Table of Contents

Series Page

Title Page

Copyright

Dedication

Series Preface

Preface

Acknowledgements

Chapter 1: Introduction to Mechanics of Discontinua

1.1 The Concept of Discontinua

1.2 The Paradigm Shift

1.3 Some Problems of Mechanics of Discontinua

References

Further Reading

Chapter 2: Methods of Mechanics of Discontinua

2.1 Introduction

2.2 Discrete Element Methods

2.3 The Combined Finite-Discrete Element Method

2.4 Molecular Dynamics

2.5 Smooth Particle Hydrodynamics

2.6 Discrete Populations Approach

2.7 Algorithms and Solutions

References

Further Reading

Chapter 3: Disc to Edge Contact Interaction in 2D

3.1 Problem Description

3.2 Integration of Normal Contact Force

3.3 Tangential Force

3.4 Equivalent Nodal Forces

Further Reading

Chapter 4: Triangle to Edge Contact Interaction in 2D

4.1 Problem Description

4.2 Integration of Normal Contact Force

4.3 Tangential Force

4.4 Equivalent Nodal Forces

Further Reading

Chapter 5: Ball to Surface Contact Interaction in 3D

5.1 Problem Description

5.2 Integration of Normal Contact Force

5.3 Tangential Force

5.4 Equivalent Nodal Forces

Further Reading

Chapter 6: Tetrahedron to Points Contact Interaction in 3D

6.1 Problem Description

6.2 Integration of Normal Contact Force

6.3 Tangential Force

6.4 Equivalent Nodal Forces

Further Reading

Chapter 7: Tetrahedron to Triangle Contact Interaction in 3D

7.1 Problem Description

7.2 Integration of Normal Contact Force

7.3 Tangential Force

7.4 Equivalent Nodal Forces

Further Reading

Chapter 8: Rock Joints

8.1 Introduction

8.2 Interaction between Mesh Entities in 2D

8.3 Joint Dilation

8.4 Shear Resistance of a 2D Rock Joint

8.5 Numerical Examples

References

Further Reading

Chapter 9: MR Contact Detection Algorithm for Bodies of Similar Size

9.1 The Challenge

9.2 Constraints of MR Contact Detection Algorithm

9.3 Space Decomposition

9.4 Mapping of Spherical Bounding Boxes onto Cells

9.5 Spatial Sorting

9.6 Quick Sort Algorithm

9.7 MR-Linear Sort Algorithm

9.8 Implementation of the MR-Linear Sort Algorithm

9.9 Quick Search Algorithm

9.10 MR-Linear Search Algorithm

9.11 CPU and RAM Performance

9.12 CPU Performance and RAM Consumption

References

Further Reading

Chapter 10: MR Contact Detection Algorithm for Bodies of Different Sizes

10.1 Introduction

10.2 Description of the Multi-Step-MR Algorithm (MMR)

10.3 Polydispersity

10.4 CPU Performance

10.5 RAM Requirements

10.6 Robustness

10.7 Applications

Further Reading

Chapter 11: MR Contact Detection Algorithm for Complex Shapes in 2D

11.1 Introduction

11.2 Contactor Circle to Target Point MR Contact Detection Algorithm

11.3 Contactor Circle to Target Edge MR Contact Detection Algorithm

11.4 Contactor Triangle to Target Edge MR Contact Detection Algorithm

11.5 Extension to Other Shapes

11.6 Reporting of Contacting Couples

Further Reading

Chapter 12: MR Contact Detection Algorithm for Complex Shapes in 3D

12.1 Introduction

12.2 Rendering Target Simplex Shapes

12.3 Sorting Target Cells

12.4 Target Cells Interrogation Tools

12.5 Searching for Contacts

Further Reading

Chapter 13: Parallelization

13.1 Introduction

13.2 Domain Decomposition Approach

13.3 Graphics Processing Units (GPU)

13.4 Structured Parallelization

Further Reading

Index

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

Ren1.1 de Borst

Perumal Nithiarasu

Tayfun E. Tezduyar

Genki Yagawa

Tarek Zohdi

Introduction to Finite Element Analysis:            Szab1.1 and Babu1.1ka           March 2011

Formulation, Verification and Validation

This edition first published 2012

© 2012 John Wiley & Sons, Ltd

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Library of Congress Cataloguing-in-Publication Data

Munjiza, Antonio A.

Computational mechanics of discontinua / Antonio A. Munjiza, Earl E. Knight and Esteban Rougier.

p. cm.—(Wiley series in computational mechanics)

Includes bibliographical references and index.

ISBN 978-0-470-97080-5 (hardback)

1. Continuum mechanics. I. Knight, Earl E. II. Rougier, Esteban. III. Title.

QA808.2.M87 2011

531–dc23

2011020576

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-97080-5

ePDF ISBN: 978-1-119-97118-4

obook ISBN: 978-1-119-97116-0

ePub ISBN: 978-1-119-97301-0

Mobi ISBN: 978-1-119-97302-7

To Cheryl, Jasna, Sole, Ignacio, Matias and Boney.

Series Preface

The series on Computational Mechanics is a conveniently identifiable set of books covering interrelated subjects that have been receiving much attention in recent years and need to have a place in senior undergraduate and graduate school curricula, and in engineering practice. The subjects will cover applications and methods categories. They will range from biomechanics to fluid-structure interactions to multiscale mechanics and from computational geometry to meshfree techniques to parallel and iterative computing methods. Application areas will be across the board in a wide range of industries, including civil, mechanical, aerospace, automotive, environmental and biomedical engineering. Practicing engineers, researchers and software developers at universities, industry and government laboratories, and graduate students will find this book series to be an indispensible source for new engineering approaches, interdisciplinary research, and a comprehensive learning experience in computational mechanics.

