computational · 2013-07-23 · introduction to finite element analysis: szabo and babu´ ˇska...

COMPUTATIONALMECHANICS OFDISCONTINUA

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

Rene de BorstPerumal NithiarasuTayfun E. TezduyarGenki YagawaTarek Zohdi

Introduction to Finite Element Analysis: Szabo and Babuska March 2011Formulation, Verification and Validation

COMPUTATIONALMECHANICS OFDISCONTINUA

Antonio A. MunjizaQueen Mary, University of London, UK

Earl E. KnightLos Alamos National Laboratory, USA

Esteban RougierLos Alamos National Laboratory, USA

A John Wiley & Sons, Ltd., Publication

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 2011531–dc23

2011020576

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

Print ISBN: 978-0-470-97080-5ePDF ISBN: 978-1-119-97118-4obook ISBN: 978-1-119-97116-0ePub ISBN: 978-1-119-97301-0Mobi ISBN: 978-1-119-97302-7

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http://www.wiley.com

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

Contents

Series Preface xi

Preface xiii

Acknowledgements xv

1 Introduction to Mechanics of Discontinua 11.1 The Concept of Discontinua 11.2 The Paradigm Shift 31.3 Some Problems of Mechanics of Discontinua 7

1.3.1 Packing 71.3.2 Fracture and Fragmentation 81.3.3 Demolition and Structures in Distress, Progressive Collapse 111.3.4 Nanotechnology 121.3.5 Block Caving 151.3.6 Mineral Processing 161.3.7 Discrete Populations in General 16References 18Further Reading 18

2 Methods of Mechanics of Discontinua 212.1 Introduction 212.2 Discrete Element Methods 21

2.2.1 Spherical Particles 222.2.2 Blocky Particles 232.2.3 Oblique and Super-Quadric Particles 232.2.4 Rigid Potential Field Particles 252.2.5 3D Real Shape Particles 252.2.6 Computer Games and Special Effects 26

2.3 The Combined Finite-Discrete Element Method 272.4 Molecular Dynamics 28

2.4.1 Common Potentials 292.5 Smooth Particle Hydrodynamics 31

viii Contents

2.6 Discrete Populations Approach 332.7 Algorithms and Solutions 35

References 36Further Reading 37

3 Disc to Edge Contact Interaction in 2D 393.1 Problem Description 393.2 Integration of Normal Contact Force 393.3 Tangential Force 443.4 Equivalent Nodal Forces 45

Further Reading 46

4 Triangle to Edge Contact Interaction in 2D 474.1 Problem Description 474.2 Integration of Normal Contact Force 474.3 Tangential Force 544.4 Equivalent Nodal Forces 55

Further Reading 56

5 Ball to Surface Contact Interaction in 3D 595.1 Problem Description 595.2 Integration of Normal Contact Force 595.3 Tangential Force 735.4 Equivalent Nodal Forces 74

Further Reading 75

6 Tetrahedron to Points Contact Interaction in 3D 776.1 Problem Description 776.2 Integration of Normal Contact Force 796.3 Tangential Force 846.4 Equivalent Nodal Forces 86

Further Reading 86

7 Tetrahedron to Triangle Contact Interaction in 3D 897.1 Problem Description 897.2 Integration of Normal Contact Force 897.3 Tangential Force 997.4 Equivalent Nodal Forces 101

Further Reading 102

8 Rock Joints 1038.1 Introduction 1038.2 Interaction between Mesh Entities in 2D 104

8.2.1 Interaction between a 2D Disk and a Straight Edge 1058.2.2 Numerical Integration of the Roller-Edge Interaction 111

Contents ix

8.3 Joint Dilation 1138.4 Shear Resistance of a 2D Rock Joint 1168.5 Numerical Examples 120


9 MR Contact Detection Algorithm for Bodies of Similar Size 1259.1 The Challenge 1259.2 Constraints of MR Contact Detection Algorithm 1259.3 Space Decomposition 1279.4 Mapping of Spherical Bounding Boxes onto Cells 1279.5 Spatial Sorting 1299.6 Quick Sort Algorithm 1309.7 MR-Linear Sort Algorithm 1359.8 Implementation of the MR-Linear Sort Algorithm 1369.9 Quick Search Algorithm 1419.10 MR-Linear Search Algorithm 1439.11 CPU and RAM Performance 1459.12 CPU Performance and RAM Consumption 151


