STUDY OF MICRO-SIZED PARTICLE DEPOSITION USING DEM, CFD-DEM AND SPH APPROACH

STUDY OF MICRO-SIZED PARTICLE DEPOSITION USING DEM, CFD-DEM AND SPH APPROACH A Thesis Presented to the Faculty of the Graduate School University of Missouri-Columbia In Partial Fulfillment of the Requirement for the Degree of Master of Science By RAIHAN TAYEB Dr. Yuwen Zhang, Thesis Supervisor JULY 016

The Undersigned, appointed by the Dean of the Graduate School have examine the thesis entitled STUDY OF MICRO-SIZED PARTICLE DEPOSITION USING DEM, CFD-DEM AND SPH APPROACH Presented By Raihan Tayeb A Candidate for the degree for the Masters of Science, and hereby certify that, in their opinion, it is worthy of acceptance Professor Yuwen Zhang Professor J K Chen Professor Stephen Montgomery-Smith

. All Praise and Thanks to the One and Only Almighty.

ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my supervisor, Dr. Yuwen Zhang, who guided me in my research and supported me both academically and financially from the first day at MU. Without his persistent guidance and support this thesis would not be possible. I would also like to thank my mentor Dr. Yijin Mao for walking me through the seemingly unknown and abysmal field of research. I would like to thank the members of my thesis committee, Dr. J. K. Chen and Dr. Montgomery-Smith for taking the time to provide valuable comments and criticism. I would like to express great gratitude to my parents whose love and encouragement help me overcome difficulties in study and life. I would also thank my friends and coworkers for all the wonderful time they shared with me. Support for this work by the U.S. National Science Foundation under grant number CBET-140448 is gratefully acknowledged. I would also like to thank the University of Missouri Bioinformatics Consortium (UMBC) for providing supercomputing resources. ii

TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii TABLE OF CONTENTS... iii LIST OF FIGURES... viii LIST OF TABLES... x NOMENCLATURE... xi ABSTRACT... xvi 1 Introduction... 1 1.1 Motivation and Objectives... 1 1. Softwares Used... 5 1..1 LAMMPS... 5 1.. LIGGGHTS... 6 1..3 OpenFOAM... 6 1..4 CFDEM... 7 Analysis of Cohesive Micro-Sized Particle Packing Structure Using History- Dependent Contact Models... 8.1 Introduction... 8. Equations of motion... 8.3 Contact forces... 9 iii

.3.1 Normal contact force... 9.3. Tangential contact force... 10.4 Van der Waals force... 11.5 Cohesive force... 1.6 Torques... 1.7 The integration scheme... 13.8 Numerical Stability... 13.9 Physical Model... 14.10 Material Properties... 19.11 Results and Discussion... 0.11.1 Porosity and Coordination Number... 0.11. Radial Distribution Function... 6.11.3 Force Distribution... 3.11.4 Comparison between contact models... 35.1 Conclusion... 36 3 Numerical Investigation of Evaporation Induced Self-Assembly of Sub-Micron Particles Suspended in Water... 37 3.1 Introduction... 37 3. Equations of motion... 37 3..1 Fluid Phase... 38 iv

3.. Solid Phase... 39 3.3 Drag Force... 39 3.4 Buoyant Force... 40 3.5 Contact Forces... 40 3.6 Energy Equation... 4 3.6.1 Fluid Phase... 4 3.6. Solid Phase... 4 3.7 Phase Change... 43 3.8 Particle Relaxation Time... 44 3.9 Algorithm... 44 3.10 Physical Model... 45 3.11 Material Properties and Simulation Parameters... 46 3.1 Results and Discussion... 48 3.13 Conclusion... 54 4 Numerical Simulation of Jamming Transition in Granular System under Cyclic Compression using Smoothed Particle Hydrodynamics... 55 4.1 Introduction... 55 4. Equations of Motions... 57 4.3 SPH Formulation... 58 4.4 Artificial Viscosity... 60 v

4.5 Tensile Instability... 61 4.6 Velocity Smoothing... 6 4.7 Equations of State... 63 4.8 Contact Model... 64 4.9 Contact Force... 65 4.10 Contact Detection... 65 4.11 Validation... 67 4.11.1 Impact of two identical disks... 67 4.1 Elasto-Plastic Theory... 70 4.1.1 The Radial Return Method... 7 4.1. Newton Raphson Method... 76 4.1.3 Implementation: Point load on a disk... 79 4.13 Boundary Treatment... 80 4.14 Physical model... 81 4.15 Simulation Parameters... 83 4.16 Results and Discussion... 84 4.16.1 Coordination Number and Packing Fraction... 84 4.16. Global Pressure Response... 85 4.16.3 Distribution of Contact Forces... 86 4.17 Conclusion... 87 vi

5 Summary and Future Work... 88 5.1 Summary... 88 5. Future Work... 90 5..1 Theoretical Development... 90 5.. Simulation Technique... 9 References... 93 VITA... 10 vii

LIST OF FIGURES Figure -1 Initial and final structure for Gaussian particles from Modified Gran-Hertz- History model with cohesion...15 Figure - Initial and final packing structure for mono-sized particles from Modified Gran-Hertz-History model with cohesion...16 Figure -3 Initial and final packing structure for uniform size particles from Modified Gran-Hertz-History with cohesion...17 Figure -4 Initial and final structure for Gaussian particles from Gran-Hooke-History model with cohesion...18 Figure -5 Effect of porosity with particle size and distribution...5 Figure -6 Effect of coordination number with particle size and distribution...6 Figure -7 RDF for particles with 75 μm radius 7 Figure -8 RDF for particles with 85μm radius 8 Figure -9 RDF for particles with 100 μm radius..9 Figure -10 RDF for particles with 110μm radius 30 Figure -11 RDF for particles with 10μm radius 31 Figure -1 Force distribution for particles with 75 μm radius and Gaussian distribution.33 Figure -13 Force distribution for particles with 75 μm radius and mono-size distribution.34 Figure -14 Force distribution for particles with 75 μm radius and uniform distribution.35 Figure 3-1 Fluid and particle configuration at time t = 0 s.46 Figure 3- Volume fraction alpha1 at (a) 0 s (b) 0.59 s (c) 1.39 s and (d).04 s..48 Figure 3-3 Arrangement of particles at (a) 1 s (b) 1.9 s (c).0 s and (d).1 s 49 Figure 3-4 Streamlines of resultant implicit forces on particles 50 viii

Figure 3-5 Distribution of coordination number at (a) 1 s (b).05 s (c).135 s and (d).167 s 51 Figure 3-6 Radial distribution function (RDF) at (a) 1 s (b).05 s (c).135 s and (d).167 s 53 Figure 4-1 The support domain of the smoothing function, W..57 Figure 4- The boundary particles indicated by color gradient, 67 Figure 4-3 Pressure distribution of two colliding disks.68 Figure 4-4 Total contact force-time histories of the impacting disks 69 Figure 4-5 Average velocity-time history of the two colliding disks 69 Figure 4-6 Hydrostatic pressure distribution 80 Figure 4-7 True stress against true strain for a point on the ball 80 Figure 4-8 One full cycle of compression and expansion showing also the force chain networks.8 Figure 4-9 A closer look of the disks and force chains..83 Figure 4-10 Average coordination number vs packing fraction 85 Figure 4-11 Global pressure vs packing fraction..85 Figure 4-1 PDF vs resultant contact force...87 ix

LIST OF TABLES Table -1 Values of the parameters used in the simulation process..19 Table - Porosity and coordination number for Modified Gran-Hertz-History model...0 Table -3 Porosity and coordination number for Gran-Hooke-History model..1 Table -4 Porosity and coordination number for Modified Gran-Hertz-History model without cohesion 1 Table -5 Porosity and coordination number for Gran-Hooke-History model without cohesion. Table -6 Magnitude of mean net contact force (N) for Modified Gran-Hertz-History model.. Table -7 Magnitude of mean net contact force (N) for Gran-Hooke-History model...3 Table 3-1 Simulation parameter and properties 46 Table 4-1 Material Properties...63 Table 4- Simulation Parameters.83 x

NOMENCLATURE d diameter of particle, m e coefficient of restitution g gravity, m/s m mass of particle, kg v velocity, m/s F force on particle, N I moment of inertia, kg m R radius of particle, m T torque, N m X position vector, m Y Young s modulus, Pa As spherical area, m c damping coefficient, s cp specific hea, J/Kg K C Cunningham correction factor Ce evaporation coefficient d diameter of particle, m D distance, m e coefficient of restitution E elastic modulus, Pa F force on particle, N g gravity, m/s xi

h heat transfer coefficient, W/m K he enthalpy of evaporation, J/kg I moment of inertia, kg m kf thermal conductivity of fluid, W/m K ks thermal conductivity of particle, W/m K m mass of particle, kg M molecular mass of water, kg n unit vector, m Nu Nusselt number p pressure, Pa Pr Prandtl number Q power, W r position vector, m R position vector, m R radius of particle, m R universal gas constant, J/mol K Re Reynold number S source term t time, s T torque, N m; temperature, K Tsat saturation temperature, K U velocity of fluid, m/s Ur relative velocity, m/s xii

up velocity of particle, m/s Vs volume of particle, m 3 Vc volume of CFD cell, m 3 A area of contact, m c sound speed, m/s C sound speed, m/s d0 diameter of particle, m e specific internal energy, J/kg E Elastic modulus, Pa F force on particle, N G Shear modulus, Pa h smoothing length, m m mass of particle, kg p hydrostatic pressure, N/m r distance vector, m R normalized distance s deviatoric stress tensor, N/m v velocity, m/s V volume of particle, m 3 x distance vector, m W kernel function Greek Symbols γ damping coefficient, s xiii

