and T<,, normalized to the same magnitude level. For example, one way to normalize these functions is j, Fg{E,l',PY,C a ,K c ) a f Q {E t >y,vtry,py,cl ry ,K? y ) [ j and T' F e {E,v,PY,C a ,K c ) e ^{E^y.u^y^Pp.C^^K^) [ There are several possible optimization algorithms that can be employed. In the present paper, we employ the method of steepest descent to drive down the objective function T along the opposite direction of its gradient. A main feature of the present work is that the objective function T in (E.5) is not an explicit function
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251 of its arguments, and thus its gradient has to be evaluated numerically. Assume that at the nth iteration the values of the objective function and the model parameters are T 71 E n z/\ Py, C", and K"; one step of the steepest descent method is as follows: At first, change each model parameter by a very small amount, then calculate the corresponding value of the objective function as follows f T n E = T((l + 6)E n ,v n ,P?,C2,K?) jT" = ?(Â£*, u n (1 + 5)P?, C a n K?) (E.8) n = ?(E,u n ,P?,{l + 6)C% t K2) where S is very small compared to 1. The gradient of the objective function with respect to the relative change of the model parameters can be approximated by dT T% T n ^NJ ** Â— r^j Â— dE dT dP Y dT dC a dT dK c ~ 6 The reason for calculating the gradient with respect to the relative change, instead of the absolute change, of the model parameters is that the values of the model parameters are in quite different levels of magnitude. For example, the Young's modulus of an aluminum alloy is about 7.0 x 10 8 N/m 2 while the Poisson's ratio is less than 0.5. One can think of the approximation in (E.9) represent the numerical gradient of the objective function with respect to the normalized model parameters. 5 FZ Â— F 1 S JP T n 6 n Â— T n 6 n Â— T n (E.9)
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252 dT In (E.9), the whole fraction Â—=; should be viewed as a complete symbol, and is oE defined in relations (2.26). In other words, the relationship between Â— a; and the ~_ dE actual gradient Â— Â— (with no overbar on E) is oE dE ::_ 8E (E.10) The same holds for the other quantities. Therefore, the new values of the model parameters for the next iteration are determined by 6E n dT E 7Z+1 ,n+l pra+1 r Y c: +l K n+l Â— Ill 9 dE = u n 9 dT dv Â— r Y 5P$ 9 dT dP Y = c%9 dT dC a K n SKI 1 dT (E.ll) Q dK c where Q is the norm of the numerical gradient defined as 9 = 'dry (dry / ar dE) + \dD) + {dP Yi fdTY + Ua + dT dK c 1/2 (E.12) The value of objective function for iteration (n+1) is therefore computed using the new values of the model parameters E n+ \ v n+ \ PÂ£ +1 CÂ£ +1 and K? +1 as follows pn+i = jr^E n+ \u n+ \P^ + \C^ + \K^ +x ) (E.13) When the difference between T n+1 and T n is very small, the optimization process is regarded as converged, i.e., when \T n+1 \
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253 1.9 1 8 1.7 "^ 1.5 1 4 1.3 i ; 1 2 i 1 1 1 1 i 1 i 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 V Figure E.5. Curve of A Y {y) vs. v for the relationship between P Y and a Y where e is a very small positive number. In this case, we set E = E n+ \ v = v n+ \ P Y = PÂ£ + \ C a = C a "+\ and K c = K" +1 (E.15) to be the values that minimize the objective function (E.5). According to Hertz theory with the von Mises yield criterion, the relationship between the yield stress a Y and the yield load P Y can be calculated using (6.19). For the contact we discussing here, we write (6.19) as follows 7T 3 R 2 (1 v 2 f Pi 6Â£ 2 \A Y {v) o Y \ (E.16) where A Y (v) is a constant dependent on the Poisson's ratio v (see VuQuoc, Zhang and Lesburg [1998]). Figure E.5 displays the plot of A Y {v) vs. v. For particles with some known properties such as E and/or v, the number of model parameters in the optimization process can be reduced. The pseudocode for the implementation of the above steepest descent algorithm, with five model parameters E, u, P Y C a and K c are presented as below.