10 MR Contact Detection Algorithm for Bodies of Different Sizes 15510.1 Introduction 15510.2 Description of the Multi-Step-MR Algorithm (MMR) 15510.3 Polydispersity 15610.4 CPU Performance 15710.5 RAM Requirements 15810.6 Robustness 15810.7 Applications 160

Further Reading 160

11 MR Contact Detection Algorithm for Complex Shapes in 2D 16311.1 Introduction 16311.2 Contactor Circle to Target Point MR Contact Detection Algorithm 163

11.2.1 Cell Size and Space Boundaries 16311.2.2 Rendering of 2D Target Points onto Cells 16611.2.3 Sorting of Target Cells 16711.2.4 Interrogation Tools for Sorted Target Cells 16711.2.5 Rendering of 2D Contactor Circles onto Cells 168

11.3 Contactor Circle to Target Edge MR Contact Detection Algorithm 17611.3.1 Rendering 2D Target Edges onto Cells 17611.3.2 Searching for Contacts 182

11.4 Contactor Triangle to Target Edge MR Contact Detection Algorithm 18411.4.1 Rendering 2D Triangles onto Cells 185

x Contents

11.5 Extension to Other Shapes 19211.6 Reporting of Contacting Couples 193

Further Reading 194

12 MR Contact Detection Algorithm for Complex Shapes in 3D 19712.1 Introduction 19712.2 Rendering Target Simplex Shapes 198

12.2.1 Rendering 3D Points onto Cells 19812.2.2 Rendering 3D Edges onto Cells 198

12.3 Sorting Target Cells 21012.4 Target Cells Interrogation Tools 21112.5 Searching for Contacts 212

12.5.1 Rendering Contactor Tetrahedron 21212.5.2 Rendering Contactor Triangular Facet 22612.5.3 Rendering Other Contactor Simplex Shapes 241Further Reading 241

13 Parallelization 24313.1 Introduction 24313.2 Domain Decomposition Approach 247

13.2.1 Communication Engine 25213.2.2 Broadcasting Engine 25413.2.3 Summing Engine 25413.2.4 Gathering Engine 25613.2.5 Distribution of Physical Objects across Processors 25713.2.6 Creating Proxies 25813.2.7 Relocating Originals 259

13.3 Graphics Processing Units (GPU) 26013.4 Structured Parallelization 262

Further Reading 263

Index 265

Series Preface

The series on Computational Mechanics is a conveniently identifiable set of books cover-ing interrelated subjects that have been receiving much attention in recent years and needto have a place in senior undergraduate and graduate school curricula, and in engineeringpractice. The subjects will cover applications and methods categories. They will rangefrom biomechanics to fluid-structure interactions to multiscale mechanics and from com-putational 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. Practicingengineers, researchers and software developers at universities, industry and governmentlaboratories, and graduate students will find this book series to be an indispensible sourcefor new engineering approaches, interdisciplinary research, and a comprehensive learningexperience in computational mechanics.

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

Preface

One of the more important breakthroughs of the modern scientific age was the develop-ment of differential calculus. The key to differential calculus is the concept of a pointwhich contains an instantaneous quantity such as point density or instantaneous velocity.Implicitly hidden is the assumption of smoothness of physical quantities, which translatesinto the assumption of a continuum. Based on this assumption, a whole range of scien-tific and engineering disciplines were developed, such as Fluid, Solid, and ContinuumMechanics. Common to all these is the existence of a set of governing partial differentialequations describing the physical problem as a continuum. With exponential advances incomputer hardware, fiber optics and related technologies, it has now become possible tosolve these governing equations using powerful computers and the associated numericalmethods of computational physics.

Modern science of the early decade of the 21st century is increasingly addressing prob-lems where the assumptions of smoothness and continuum are no longer true. The bestexample is Nano-Science and Nanotechnology where length scales are so small that thecontinuum assumption is simply not valid. Other examples include complex systems suchas biological systems, financial systems, crowds, hierarchical materials, mineral process-ing, powders, and so on. In these systems it is the presence of the interaction of a largenumber of individual atoms, molecules, particles, organisms, market players, individualpeople in the crowd or other individual building blocks of a complex system that producenew emergent properties and emergent phenomena such as a droplet of liquid, marketcrash, crowd stampede, and so on. A common feature of all of these is the departure fromthe continuum assumption towards an explicit adoption of the discontinuum. The newscientific discipline that has therefore emerged is called Mechanics of Discontinua.

