title: Granular materials: an extensive numerical study about the effect of grain shape and polydispersity on collective granular mechanics
Creators: name: Richters-Finger, Steffen
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2023
Subjects: subject: 530 Physics

description: Die vorliegende Arbeit beschäftigt sich mit der Untersuchung des Einflusses der Materialzusammensetzung (Teilchenform und Polydispersität) auf die kollektiven mechanischen Eigenschaften eines Granulats. Hierfür werden vorwiegend numerische Verfahren in Kombination mit einer angepassten Modellierung der Kontaktkräfte eingesetzt, um eine breite Menge von Materialtypen zu untersuchen. Insbesondere werden dabei die Packungseigenschaften, die Kompaktion durch vertikales Klopfen sowie der Widerstand gegenüber Scherkräften im Detail betrachtet. Im Fokus stehen dabei unter anderem die Hohlraumverteilung im Granulat und die Organisation des Kontakt- und Kräftenetzwerks.
description: The present thesis addresses the influence of material composition (grain shape and polydispersity) on the collective mechanical properties of a granular system. To this end, predominantly numerical methods in combination with a suitable model for the contact forces are deployed in order to consider a wide variety of different types of materials. In particular, the packing properties, the compaction induced by vertical tapping, as well as the resistance to shearing are investigated with great detail. The analysis is, among other properties, focused on the distribution of voids and the differences in the contact fabric and force network.
description: Inhaltsangabe ..... i Abstract ..... iii List of Symbols ..... v 1. Introduction ..... 1 2. Numerics ..... 5 2.1. Discrete Element Method ..... 6 2.1.1. Basics ..... 6 2.1.1.1. Overview ..... 9 2.1.1.2. Spatial Sort ..... 10 2.1.1.3. Collision Detection ..... 13 2.1.1.4. Force and Torque Application ..... 15 2.1.1.5. Spatial Periodicity ..... 15 2.1.1.6. Time Step Algorithm ..... 16 2.1.1.7. Parallelization ..... 22 2.1.2. Non-Spherical Shape Extension ..... 23 2.1.2.1. Overview ..... 23 2.1.2.2. Collision Detection ..... 26 2.1.2.3. Sample Placement ..... 33 2.1.2.4. Further Optimization ..... 38 2.1.2.5. Force and Torque Application ..... 45 2.1.3. Container Walls ..... 46 2.2. Continuum Models ..... 47 3. Contact Force Model ..... 49 3.1. Linear Spring-Dashpot Model ..... 49 3.1.1. Cundall and Strack ..... 49 3.1.2. Schwager and P¨oschel ..... 53 3.2. Non-relaxing States ..... 54 3.2.1. Phenomenology ..... 55 3.2.2. Small Scale Energy Generation ..... 58 3.3. Modified Linear Spring-Dashpot Model ..... 60 3.3.1. Work-Dependent Loading of Tangential Spring ..... 60 3.3.2. Release of Excess Energy ..... 62 3.3.3. Relaxation Curves ..... 65 3.3.4. High Sensitivity Simulations ..... 68 3.4. Contact Force Parameters ..... 69 3.4.1. Force Parameters from Time Step Length ..... 70 3.4.2. Time Step Length from Force Parameters ..... 72 3.4.3. Parameter Selection in this Thesis ..... 72 4. Selection of Shapes ..... 75 4.1. Spherical Grain-Systems ..... 76 4.2. RIS-Systems: Regular and Identical Shapes ..... 77 4.2.1. Overview ..... 78 4.2.2. Transition Paths ..... 80 4.3. MIS-Systems: Mixtures of Irregular Shapes ..... 80 5. Packing Properties ..... 85 5.1. Quantification of Granular Packings ..... 86 5.1.1. Geometric Properties of Shapes ..... 86 5.1.2. Packing Fraction ..... 88 5.1.3. Void Distribution ..... 90 5.1.4. Grain Orientation ..... 95 5.1.5. Contact Fabric ..... 97 5.1.5.1. Coordination Number ..... 100 5.1.5.2. Contact Orientation, Fabric Tensor ..... 100 5.1.5.3. Contact Position Distribution ..... 104 5.1.6. Force Network ..... 104 5.1.6.1. Fraction of Critical Contacts ..... 109 5.1.6.2. Force Distribution ..... 111 5.2. Spherical Grain-Systems ..... 117 5.2.1. Packing Fraction ..... 118 5.2.2. Void Distribution ..... 122 5.2.3. Contact Fabric ..... 125 5.2.3.1. Coordination Number ..... 125 5.2.3.2. Contact Orientation ..... 126 5.2.4. Force Network ..... 129 5.2.4.1. Angular Force Distribution ..... 129 5.2.4.2. Force Magnitude Distribution ..... 133 5.2.4.3. Critical Contacts ..... 134 5.2.5. Summary ..... 135 5.3. Single Shape-Systems ..... 136 5.3.1. Packing Fraction ..... 137 5.3.2. Void Distribution ..... 140 5.3.3. Grain Orientation ..... 148 5.3.4. Contact Fabric ..... 149 5.3.4.1. Coordination Number ..... 149 5.3.4.2. Contact Position ..... 152 5.3.4.3. Contact Orientation ..... 155 5.3.5. Force Network ..... 157 5.3.5.1. Angular Force Distribution ..... 159 5.3.5.2. Force Magnitude Distribution ..... 160 5.3.5.3. Critical Contacts ..... 