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254 Algorithm D.l. Leastsquare extraction of elastoplastic model parameters by steepest descent method. ; Goal : Find E, vPy, C a and, K c that minimize the objective function (Â£7.5). 2 Data : Particle radius R, and mass m. 3 Data : Experimental data : P exp ,i, P exp ,2 Â• P e x P ,mi, and a e x P ,u o> exp ,2 "exp.mi4 Data : Experimental data : v iny i,v int 2 Â• w m m 2, and ei,e 2 Â• Â• Â• e m2 5 Initial guess : E try v try Py ry C^ K^ e Optimization parameters : u a ,uj e ,e, and 8. 9 Calculate J* using E try v tTy ,Py y C*v, K** via (Â£7.5). 10 Set E M = E try ; n Set u old = u try ; 12 Set P Y ld = P[J y ; is Set Cf d = C*; 14 Set K ld = K**\ is Set P M = PTy 16 Set converg = false. is while converg = false, 19 Calculate E new v new P^ ew CÂ™ w and KÂ™ w via (E.ll). 20 Calculate T ntw with E new u new PÂ™ w CÂ™ w KÂ™ w via (E.b). 21 if T new > T old 22 Shrink the step size by 23 Set 8 = 0.8 5; 24 Set e = 0.8 e; 25 elseif T new < T old 26 jf (Â£7.14) is true, 27 Set converg = true. 28 (D elseif (Â£7.14) is false,
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255 29 bet Â£,""* = jb'""" ; 30 Set v old = v new ; SI Set PÂ§! d = P$ ew ; 32 Set C ld = CÂ™ w ; 33 Set K ld = KÂ™ w ; 34 Set F M = T new 35 endif 36 endif 37 endwhile 39 Set E = jpnew. 40 Set v = u new \ 41 Set P Y = P$ ew ; 42 Set C a = CÂ™ w ; 43 Set K c = KÂ™ w ; 45 Calculate ay via (6.19). E.3. Numerical Example: Application to Soybeans E.3.1. Extraction of Mechanical Properties The experimental data shown in Table E.l and Figure E.4 are employed to carry out an extraction of the mechanical properties of soybeans. Recall that the curve in Figure E.4 was obtained from a least square fit of experimental data from singleparticle compression tests. For the extraction of soybean's mechanical properties, we select ten points on this curve for use as experimental data. It should be noted that one could also use the actual experimental data points. The shape of a soybean can be approximated by an ellipsoid with the following averaged measured
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256 half principal axes: length I = 3.65 x 1G 13 m, width w Â— 3.05 x 10 3 m, and height h = 2.75 x 10 3 m. The compression tests were performed on the "flat" side of soybeans, i.e., in the direction of the height h, i.e., the shortest principal axis. As a result, the average radius of curvature at the contact point for these compression tests is 111) R = Â— = 4.05 x 10" 3 m (E.17) h This average radius of curvature is used for the extraction of the mechanical properties of soybeans. The average mass of a single soybean at moisture content of 10.7 % (d.b.) 9 from our measurement is m = 1.49 x 10 4 kg. The Young's modulus and Poisson's ratio of soybeans given in the ASAE [1996] are E = 126 MP a = 1.26 x 10 8 N/m 2 and v = 0.4, respectively. The conditions such as the moisture content of the soybeans under which the above data are obtained are, however, not given. Since the stiffness of soybeans highly depends on the moisture content, the values from the ASAE [1996] may not be the Young's modulus and Poisson's ratio of soybeans under the condition when the experiments were carried out. In the present work, the values of E and v represent values averaged over the whole soybean, and are part of a larger set of parameters designed to produce accurate contact elastoplastic normal forcedisplacement (NFD) model compared to experiments. To compare the optimal model parameters, we carried out the optimization as presented in Section E.2, with the following trial values: E try Â— 126 MPa = 1.26 x 10 8 N/m 2 v = 0.4, P? y = 0.40 N, C%* = 2.00 x lO" 5 m/N, and Kl ry = 3.00 x 10~ 6 1/JV, and with the convergence tolerance e = 1.0 x 10~ 3 The value for the weighting coefficients in the objective function T are set to: u> a = 0.1 and u e = 0.9. The optimization process converged after 18 iterations, with 9 Dry basis, i.e., the percentage of the weight of moisture to the weight of the dry material.