θ rotational angle, rad μs sliding friction coefficient μr rolling friction coefficient ξn normal direction displacement, m ξt tangential displacement, m ρ particle density, kg/m 3 σ standard deviation σp Poisson ratio ω angular velocity, rad/s αf volumetric fraction of fluid γ surface energy density, J/m μ fluid viscosity, Pa s δn normal direction displacement, m δt tangential displacement, m λ mean free path of water, m μs sliding friction coefficient μr rolling friction coefficient ρ fluid density, kg/m 3 σ Poisson ratio; surface tension, N/m τ shear stress tensor, Pa αd normalization factor β constant Γ EOS parameter xiv

ε tensile instability coefficient η EOS parameter θ λ angle between normal vectors penalty parameter Π artificial viscosity term ρ density, kg/m 3 σ stress tensor, N/m ψ color function xv

ABSTRACT Self-assembly and packing of colloids and micro or nano scale particles has become a subject of great interest due to widespread advancement of micro-scale technologies. In this thesis, several numerical analyses are performed to study the packing or self-assembly of micro or nano sized particles under dry or wet condition. Part one of the thesis is concerned with DEM simulation of micro-sized cohesive granular particles using two history dependent contact models. The simulation results are presented using porosity, coordination number, RDF and force distribution. It was observed that the particles with Gaussian distribution always have the lowest packing density while the mono-sized particles normally have the highest packing density. For cohesive particles, size distributions result in the same tendency of packing density but has much less variation with particle size. There is no significant effect of cohesion on coordination number but particle size and size distribution do influence the result. The differences in porosity, coordination number, RDF and magnitude of mean net force between the two models used are not substantial which show that any of the models can be used for simulation of particle packing. However, the Gran-Hooke-History model is found to be more efficient than the Modified Gran-Hertz-History model. In the second part self-assembly of micro-sized particles induced by evaporation is numerically investigated. The problem involves interaction between solid and fluid as well as interaction between fluids. The problem also involves phase change. A coupled CFD-DEM method is used to simulate the multiphase system. In the simulation liquid water film evaporates and leaves the particles at the xvi

container alone. Interesting patterns are seen to emerge as the liquid water film evaporates. The resulting packing structure analyzed in terms of the range of coordination number and radial distribution function also indicate the self-assembly of the particles. In the third part of the thesis low velocity SPH method developed by Seo et. al is used to simulate the cyclic packing of deformable two dimensional disks and study their packing behavior. The results obtained show that the average coordination number varies with packing fraction during jamming by conforming to the isostatic conjecture. Stress relaxation is seen to occur after several compression cycles which is marked by a decrease in coordination number and global pressure. Force distribution shows similar exponential behavior as the average force on the system is increased. The SPH method is also adapted to include the elasto-plastic behavior of materials. In future, the present work can be extended to include the contact friction of the particles. A method to achieve this is also shown by applying virtual work principal and penalty method. xvii

1 Introduction 1.1 Motivation and Objectives Self-assembly and packing of colloids and micro or nano scale particles has become a subject of great interest due to widespread advancement of micro-scale technologies. Fabrication of micro-scale structures essential for many industrial applications such as optics [1-3], catalysis [4, 5] and printing technology [6] can be achieved by assembling micro-sized particles. A complete numerical analysis of packing phenomena in particle laden multiphase systems can be of great importance by providing deeper understanding of the packing process. Part one of the thesis numerically investigated packing of granular material, a process that has many industrial applications ranging from manufacturing raw materials to developing advanced products. The impact of particle properties on their packing structures is of the prime importance to the entire packing process and is always essential for fabrication. A better understanding of packing is beneficial to optimize and to improve the industrial applications. This topic has been intensively studied in the past decades; many of them focused on the micro level packing [7-9] where packing density, which is equals to unity minus its porosity, is used as their main indicator to measure and evaluate quality of packing structure [10]. Among those works, researchers, by varying particle sizes, size distributions or forces involved in the packing process, obtained detailed information of packing structures and revealed weighted influences from different parameters [11-15]. Some of them are interested in cohesive effect that is substantially caused by cohesive forces such as van der Waals force, capillary force that is associated with wet particles and electrostatic force that can be important for finer particles. Cohesive effect turns out to be 1

of importance in particular situations, for example, when packing containers are no longer rigid but are kind of material that has similar properties like dry sand, cement or wet soil, or it is not even solid just like settling particles in the fluid where effect of gravity will reduce and cohesive effect will become significant [16-18]. The studies on behaviors of cohesive particles are usually carried out by changing particle sizes, mixture component percentage if particles are not made with the same material or fluid density if particles are settled down into a fluid. Effects of particle diameters (mean diameters for mixed particle cases) are considered to be a great factor that influences the packing structures thereby worth more attention. Boundary condition is another important factor that alters the force and deformation of the packing structure. Previously, researchers mostly adopted the periodic boundary conditions for the packing process where particles that exit the simulation box will come back in opposite direction in order to maintain the number of particles in the simulation box [19]. This approach allows that the simulation can be carried out smoothly, because of lower chances of losing systemic energy and generating huge interactive force. In this work, the cohesive effects associated with size distributions, which include mono-sized, uniform, and Gaussian distributions, will be investigated by using two different history-dependent contact models. It is worth to point out that the uniform distribution of particle size indicates the sizes of particles linearly increase from the minimum to the maximum. The range for uniform distribution is kept at a constant of 40 μm. For Gaussian distribution, the STD (standard deviation), σ, is set as 13.33 μm for all cases. In addition, fixed boundary conditions are applied to all sides of the simulation box for all the cases such that the particles may collide with boundaries during the packing

process which will cause energy losses due to friction. In order to understand the fundamentals that govern the cohesive particle packing, a series of well-designed programs are developed based on the Discrete Element Method [0-3]. LIGGGHTS[4] that is based on LAMMPS [5], providing a simulator of solving particle related problems from industrial applications, is employed to resolve the packing process. The simulation results, including radial distribution function (RDF) that indicates how number density changes with distance from a selected reference point, force distributions that give a view on the magnitude of forces acting on the particles, porosities and coordination numbers, are presented in this paper. The second part of the thesis provides a complete numerical analysis of self-assembly of micro-scale particles induced by evaporation using CFD-DEM approach. Many numerical methods such as Lattice Boltzmann method (LBM) [6, 7], Immersed Moving Boundary (IMB) method [8] are used to simulate the interaction between particle and fluid. However, these methods have the disadvantage that if the number of particle is large computational efficiency increases enormously. The method used in this paper CFD-DEM is a robust and efficient method for simulating interactions among fluid and large number of particles. In CFD-DEM, particle to particle interaction is computed by Discrete Element method (DEM) which is a highly appreciated Lagrangian approach for particle interaction simulation. The fluid phase is governed by Navier Stokes equation and finite volume method is used for the CFD part. In this study the final packing structure of 4500 mono-sized particles with radius of 5 µm is analyzed after they are settled down and the liquid medium in which the particles are suspended is evaporated. While most researchers focus on the momentum exchange 3

between particle and fluid [9-31], in this study the heat transfer between particles and fluid and among particles are also taken into account. The problem in this case gets more challenging as evaporation or phase change is also included. The final packing structure of the particles is analyzed using distribution of coordination number and radial distribution function (RDF). The results showed that just before the liquid evaporates the particles agglomerate and form a pattern. The third part of the thesis deals with another particular aspect of granular material which is the phenomenon of jamming where randomly organized system of particles changes from mechanically unstable states to stable states. Jamming phenomena is also observed in colloids, foams and glass transition in molecular liquids. In most of these cases the system starts from an unjammed state and gradually transition to jammed state. Sometimes the system undergoes several transitions between jammed and unjammed states. In this paper the jammed states of two dimensional granular materials is numerically investigated by subjecting the system to consecutive compression cycles. There are a number of simulation methods developed so far in order to capture the behavior of granular materials. These methods can be divided into two categories discrete and continuum. Discrete method is by far the most widely used method for simulating granular media. In discrete methods, such as DEM (Discrete Element Method) as discussed earlier, behavior of each particle that makes up the granular system is captured by directly simulating the interaction between them. However, DEM cannot provide enough resolution to study the stresses developed in individual particles. In this thesis, Smoothed Particle Hydrodynamics (SPH) method is used to simulate the interaction of granular particles. In SPH, each granular particle is made up of several SPH particles which can accurately 4

account for the stress generated within each granular particle. In addition, contact forces are calculated using an advanced formulation that accounts for the contact geometry and force distribution around the contact area. SPH method, introduced by Lucy [3], Gingold [33, 34] and Monaghan [35, 36], was originally developed for solving astrophysics problem. This method has the ability to handle large deformation especially when comes to solid bodies. Hence it has been applied to hyper-velocity impact [36], explosion [37] and fluid dynamics [38]. When SPH is applied to high velocity impact problems boundary and contact properties on the contact surface are ignored because of their insignificant contribution. However, for low velocity impact or small deformation, as mentioned in [39], neglecting contact conditions can arise several problems. Some of them include ghost stress, which arises due to incorrect inclusion of boundary particles within a smoothing length, false tensile force during separation and normalization of kernel not satisfied. In order to avoid such problems, [39] developed a method that addresses problems involving small deformation as well as large deformation for low velocity impact. In this paper their method is adopted for simulation of granular particles. 1. Softwares Used 1..1 LAMMPS LAMMPS, which stands for Large-scale Atomic/Molecular Massively Parallel Simulator, is an open-source molecular dynamics software developed and distributed by Sandia National Laboratories. LAMMPS can run on single or multiple processors using MPI for parallel communications. LAMMPS comes with a package called User-SPH that enables users to install and run SPH simulations. In this work the User-SPH package of 5