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

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

xiv Preface

This book aims to provide a comprehensive introduction to the subject including adetailed description of state of the art computational techniques. As such it is a mustread for both experimental and theoretical researchers or practitioners involved in fastdeveloping areas such as nano-science, nanotechnology, medical sciences, pharmaceu-ticals, material sciences, mineral processing, complex biological and financial systems.The book comes with open source 3D discontinua computer software and also withopen source MR nano-science computational tools, which are available on the companionwebsite: 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 colleaguesand 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 studentsas well as Postdoctoral researchers. Special thanks go to Professor J.R. Williams fromMIT, Professor Bibhu Mohanty from University of Toronto, Professor Graham Mustoefrom Colorado School of Mines, our colleagues at Los Alamos National Laboratory(Robert P. Swift, Theodore C. Carney, Christopher R. Bradley, Wendee M. Brunish, DavidW. Steedman, Doran R. Greening and others), Professor F. Aliabadi from Imperial CollegeLondon, Dr. Ing Harald Kruggel-Emden from Ruhr-University, Bochum, Germany, andDr. Paul Cleary, CSIRO, Australia. Many thanks must also go to Dr. Nigel John for allthe help he has provided.

1Introduction to Mechanicsof Discontinua

1.1 The Concept of Discontinua

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

This velocity change can be written as

a = �ν

�t(1.1)

where a is the acceleration. From his observations one can say that Galileo nearly discov-ered differential calculus. The discovery of differential calculus would have naturally ledhim to the laws of motion and the story of classical mechanics would have come muchearlier in history.

In reality it was Newton who extrapolated the concept of

a = lim�t→0

�ν

�t(1.2)

to the case that when �t is nearly zero, instantaneous acceleration is achieved, as shownin Figure 1.2.

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

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

v = drdt

(1.3)

Computational Mechanics of Discontinua, First Edition. Antonio A. Munjiza, Earl E. Knight and Esteban Rougier.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

2 Computational Mechanics of Discontinua

Δt Δt

Δv

Δv

v

t

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

Δt → 0

Δv

v

t

Figure 1.2 Graphical representation of instantaneous acceleration.

which is the instantaneous velocity or

ρ = dm

dV(1.4)

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

ρ = �m

�V(1.5)

where �V is some finite volume and �m is the mass of that volume.This concept can be expanded to any distributed quantity such as load

p = lim�x→0

�f�x

(1.6)

where p is the value of the distributed load at a specific point, as shown in Figure 1.3.Of course, in these extrapolations a hidden assumption is made: Qualitatively nothing

changes as �x or �V gets smaller and smaller . This is the standard continuum assump-tion and it is both true and not true. It is true if one is solving a problem where oneis really interested in results in terms of average quantities, that is, the physics of theproblem are contained at relatively large finite time, length, volume or similar scales.

One of the first surprises engineers encountered regarding solutions of governingequations on the theory of elasticity involved failures of structural components at stress

Introduction to Mechanics of Discontinua 3

Δx

p

Figure 1.3 Distributed load.

levels much smaller than the obtained stresses and strains would indicate. This was espe-cially pronounced with brittle materials, while ductile materials such as aluminum weremore resistant to sudden failure. Nevertheless, in the 1950s there was an infamous story ofa British-made de Havilland DH 106 Comet passenger jet having a catastrophic structuralfailure in mid-air.

This was actually due to the development of “brittle like” dynamic fatigue cracks. Theprocess of developing a brittle crack occurs basically at the micromechanical level ofmicro-structural elements of material (crystals, fibers, even atoms and molecules).

By extrapolating the continuum formulation to almost zero length and volume scales,for:

σ = lim�a→0

f

�a(1.7)

where σ is the axial stress, f is the axial load and �a is the cross section area, the wholemicro-structure of a given material is automatically eliminated together with all emergentphenomena and emergent properties originating from this microstructure. Two of theseeliminated phenomena are brittle and fatigue crack.

That said, problems of brittle fracture and fatigue have been addressed in a semi-empirical fashion that makes it possible to use continuum-based stress analysis in a designprocess.