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257 E (N/m 2 ) (a) T vs. E. (b) T vs. v. Figure E.6. Convergence of objective function. from the experimental data are < 2.373 x 10" 4 The final values of E, v, P Y C a and K c extracted E = 1.288 x 10 8 N/m 2 = 128.8 MPa v = 0.4134 (E.18) and P Y 0.3952 N C a = 2.202 x 10~ 5 m/N K c = 2.970 x 10~ 6 1/N (E.19) The corresponding yield stress o Y of soybeans calculated using (6.19) is a Y = 2.750 x 10 6 N/m 2 = 2.750 MPa (E.20) Recall that due to the inhomogeneity inside a soybean (see Figure E.l), the values of E, u, ay are of some average sense, and together with the other model parameters (C a and K c ) provide a good correlation between the new NFD model of VuQuoc and Zhang [19986] and experimental data. Note that for an aluminum alloy, we have E = 70,000 MPa and a Y = 100 MPa (see VuQuoc and Lesburg [1998] for finite element analysis of contact problems involving spheres of aluminum alloy).
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258 Â§ * Â• Â•* 4* CJ 90d I
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259 Figures E.6(a), E.6(b), E.7(c), E.7(a), and E.7(b) show the convergence of the optimization process, i.e., the decrease of the objective function with the changes in the model parameters E, v, Py, C a and K c respectively. Even though the curve shown in Figure E.6(a) displays some oscillations in the values of E in the final stage of the convergence of the objective functions, which continues to decrease. All other figures display a monotonic convergence of the model parameters (other than E). E.3.2. Simulation Results Using the VuQuoc and Zhang [19986] NFD model, with the extracted elastoplastic properties of soybeans as shown in (E.18) and (E.19), we generate the NFD relation during compression of a soybean against a rigid surface. In addition, we simulate the single soybean drop tests using DEM based on the VuQuoc and Zhang [19986] NFD model. 1.5 E = 1.288 x 10 8 N/m 2 v = 0.4134 / Py = 0.3952 N ,a Y = 2.750 x 10 6 N/jm 2 / C a = 2.202 x 10~ 5 m/N K c = 2.970 x 10~ 6 l/N NFD model Hertz + Experiment Figure E.8. Comparison of simulation results with experimental data: Normal force P vs. normal displacement a.
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260 1.05 1 99 a .2 0.95 [ 0.9EC i. Pi 0.85 Ch 0.8 U &go.75O O 0.7 h 0.65 1.6 1 1 1 I [ 1 NFD model Experiment Hertz x """i^ 1 1 1 1 1 X i 1.8 2 2.2 2.4 2.6 2.8 3 Incoming Velocity Vi n (ra/s) 3.2 Figure E.9. Comparison of simulation results with experimental data: Coefficient of restitution e vs. incoming velocity Vi n Figure E.8 shows a comparison of the NFD relations (P vs. a) coming from various sources. The dotted line is the curve fitting result from the experimental data. The '+' sign on the dotted line are those data points that were used for the optimization. The NFD curve produced by the VuQuoc and Zhang [19986] NFD model, with the elastoplastic properties shown in (E.18) and (E.19), is represented by the solid line in Figure E.8. Simulation results thus agree well with experimental data. The dashed line is the NFD curve produced by Hertz theory for elastic contact, which is stiffer than the results for elastoplastic contact. Figure E.9 shows the coefficient of restitution e vs. the incoming velocity Vi n obtained from the DEM simulation of single soybean dropping tests using the VuQuoc and Zhang [19986] NFD model, with the extracted properties of soybeans in (E.18) and (E.19). Points represented by the '*' symbols are the experimental data used for our optimization. The dashed line represents the results from our simulation. The error on the coefficient of restitution is only about 5%, which is very
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261 small when compared to experimental data. The model thus agrees quantitatively well with experiments. The difference may come from various reasons, one of which is experimental error. Another reason could be the model itself. On the other hand, we have shown in VuQuoc and Zhang [19986] that the new NFD model out performs existing models, even for the coefficient of restitution vs. incoming velocity. For Hertz theory, due to elastic contact, the coefficient of restitution remains at e = 1.0 for all magnitude of the incoming velocity. Figures E.10(a), E.10(b), and E. 10(c) show the relation between the normal contact force P and the normal displacement a, as obtained from the simulated drop tests with incoming velocities v in = 1.705 m/s, v in = 2.381 m/s, and v in = 3.156 m/s, respectively. From these results, we observe that the highest normal contact force level depends on the magnitude of the incoming velocity. The higher the incoming velocity, the larger the plastic deformation, and hence larger the energy dissipation caused by the plastic deformation during the collision. Therefore, the results agree with experimental observation (see Goldsmith [I960]) qualitatively.