LAMMPS is extended to include solid-solid, liquid-liquid and solid-liquid multiphase simulations. LAMMPS has a very efficient neighbor searching and neighbor list creating algorithm which makes it suitable for SPH calculation. LAMMPS s codes are written in C++ language and can be easily modified. 1.. LIGGGHTS LIGGGHTS stands for LAMMPS improved for general granular and granular heat transfer simulations. LIGGGHTS is an extension of LAMMPS for granular particle simulation using Discrete Element Method. LIGGGHTS inherits all the useful features of LAMMPS including open source code, efficient parallelization, particle insertion and deletion, importing and handling of complex wall geometries from CAD, heat conduction between particles in contact etc. LIGGGHTS is also written in C++ language and its code can be easily modified. In this work a new contact force model is added to the existing Gran-Hertz-History contact model by including van der Waals force as one of the contact force for granular interaction. 1..3 OpenFOAM OpenFOAM is a free, open source computational fluid dynamics (CFD) software developed primarily by OpenCFD Ltd. OpenFOAM is written in C++ and it uses finite volume method to solve continuum mechanics problems, including CFD. OpenFOAM has an extensive range of features to solve anything from complex fluid flows involving chemical reactions, turbulence and heat transfer, to solid dynamics and electromagnetics. OpenFOAM provides users tools for meshing and for pre- and post-processing. It can be run on single or multiple processors for parallel computation. OpenFOAM provides users complete freedom to customize and extend its existing functionality. In this study a new 6

OpenFOAM solver is created, named InterEvapDEMFoam, which includes interface evaporation model to model evaporation and also includes volume averaged transport equations for solid-liquid multiphase flow. The heat transfer between solid and fluid is also taken into account by adding a source term to the energy equation. In this work OpenFOAM is used to solve the CFD part of the micro scale nanoparticle self-assembly problem discussed in Chapter 4. 1..4 CFDEM CFDEM stands for Computational Fluid Dynamics (CFD) -Discrete Element Method (DEM) coupling. It is in essence an extension of library based on OpenFOAM source code, by integrating LIGGGHTS thus enabling simulation of particle fluid multiphase system. CFDEM allows users to expand standard CFD solvers of OpenFOAM to include a coupling to the DEM code LIGGGHTS. In this work CFDEM is used to simulate the micro sized particle suspension undergoing evaporation 7

Analysis of Cohesive Micro-Sized Particle Packing Structure Using History-Dependent Contact Models.1 Introduction Discrete Element Method (DEM) also known as Distinct Element Method has been used for the simulation of granular system. This method was originally developed by Cundall for rock mechanics problem. Cundall and Strack later provided the basic formulation of DEM using spherical or cylindrical particles. DEM has been showed by Cundall and Hart to be a better tool tool for modelling discontinuous media compared to other methods like finite element method. DEM is a lagrangian method where trajectory of each particle is calculated separately using Newton s equation of motion. Collisions among particles and between particles and walls are accounted for using contact force models. Both normal contact force and tangential contact force are accounted for in the contact model. Other forces such as cohesive force, van der Waal force can be considered in the equations of motion.. Equations of motion It is well known that any motions of a rigid particle can be decomposed to two parts: translational and rotational motions. Referring to the Newton s second law, the governing equations for each particle during this packing process can be written as: X mi t i F i (.1) I i d dt θ i (.) T i 8

F F F (.3) n t i ij ij T T T (.4) r t i ij ij where mi is the mass of the i th particle, Xi is the position vector of the i th particle, Ii is the moment of inertia that equals to 0.4miRi and the rotated angle of particle i is represented by θi. The symbol Fi in Eq. (1.1) is the resultant contact force generated by two collided particles, i and j. This force can be decomposed further into two components: one is contact n force in normal direction F ij and the other is contact force in tangential direction F t ij, as shown in Eq. (1.3). The symbol Ti in Eq. (1.) is the resultant torque acting on the i th particle. It can also be decomposed into two components: torques caused by rolling friction and tangential force, respectively, as given in Eq. (1.4)..3 Contact forces The two contact models adopted in this work are both history deformation dependent. The difference is from the relationship between deformation and contact force. Gran-Hertz- History model describe a nonlinear relationship between contact force and overlap distance, while Gran-Hooke-History model gives a linear relationship. The open-source software package LIGGGHTS provides both of these models. However, the Gran-Hertz- History model is modified to include van der Waals force, which can be significant for small particles, and thereby refer to as Modified Gran-Hertz-History model..3.1 Normal contact force n The normal contact force F ij can be determined by [40-4], n F ij K n ξn γ n vij. n ij nij (.5) 9

where in Modified Gran-Hertz-History model, parameters are given by K n 4 Yeff Rξ n, 3 5 Sm, 6 γ n eff n eff eff ln e e ln, Sn Yeff Rξn. and in Gran-Hooke-History model, 0.8 16 n eff eff ch K 15 R Y m v 0., γ n 4m K eff n 1 ln e. and v ij represents the velocity of the particle i relative to velocity of the particle j, nij is the unit vector point from particle i to particle j, e is the coefficient of restitution of the particles, R RiR j / Ri R j is the effective radius that represent the geometric mean diameter of the i and j particle, Y eff 1σ 1σ 1/ Y1 Y 1 is the effective Young s modulus that is calculated in terms of individual Young s modulus and Poisson ratio accordingly, ξn Ri R j R is the overlap in normal direction and ij m eff mm i j is the effective m m i j masses of the particles. Characteristic velocity v ch is taken as unity in Gran-Hooke-History model..3. Tangential contact force The contact force in tangential direction is calculated by [43], t n F ij min μ F ij, K t ξt tij γ t vt t ij tij (.6) where in Modified Gran-Hertz-History model, parameters are determined by, K t, 8Geff Rξn 5 Sm, S 8 Rξ 6 t Geff n, γ t eff t eff 10

G eff σ σ σ σ 1 1 1/ Y1 Y 1 1. and in Gran-Hooke-History model, K t K, γ γ n t n where ξ t t vtdt represents the tangential displacement vector between the two spherical t0 particles, vt [ vi vj tij] tij ωi Ri ωj R j is the tangential relatively velocity, tij is the unit vector along the tangential direction, t0 is the time when the two particles just touch and have no deformation, t is the time of collision, ωi or ωj is the angular velocities of particles i or j and Ri or Rj is the vector running from the center of particle i or j to the contact point of the two particles..4 Van der Waals force The van der Waals forces among particles are included only in the Modified Gran- Hertz-History model. The van der Waals force, F v ij between particles i and j is given by [44], F v ij 3 3 64Ri Rj h Ri Rj h Rih Rjh h Rih R jh Ri R j Ha 6 4 (.7) where Ha is the Hamaker constant, and h is the separation of surfaces along the line of the centers of particles i and j. A minimum separation distance h min is considered to prevent F v ij becoming infinity when h goes to zero. The Hamaker constant is related to the surface energy density by [45]: Ha 4 kh (.8) min 11

.5 Cohesive force The cohesive force is included in both Modified Gran-Hertz-History model and Gran- Hooke-History model. For the cohesive force, Johnson-Kendall-Roberts (JKR) model [46] based on Hertz elastic theory is used to estimate the cohesive behavior of the particles. In Hertz elastic theory, the normal pushback force between two particles is proportional to the area of overlap between the particles. Based on Hertz elastic assumption and meanwhile considering the contact surface as perfectly smooth, the JKR model here is satisfactorily accurate to determine the cohesive force. In fact, the basic idea is that if two particles are in contact, it adds an additional normal force tending to maintain the contact, F ka (.9) where k is the surface energy density and A is the particle contact area. For sphere-sphere contact [47], contact area A is evaluated by, ( dist Ri Rj )( dist Ri Rj )( dist Ri Rj )( dist Ri Rj ) A (.10) 4 dist where dist is the central distance between the i and j particles. Ri and Rj are the radius of the i th and j th particle, respectively..6 Torques The torque due to tangential contact force and the torque due to rolling friction are calculated in the same way for both models [48]: T R F (.11) t t ij i ij ω t T μ R ξ t (.1) r ij ij ij r K n n ij ωij 1

whereωij ωi ωj is the relative angular velocity..7 The integration scheme In simulating discontinuous media as in DEM explicit time integration scheme is preferred over implicit one due to massive memory requirement for implicit scheme. In this study, the Central Difference scheme also known as the Velocity Verlet scheme is used for time integration of the equation of motions. Accuracy of the integration scheme is of second order. Time integration operation for the n-th time step is as follows: a n i F m n i i (.13) v v a t (.14) n 1/ n 1/ n i i i x x v t (.15) n 1 n n 1/ i i i Here x is the displacement, v is the velocity, a is the acceleration, and n indicates the time step position. t is the time step.8 Numerical Stability Although explicit time integration scheme enjoys higher computational efficiency for discontinuous and large systems than implicit ones, it has one inherent disadvantage in that the numerical stability of the integration depends on time step, t. Considering the fact that the contact force is mainly related to the particle deformation, the time-step must be sufficiently small to prevent any unrealistic overlap [49]. In this work, the time step is set to be 1 10-8 s for all simulation cases. It should be pointed out that the velocity of each 13

particle will hardly reach zero completely but the magnitude of velocity will approach to an extremely small value. In this work, the particles are considered to be completely stationary when their mean velocities are below 1 10-8 m/s..9 Physical Model For each DEM simulation, 4,500 particles are settled in a simulation box having length and width equal to 0.006m and the particles have no initial physical contact among them. The initial porosity is kept constant at 0.75. Figure -1 shows the initial state of Gaussian particle packing. As the simulation time increases, the particles begin to fall down due to gravity and then collide with other particles or with the boundaries. In this work, all six sides of the simulation box are considered as physically stationary. In this work, the time step is set to be 1 10-8 s for all simulation cases. It should be pointed out that the velocity of each particle will hardly reach zero completely but the magnitude of velocity will approach to an extremely small value. In this work, the particles are considered to be completely stationary when their mean velocities are below 1 10-8 m/s. Sixty scenarios are studied in this work: five different mean radius (75µm, 85 µm, 100 µm, 110 µm, and 10 µm) and three different size distributions (mono-sized, uniform and Gaussian) for two contact models (Modified Gran-Hertz-History model and Gran-Hooke- History model) with and without cohesion. It should be noted that the deformation calculation is very important for packing simulation since the oversimplified model of calculating overlap distance is always the main reason that leads to the simulation crash by introducing unrealistic energy. Two basic rules are applied to these packing simulations: one is that particles are always considered as rigid body even though a deformation is considered by the chosen model, and the other is that the critical central distance is set for 14

particle deformation. The critical distance is 1.01(d1+d)/ where d1 and d are the diameters of the two particles. It means when the central distance of two particles is less than the critical distance the two particles are considered to be in direct contact [1]. (a) Particles at t = 1 10-8 sec (b) Particles at t = 0. sec Figure -1 Initial and final structure for Gaussian particles from Modified Gran-Hertz-History model with cohesion. 15