A similar situation involving structural failure occurred in the well known collapseof the Malpasset Dam in 1959. The failure there happened due to a discontinuity inthe rock mass under the foundation of the dam. This catastrophic collapse triggered thedevelopment of a whole set of science on discontinuous rock masses.

However, with the advent of present day science, the field of problems that are difficultto describe using governing differential equations has grown exponentially in recent years.These include traditional research and engineering problems such as mining, mineralprocessing, pharmaceuticals, medicine delivery techniques, fluid flow problems, problemsof astrophysics, and also problems of nano-science and nano-technology, social sciences,biological sciences, economics, marketing, etc.

1.2 The Paradigm Shift

With the discovery of differential calculus a complete paradigm shift in how scientificproblems were approached occurred. Differential calculus became a powerful enabling


technology in the hands of scientists and engineers that enabled the formulations of themost challenging problems in terms of mathematical equations. The first beneficiary ofthis discovery was, of course, Newtonian Mechanics.

In the course of dealing with differential problems further insights into powerful math-ematical tools were gained. In dealing with stress analysis problems, it was soon realizedthat at a single material point P, as shown in Figure 1.4, stress can be described as adistributed internal force per unit area of a particular internal surface. The problem is thatone can put many internal surfaces through the same point. Thus the stress at point Pdepends on what surface is chosen; in general, the stress on surface n1 is different fromthe stress on surface n2.

To solve this stress puzzle, the work of Cauchy and others led to the concept of atensor and thus, tensorial calculus was developed. During the 1960s, Truesdell and Gurtingeneralized the concept of a tensor and redefined it as a linear mapping that maps onevector into another. Stress then becomes a mapping from the vector space of surfacesto the vector space of forces. In a sense, for a given surface a particular internal forceis assigned and if the surface doubles in size, so does the force – thus linear mapping.One could in theory represent this mapping using a spreadsheet; but this mapping ismore conveniently represented using vector bases and matrices that describe the mapping(tensor) in a given vector base. Tensors as physical quantities became a powerful conceptin describing stresses, strains, gravity, etc.

Formulating an engineering or scientific problem in terms of differential equations ismuch easier than solving these equations. In fact solutions in a closed analytical formrarely exist and engineers and scientists are forced to use approximate solutions of thegoverning differential equations.

A real revolution in our ability to solve governing partial differential equations occurredwith the arrival of affordable silicon-chip-based computers. One could argue that thedevelopment of affordable computing hardware together with the accompanying computerlanguages was a milestone as important as the discovery of differential calculus itself andin modern science the two go together; one enables the formulation of the problem, whilethe other enables solving of the actual equations.

The problem is that all of these are based on the continuum assumption as explainedabove; there is a whole diverse field of problems, especially in modern science, that donot subscribe to this assumption. The crack propagation problem is one of the simplestof these. Another problem that does not subscribe to the assumption is flow through avery small diameter tube or a nano-tube. The diameter of the nano-tube is comparable to

P

n1

n2

n3

Figure 1.4 Distributed internal force for different surfaces.


the size of individual atoms, as shown in Figure 1.5, thus any smoothing of the micro-structure through the continuum assumption automatically eliminates the most interestingphysical phenomena occurring at this length scale.

One could argue that this should be expected for very small scale problems. However,even cosmic scale problems have similar properties to these smaller scale problems; inrarefied gases the mean free path of the molecules is comparable to the physical scale ofthe problem, therefore requiring one to account for discontinua effects.

Other examples where one must account for discontinua effects are: granular flow, rockslides, spontaneous stratification, spraying, milling, shot pinning, mixing and other similarindustrial engineering and scientific problems. Even problems such as fire evacuation andcrowd control have the same discrete aspects.

In all these cases the physical behavior originates from interaction between individualentities such as atoms, molecules, grains, particles, members of a crowd, etc. What oneneeds to describe is the structure of the problem (that is, the individual entities such as aninitial position for each person in the crowd), the behavior of each individual entity (eachperson in the crowd which may have different psychological aspects) and the interactionbetween the individual entities (for example, individual members of the crowd pushingeach other).

To analyze a problem formulated in the above way one would require a computerin order to solve the problem in this lifetime. However, there are no continuum basedgoverning equations based on averaging properties such as

ρ = dm

dV(1.8)

The flowchart of formulating a discontinua problem is shown in Figure 1.6. In essence,one can say that there is another fundamental paradigm shift in how some scientificproblems are approached. This paradigm shift can be proven as important and as essentialfor the future of science as differential calculus was for past science.