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262 iW = 1.296 (m/s) e S im = 0.7457 a (m) P alpha curve for collision: Vln 2.381 (nvsoc) 1 1 vent = 1702 {m/s) e sim = 0.7009 e exp = 0.70 r 02 04 0.6 08 a (m) (a) NFD relation from simulation: Incoming (b) NFD relation from simulation: Incomvelocity Vi n = 1705 (m/s). ing velocity Vi n = 2.381 (m/s). P alpha curve lor collision: Vin = 3.156 (m/sec) v out = 2.131 (m/s) e S im = 0.6615 (c) NFD relation from simulation: Incoming velocity Vi n = 3.156 (m/s). Figure E.10. Forcedisplacement curves for the collisions.
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REFERENCES ABAQUS [1995], Version 5.61, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI. Aldrich, S. R. and Scott, W. O. [1983], Modern Soybean Production, S&A Publications, Inc., Champaign, IL. Allen, M. P. and Tildesley, D. J. [1987], Computer Simulation of Liquids, 1st ed, Clarendon Press, Oxford, England. American Society of Agronomy, Inc. [1973], Soybeans: Improvement, Production and Uses, American Society of Agronomy, Inc., Madison, WI. ASAE [1996], Compression test of food material of convex shape, in ASAE Standards 1996, ASAE, pp. 500504. Baraff, D. [1995], Interactive simulation of solid rigid bodies, Computer Graphics 15, 6375. Bishara, A. G., ElAzazy, S. S. and Huang, T.D. [1981], Practical analysis of cylindrical farm silos based on finite element solutions, ACI Journal 78, 456462. Cao, Y. [1996], An efficient detection algorithm for arbitrary rigid shapes, Master's thesis, University of Florida, Gainesville, FL. Cattaneo, C. [1938], Sul contatto di due corpi elastici: distribuzione locale degli sforzi, Accademia dei Lincei, Rendicotti 27(6), 342348. Cohen, J. D., Lin, M. C, Manocha, D. and Ponamgi, M. K. [1994], Interactive and exact collision detection for largescaled environments, Technical report, Computer Science Department University of North Carolina, Chapel Hill, NC. NSF MIP9306208. Cundall, P. and Strack, O. [1979], A discrete numerical model for granular assemblies, Geotechnique 29(1), 4765. Ennis, B. J., Gree, J. and Davies, R. [1994], Partical technology: The legacy of neglect in the U.S., Chemical Engineering Progress 90, 3243. Evans, D. J. and Murad, S. [1977], Singularity free algorithm for molecular dynamics simulation of rigid polyatomics, Molecular Physics 34(2), 327331. Goldsmith, W. [1960], Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold, Ltd., London. Goldstein, H. [1981], Classical Mechanics, 2nd ed, AddisonWesley Publishing Company, Reading, MA. Gurtin, M. E. [1981], An Introduction to Continuum Mechanics, Academic Press, San Diego. 263
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264 Hertz, H. [1882], Uber die Beriihrung fester elastischer Korper (On the contact of elastic solids), J. Reine und Angewandte Mathematik 92, 156171. Hockney, R. W. and Eastwood, J. W. [1988], Computer Simulation using Particles, Institute of Physics Publishing, Bristol and Philadelphia. Johnson, K. L. [1985], Contact Mechanics, 2nd ed, Cambridge University Press, New York. Kalker, J. J. [1990], ThreeDimensional Elastic Bodies in Rolling Contact, Kluwer Academic Publishers, Boston. Lemaitre, J. and Chaboche, J. L. [1990], Mechanics of Solid Materials, Cambridge University Press, New York. Lesburg, L. [1997], Toward a realistic and consistent forcedisplacement model for discrete element simulation of dry particle systems, Master's thesis, University of Florida, Gainesville, FL. Lin, X. and Ng, T.T. [1994], Numerical modeling of granular soil using random arrays of three dimensional elastic ellipsoids, in 8th International Conference on Computer Methods and Advances in Geomechanics, Vol. 1, Morgantown, WV. LoCurto, G. J. [1997], Friction effects on chute flow of