(a) Particles at t = 1 10-8 sec (b) Particles at t = 0. sec Figure - Initial and final packing structure for mono-sized particles from Modified Gran- Hertz-History model with cohesion. 16

(a) Particles at t = 1 10-8 sec (b) Particles at t = 0. sec Figure -3 Initial and final packing structure for uniform size particles from Modified Gran- Hertz-History with cohesion. 17

(a) Particles at t = 1 10-8 sec (b) Particles at t = 0. sec Figure -4 Initial and final structure for Gaussian particles from Gran-Hooke-History model with cohesion. 18

.10 Material Properties Table -1 Values of the parameters used in the simulation process Parameters Values Particle density ρ 7870 kg/m 3 Young s modulus Y 00 10 9 N/m Restitution coefficient e 0.75 Sliding friction coefficient μ s 0.4 Rolling friction coefficient μ r 10-4 Poison ratio σ P 0.9 Hamaker constant, H a Minimum separation distance, h min 1.1 10-0 J 1 10-10 m Surface energy density, k 0.80 J/m 19

.11 Results and Discussion.11.1 Porosity and Coordination Number Table - Porosity and coordination number for Modified Gran-Hertz-History model Porosity Coordination number Radius Uniform Gaussian Monosized Monosized Uniform Gaussian 75μm 0.615 0.695 0.713 4.04 3.50 3.14 85μm 0.656 0.685 0.660 4.84 4.49 4.01 100μm 0.583 0.615 0.637 5.7 5.9 5.14 110μm 0.575 0.505 0.557 5.41 5.56 5.37 10μm 0.485 0.574 0.487 5.36 5.46 5.45 0

Table -3 Porosity and coordination number for Gran-Hooke-History model Porosity Coordination number Radius Uniform Gaussian Monosized Monosized Uniform Gaussian 75μm 0.599 0.695 0.74 4.00 3.59 3.11 85μm 0.656 0.685 0.643 4.83 4.50 4.00 100μm 0.566 0.615 0.476 5.5 5.5 5.13 110μm 0.55 0.505 0.591 5.50 5.55 5.43 10μm 0.476 0.574 0.651 5.33 5.47 5.36 Table -4 Porosity and coordination number for Modified Gran-Hertz-History model without cohesion Porosity Coordination number Radius Uniform Gaussian Monosized Monosized Uniform Gaussian 75μm 0.454 0.485 0.505 4.05 3.55 3.46 85μm 0.436 0.478 0.493 4.85 4.48 4.08 100μm 0.415 0.4 0.439 5.8 5.7 5.1 110μm 0.430 0.41 0.437 5.33 5.47 5.35 10μm 0.49 0.46 0.438 5.43 5.50 5.38 1

Table -5 Porosity and coordination number for Gran-Hooke-History model without cohesion Porosity Coordination number Radius Uniform Gaussian Monosized Monosized Uniform Gaussian 75μm 0.407 0.483 0.458 4.01 3.5 3.60 85μm 0.439 0.460 0.47 4.8 4.44 4.00 100μm 0.419 0.47 0.453 5.8 5.31 5.13 110μm 0.47 0.45 0.448 5.37 5.44 5.33 10μm 0.419 0.435 0.436 5.36 5.55 5.46 Table -6 Magnitude of mean net contact force (N) for Modified Gran-Hertz-History model Radius Cohesion No Cohesion Mono-sized Uniform Gaussian Mono-sized Uniform Gaussian 75μm 1.91 10-7 3.00 10-7.0 10-6 1.94 10-7.87 10-7.5 10-7 85μm 3.75 10-7 4.86 10-7 4.13 10-7 3.76 10-7 4.74 10-7 4. 10-7 100μm 8.50 10-7 1.07 10-6 9. 10-7 8.60 10-7 1.03 10-6 9.09 10-7 110μm 1.41 10-6 1.59 10-6 1.5 10-6 1.41 10-6 1.59 10-6 1.50 10-6 10μm.30 10-6.41 10-6.30 10-6.3 10-6.40 10-6.30 10-6

Table -7 Magnitude of mean net contact force (N) for Gran-Hooke-History model Radius Cohesion No Cohesion Mono-sized Uniform Gaussian Mono-sized Uniform Gaussian 75μm 1.96 10-7.93 10-7.3 10-7.00 10-7.85 10-7.3 10-7 85μm 3.9 10-7 4.94 10-7 4.30 10-7 3.94 10-7 5.46 10-7 4.10 10-7 100μm 8.75 10-7 1.00 10-6 9.8 10-7 8.61 10-7 1.03 10-6 8.87 10-7 110μm 1.50 10-6 1.59 10-6 1.45 10-6 1.39 10-6 1.58 10-6 1.49 10-6 10μm.4 10-6.35 10-6.31 10-6.19 10-6.44 10-6.48 10-6 Figure -5 and Figure -6 present the porosities and coordination numbers for different cases. It can be seen that the porosity decreases along with the increasing particle radius for all distributions when cohesive forces are considered. Similar trend was observed in the work of previous researchers [50]. This decrease in porosity with increase in radius is expected since with increase of radii or masses of the particles the initial supplied energy (gravitational potential) also increases. So the effect of cohesion in the packing of particles decreases and the porosity values become closer to that for Random Loose Packing [8, 51]. This also explains the decrease in differences between different size distributions in terms of porosity when the radius increases. Among the three distributions considered, Gaussian distribution has the highest porosity and mono-size has the lowest. The porosity values for the two models, Modified Gran-Hertz and Gran-Hooke are slightly different but both show the same trend. As for the non-cohesion case porosity also decreases with increase in 3

particle radius, but the porosity values are much smaller. Figure 5 also shows that the rate of decrease of porosity with radius for non-cohesion case is much smaller. For mono-sized distribution without cohesion, porosity remains almost constant for both Modified Gran- Hertz-History model and Gran-Hooke-History model. Since there is no cohesion the dissipative forces are smaller and particles can pack more closely. Again the difference between the two models in non-cohesion cases is very small. For the coordination number, the trends for three distributions with cohesion are similar. It can be observed that the coordination number increases as particle radius increases which is exactly the opposite of the trend of porosity. Unlike porosity, Gaussian distribution now has the lowest coordination number and mono-size distribution has the highest. Interestingly, it is found that there is no significant change in coordination number whether or not cohesion is included. However, one can expect that coordination number should be smaller when there is no cohesion (porosity is larger). This can be explained as follows. When there is cohesion, particles tend to clump together and form clusters. These clusters have void spaces in them. Due to this formation of clusters in some region particles have high coordination number and in some region the coordination number is small. The coordination numbers given in Table - - 5 and Figure -6 are average of coordination numbers for all particles. It can be seen that the coordination numbers for cohesion and non-cohesion cases are similar. 4

(a) Modified Gran-Hertz-History Model (b) Gran-Hooke-History Model Figure -5 Effect of porosity with particle size and distribution 5

(a) Modified Gran-Hertz-History Model (b) Gran-Hooke-History Model Figure -6 Effect of coordination number with particle size and distribution.11. Radial Distribution Function Figure -7-11 show the RDF for particle systems with mean radius of 75 µm, 85µm, 100 µm, 110 µm and 10 µm and associated with three different size distributions (monosized, uniform and Gaussian). For the cases where the particles have the same radius, three main apparent peaks appear. The first peak is sharply at r which is for the initial one to one contact, the second and the third are at around r and 4r, respectively which 6

corresponds to the two characteristic particle contact types, namely edge-sharing-in-plane equilateral triangle and three particles centers in a line (the three contact types are illustrated in Figure -7 (a)). The second and third peaks merge into a single second peak for other distributions. The particle systems with mono-size distribution usually have the highest peak values among all three cases. The peak values for Hertz model are close to that for Hooke model. Also the peak values of RDF are almost same for cohesion and noncohesion cases. (a) Modified Gran-Hertz-History with cohesion (b) Modified Gran-Hertz-History without cohesion (c) Gran-Hooke-History with cohesion (d) Gran-Hooke-History without cohesion Figure -7 RDF for particles with 75 μm radius 7

.11.3 Force Distribution Figure -1-14 show the force distribution results after the particles are completely packed for particle systems with mean radius of 75µm. The force distribution graphs for other particle radii look similar. For same distribution the counts for each force magnitude are not exactly the same but close. However, as the particle radius increases, the force magnitudes increase as a response. Table -6 and Table -7 give the mean net force for all the cases when the particles are finally packed. It has to be pointed out that the resultant force here does not represent gravity, since effect of gravity is small (of the order of 10-1 N). It can also be observed that the net force does not vary much even when the cohesion is included. The mean net force increases with the size of the particles, and it can also be seen that this force has the largest value if the particle size follow uniform distribution. Particles with Gaussian distribution have the secondary magnitude of force, while the mono sized particles have the smallest net force. The difference in magnitude of mean net force between the two different contact models is negligible. 3

