Mechanics of Granular Matter

Mechanics of Granular Matter

Mechanics of Granular Matter Qicheng Sun & Guangqian Wang Tsinghua University, Beijing, China

Qicheng Sun & Guangqian Wang Tsinghua University, Beijing, China Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: witpress@witpress.com http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: infousa@witpress.com http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-644-8 eisbn: 978-1-84564-645-5 Library of Congress Catalog Card Number: 2011936267 No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications. The original Chinese language work has been published by SCIENCE PRESS, Beijing. Science Press 2013. All rights reserved. Printed by Lightning Source, UK. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Contents Preface xi Chapter 1 Behaviors of granular materials 1 1.1 Introduction 1 1.2 Static behaviors 4 1.2.1 Coulomb friction law 4 1.2.2 Janssen effect 5 1.2.3 Effective stress 7 1.2.4 Rowe stress dilatancy relation 8 1.3 Dynamic behaviors 10 1.3.1 Granular flows 10 1.3.2 Faraday circulation 12 1.3.3 Reynolds dilatancy 13 1.3.4 Clustering in granular gas 14 1.4 Force measurement and internal structure recognition 14 1.4.1 X-ray radiography 14 1.4.2 Photoelastic stress analysis 16 1.5 Granular physics 18 1.5.1 Granular solid hydrodynamics 19 1.5.2 Statistical mechanics 21 1.5.3 Granular gas 21 Chapter 2 Contact mechanics of spherical particles 25 2.1 Nonadhesive contact 26 2.1.1 Normal force (Hertz model) 26 2.1.2 Tangential force (Mindlin-Deresiewicz model) 27 2.2 Adhesive contact 30 2.2.1 Bradley model and DMT model 30

2.2.2 JKR model 32 2.2.3 Maugis Dugdale model 34 2.2.4 Thornton model 37 2.2.4.1 From contact initiation to a critical peeling state 38 2.2.4.2 After the critical peeling state 39 Chapter 3 Soft-sphere approach and hard-sphere approach 43 3.1 Soft-sphere approach 43 3.1.1 Calculation of contact force 45 3.1.2 Spring stiffness 46 3.1.3 Damping coefficient 47 3.2 Hard-sphere approach 49 3.2.1 One-dimensional collision 49 3.2.2 Two- and three-dimensional collisions 50 3.2.3 Normal restitution coefficient 51 3.2.4 Tangential restitution coefficient 53 3.3 Comparisons 56 Chapter 4 Liquid bridge forces 59 4.1 Liquid distribution 59 4.2 Static liquid bridge force 60 4.2.1 Separation distance 63 4.2.2 Multiple liquid bridge force 65 4.3 Dynamic liquid bridge force 66 4.3.1 Normal force of Newtonian fluid 66 4.3.2 Normal force of power-law fluid 67 4.3.3 Tangential resistance of Newtonian fluid 69 4.3.4 Tangential force of power-law fluid 69 Chapter 5 Discrete element method 73 5.1 Contact searching 73 5.2 Rigid-sphere-based DEM 74 5.3 Soft-sphere-based DEM 75 5.3.1 Dynamic relaxation 75 5.3.2 Numerical scheme 76 5.3.2.1 Euler method 76 5.3.2.2 Verlet integration 77 5.3.3 System evolution 77 5.3.4 Time step 78 5.4 Large-scale parallel computing 81

Chapter 6 Force chains 85 6.1 Formation of force chain 85 6.2 Measurements of contact force 87 6.2.1 Photoelastic stress analysis 87 6.2.2 Carbon paper method 89 6.2.3 Electronic balance weighing method 90 6.2.4 Discrete element method 91 6.3 Bulk contact stress 93 6.4 Bulk friction 98 6.5 Bulk restitution coefficient 99 6.6 Bulk elasticity 100 6.7 Correlation of force network with mechanical properties 101 6.8 Multiscale mechanics strategy 103 6.9 Characteristic time scales 104 6.9.1 The macroscopic time scale t c 105 6.9.2 Three dimensionless numbers 106 Chapter 7 Jamming and structure transformations 109 7.1 Introduction 109 7.2 Frictionless soft sphere systems 111 7.2.1 Coordination number of an isostatic system 112 7.2.2 Elastics modulus 112 7.2.3 Vibrational density of states 113 7.2.4 Microscopic criterion for stability under compression 115 7.2.5 Pair-correlation function 115 7.3 Frictional soft sphere systems 117 7.3.1 Critical coordination number 117 7.3.2 Generalized isostaticity 118 7.3.3 Z φ phase diagram 119 7.3.4 Characteristic frequency of density of state and the modulus ratio G/K 120 7.4 Jamming of other disordered systems 120 7.4.1 Jamming of nonspherical particles 120 7.4.2 Jamming of foams under shear 122 7.4.3 Glass-like transition of rigid granular fluid 122 7.5 Structural transformation in a frictional system 123 7.5.1 Numerical simulations 124 7.5.2 Pair-correlation function g (r) 125 7.5.3 Force force correlation and position position correlation 126 7.5.4 Unjamming process 127 7.6 Conclusions 129