soybeans, Master's thesis, University of Florida, Gainesville, FL. LoCurto, G. J., Zhang, X., Zakirov, V., Bucklin, R. A., VuQuoc, L., Hanes, D. M. and Walton, O. R. [1997], Soybean impacts: Experiments and dynamic simulations, Transactions of the American Society of Agricultural Engineering (ASAE) 40(3), 789794. Lu, Z., Negi, S. C. and Jofriet, J. D. [1995], 'A hybrid FEM and DEM model for numerical analysis of granular material flow', An ASAE meeting presentation, Paper No. 954450. Lubliner, J. [1990], Plasticity Theory, Macmillan Publishing Company, New York. Malvern, L. [1969], Introduction to the Mechanics of a Continuous Medium, PrenticeHall, Inc., Englewood Cliffs, New Jersey. Maw, N., Barber, J. and Fawcett, J. [1976], The oblique impact of elastic spheres, Wear 38(1), 101114. Mindlin, R. D. [1949], Compliance of elastic bodies in contact, ASME Journal of Applied Mechanics 16, 259268. Mindlin, R. D. and Deresiewicz, H. [1953], Elastic spheres in contact under varying oblique forces, ASME Journal of Applied Mechanics 20, 327344. Mishra, B. K. [1995], Ball charge dynamics in a planetary mill, KONA Powder and Particle 13, 151158.
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265 Ning, Z. and Thornton, C. [1993], Elasticplastic impact of fine particles with a surface, in C. Thornton, ed., Powders & Grains, Balkema, Rotterdam, Netherlands, pp. 3338. Simo, J. C. and Wong, K. K. [1991], Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum, International Journal for Numerical Methods in Engineering 31(1), 1952. Thornton, C. [1997], Coefficient of restitution for collinear collisions of elastic perfectly plastic spheres, ASME Journal of Applied Mechanics 64, 383386. Timoshenko, S. P. and Goodier, J. N. [1970], Theory of Elasticity, 3rd ed, McGrawHill, New York. Ting, J. M., Khwaja, M., Meachum, L. R. and Rowell, J. D. [1993], An ellipsebased discrete element model for granular materials, International Journal for Numerical and Analytical Methods in Geomechanics 17(9), 403423. Tsuji, Y., Tanaka, T. and Ishida, T. [1992], Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe, Powder Technology 71, 239250. VuQuoc, L. and Lesburg, L. [1998], Contact forcedisplacement relations for spherical particles accounting for plastic deformation. To be submitted. VuQuoc, L. and Zhang, X. [1998a], An accurate and efficient tangential forcedisplacement model for elasticfrictional contact in particleflow simulations, Mechanics of Materials. To appear. VuQuoc, L. and Zhang, X. [19986], An elastoplastic contact forcedisplacement model in the normal direction: Displacementdriven version, Proceedings of the Royal Society: Series A. Submitted. VuQuoc, L. and Zhang, X. [1998c], A new tangential forcedisplacement model for elastoplastic frictional contact: Displacementdriven version, Proceedings of the Royal Society: Series A. Submitted. VuQuoc, L., Lesburg, L. and Zhang, X. [1998], A tangential forcedisplacement model for contacting spheres accounting for plastic deformation: Forcedriven formulation, Journal of the Mechanics and Physics of Solids. In preparation. VuQuoc, L., Zhang, X. and Lesburg, L. [1998], A normal forcedisplacement model for contacting spheres accounting for plastic deformation: Forcedriven formulation, ASME Journal of Applied Mechanics. Submitted. VuQuoc, L., Zhang, X. and Walton, O. R. [1997], A 3D discrete element method for dry granular flows of ellipsoidal particles, Computer Methods in Applied Mechanics and Engineering. Invited paper for the special issue on Dynamics of Contact/Impact Problems. To appear. Walton, O. R. [1993], Numerical simulation of inelastic, frictional particleparticle interactions, in M. C. Roco, ed., Particulate TwoPhase Flow, ButterworthHeinemann, Stoneham, MA, chapter 25, pp. 884911.