(a) Modified Gran-Hertz -History with cohesion (b) Modified Gran-Hertz -History without cohesion (c) Gran-Hooke-History with cohesion (d) Gran-Hooke-History without cohesion Figure -14 Force distribution for particles with 75 μm radius and uniform distribution.11.4 Comparison between contact models By comparing the two models, Modified Gran-Hertz-History and Gran-Hooke-History, it can be seen that the difference between them is not significant in terms of porosity, coordination number and mean net force. Both of these models assume that the particles are viscoelastic and have a stiffness term and dissipation term. As pointed out by [5] the linear Gran-Hooke model can be as accurate as the non-linear Modified Gran-Hertz model if the stiffness constants, Kn and Kt, and damping coefficients, γn and γt, are evaluated carefully. In this study, even though cohesion is included, the results obtained from the two models are still close. Van der Waals force included in the Modified Gran Hertz model did not seem to play a great role in the packing process. This might be because particle sizes 35

are too large for van der Waals force to take effect. When the efficiency of the two models are considered, the simulations with the Gran-Hooke-History model ran faster than the simulation with the Modified Gran-Hertz-History model. So the linear Gran-Hooke- History model is more efficient than the Modified Gran-Hertz-History model..1 Conclusion A study on packing structures of particle system with different radii and size distributions using two different models are carried out by the Discrete Element Method. The simulation results including RDF and force distribution, porosity and coordination number are presented. It was observed that the particles with Gaussian distribution always have the lowest packing density while the particles with uniform size distribution have the medium packing density and mono-sized particles normally have the highest packing density. For the particles packing under cohesive effect, size distributions result in the same tendency of packing density but has much less variation with particle size. Coordination number is not affected by cohesion significantly but particle size and size distribution do influence the result. The differences in porosity, coordination number, RDF and magnitude of mean net force between the two models used are not substantial which show that any of the models can be used for simulation of particle packing. However when efficiency is considered the Gran-Hooke-History model is found to be more efficient than the Modified Gran-Hertz-History model. Therefore, Gran-Hooke-History model can be the model of choice for simulating micro-sized particles. 36

3 Numerical Investigation of Evaporation Induced Self-Assembly of Sub-Micron Particles Suspended in Water 3.1 Introduction In many applications granular materials are surrounded by fluid and the solid particles and fluid influence each other s behavior. Fluid behavior can be predicted by Computational Fluid Dynamics (CFD) simulation whereas DEM can be used to model particle behavior. For a system with both granular material and fluid a coupling between the two method called CFD-DEM is desirable to simulate the particle-fluid system. CFD- DEM method can further be classified into two categories: resolved CFD-DEM and unresolved CFD-DEM. In resolved CFD-DEM the particles are larger than the fluid control volume and meshing for CFD is done around the particles. The method provides a better resolution of the fluid field and the force on each particle is calculated individually. This method is only suitable for cases with small number of particle due to computational expense required for large number of particles. In contrast, the unresolved CFD-DEM, which is used in this work, is capable of handling large number of particles. The particles are significantly smaller than the mesh cells and there can be more than one particle in a cell. The particle phase is characterized by void fraction which is the ratio of the fluid volume in a cell to the cell volume. 3. Equations of motion The governing equation for the fluid phases consists of a set of volume averaged mass and momentum balance equations in an Eulerian description. The fluid phases are 37

considered incompressible and in the CFD domain the particle phase is characterized by void fraction field αf. The liquid and the vapor phases are denoted by α1 and α respectively. The mass and momentum equations for the fluid and solid phases are given in the following sections. 3..1 Fluid Phase t f f U 0 ( 1 f ) 1 U 1 U 11 S t t f f r v p K1 U U U g r F f f f gh f f f d (3.1) (3.) (3.3) where (3.4) 1 f f f (3.5) 1 1 p p gr (3.6) f gh f f T U U U I 3 (3.7) where ρ is the average fluid density, U is the average velocity of fluid, S v represents source terms due to phase change, σk is the surface tension effect at the interface, prgh is the dynamic pressure, Fd is the drag force term, µ is the mean viscosity and g is the acceleration due to the gravity. The third term on the left side of equation (3.) is the compressibility term added to obtain sharp interfaces [53] and Ur is the relative velocity between two fluid phases. 38

3.. Solid Phase m u F F F m g F F (3.8) n t i ip, ij ij coh i d b r t I ω T T (3.9) i i ij ij where mi is the mass of the i th particle, up,i is the velocity of the i th particle, Ii is the moment of inertia of the particle and ωi is the angular velocity of the i th particle. Fij n and Fij t are normal and tangential force due to contact, Fcoh is the cohesive force between two particles, mig is force of gravity, Fd is the drag force due to relative fluid motion and Fb is the buoyant force. Tij r is the torque due to rolling friction and Tij t is the torque due to tangential force. 3.3 Drag Force The drag force accounts for the force acting on the particle due to relative motion between particle and fluid. The drag model used is suitable for particles in the near submicron range [54]. F d 18 dc c U u p (3.10) where Cc d 1.1 d / 1 1.57 0.4e (3.11) where λ is the mean free path of water which is.5 10-10 m and d is the diameter of the particle. 39

3.4 Buoyant Force F b gv (3.1) s where V s is the volume of the particle. 3.5 Contact Forces A contact force model is implemented to take into account the interaction among particles and between particle and wall. The model used is the Gran-Hertz-History model where a non-linear relationship is assumed between overlap distance during collision and normal contact force. Other force that are included are viscous damping force and cohesive force. Tangential force during collision is also taken into account. The normal contact force during collision Fij n is given by [14, 40-4]: n F.ˆ ˆ ij Yn n cn vij n ij nij (3.13) where Y 4 E R / 3, e 5 B m /6 n eff n c, ln / ln n eff n eff e e e eff, B E. n eff R n where nˆ ij is the unit vector pointing from particle i to particle j, e is the coefficient of restitution of the particles, R = R i R j /(R i + R j ) represents the geometric mean radius of the i and j particle, vij represents the velocity of the particle i relative to velocity of the particle j, Eeff = [(1-σ1 )/E1 + (1-σ )/E] -1 is the effective Elastic modulus computed by each particle s Elastic modulus and Poisson ratio, δn=ri+rj- Rij is the displacement in normal direction and meff = mimj/mi+mj is the effective masses of the particles. 40

The contact force in tangential direction is given as [43] t n F min F, tˆ c v tˆ tˆ s Y t t ij ij t ij t ij ij (3.14) where Yt 8Seff Rn, e 5 B m /6, Bt 8Seff Rn, c t eff t eff σ σ σ σ Seff 1/ 1 1 1 / E1 1 / E where δt is the tangential displacement vector given by δ t = v t dt, t ij is the unit vector along the tangential direction, t0 is the time when the two particles just contact without deformation, t is the time of collision. The tangential relative velocity is given by v t = [(v i v j ) t ij ]t ij + (ω i R i ω j R j ) where ωi or ωj is the angular velocities of particles i or j and Ri or Rj is the radial vector. For the cohesive force Johnson-Kendall-Roberts (JKR) model [46] which is based on Hertz elastic theory is used. The mathematical expression is given as follow: t t 0 F coh ( D Ri R j )( D Ri R j )( D Ri R j )( D Ri R j ) / 4D (3.15) where γ is the surface energy density and D is the central distance between the i and j particles. The torque due to rolling friction and tangential force is given by [48]: T R F (3.16) t t ij i ij ω ˆ ij t r ij T ˆ ij rry n n tij (3.17) ω ij where µr is the rolling friction coefficient and ωij = ωi - ωj is the relative angular velocity. 41

3.6 Energy Equation 3.6.1 Fluid Phase The energy equation for fluid phase is [55]: t ha T T (3.18) V s s cp ftf cp futf f k f Tf Sq f s c where cp is the mean specific heat capacity of the fluid, kf is the mean thermal conductivity of the fluid, S q is the source term due to evaporation, hs is the mean convection heat transfer coefficient for particle, As is the surface area of the particle, Vc is the volume of the CFD cell and Tf and Ts is the temperature of fluid and particle respectively. The convection heat transfer coefficient hs is calculated using the Nusselt number correlation for particles in a stream of fluid [55]. Nu 0.6 Re Pr Res < 00 (3.19) n 1/ 1/3 s f s Nu 0.5 Re Pr 0.0 Re Pr 00 < Res <1500 (3.0) n 1/ 1/3 n 0.8 1/ 3 s f s f s Nu s n 0.000045 Re Res > 1500 (3.1) f 1.8 s where n = 3.5, Nus = hsd/kf, Res = ρd U-up /µ and Pr = µcp/kf. 3.6. Solid Phase Heat transfer between particles is also taken into consideration [43]. Q h T (3.) cond, i j c, i j i j h 4kk si sj c, ij contact, i j ksi ksj A 1/ (3.3) 4

where hc,i-j is heat transfer coefficient due to conduction, ΔTi-j is the temperature difference between particle i and j, ksi is the thermal conductivity of particle i and Acontact, i-j is the contact area for particle i and j. The particle temperature is finally calculated as follow: dt m c h A T T Q (3.4) si, s ps s s f s cond, i j dt contacts i j 3.7 Phase Change The heat flux at the phase boundary for deviation of interfacial temperature Ti from Tsat is given as [56], ev ev i sat q h T T (3.5) where qev is the heat flux due to evaporation and hev is the interfacial vaporization heat transfer coefficient. It is given by [56], h ev Ce M (3.6) h C e e 3/ R Tsat where Ce is the evaporation coefficient, he is the enthalpy of vaporization, ρ is the density of vapor, R is the universal gas constant and M is the molecular mass of water. The gradient of the volume fraction field α1 is zero everywhere except at the interface. The interfacial area can be calculated by taking the volume integral of the magnitude of the α1 field over a region encompassing the interface [56]. This gives us a way to define the evaporation source term S q and S v based on equation (). And the temperature source term due to evaporation is 43

