Micromechanics of Granular Materials E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 518

Veröffentlichungsjahr: 2013

Beschreibung

Nearly all solids are compised of grains. However most studies treat materials as a continious solid. The book applies analysis used on loose granular materials to dense grainular materials. This title's main focus is devoted to static or dynamic loadings applied to dense materials, although rapid flows and widely dispersed media are also mentioned briefly. Three essential areas are covered: Local variable analysis: Contact forces, displacements and rotations, orientation of contacting particles and fabric tensors are all examples of local variables. Their statistical distributions, such as spatial distribution and possible localization, are analyzed, taking into account experimental results or numerical simulations. Change of scales procedures: Also known as "homogenization techniques", these procedures make it possible to construct continuum laws to be used in a continuum mechanics approach or performing smaller scale analyses. Numerical modeling: Several methods designed to calculate approximate solutions of dynamical equations together with unilateral contact and frictional laws are presented, including molecular dynamics, the distinct element method and non-smooth contact dynamics. Numerical examples are given and the quality of numerical approximations is discussed.

Leseprobe

Contents

Introduction

Chapter 1. Experimental and Numerical Analysis of Local Variables in Granular Materials

1.1. Introduction

1.2. Description of granular texture

1.3. Granular kinematics

1.4. Force transmission

1.5. Conclusion

1.6. Bibliography

Chapter 2. The Stress Tensor in Granular Media and in other Mechanical Collections

2.1. Introduction

2.2. Efforts and virtual power

2.3. Equilibrium

2.4. Comparison with the pair-by-pair approach

2.5. Directions of cut

2.6. Coarse graining the equation of Statics

2.7. One step into Dynamics

2.8. Bibliography

Chapter 3. Multiscale Techniques for Granular Materials

3.1. Introduction

3.2. Scale change and fabric tensors

3.3. Change of scale for static variables

3.4. Change of scale for kinematic variables in granular materials

3.5. Statistical homogenization in granular materials

3.6. Bibliography

Chapter 4. Numerical Simulation of Granular Materials

4.1. Introduction

4.2. The actors of a contact problem

4.3. Kinematic relations

4.4. The dynamical equation

4.5. Frictional contact laws

4.6. The equations governing a collection of contacting bodies

4.7. Preparingnumerical samples

4.8. Smooth DEM numerical methods

4.9. Non-smooth DEM numerical methods

4.10. Some illustrating examples

4.11. Quasi-static evolutions, equilibrium dedicated methods

4.12. Accuracy criteria

4.13. Indetermination in granular materials

4.14. Stability

4.15. Numerical integration schemes

4.16. More non-smooth DEM methods

4.17. Signorini μ-Coulomb derived laws

4.18. Conclusion

4.19. Appendix: basic convex analysis

4.20. Bibliography

Chapter 5. Frictionless Unilateral Multibody Dynamics

5.1. Introduction

5.2. The dynamics of rigid body systems

5.3. The dynamics of rigid body systems with perfect bilateral constraints

5.4. The dynamics of rigid body systems with perfect unilateral constraints

5.5. Bibliography

List of Authors

Index

First published in France in 2001 by Hermes Science/Lavoisier entitled: Micromécanique des matériaux granulaires © Hermes Science Ltd, 2001

First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd, 2009

The rights of Bernard Cambou, Michel Jean and Farhang Radjaï to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Micromécanique des matériaux granulaires. English.

Micromechanics of granular materials / edited by Bernard Cambou, Michel Jean, Farhang Radjai.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-075-2

1. Granular materials--Mechanical properties. 2. Granular materials--Microstructure. 3. Particles. I. Cambou, Bernard. II. Jean, Michel. III. Radjai, Farhang. IV. Title.