160 5.3.6. Influence of Polydispersity ..... 165 5.3.6.1. Grain Configurations ..... 166 5.3.6.2. Packing Fraction ..... 167 5.3.6.3. Grain Orientation ..... 170 5.3.6.4. Contact Fabric ..... 173 5.3.6.5. Critical Contacts ..... 175 5.3.7. Summary ..... 175 5.4. Mixture-Systems ..... 177 5.4.1. Packing Fraction and Void Distribution ..... 177 5.4.2. Grain Orientation ..... 183 5.4.3. Contact Fabric ..... 184 5.4.4. Force Network ..... 185 5.4.4.1. Size Segregation and Strong Network ..... 186 5.4.4.2. Force Distribution ..... 190 5.4.4.3. Critical Contacts ..... 192 5.4.5. Summary ..... 193 6. Compaction Properties ..... 195 6.1. Process Definition ..... 196 6.2. Compaction Parameter Estimation ..... 199 6.2.1. Problem Description ..... 199 6.2.1.1. Numerical Instability ..... 199 6.2.1.2. Origin of the Numerical Instability ..... 200 6.2.2. Improved Estimation Procedure ..... 204 6.2.2.1. Generation of Reliable Test Data ..... 204 6.2.2.2. Description of the Fitting Procedure ..... 205 6.2.3. Test and Error Estimation ..... 211 6.2.3.1. Test ..... 214 6.2.3.2. Error Estimation ..... 216 6.2.3.3. Averaging of Results ..... 217 6.3. Spherical Grain-Systems ..... 218 6.3.1. Packing Fraction ..... 218 6.3.2. Void Distribution ..... 223 6.3.3. Contact Fabric ..... 228 6.3.4. Force Network ..... 231 6.3.5. Summary ..... 234 6.4. Single Shape-Systems ..... 235 6.4.1. Packing Fraction ..... 235 6.4.2. Void Distribution ..... 238 6.4.3. Grain Orientation ..... 242 6.4.4. Contact Fabric and Force Distribution ..... 243 6.4.5. Influence of Polydispersity ..... 244 6.4.5.1. Phenomenology and Packing Fraction ..... 246 6.4.5.2. Void Distribution ..... 249 6.4.6. Summary ..... 251 6.5. Mixture-Systems ..... 252 6.5.1. Packing Fraction ..... 252 6.5.2. Void Distribution ..... 253 6.5.3. Contact Fabric ..... 255 6.5.4. Summary ..... 256 6.6. Modeling the Compaction Process ..... 257 6.6.1. Iterative Compaction Model ..... 257 6.6.2. Irreversible Two-State Compaction ..... 259 6.6.2.1. Slow Density Relaxation ..... 260 6.6.2.2. Application to Empirical Data ..... 264 6.6.2.3. Relation with the Heuristic Model ..... 267 6.6.3. Generalizations of the Two-State Model ..... 270 6.6.3.1. Reversible Two-State Model ..... 270 6.6.3.2. Changes in Process Parameters ..... 273 6.7. Summary ..... 275 7. Stability Properties ..... 277 7.1. Chute Problem ..... 277 7.1.1. Setup ..... 277 7.1.2. Parameters ..... 279 7.2. Concept of Stability ..... 280 7.2.1. Rearrangement Events ..... 281 7.2.2. Angles of Stability ..... 283 7.2.2.1. Critical Angle ..... 283 7.2.2.2. Angle of Repose ..... 285 7.2.2.3. Hysteresis ..... 286 7.2.3. Inclination Parameter ..... 287 7.3. Spherical Grain-Systems ..... 288 7.3.1. Angles of Stability ..... 288 7.3.2. Rearrangements ..... 289 7.3.3. Packing Fraction ..... 292 7.3.4. Contact Fabric ..... 293 7.3.5. Force Network ..... 293 7.3.5.1. Force Distributions ..... 295 7.3.5.2. Critical Contacts ..... 298 7.3.6. Summary ..... 300 7.4. Single Shape-Systems ..... 301 7.4.1. Angles of Stability and Rearrangements ..... 302 7.4.2. Force Network ..... 305 7.4.2.1. Force Distributions ..... 305 7.4.2.2. Critical Contacts ..... 309 7.4.3. Influence of Polydispersity ..... 310 7.4.4. Summary ..... 311 7.5. Mixture-Systems ..... 313 7.5.1. Angles of Stability ..... 313 7.5.2. Force Network ..... 313 7.5.3. Summary ..... 317 8. Summary and Outlook ..... 319 8.1. Summary ..... 319 8.1.1. Simulation of Non-Spherical Grains ..... 319 8.1.2. Contact Force Model ..... 320 8.1.3. Packing Properties ..... 320 8.1.4. Compaction Properties ..... 321 8.1.5. Stability Properties ..... 322 8.2. Outlook ..... 323 8.2.1. Generalization to Three Spatial Dimensions ..... 323 8.2.1.1. Generalized RIS-Materials ..... 324 8.2.1.2. Generalized MIS-Materials ..... 325 8.2.2. Chute Flow Properties ..... 327 8.2.2.1. Extension of the Range of Considered Materials ..... 330 8.2.3. Additional Forms of Grain Interactions ..... 331 8.2.4. Effect of Material Composition on Other Processes ..... 331 8.2.5. Pattern Formation ..... 332 8.2.6. Anomalous Behavior of Elongated Grains ..... 332 Appendix ..... 333 A. Voronoi Tessellation ..... 333 B. Kernel Density Estimation ..... 333 B.1. Basics ..... 333 B.2. Bandwidth Selection ..... 335 B.3. Bounded Domains ..... 336 B.3.1. Boundary Kernel for Support with Lower Bound ..... 336 B.3.2. Boundary Kernel for Support with Upper Bound ..... 339 B.3.3. Boundary Kernel for Support with Lower and Upper Bound ..... 340 B.4. Periodic Domains ..... 342 B.5. Weighted KDE ..... 342 C. Generation of Three-Dimensional Irregular Shapes via Mathematica ..... 343 Bibliography ..... 347