This paradigm shift can be described as a move towards describing physical systemsusing “discrete populations”. A discrete population can be a space occupied by individualatoms that interact with each other. Motion of these atoms may produce a crystal or adroplet of liquid or a gas phase of matter, where pressure consists of interaction betweenindividual atoms. The duration of these interactions are measured in femtoseconds.

1 nm

0.376 nm

0.340 nm

Carbon atom

Argon atom

Figure 1.5 Schematic representation of a cross section of Argon gas flow through a nano-tube.


Assemble discrete populations(individual entities)

Define interactionlaws

Perform a virtualexperiment

Observe the emergentproperties and derive

conclusions

Figure 1.6 Flowchart for the formulation of a Mechanics of Discontinua problem.

In a similar way a discrete population can be a sports arena that is occupied by individualspectators. Interactions between these spectators in emergency evacuations may produceemergent properties such as panic and stampede or a bottleneck.

A discrete population can also be composed of individual terrestrial bodies thatattract each other through gravity which then leads to particle accretion and/or granulartemperature.

A discrete population can be composed of the individual components of a tall building.An emergent property may be the progressive collapse produced by individual componentsfalling and hitting other components, thus causing a domino effect.

Further generalization of the discrete population concept could be applied to a pop-ulation of market participants interacting with each other through trading. An emergentproperty would be the stock market crash or property bubble.

A discrete population can be a biological ecosystem with interaction between individualplayers. An emergent property would be extinction or stock depletions.

In all the above presented cases, one arrives at the concept of discontinuum. The conceptof continuum is based on smoothing out all the complexities of the micro-structure throughan averaging process that “in the limit” produces instantaneous velocity or density at thepoint. This averaging process uses instantaneous or point limit averaged quantities and isnaturally described using differential equations.

As opposed to the continuum concept, the discontinuum concept emphasizes the micro-structure of certain length scales. For example, it uses individual terrestrial objects ratherthan smoothing them through a density field, etc.

The discontinuum approach always concentrates at certain levels of discontinua such asthe level of individual atoms, or the level of individual particles, or the level of individualhumans, or the level of individual spectators, etc.

The applied science that deals with the formulation, simulation and solving of theproblems of discontinua is called Mechanics of Discontinua. Mechanics of Discontinuais a relatively new discipline based on emphasizing the discontinuous nature of a givenproblem. An essential part of Mechanics of Discontinua is computer simulation leadingto emergent properties.


1.3 Some Problems of Mechanics of Discontinua

1.3.1 Packing

Taking into account that the mean free path of molecules for most engineering materialsis very small in comparison to the characteristic length of most engineering problems,one may arrive at a conclusion that engineering materials are well represented by a hypo-thetical continuum model. That this is not the case is easily demonstrated by the followingproblem:

A square base glass container is filled with particles of varying shapes andsizes as shown in Figure 1.7. The particulate is left to fall from a given height.During the fall under gravity, the particles interact with each other and withthe walls of the container. In this process energy is dissipated and finally allparticles come to a state of rest. The question to be answered is what is the totalvolume occupied by the particulate after they have come to their state of rest?

This problem is referred to as the container problem. It is self evident that the definitionof density ρ given by

ρ = dm

dV(1.9)

and the definition of mass m given by

m =∫

V

ρdV (1.10)

are not valid for the container problem.

Figure 1.7 Gravitational deposition of spherical particles inside a rigid container.


For the problem shown in Figure 1.7, the total mass of the system is given by

m =N∑

i=1

mi (1.11)

where N is the total number of particles in the container and mi is the mass of the individ-ual particles. In other words the total mass is given as a sum of the masses of the individualparticles. It is worth mentioning that the size of the container is not much larger than thesize of the individual particles. The way the particles pack in the container and the mass ofthe particles in the container is a function of the size of the container, shape of individualparticles, size of individual particles, deposition method, deposition sequence, etc.

For example, if the size of the particles is changed significantly, then the final densityprofile obtained inside the container is drastically affected, as shown in Figure 1.8.