Discrete element methods are used in a wide variety of applications – ranging from fragmentation and mineral processing in engineering to the simulation of the dynamics of galaxies in astrophysics. This book – written by leading experts in the field – provides a comprehensive overview of discrete element methods with an emphasis on algorithmic and implementation aspects. A unique feature is the in-depth treatment of accurate and fast methods for contact detection between particles, which is of pivotal importance for the efficiency of discrete element methods. Starting from basic concepts in discontinua, the book further touches upon molecular dynamics simulations, smooth particle hydrodynamics and the combination of discrete element with finite element methods, and discusses parallel implementations of discrete element methods.

Preface

One of the more important breakthroughs of the modern scientific age was the development of differential calculus. The key to differential calculus is the concept of a point which contains an instantaneous quantity such as point density or instantaneous velocity. Implicitly hidden is the assumption of smoothness of physical quantities, which translates into the assumption of a continuum. Based on this assumption, a whole range of scientific and engineering disciplines were developed, such as Fluid, Solid, and Continuum Mechanics. Common to all these is the existence of a set of governing partial differential equations describing the physical problem as a continuum. With exponential advances in computer hardware, fiber optics and related technologies, it has now become possible to solve these governing equations using powerful computers and the associated numerical methods of computational physics.

Modern science of the early decade of the 21st century is increasingly addressing problems where the assumptions of smoothness and continuum are no longer true. The best example is Nano-Science and Nanotechnology where length scales are so small that the continuum assumption is simply not valid. Other examples include complex systems such as biological systems, financial systems, crowds, hierarchical materials, mineral processing, powders, and so on. In these systems it is the presence of the interaction of a large number of individual atoms, molecules, particles, organisms, market players, individual people in the crowd or other individual building blocks of a complex system that produce new emergent properties and emergent phenomena such as a droplet of liquid, market crash, crowd stampede, and so on. A common feature of all of these is the departure from the continuum assumption towards an explicit adoption of the discontinuum. The new scientific discipline that has therefore emerged is called Mechanics of Discontinua.

While Continuum Mechanics smears out all the complex processes occurring at a certain length and time scale, Mechanics of Discontinua emphasizes these processes. Solving equations of Continuum Mechanics produces numerical simulations which quantify “a priori” described physical quantities. In contrast, solving equations of Mechanics of Discontinua produces a virtual experiment that generates new qualities and properties, thus surprising the observer; for instance, from individual atoms a droplet of liquid or a crystal may appear; from individual market players a market crash may happen; from the behaviour of individual people a stampede may occur.

Mechanics of Discontinua is a fundamental paradigm shift from the science that measures “a priori” defined properties to the science that produces these as emergent properties and emergent phenomena.

This book aims to provide a comprehensive introduction to the subject including a detailed description of state of the art computational techniques. As such it is a must read for both experimental and theoretical researchers or practitioners involved in fast developing areas such as nano-science, nanotechnology, medical sciences, pharmaceuticals, material sciences, mineral processing, complex biological and financial systems. The book comes with open source 3D discontinua computer software and also with open source MR nano-science computational tools, which are available on the companion website: www.wiley.com/go/munjiza.

Acknowledgements

The authors would like to express their gratitude to the publishers, John Wiley & Sons, Ltd, for their excellent support. We would also like to thank our numerous colleagues and research collaborators from all over the world: the USA, China, Japan, Germany, Italy, Canada, and the UK. Our thanks also go to current and previous PhD students as well as Postdoctoral researchers. Special thanks go to Professor J.R. Williams from MIT, Professor Bibhu Mohanty from University of Toronto, Professor Graham Mustoe from Colorado School of Mines, our colleagues at Los Alamos National Laboratory (Robert P. Swift, Theodore C. Carney, Christopher R. Bradley, Wendee M. Brunish, David W. Steedman, Doran R. Greening and others), Professor F. Aliabadi from Imperial College London, Dr. Ing Harald Kruggel-Emden from Ruhr-University, Bochum, Germany, and Dr. Paul Cleary, CSIRO, Australia. Many thanks must also go to Dr. Nigel John for all the help he has provided.

Chapter 1

Introduction to Mechanics of Discontinua

1.1 The Concept of Discontinua

It was Galileo who noticed that the velocity of a free falling body increases by a constant amount in a given fixed increment of time. The more general case of this is the variable change of velocity, as shown in Figure 1.1.

Figure 1.1 Nonlinear change of velocity as a function of time.

This velocity change can be written as

1.1 1.1

where a is the acceleration. From his observations one can say that Galileo nearly discovered differential calculus. The discovery of differential calculus would have naturally led him to the laws of motion and the story of classical mechanics would have come much earlier in history.

In reality it was Newton who extrapolated the concept of

1.2 1.2

to the case that when Δt is nearly zero, instantaneous acceleration is achieved, as shown in Figure 1.2.

Figure 1.2 Graphical representation of instantaneous acceleration.

Leibnitz took the concept even further and generalized it, thus developing what is now called “differential calculus”. One could argue that differential calculus is the most important discovery of modern science. It has enabled scientists and engineers to describe physical problems in terms of governing equations. The governing equations are usually a set of partial differential equations that describe a particular engineering or scientific problem. Examples of these include equilibrium equations of the linear theory of elasticity and also the Navier-Stokes equations describing the flow of Newtonian fluids.

All of these are based on the concept of instantaneous, point or distributed quantity such as

1.3 1.3

which is the instantaneous velocity or

1.4 1.4

which is the point density. Without differential calculus one would have to use the average density given by

1.5 1.5

where ΔV is some finite volume and Δm is the mass of that volume.

This concept can be expanded to any distributed quantity such as load

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