Mechanics of Granular Matter

Transcription

1 Mechanics of Granular Matter

2

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

4 Qicheng Sun & Guangqian Wang Tsinghua University, Beijing, China Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) ; Fax: 44 (0) For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: ; Fax: British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: eisbn: Library of Congress Catalog Card Number: 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 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.

5 Contents Preface xi Chapter 1 Behaviors of granular materials Introduction Static behaviors Coulomb friction law Janssen effect Effective stress Rowe stress dilatancy relation Dynamic behaviors Granular flows Faraday circulation Reynolds dilatancy Clustering in granular gas Force measurement and internal structure recognition X-ray radiography Photoelastic stress analysis Granular physics Granular solid hydrodynamics Statistical mechanics Granular gas 21 Chapter 2 Contact mechanics of spherical particles Nonadhesive contact Normal force (Hertz model) Tangential force (Mindlin-Deresiewicz model) Adhesive contact Bradley model and DMT model 30

6 2.2.2 JKR model Maugis Dugdale model Thornton model From contact initiation to a critical peeling state After the critical peeling state 39 Chapter 3 Soft-sphere approach and hard-sphere approach Soft-sphere approach Calculation of contact force Spring stiffness Damping coefficient Hard-sphere approach One-dimensional collision Two- and three-dimensional collisions Normal restitution coefficient Tangential restitution coefficient Comparisons 56 Chapter 4 Liquid bridge forces Liquid distribution Static liquid bridge force Separation distance Multiple liquid bridge force Dynamic liquid bridge force Normal force of Newtonian fluid Normal force of power-law fluid Tangential resistance of Newtonian fluid Tangential force of power-law fluid 69 Chapter 5 Discrete element method Contact searching Rigid-sphere-based DEM Soft-sphere-based DEM Dynamic relaxation Numerical scheme Euler method Verlet integration System evolution Time step Large-scale parallel computing 81

7 Chapter 6 Force chains Formation of force chain Measurements of contact force Photoelastic stress analysis Carbon paper method Electronic balance weighing method Discrete element method Bulk contact stress Bulk friction Bulk restitution coefficient Bulk elasticity Correlation of force network with mechanical properties Multiscale mechanics strategy Characteristic time scales The macroscopic time scale t c Three dimensionless numbers 106 Chapter 7 Jamming and structure transformations Introduction Frictionless soft sphere systems Coordination number of an isostatic system Elastics modulus Vibrational density of states Microscopic criterion for stability under compression Pair-correlation function Frictional soft sphere systems Critical coordination number Generalized isostaticity Z φ phase diagram Characteristic frequency of density of state and the modulus ratio G/K Jamming of other disordered systems Jamming of nonspherical particles Jamming of foams under shear Glass-like transition of rigid granular fluid Structural transformation in a frictional system Numerical simulations Pair-correlation function g (r) Force force correlation and position position correlation Unjamming process Conclusions 129

8 Chapter 8 Point loading response and shear band evolution Point loading transmission Numerical experiment setup Point loading transmission Force network under uniaxial compression Criteria of force chains Lateral pressure coefficient Shear bands Macroscopic phenomena Mesoscale analysis on shear bands Force-chain structures Energy transformations Introduction Simulation setup Energy analysis Elastic energy and critical sensitivity Kinetic energy Energy dissipation Discussions 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 Bagnold number Inertial number and contact stress Flow regimes Constant-volume granular flows Constant-stress granular flows Regime transition Macrostress Contact time number Coordination number Constitutive relations Plane shear flow with zero gravity Slope flow under gravity 182 Appendix A. Internal parameters of granular flows 183

9 Chapter 10 Preliminary multiscale mechanics Macroscopic stress and strain Macro micro relations Multiscale mechanics 190 Index 193

10

11 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