TA418.78.M5213 2009

620’.43--dc22

2009010452

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-075-2

Introduction

After the Second World War, all the developed countries began ambitious infrastructure plans in the domains of transport, energy and water supply. Many large dams, in particular rock-fill dams, were built at that time. The design of such large works was a matter of what we call in France ‘l’Art de l’Ingénieur’, an efficient mixture of science, technology and personal experience. This design was based on classical continuum mechanics. The discrete nature of rock-fill, i.e. the fact that the dam material was an assembly of rock blocks, was usually completely ignored. The dimensions of the constitutive blocks may reach one meter, however; not quite negligible when compared to the dimensions of the whole work.

The important challenge guiding these works in terms of social and economical development was certainly one of the main reasons why a number of engineers and researchers involved in the design, using classical rules, tried to go further by considering the discrete nature of the materials. While the continuum mechanics seemed at that time a well-establisheld field, considering these materials not as a continuum but as they really are (i.e. assemblies of more or less rigid elements contacting each other) was a new and attractive domain of research.

Among the few people who worked on this subject early on, Dantu discovered the strong heterogenity of the distribution of contact forces in granular materials by means of the photoelasticity analysis of 2D analog materials (piles of translucid disks). Biarez, analyzing 2D analog materials (piles of disks), introduced the concept of fabric anisotropy related to the distribution of contact orientation. This was generalized later on by Satake who defined the fabric tensor. In 1966, Weber exhibited a now classical formula relating the local contact forces to a particular Cauchy-like stress tensor. On the basis of particle considerations, Rowe proposed the well-known stress-dilatancy relation which has been extensively used in many phenomenological models in soil mechanics. Horne, Caquot and other researchers tried to relate the classical internal angle of friction used in soil mechanics to the angle of friction measured at the local scale. This isssue is still open today. Marsal, performing triaxial tests at a very large scale with an assembly of blocks, found that the failure of blocks may play an important role in the deformation of the whole assembly.

All these highly innovative results for their time were obtained mainly from tests performed on analog materials equipped with measurement devices; some tests were also performed on actual materials but were more difficult to analyze. Some theoretical attempts relying on idealized particle arrays were also made. At the end of the 1960s, all these attempts reached a dead end due to the lack of mechanical data at the local scale. In 1971, Cundall developed a new approach: the Distinct Element Method (DEM). The method was named in order to highlight the fact that the material is not considered as a continuum, as in the classical Finite Element Method (FEM), but that all grains of the material are considered to be individual rigid bodies interacting through frictional contact laws governed by the dynamical equation. It was also a step towards departing from the quasistatic turn of mind. Some more simple methods in the spirit of Molecular Dynamics were also applied to granular materials. The ideas of J.J. Moreau based on Convex Analysis allowed the proposal of a consistent frame for the non-smooth laws involved in frictional contact, i.e. unilaterality and dry friction. These ideas suggested numerical algorithms known as Non-Smooth Contact Dynamics (NSCD), whereas grains are considered as distinct bodies as they are in the DEM method.

Sophisticated 2D or 3D samples with grains of all shapes with all kinds of interacting laws are implemented today in the numerical software. In the light of discrete numerical simulations and new imaging techniques, the last twenty years of research on granular materials have been marked by an ever-growing interest in the granular microstructure and its link with macroscopic behavior.

The purpose of this book is to provide an overview of some major concepts and analysis tools developed during the last twenty years in the field of granular materials:

– Chapter 1 presents basic definitions and methods to characterize the granular microstructure. Relying on both experimental and numerical data, the distributions of grain velocities and contact forces are analyzed both by their average and in terms of their spatio-temporal fluctuations.

– Chapter 2 states rigorous mechanical concepts allowing the construction of the stress tensor for quite general assemblies of bodies. Numerical examples illustrate the properties of the stress tensor.

– Chapter 3 is concerned with the question of the behavior of granular materials at the scale of a representative sample. Engineers need relevant laws to use in the frame of continuum mechanics (it is impossible to take into account the infinitely numerous degrees of freedom of all grains in a real-size structure). The question of how to deduce the macroscopic behavior of a sample from contact data at the local scale is raised, which belongs to the field of homogenization techniques. This chapter provides methods and concepts to deal with this issue and presents several results.