S (3.7) q qev 1 dv / V V The mass source term due to evaporation is S S / h (3.8) v q f e 3.8 Particle Relaxation Time In coupled calculations the ability of a particle to follow the fluid flow also needs to be considered when deciding the time step. The particle relaxation time provides a measure for this quantity. p sd 18 f 1 0.15Re 0.687 s 1 (3.9) Crowe et al [57] derive this time value by considering the equation of motion of a spherical particle in the limit of low Reynolds number flow. 3.9 Algorithm The algorithm for solving the coupled CFD-DEM is as follows: 1. The particles positions, velocities and temperatures are determined by DEM.. Void fraction and mean particle velocities for each cell in the CFD mesh are determined. 3. Based on the information of α f and particle fluid relative velocity the momentum exchange term is calculated. 4. CFD solver solves the fluid flow. 44

a. The alpha transport equation is first solved. b. The momentum equation matrix is constructed. c. The velocity field is predicted without considering the pressure gradient at this stage. d. The pressure equation is solved. e. The velocity field is corrected with the new pressure field and the phase fraction alpha is updated. f. The transport equations for turbulence quantities are solved. 5. The fluid forces acting on the particles are calculated and sent to the DEM solver. 6. At the same time the data for the heat transferred to the particles is sent to DEM. 7. The whole process is repeated from 1. 3.10 Physical Model In this study 4500 particles each with a radius of 5 µm are initially suspended in liquid water film without touching each other. The size of the CFD domain is 6 mm 6 mm 6 mm. Liquid water filled the container to a height of 0.5 mm. To prevent the influence of side walls, periodic boundary condition is applied in lateral directions. The time step for the DEM case is 1 10-8 s whereas the time step for the CFD case is 1 10-5 s which means that for each CFD iteration DEM runs for 1000 iterations. Smaller time step for DEM is applied to prevent any unrealistic overlap between particles during collisions. The entire CFD domain is kept at a temperature of 373.15 K initially with the exception of bottom wall which is kept at a temperature of 378.15 K for all the time. Figure 3-1 shows the initial configuration of the fluids and particles. 45

Figure 3-1 Fluid and particle configuration at time t = 0 s 3.11 Material Properties and Simulation Parameters Table 3-1 Simulation parameter and properties Parameters Values CFD domain size, mm mm mm 6 6 6 CFD cell size, mm mm mm 0.1 0.1 0.1 CFD time step 1 10-5 s DEM time step 1 10-8 s Particle number 4500 Particle radius 5 10-6 m Particle density, ρ s 7.87 10 3 kg/m 3 Water (liquid) density, ρ 1 958.4 kg/m 3 Vapor density, ρ 0.586 kg/m 3 46

Water (liquid) thermal conductivity, k f1 0.691 W/m K Vapor thermal conductivity, k f 0.046 W/m K Particle thermal conductivity, k s 80 W/m K Water (liquid) viscosity, µ 1.8 10-4 N s/m Vapor viscosity, µ 1.7 10-5 N s/m Molecular mass of water, M 18.05 10-3 kg Universal gas constant, R 8.314 J/mol K Mean free path of water, λ.5 10-10 m Specific heat of water (liquid), c p1 419 J/kg K Specific heat of vapor, c p 060 J/kg K Specific heat of particle, c ps 450 J/kg K Surface tension of water, σ 0.07 N/m Evaporation coefficient, C e 0.1 Saturation temperature, T sat 373.15 K Young s modulus, E 00 10 9 N/m Restitution coefficient, e 0.75 Sliding friction coefficient, μ s 0.4 Rolling friction coefficient, μ r 10-4 Poison ratio, σ 0.9 Surface energy density, γ 0.80 J/m 47

3.1 Results and Discussion Figure 3- shows evaporation of water film from the surface. Liquid water is found to be completely evaporated at about.17 sec which means that evaporation still continues after the particles settle down. Particles are found to be completely settled at around 1 s which is much longer than the theoretical value to fall down if there is no water film. This is expected since there are drag force and buoyant force acting on the particles. (a) (b) (c) (d) Figure 3- Volume fraction alpha1 at (a) 0 s (b) 0.59 s (c) 1.39 s and (d).04 s 48

Figure -13 shows the packing structure of the particles after they settled down at the bottom of the container. It can be seen that packing configuration tends to form a pattern with the particles agglomerated in a special way. This clustering of particles continues as the water continues to evaporate. (a) (b) (c) (d) Figure 3-3 Arrangement of particles at (a) 1 s (b) 1.9 s (c).0 s and (d).1 s 49

Figure 3-4 Streamlines of resultant implicit forces on particles Figure 3-4 shows the streamlines (blue lines) of the resultant forces acting on the particles at s. It can be seen that lines are prominent around the particle clusters. To characterize the packing structure of the particles, the distribution of coordination number and radial distribution function (RDF) are taken into account. To define particles that are touching each other a critical distance equal to 1.01(r1+r), where r1 and r are the radii of two particles, is set. Coordination numbers of the particles are calculated based on the assumption that if the center to center distance between two particles is less than this critical distance then the particles are considered as touching each other. 50

(a) (b) (c) (d) Figure 3-5 Distribution of coordination number at (a) 1 s (b).05 s (c).135 s and (d).167 s Figure 3-5 shows the distribution of coordination number at various times. At time equal to 1 s particles settled down at the bottom but they are barely touching each other. This can be easily seen from the Figure 3-5 (a) where most particles are shown to have a zero coordination number. At time equal to.05 s there is a gradual rise in coordination number which shows that some particles are touching and coming closer to each other. At times equal to.135 s and.167 s coordination numbers as high as 8 or 9 are observed. This means that some particles are sitting in a closed packed region. 51

Another parameter which is used to analyze the packing structure is the radial distribution function (RDF). It describes the probability to find a particle in a shell dr at a distance r from another reference particle. V dn(r) g(r) N 4 r dr (3.30) where V represents the total volume occupied by particles, N is the total number of particles, and dn(r) is the number of particles in a shell of width dr at distance r from the reference particle. 5

(a) (b) (c) (d) Figure 3-6 Radial distribution function (RDF) at (a) 1 s (b).05 s (c).135 s and (d).167 s RDF at time equal to 1 s shown in the Figure 3-6 (a) shows no peak since the particles are not touching each other. At time.05 s RDF shows peaks indicating the fact that particles are now forming clusters. A closer look at the peaks reveals that there are actually three peaks. The first peak is at r which corresponds to the one to one contact configuration. The second and third peak are at r and 4r which correspond to the edgesharing-in-plane equilateral triangle and three particles centers in a line contact type. The peak values of RDF continue to change as the particle packing structure changes. 53

3.13 Conclusion A numerical investigation on evaporation induced self-assembly of sub-micron particles is carried out using a coupled CFD-DEM approach. The interaction between fluid and particle is thoroughly considered by taking into account the momentum exchange and heat transfer between particle and fluid. In the simulation liquid water film is allowed to evaporate and leave the particles at the container alone. Interesting patterns are seen to emerge as the liquid water film evaporates. The resulting packing structure is analyzed in terms of the range of coordination number and radial distribution function which also indicate the self-assembly of the particles. 54

4 Numerical Simulation of Jamming Transition in Granular System under Cyclic Compression using Smoothed Particle Hydrodynamics 4.1 Introduction SPH method, introduced by Lucy [3], Gingold [33, 34] and Monaghan [35, 36], was originally developed for solving astrophysics problem. This method has the ability to handle large deformation especially when comes to solid bodies. Hence it has been applied to hyper-velocity impact [36], explosion [37] and fluid dynamics [38]. When SPH is applied to high velocity impact problems boundary and contact properties on the contact surface are ignored because of their insignificant contribution. However, for low velocity impact or small deformation, as mentioned in [39], neglecting contact conditions can arise several problems. Some of them include ghost stress, which arises due to incorrect inclusion of boundary particles within a smoothing length, false tensile force during separation and normalization of kernel not satisfied. In order to avoid such problems, [39] developed a method that addresses problems involving small deformation as well as large deformation for low velocity impact. In this thesis their method is adopted for simulation of granular particles. In SPH, the continuum is decomposed into a set of arbitrarily distributed particles or integration points. Those particles are then advected in a Lagrangian sense. Field variables such as density, velocity, etc are calculated from the exact integral interpolant of the field quantity. ' f r f r r r ' d r ' (4.1) 55

where f (r) is a field quantity determined at the position vector r, while δ is the Dirac delta function. The above equation can be approximated by f r f r ' W r r ', h d r ' (4.) where W (r r, h) is called the kernel or smoothing function and f (r) is now determined by weighted interpolation of surrounding field quantities. The smoothing length, h defines the domain of influence of the kernel function, W. The kernel function, W is chosen such that it satisfies two condition. The first is the normalization condition that states W r r ', h d r ' 1 (4.3) The integration of the smoothing function should produce unity. The second condition is the Delta function property which states that the kernel function approaches the delta function as smoothing length, h approaches zero. h0 r r r r r lim W ', h d ' ' (4.4) For numerical efficiency a third condition is imposed on the kernel function which is the compact condition. W r r ', h 0 when r r ' kh (4.5) The quantity kh define the effective (non-zero) area of the smoothing function. This area is called the support domain. The parameter k is chosen such that sufficient amount of the function falls within the support domain and the error due to interpolation is small. 56