Chapter 8 Point loading response and shear band evolution 133 8.1 Point loading transmission 133 8.1.1 Numerical experiment setup 134 8.1.2 Point loading transmission 135 8.2 Force network under uniaxial compression 137 8.2.1 Criteria of force chains 138 8.2.2 Lateral pressure coefficient 139 8.3 Shear bands 140 8.3.1 Macroscopic phenomena 141 8.3.2 Mesoscale analysis on shear bands 143 8.3.3 Force-chain structures 146 8.4 Energy transformations 148 8.4.1 Introduction 148 8.4.2 Simulation setup 150 8.4.3 Energy analysis 152 8.4.3.1 Elastic energy and critical sensitivity 152 8.4.3.2 Kinetic energy 153 8.4.3.3 Energy dissipation 154 8.4.4 Discussions 155 8.4.5 Outlook 157 Appendix A. Formulations of energies in granular systems 157 A.1 Elastic energy 157 A.2 Kinetic energy 158 A.3 Dissipated energy due to friction 158 Chapter 9 Granular flows 9.1 Coulomb friction 161 9.2 Bagnold number 162 9.3 Inertial number and contact stress 165 9.4 Flow regimes 166 9.5 Constant-volume granular flows 171 9.6 Constant-stress granular flows 175 9.7 Regime transition 176 9.7.1 Macrostress 176 9.7.2 Contact time number 178 9.7.3 Coordination number 179 9.8 Constitutive relations 180 9.8 Plane shear flow with zero gravity 182 9.8.2 Slope flow under gravity 182 Appendix A. Internal parameters of granular flows 183

Chapter 10 Preliminary multiscale mechanics 187 10.1 Macroscopic stress and strain 188 10.2 Macro micro relations 189 10.3 Multiscale mechanics 190 Index 193

Preface Granular materials are intrinsically athermal since their dynamics always occur at a state far from equilibrium. Quasi-static granular solids and granular flows are of great engineering importance, and innumerable equations have been presented to fit test data. However, their mechanical behaviours are still rather poorly understood, in contrast with the theoretical successes in studying highly excited granular gases. Granular systems exhibit distinct characteristics on multiple spatial and temporal scales. A constituent particle is of course a solid, but granular materials may behave differently from ordinary solids, liquids and gases. This book focuses on the basic mechanics and underlying physics of granular materials. It starts with an introduction of contact mechanics of individual particles. It then discusses the structure of force chains network, and the influence on bulk mechanical properties of granular solids and granular flows. A preliminary multiscale framework is proposed for the nonlinear mechanics and strain localization in granular materials. Special thanks are due to the following for their assistance: Prof. Jinghai Li encouraged Q.S. to conduct a multiscale analysis on granular materials. Prof. Feng Jin supported Q.S. during his difficult times. Dr Guohua Zhang revised the statistical mechanics. Dr Zhongwei Bi simulated the shear band initiation and development. Dr Xia Li and Dr Jianmin Qin provided the new derivations of stress and strain. Dr Shunying Ji and Dr Gongdan Zhou discussed the granular flow studies, and Mr. Jianguo Liu prepared photoelastic tests. The support of the National Key Basic Research Program of China, the Natural Science Foundation of China, and the State Key Laboratory of Hydroscience and Engineering, Tsinghua University, is also acknowledged. Qicheng Sun & Guangqian Wang Tsinghua University, 2013