The Physics of the Deformation of Densely Packed Granular Materials

226 Pages·2020·14.285 MB·English

by  Maarten Anton Koenders

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Published by
World Scientific Publishing Europe Ltd.
57 Shelton Street, Covent Garden, London WC2H 9HE
Head office: 5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
Library of Congress Cataloging-in-Publication Data
Names: Koenders, M. A. C., author.
Title: The physics of the deformation of densely packed granular materials / M.A.C. Koenders.
Description: New Jersey : World Scientific, [2020] | Includes bibliographical references and index.
Identifiers: LCCN 2019053417 | ISBN 9781786348234 (hardcover) | ISBN 9781786348241 (ebook) |
ISBN 9781786348258 (ebook other)
Subjects: LCSH: Granular materials. | Deformations (Mechanics)--Mathematical models.
Classification: LCC TA418.78 .K63 2020 | DDC 620/.43--dc23
LC record available at https://lccn.loc.gov/2019053417
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2020 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
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For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/Q0244#t=suppl
Desk Editors: Herbert Moses/Shi Ying Koe
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
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Preface
In the early 1980s, the idea first took hold that the mechanical response of
a dense granular medium can be understood from a basis of the inter-
particle contact properties. The initial efforts, a mean-field theory, had
very poor results and papers on ‘micro-mechanics’ were usually relegated
to the last section of conference proceedings. Gradually, the insight came
about that a granular medium cannot be captured in a mean-field theory
and that some form of non-homogeneity in the fabric properties has to be
accounted for. The beginnings of this concept were implemented using the
available continuum theories on heterogeneity and a few papers came out
in the early 90s showing that in certain special cases the mechanical
response was captured, but — irritatingly — not all cases. Highly
anisotropic packed beds, for example, could not be accounted for and the
failure of a granular medium at high stress ratio remained a mystery.
While progress since then has been slow, it is now clear that a proper
theory of granular deformation must include a method that deals with
heterogeneity that is particularly applicable to a system of particles. This
turns out to be the theory of ‘connected media’, which captures the physics
of contacting particulates in an appropriate manner. It has also been
extended to anisotropic cases. A rigorous approach to Coulomb friction as
an inter-particle interaction is required as well. Together these
developments can now be implemented with great success.
To preserve analytical insight it is advantageous to use simplified
models with round particles and on occasion do a two-dimensional
calculation, rather than a three-dimensional one. This does not matter for
the understanding of the physics that is at play. The theories have also been
applied successfully to other fields where the inter-particle interaction has
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vi Preface

a more chemical character. Filter cake formation (relevant to chemical
engineering) is an example. Due to the large number of natural
occurrences and applications of dense granular matter there is relevance in
a variety of disciplines.
This book presents a detailed exposition of all the concepts and
mathematical techniques that are necessary to understand the current state
of the subject. The student from a non-mathematical background may
initially have to put in a certain amount of work to grasp the intricacies of
the line of argument. This is a very algebraic subject; there is not much
one can do about that. However, a mathematical appendix and an
introductory chapter on continuum mechanics and Cartesian tensor
calculus are provided to make the journey easier.

Curt Koenders
Canterbury, 2019
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About the Author
M.A.C. Koenders is a physicist who has worked in industry, academia
and as a consultant. His expertise is in the mechanics of granular media.
He has collaborated with physicists, civil engineers, chemical engineers,
mathematicians and geologists. He has some 200 (co)-authored papers in
refereed journals, book chapters and conference proceedings relevant to
these subjects. He has also been a substantial fund-raiser to benefit the
progress of the subject and collaborated on many projects to further the
understanding of aspects of granular mechanics.
In industry he was active in drawing up geotechnical filtration rules for
the soil mechanics community. In other work on civil engineering he has
had a long-standing collaboration with the Bundesanstalt für Wasserbau,
Karlsruhe to provide a theory on the mechanics of unsaturated soils.
In chemical engineering he collaborated to describe and improve
filtration processes, especially the mechanics of cake formation. He was
awarded the gold medal of the Filtration Society (2003) for his
contributions to oscillated filtration.
In geology he contributed to the understanding of processes under
volcanoes, especially co-authoring papers on magma infiltration into
dilatant granular layers.
In mathematics he introduced the concepts of structures formation in
granular materials and non-Newtonian flow processes through granular
masses.
He has supervised 13 PhD students. He is a member of the Institute of
Physics and currently associated with the University of Southampton.

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Contents
Preface v
About the Author vii
Chapter 1 General Concepts 1
1.1 Introduction 1
1.2 The isostatic state and jamming 6
1.3 The statically indeterminate case and
computer simulations 10
1.4 Contact laws 14
1.5 The frictional interaction 15
1.6 Contact laws in terms of material parameters 21
1.7 Interaction for small particles in a fluid
environment 24
Chapter 2 Continuum Mechanics and Cartesian Tensor
Calculus 33
2.1 Initial considerations 33
2.2 Rotations 34
2.3 The strain tensor 36
2.4 The stress tensor 40
2.5 Tensors 44
2.6 Material response 45
2.7 Isotropic materials 47
2.8 Elastic behaviour 48
2.9 Anisotropic materials 50
2.10 Coaxiality 51
2.11 Objectivity and pre-stressed materials,
the Jaumann derivative 52
2.12 Frictional materials 55
ix