Figure 4-1 The support domain of the smoothing function, W By discretizing the entire system into a finite number of particles that carry individual mass and occupy individual space, the integral equation can be approximated by summation interpolant. where m j and of f is given as r r r, m N j fi f j W j h (4.6) j1 j j are the mass and density of particle j at r j. The gradient and divergence r r r, m N j fi f j iw j h (4.7) j1 j N mj f f r Wr r, h (4.8) i j i j j1 The above SPH formulation are used to discretize the governing equations. j 4. Equations of Motions The mass, momentum and energy conservation equations from continuum mechanics are given as, 57

D v (4.9) Dt x Dv Dt 1 x (4.10) De v Dt x (4.11) where the density, v the velocity component, the total stress tensor and e is the specific internal energy. The total stress tensor is made up of two parts, the hydrostatic pressure p and the deviatoric shear stress s. p s (4.1) a The stress rate obtained from the constitutive relation must be invariant with respect to rigid body rotation when large deformation is involved. In this paper, the Jaumann stress rate is adopted for this purpose as, s s R s R Ge (4.13) where R is the rotation rate tensor defined as R 1 v x v x (4.14) 4.3 SPH Formulation The SPH or Smoothed Particle Hydrodynamics is a meshless method where the governing equations are solved by discretizing the entire system using finite number of 58

particles that carry individual mass and occupy certain space. These particles are the mathematical interpolation point themselves. The material properties of the particles are calculated from their relationship with neighboring particles using the kernel function for interpolation. In this paper the elastic SPH model formulated by [58] are used. The SPH approximation for mass, momentum and energy equation take the following form, Di Dt W N a a ij m j vi v j (4.15) j 1 x i Dv Dt i N i j Wij mj j1 i (4.16) j xi Dei Dt N 1 j W i a a ij mj vi v j j1 i (4.17) j xi where m is the mass of individual particle and W ij is the kernel function. Several different kernel functions have been used in SPH literature. The one used in this paper is the most popular one, the cubic spline function proposed by [35], which has the following form: 1 3 R R 0 R 1 3 1 3 W R, h d R 1 R 6 0 R (4.18) where d is the normalization factor which is 15/ 7 h in two-dimension, h is the smoothing length and R is the distance between particles i and j normalized as R r / h. 59

4.4 Artificial Viscosity In SPH literature a dissipative term, ij, which is also called artificial viscosity, is introduced into the governing equation to prevent large unphysical oscillation in the numerical solution and to improve numerical stability. The artificial viscosity is incorporated into the momentum equation in the following way: Dv Dt i N i j W ij mj ij j1 i (4.19) j xi paper. The most widely applied artificial viscosity derived by Monaghan [59] is used in this cijij ij vij xij 0 ij ij 0 vij xij 0 (4.0) where v x 1 1 h, c c c, ij ij ij ij i j ij i j r ij h (4.1) x x x, v v v (4.) ij i j ij i j In the above equation, and are constants and taken as unity, c is the sound speed and is 0.01. The energy equation takes the form of Dei Dt N 1 j W i a a ij mj ij vi v j j1 i (4.3) j xi 60

4.5 Tensile Instability In SPH, for solid body deforming under tension numerical instability arises and material can fail unrealistically. This happens because under tension the SPH particles attract each other and tend to clump together. Monaghan [60] and Grey et al. [61] successfully introduced artificial stress method to overcome this problem of tensile instability. Their method involves adding another term to the momentum equation. The additional term comes from the dispersion relations. dvi dt j W i n ij mj ij Ri Rj f ij j i (4.4) j xi where n is the exponent which depends on the smoothing kernel, f ij is the repulsive force term which is given in terms of the kernels by [60] as: f ij W W ij p, h (4.5) where W p is the initial particle spacing. In this paper, h is assumed to be constant so that p, h is also constant. It is found that best result can be obtained by setting h1.5p. The [61] suggested a value of 4 for n for best result so that value is used in this work. tensor For the two-dimensional case used in this paper, the components of the artificial stress R i for particle i in the reference coordinate system (x, y) are computed from the principal components xx R i and R by the coordinate transformation: yy i R R cos R sin (4.6) xx xx yy i i i i i R R sin R cos (4.7) yy xx yy i i i i i 61

xy xx yy Ri R i R i sini cosi (4.8) where the angle i is defined as, xy i tan i xx yy i i (4.9) xx where i, yy i and xy i are components of stress tensor of particle i in the reference frame (x, y). The diagonal components of the artificial stress tensor in principle axes are calculated as [61], R xx i xx xx if 0 i i i (4.30) 0 otherwise R yy i yy yy if 0 i i i (4.31) 0 otherwise where is a constant and is taken as 0.3 which as suggested by [61] is the best value for elastic solid. xx and yy are the principal stresses of particle i. They are obtained as, i i cos sin cos sin (4.3) xx xx xy yy i i i i i i i i sin sin cos cos (4.33) yy xx xy yy i i i i i i i i 4.6 Velocity Smoothing The velocities v of the particles obtained by time integration of the momentum equation are corrected in order to smooth out any unexpected numerical peaks. The correction is done by [6], 6

m v v v v W N i i i ij (4.34) j i j j 1 0.5 j The corrected velocities are used to update the position of the particles, while the uncorrected velocities are used for time integration of the momentum equation at the following step. 4.7 Equations of State Pressure, density and internal energy of a material are related by the equation of state of the system. In this paper the Mie-Gruneisen equation for solids [58] is used to calculate the pressure arising from the deformation of the material. Table 4-1 gives the material properties and constants for Mie-Gruneisen EOS of lead. 4 c 10 m / s ( g / cm 3 ) 0 Table 4-1 Material Properties S G( GPa ) E( GPa ) Lead 1.134 1.19 1.80 5.6 16.00 The required equations are 1 p 1 ph e (4.35) where 1 (4.36) 0 p H 3 a0 b, 0 0 c 0 a, 0 0 (4.37) 63

a 0 0c (4.38) b a 1 S 1 0 0 (4.39) c0 a 0 S 1 3 S 1 (4.40) 4.8 Contact Model The contact model used here is that derived by [39] for SPH low velocity impact problem. They derived the weak form of the contact problem using the virtual work principal and solved it by penalty method that involves both a penetration and a penetration rate. The SPH form of the variational equation for the contact problem is given by [39], N v i iwijv jvi j WijV jvi 1p n pn niwij AV i j vi 0 (4.41) i jm i jm i jm i where M i represents those particles within a distance h of a particle i, particles within a distance h of particle i, and A i is the contact area. B M those boundary i p n is the penetration and p n is the penetration rate. 1 denotes the penalty parameter for the penetration rate and is the penalty parameter for the penetration. The inside of the bracket must be zero for allowable virtual velocity v i. Therefore the momentum equation for two bodies in contact can be written as [39], m (4.4) int v F F con i i i i where 64

m F int i j ij j i jmi F W V V i i ij j i jmi W V V p p nw AV con i 1 n n i ij i j jmi (4.43) 4.9 Contact Force The [39] Uses the one dimensional elastic wave theory to obtain the expression for contact force. Their equation for penalty parameters are, jc j Ei E j 1 1p p ici p p jc j ic i Ei E j d o (4.44) where i and j are densities of SPH particle i and j in contact, C i and C j are their respective sound speeds, E i and E j are their respective elastic moduli and d o is the diameter of the SPH particles. 4.10 Contact Detection To find the boundary particles a color parameter, i is introduced for each SPH particles. A particle will be designated as a boundary particle if the summation of i is less than 0.85~0.90 of the original index value [38, 39]. i jmw j ij / j jn i (4.45) The boundary normal vector is obtained from the gradient of the color parameter. n / (4.46) i i where 65

m W / i jmi j j ij ij (4.47) The following criterion is used to detect the contact [39]. d d j d i i d j r ij Max, (4.48) where r ij is the center to center distance between two SPH particles and d i and their respective diameters. d j are To obtain the penetration and penetration rate for two SPH particles in contact by taking into account the curvature of the surface [39] proposes a way to find the average normal vector for the two surfaces in contact. If is the angle formed with the normal vectors of particle i and j then the average normal vector for the two particles in contact is given by, ii n,n i j ni n i if Max c rij n i, rij n j, n i n c 70 o i j then n av n i, n j, n i n j n n i j c nav ni n j i j then n av n i, n j, n i n j n n n n j j (4.49) where c is the critical angle. The penetration and penetration rate for contact between particles is given by 66

p R R r n p nav i j ij v v n nav i j av av (4.50) The force of contact is then obtained by, F con i 1 nav nav i ij i j jmi p p nw AV (4.51) 4.11 Validation 4.11.1 Impact of two identical disks The above SPH contact model is used to simulate impact of two identical disks each of radius 50 mm and the number of SPH particle in each disk is 91. The material is steel whose properties are given in Table 4-1. The Mie-Gruneisen equation of state is used. The disks move with an equal but opposite initial velocity of 10 m/s. First the contact detection algorithm is tested to see if the boundary particles can be single out from the inner particles. Figure 4- shows that the boundary particles have a much larger color gradient, than the particles inside. Figure 4- The boundary particles indicated by color gradient, 67

Figure 4-3 Pressure distribution of two colliding disks Figure 4-4 shows the time history of the total contact force on the particles. The current result is compared with the Hertz solution for collision of two disks. The total force in Hertz model is a function of deformation given by The total force in Hertz model is a function of deformation given by: where P 3/ n (4.5) n k 16 9 RR 1 k k R R 1 1 1 1 1 1, k E1 E (4.53) where is the deformation. The maximum value of is 5v 1 4 nn1 m1 m n1 mm 1 /5 (4.54) The time for collision between two spheres with same radius and material properties from the dynamic Hertzian solution is given as, 68

/5 5 1 R t.94 4 E v 1/5 (4.55) The result shows that the solution of the Hertz contact is very close to the simulated solution. Figure 4-4 Total contact force-time histories of the impacting disks Shows the average velocity-time history of the two colliding disks. It can be seen that the final rebound velocity is equal to the initial velocity of impact. Therefore the kinetic energy is conserved before and after the collision. Figure 4-5 Average velocity-time history of the two colliding disks 69