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266 Walton, O. R. and Braun, R. L. [1986], Viscosity, granulartemperature, and stress calculations for shearing assemblies of inelastic, frictional disks, Journal of Rheology 30(5), 949980. Walton, O. R. and Braun, R. L. [1993], Simulation of rotarydrum and repose tests for frictional spheres and rigid sphere clusters, Technical report, Lawrence Livermore National Laboratory, Livermore, CA. UCRLJC115749. Walton, O. R., Brandeis, J. and Cooper, J. M. [1984], Modeling of inelastic, frictional, contact forces in flowing granular assemblies, in 21st Annual Meeting of The Society of Engineering Science, Blacksburg, VA, p. 257. Williams, J. R. and O'Connor, R. [1995], A 3D representation scheme for fast contact detection in multibody dynamics, Technical report, Intelligent Engineering Sysltems Laboratory (IESL) Department of Civil and Environmental Engineering, MIT, Cambridge, MA. Contract No: F2960191C0029. Yu, M. M.H., Moran, B. and Keer, L. M. [1995], A direct analysis of threedimensional elasticplastic rolling contact, Journal of Tribology 117, 234243. Zhang, X. and VuQuoc, L. [1998a], Computer simulation of granular flow using elastoplastic contact forcedisplacement models: With comparison to experiments, Granular Matter. Submitted. Zhang, X. and VuQuoc, L. [19986], A method to extract the mechanical properties of particles in collision based on a new elastoplastic normal forcedisplacement model, International Journal of Plasticity. Submitted. Zhang, X. and VuQuoc, L. [1998c], Modeling the dependence of the coefficient of restitution on the impact velocity in elastoplastic collisions, International Journal of Impact Engineering. Submitted. Zhang, X. and VuQuoc, L. [1998d], Simulation of chute flow of soybeans using an improved tangential forcedisplacement model, Mechanics of Materials. Submitted.
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BIOGRAPHICAL SKETCH Xiang Zhang was born on 11 June, 1966, in Jian, Jiangxi Province, P. R. China. He received his Bachelor of Engineering degrees in engineering mechanics and in civil engineering from Tsinghua University in July 1987. He continued at Tsinghua University with graduate studies, and received his Master of Engineering degree in engineering mechanics in Dec. 1989. During the years from 1990 to 1994, he worked as a structural engineer with Shougang Corp. in Beijing. Currently, he is pursuing the Doctor of Philosophy degree in engineering mechanics and the Master of Science degree in computer & information science at the University of Florida. He is expecting to receive his Doctor of Philosophy degree in Dec. 1998. His research interests include finite element method, computer simulation and visualization, contact mechanics, engineering structure design and analysis, and computer software development and application. 267
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. VuQj*9^A*t^W Ibrahim KTEbcioglu Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (LLiLL +fs ChenChi Hsu Professor of Aerospace Engineering, Mechanics, and Engineering Science
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sartaj K. Sahni Distinguished Professor of Computer and Information Science and Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1998 Winfred M. Phillips Dean, College of Engineering M. J. Ohanian Dean, Graduate School
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LD 1780 199& .Zu% UNIVERSITY OF FLORIDA 3 1262 08555 1249
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LD 1780 199& .Zu% UNIVERSITY OF FLORIDA 3 1262 08555 1249