The container problem is a typical problem where continuum-based models cannotbe applied. This problem also demonstrates that discontinum-based simulations recovercontinuum formulation when the size of the individual discrete elements (diameter ofsphere in the problem described above) becomes small in comparison to the characteristiclength of the problem being analyzed. In the container problem, the characteristic lengthof the problem is the length of the smallest edge of the box. Thus, the continuum-basedmodels are simply a subset of more general discontinum-based formulations; applicablewhen microstructural elements of the matter comprising the problem are very small incomparison to the characteristic length of the problem being analyzed.

1.3.2 Fracture and Fragmentation

Fracture, and especially brittle fracture, was one of the early problems that required thecontinuum assumption to be mended in order to take into consideration the complexmicrostructural processes that produce fracture.

Figure 1.8 Gravitational packs for different size distributions.


One of the amended theories was Griffith’s fracture energy release rate theory. Anotherapproach is based on stress intensity factors. These are quite good in predicting sin-gle crack propagation mechanisms. However, when it comes to multiple cracks and/orcomplex fracture patterns these theories face numerous problems in describing the firstprinciple physics that occur.

In many industrial engineering and scientific applications fracture, fracture patterns,and very often fragmentation play a key role in understanding these processes. In thesecases, continuous based approaches are generally inadequate.

However, discontinuum based approaches have provided good predictions of even themost complex experimental fracture patterns such as those illustrated in Figures 1.9, 1.10and 1.11. The results shown in these figures clearly demonstrate that the methods used

Figure 1.9 Symmetric triangular pulse of peak pressure 80 MPa and duration 1 ms.

Figure 1.10 Symmetric triangular pulse of peak pressure 200 MPa and duration 0.2 ms.


Figure 1.11 Symmetric triangular pulse of peak pressure 500 MPa and duration 0.05 ms.

Figure 1.12 Detonation gas driven fracture pattern.


in mechanics of discontinua are able to describe fracture and fragmentation processesaccurately, including complex dynamic fracture patterns.

In Figure 1.9 a disk with a hole in its center is shown. The hole is subject to a symmetrictriangular pulse of peak pressure of 80 MPa. Discontinua-based simulation produces afracture problem that compares well with experimental results.

In addition, the discontinua-based simulation is able to capture the changes in thefracture pattern due to the changes in pressure pulse as shown in Figures 1.10 and 1.11.The fracture patterns shown in Figure 1.11 are quite different from the fracture patternsshown in Figure 1.9, even though the only differences between the two problems are thepeak pressure and pulse duration.

In these cases the combined Finite-Discrete Element Method (FEM-DEM) was used toobtain the fracture patterns. The same method can be used to obtain complex fracture pat-terns driven by fluid flow through the cracks as shown in Figure 1.12, where a detonationgas driven fracture process is illustrated.

There are many important applications where these discontinua processes play a keyrole: petroleum engineering, geothermal energy, carbon sequestration, mining, powderceramics, construction, excavation, demolition, rock crashing, etc.

1.3.3 Demolition and Structures in Distress, Progressive Collapse

With recent worldwide changes in design codes a decisive shift towards limit state analysisbased on random sets has been made. Ultimately, in an ideal world one would take a givenbuilding or a structure and apply “what if” scenarios using computer based simulationsand input parameters (loads, material properties, etc.) that are random variables. From theresults of these simulations one would obtain the probability of a given structure failingor collapsing. These types of analysis could ultimately replace the traditional industrialsafety factors.

Due to computer hardware constraints and other factors, these kinds of simulations arestill within the research domain. Nevertheless, there are, even at present, needs to analyzethe collapsing process of a structure. In Figure 1.13 a simulation of a demolition of achimney stack is shown. Very often, these types of demolitions have to be executed quitecarefully due to the spatial constraints for the falling debris.

Another case when a structure has to be analyzed in terms of the collapse processis the progressive collapse. This is best illustrated by the progressive collapse of tallbuildings. The upper floors collapse and their kinetic energy progressively breaks thefloors underneath them.

The collapse of the upper floors can be triggered by impact, explosion, fire or a similarcatastrophic event. This event can be a relatively small energy process that in normalcircumstances would only produce limited localized damage to the building. However,in a progressive collapse situation, this event acts as a trigger that initiates an energyrelease process that is orders of magnitude greater than the triggering event. This processis simply the transfer of the upper floor’s potential energy into kinetic energy. This kineticenergy acts as a sledge hammer that destroys the parts of the building that it hits. In turn,the newly broken parts start falling, accumulating extra kinetic energy and the process

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