4.1 Elasto-Plastic Theory For an elastic perfectly plastic material, can be decomposed into two parts: elastic strain rate tensor e and the other part is plastic strain rate tensor p. (4.56) e p law. The elastic strain rate tensor e is normally calculated by the generalized Hooke s s 1 (4.57) G 3E e where s is the deviatoric shear stress rate tensor; v is Poisson s ratio; E is Young s modulus; G is the shear modulus and is the sum of the three normal stress components, i.e. xx yy zz (4.58) The plastic strain rate tensor can be computed by using the plastic flow rule g (4.59) p where is the rate of change of the plastic multiplier which depends on the stress state and load history. Here g is the plastic potential function, which specifies the direction to which the plastic strain develops. If the plastic potential function g coincides with the yield function f of the material, the flow rule is then called the associated type; otherwise it is called the non-associated type. The plastic multiplier has to satisfy the following conditions of the yield criterion 1. = 0 if f < 0 or f = 0 and d f < 0. This corresponds to elastic or plastic unloading.. > 0 if f = 0 and d f = 0. This corresponds to plastic loading. 70

The value of plastic multiplier can be calculated by using the consistency condition, which states that f df d 0 This equation assures that the new stress state the yield criterion: d (4.60) after loading still satisfies f d f df f (4.61) For the yield function, f the von Mises yield criterion is adopted which expresses the relationship between second invariant of stress tensor, J and yield stress, Y. The von Mises yield criterion is s s Y 3 (4.6) 1 Now, J s s which is the second invariant of stress tensor. Therefore, 3J Y (4.63) 3 J Y 0 (4.64) Hence, f 3 J Y 0 (4.65) The idea is to solve the elasto-plastic constitutive relation iteratively to obtain strain increment from stress increment. In this study the Johnson-Cook model for elasto-plastic contact problem is to used. Y B C T 0 0 n eq * eq 1 ln 1 m (4.66) 71

term 0 This equation has five experimentally determined parameters (, B, C, n, m). The * T is computed as follows, T T T T T * r m r (4.67) eq and eq are the equivalent plastic strain and equivalent plastic strain rate given by, t eq :, eq initial eq 3 dt (4.68) 0 The Johnson-Crook model has shown to be highly successful for a wide range of materials. 4.1.1 The Radial Return Method In case of plastic loading the stress state calculated for a strain increment may not satisfy the yield criterion. The radial return method is one of the ways to meet the plastic loading condition. The method here is adopted from [63] The radial return method ensures that the resulting stress is always on the yield surface. Starting from the Hooke s law for elastic strain increment, d D dt e d D dt d D dt D dt p p (4.69) 7

a Gee K e 1 a a G e e K e 3 1 a dt G e dt 3K G e dt 3 1 a dt G p dt 3K G p dt 3 1 a 1 a dt G dt 3K G dt G p dt 3K G p dt 3 3 (4.70) Given that the plastic strain rate tensor can be computed by using the plastic flow rule g dt (4.71) p dt 1 a dt G dt 3K G dt 3 g 1 g a G dt 3K G dt 3 (4.7) Now applying the Jaumann strain rate correction dt R dt R dt G dt 1 a g 1 g a 3K G dt G dt 3K G dt 3 3 (4.73) Hence the plastic part of the stress tensor can be written as g 1 g a p dt G dt 3K G dt 3 (4.74) Now assume A is the initial stress state, B is the stress state after adding the elastic part and C is the desired final stress state that satisfies the yield condition. 73

f f C f B p 0 f C B p 0 f f B f p B B (4.75) Now putting the expression for dt p instead of p g 1 g a f B G dt 3K G dt 3 f f B f g 1 g a G dt 3K G dt 3 B B (4.76) As given earlier, f 3 J Y Therfore, f f J 1 f 3 s J J J J s (4.77) Using associated flow rule g = f, J g g 1 g 3 s s J J J J (4.78) Hence the equation for becomes, 74

f f g 1 g a G dt 3K G dt 3 3 3 1 a Gdts s 3K G s dts 4J 3 J 3 J Y 3 3 1 3 a s G s dt 3K G s dt J 3 J J Y (4.79) Since s a 0 (4.80) 3 J Y 3 4GdtJ 4J 3 J 3Gdt Y (4.81) In the above equation dt is the time step. So can be estimated using the above equation. Algorithm 1. Start with initial 0. Iteration j = 0,, n 3. Find using 3 J 3Gdt Y g 1 g a p G dt K G dt 3 4. Find the plastic corrector 3 5. Add the plastic corrector to the initial stress. 1 6. Continue until f j 1 j j p 75

The convergence rate is found to be quite good for this method. However, to increase the accuracy of the solution a fully implicit Newton Raphson method is also considered. 4.1. Newton Raphson Method The equations to be solved are Now differentiating with respect to p D Dg 0 (4.8) 0 q f (4.83), p D Dg D g 0 (4.84) Applying the Jaumann stress rate, 1 a p dt R dt R dt G dt 3K G dt 3 g g G dt 0 1 g g 3 a K G dt 3 (4.85) Then differentiating with respect to, p I D g p Dg p g 1 g a I G dt 3K G dt dt 3 p g 1 g a G 3K G 3 dt (4.86) The second equation can be written as, 76

q f 0 f q f 0 q q f f (4.87) (4.88) q q dt f Y dt dt 3 J s (4.89) Now from the Johnson-Cook model, Y B C T 0 T * 0 n eq * eq 1 ln 1 t p p p eq p p eq initial eq 3 0 T Tr, :, dt T T m r m (4.90) The plastic strain and plastic strain rate can be shown as, p g g eq p p 3 3 p g g 3 s eq 3 J 3 p 3 4 J eq J 3 p eq dt s (4.91) Then, 77

Y B C T If p n 0 * m p eq 1 ln 1 eq 0 Y Bn C B T dt 0 dt eq p eq 0 1 p eq C 0 p * 1 ln 1 eq p eq n p 1 1 eq Y B T Y Bn dt 0 * n1 p * 1 1T eq m p n n m m (4.9) So the expression for Meanwhile, Y dt can be put in equation, g 3 s J g 3 1 3/ 1 s J s s J xx 1 xx yy zz s xx 3 yy 1 xx yy zz yy 3 zz 1 xx yy zz zz 3 s s s s s s (4.93) Therefore, 78

xx 1 yy zz yy 1 xx zz s xx yy 3 3 3 3 zz 1 xx yy zz 3 3 s s s s s s (4.94) 1 1 1 1 1 1 8 3 3 3 j p p j dt dt dt j p j q q dt q dt dt g 3 1 8 3 7 J J J g 1 g a g 1 g a I G dt 3K G dt G 3K G 3 3 3 Y s J dt j dt j p j dt q j (4.95) (4.96) 1 dt j j j (4.97) j j dt 1 j The iteration continues up to convergence. (4.98) f j 1 (4.99) 4.1.3 Implementation: Point load on a disk Figure 4-7 shows the graph of true stress plotted against true strain for a ball with a point force of 5000 N on its top. The stress-strain plot shows the characteristic non-linear part for the plastic region followed by a linear elastic part at the beginning. When the stress crosses 500 MPa plastic deformation begins where the yield stress is 500 MP. 79

Figure 4-6 Hydrostatic pressure distribution Figure 4-7 True stress against true strain for a point on the ball 4.13 Boundary Treatment The usual practice in SPH is to update the density using the summation density approach. N mw (4.100) i j ij j1 However, if (4.101) is used then the kernel normalization condition will be dissatisfied at the boundary surface and the particles near the surface will have lower densities. The pressure calculated from the equation of state will be incorrect and the numerical result 80

will be degraded. To overcome the problem the rate of change of density is calculated from the continuity equation as discussed in Monaghan [59]. All particles are assigned the same initial density which changes when the particles are in relative motion. The continuity equation can be written in the form D v v Dt x x (4.10) which gives equation 4.7 when converted to SPH form. 4.14 Physical model The simulation domain includes a square container with,400 mono-sized disks within the box. The box size, disk size and their material properties are given in Table 4-1 and Table 4-. The disk are placed such that there is no initial contact with the wall of the container or among the disks. The initial packing fraction for the system is 0.8. Gravity is neglected in this study. Two sides of the container are kept stationary while the two other sides are displaced with a linear velocity. As can be seen from Figure 4-8 the left and bottom walls move backward and forward with an amplitude of 0 mm in each cycle. The time period for the cycle is.5 millisecond. However, each cycle consists of several steps where the walls are kept stationary to relax the system. The time step used for the simulation is 0.1 microsecond. The moving walls perform a series of 10 cycles where the system is quasi-statically compressed or expanded. 81

(a) Time = 0s (b) Time = 7.5 10-4 s (c) Time = 1.5 10-4 s (d) Time = 17.5 10-4 s (e) time = 5 10-4 s Figure 4-8 One full cycle of compression and expansion showing also the force chain networks Figure 4-9 gives a closer view of the disks and the force chains. Each disk consists of 61 SPH particles arranged in an optimum packing configuration. The radius of each SPH particle is 0.34868 mm. The walls are also made of SPH particles. For the case of walls square lattice configuration is used and the lattice spacing is twice that of the SPH particle radius. Each wall has two layers of SPH particles. 8

Figure 4-9 A closer look of the disks and force chains 4.15 Simulation Parameters Table 4- Simulation Parameters Parameters Values Box size, m m 0.94 0.94 Time step 1 10-7 s Granular particle numbers 400 Granular Particle radius 3.06 10-3 m Number of SPH particles per disk 61 SPH Particle radius 3.4868 10-4 m 83