Computational Mechanics of Discontinua

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

1.6 1.6

where p is the value of the distributed load at a specific point, as shown in Figure 1.3.

Figure 1.3 Distributed load.

Of course, in these extrapolations a hidden assumption is made: Qualitatively nothing changes as Δx or ΔV gets smaller and smaller. This is the standard continuum assumption and it is both true and not true. It is true if one is solving a problem where one is really interested in results in terms of average quantities, that is, the physics of the problem are contained at relatively large finite time, length, volume or similar scales.

One of the first surprises engineers encountered regarding solutions of governing equations on the theory of elasticity involved failures of structural components at stress levels much smaller than the obtained stresses and strains would indicate. This was especially pronounced with brittle materials, while ductile materials such as aluminum were more resistant to sudden failure. Nevertheless, in the 1950s there was an infamous story of a British-made de Havilland DH 106 Comet passenger jet having a catastrophic structural failure in mid-air.

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

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

1.7 1.7

where σ is the axial stress, f is the axial load and Δa is the cross section area, the whole micro-structure of a given material is automatically eliminated together with all emergent phenomena and emergent properties originating from this microstructure. Two of these eliminated 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 design process.

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

However, with the advent of present day science, the field of problems that are difficult to describe using governing differential equations has grown exponentially in recent years. These include traditional research and engineering problems such as mining, mineral processing, pharmaceuticals, medicine delivery techniques, fluid flow problems, problems of 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 scientific problems were approached occurred. Differential calculus became a powerful enabling technology in the hands of scientists and engineers that enabled the formulations of the most challenging problems in terms of mathematical equations. The first beneficiary of this discovery was, of course, Newtonian Mechanics.

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

Figure 1.4 Distributed internal force for different surfaces.

To solve this stress puzzle, the work of Cauchy and others led to the concept of a tensor and thus, tensorial calculus was developed. During the 1960s, Truesdell and Gurtin generalized the concept of a tensor and redefined it as a linear mapping that maps one vector into another. Stress then becomes a mapping from the vector space of surfaces to the vector space of forces. In a sense, for a given surface a particular internal force is 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 is more conveniently represented using vector bases and matrices that describe the mapping (tensor) in a given vector base. Tensors as physical quantities became a powerful concept in describing stresses, strains, gravity, etc.

Formulating an engineering or scientific problem in terms of differential equations is much easier than solving these equations. In fact solutions in a closed analytical form rarely exist and engineers and scientists are forced to use approximate solutions of the governing differential equations.

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

The problem is that all of these are based on the continuum assumption as explained above; there is a whole diverse field of problems, especially in modern science, that do not subscribe to this assumption. The crack propagation problem is one of the simplest of these. Another problem that does not subscribe to the assumption is flow through a very small diameter tube or a nano-tube. The diameter of the nano-tube is comparable to 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 interesting physical phenomena occurring at this length scale.

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

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; in rarefied gases the mean free path of the molecules is comparable to the physical scale of the problem, therefore requiring one to account for discontinua effects.

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

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

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

1.8 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 scientific problems are approached. This paradigm shift can be proven as important and as essential for the future of science as differential calculus was for past science.

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

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

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

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

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 components falling and hitting other components, thus causing a domino effect.

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

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

In all the above presented cases, one arrives at the concept of discontinuum. The concept of continuum is based on smoothing out all the complexities of the micro-structure through an averaging process that in the limit produces instantaneous velocity or density at the point. This averaging process uses instantaneous or point limit averaged quantities and is naturally 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 rather than smoothing them through a density field, etc.

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

The applied science that deals with the formulation, simulation and solving of the problems of discontinua is called Mechanics of Discontinua. Mechanics of Discontinua is a relatively new discipline based on emphasizing the discontinuous nature of a given problem. An essential part of Mechanics of Discontinua is computer simulation leading to 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