– Chapter 4 is devoted to numerical simulation methods, mainly the smooth DEM methods and the NSCD methods. The advantages and disadvantages of the different methods are analyzed. Attention is drawn to the fact that a numerical algorithm is in itself a model. Monitoring of numerical simulations must be carried out as in physical experiments and results must be considered shrewdly.

– Chapter 5 will interest mathematically inclined readers. It illustrates the fact that the difficulties met in investigating frictional contact problems have deep mathematical origins.

This book is a completely revised and augmented translation of the former Micromécanique des matériaux granulaires by B. Cambou and M. Jean (Hermes Science), 2001.

Bernard CAMBOU, Michel JEAN, Farhang RADJAÏ

Chapter 1

Experimental and Numerical Analysis of Local Variables in Granular Materials1

1.1. Introduction

Granular materials consist of densely packed solid particles and a pore-filling material which can be a fluid or a solid matrix. The particles interact via elastic repulsion, friction, adhesion and other surface forces. By nature, the length scales involved in these contact interactions are much smaller than the particle size. External loading leads to particle deformations as well as cooperative particle rearrangements. The particle deformations are of particular importance in powder metallurgy, for example, but the particles may be considered as quasi-rigid beyond the elastic response times.

The contact network and pore space are the two facets of the microstructure of granular materials to which we will refer, in this chapter, as granular texture. At the particle scale, the granular texture involves three basic vectors from which other local geometrical variables can be defined: (1) the branch vector joining the centers of contacting particles; (2) the contact orientation vector (contact normal) defined as the unit vector normal to the particle boundary at the contact zone ; and (3) the contact vectors joining the particle centers to the contact point; see Figure 1.1. The reaction forces and acting on two particles at their contact zone have a unique application point. This point may be considered as their contact point in the case of extended contacts between two polyhedral particles.

Figure 1.1.Local vectors at the contact between two particles 1 and 2: branch vectorcontact normalcontact forceand contact vectors

Two different local frames can be associated with a pair of contacting particles: (1) the frame defined by the contact normal and two orthogonal unit vectors in the contact plane (tangential to the two particles at the contact point); and (2) the frame defined by the radial unit vector and two orthogonal unit vectors in a orthoradial plane (orthogonal to the branch vector). These two frames coincide in the case of spherical particles. In two-dimensions (2D), the local frame is uniquely defined by a single tangent unit vector t or t.

The granular texture is disordered with many different variants depending on the composition (particle shapes and sizes), interactions and assembling procedure. The granular disorder is essentially characterized by the fact that, as a result of geometrical exclusions among particles, the local vectors vary discontinuously from one contact to another. In other words, the local environments fluctuate in space. The contact network evolves with loading so that the local environments also fluctuate in time. The highly inhomogenous distribution of contact forces reflects granular disorder in static equilibrium. In particular, the force chains reveal long-range correlations whereas the presence of a broad population of very weak forces results from the arching effect. The force and fabric anisotropies are two complimentary aspects of stress transmission. They can be employed in a local (particle-scale) description of granular media in the quasi-static state.

The geometrical changes of granular texture are at the origin of the complex rheology of granular materials. These changes are highly nonlinear, involving creation and loss of contacts, rotation frustration and frictional sliding. They depend on the dissipative nature of contact interactions and steric exclusions among particles. In quasi-static deformation, various features of the plastic behavior such as shear strength and dilatancy can be traced back to the evolution of granular texture. Two issues are of primary interest in microscopic modeling of granular plasticity: (1) what is the lowest level of textural information, and to what extent does it control the effective properties of the material? and (2) how do the effective properties depend on higher order textural information?

In this chapter, we introduce several concepts and tools for the description of granular texture, kinematics and force transmission with examples and illustrations from discrete element simulations (molecular dynamics and contact dynamics, see Chapter 4) and experiments. We first consider the description of granular texture in terms of particle positions and contact orientations. The kinematics and mechanisms of plastic deformation are then analyzed. Finally, we focus on stress transmission and its link with granular texture.