Chapter 1
The Fiber Bundle Model

Dear reader, if differential geometry is your field, please put this book back on the shelf. It is not for you. The fiber bundles that we deal with here are not spaces, but bundles of breakable fibers. Fibers that stretch and fail. They belong to the realm of engineers, physicists, and statisticians. They are models for how materials fail under duress.

Most materials do not consist of fibers. But materials are prone to failure under loading. Keeping material failure under control is one of the most important tasks of engineering. We need to be able to trust that our buildings will not collapse, our airplanes do not disintegrate in mid-air, our tankers do not rupture at sea.

Given the variety of materials and configurations they are used in, it seems a daunting task to attempt constructing a general theory of fracture and failure. Such a theory exists, however, and goes under the name of linear elastic fracture mechanics (LEFM) [1]. This has become a very refined theory over the years, and there is no doubt that it has been successful. Linear elastic fracture mechanics has as a starting point the theory of elasticity. This is a theory that treats materials as continuous, and as a result, linear elastic fracture mechanics is a top-down approach.

A completely different approach has come to life over the past couple of decades: atomistic modeling [2]. This approach hinges on the advent of the computer as a serious research tool. It is now possible to model materials with (fairly) realistic forces between the atoms in such quantities that it is possible to hook the results up with those approaches that start from a continuum description: top-down meets bottom-up.

Is there then any room for simplified fiber bundle models in the middle? Our answer is yes, and we will use the next couple of hundred pages or so to convince you, dear reader.

1.1 Rivets Versus Welding

Here is a couple of examples of failures that seem to be opposite of each other in order to highlight the complexity of the central problem: how to ensure that structures do not fail.

We zoom in on the failure of the hull of a Boeing 737 airplane during Aloha Airlines flight 243 on April 28, 1988, where a part of the fuselage of the airplane was ripped away mid-air; see Figure 1.1. Amazingly, the pilots were able to land the aircraft with 89 passengers and 6 crew members. The failure process had started long before as a small crack near a rivet due to metal fatigue initiated by crevice corrosion. The crack grew due to the cyclic pressure loading from flying and being on the ground. As the length of the crack grew, the stresses in front of it increased, and at some point, it became unstable, opening up the fuselage by moving from rivet to rivet in the way perforated paper fails. Clearly, understanding what happened and how it can be prevented from happening again belongs to the realm of engineering. However, the growth of the initial crack and how it went unstable are just as much a problem for fundamental science: what are the underlying mechanisms and how do they manifest themselves? In the AA flight 243 incident, the rivets played a crucial role.

Figure 1.1 The Boeing 737 after the explosive decompression that occurred during flight on April 28, 1988, in Hawaii. (Photo credit: National Transportation Safety Board)

The American Liberty cargo ships produced during World War II were the first ships that had hulls that were welded rather than riveted. Yet, 12 out of the 2710 ships built broke in half without warning. Cracks formed, grew slowly, went unnoticed, and at some point, they became unstable, breaking the ship apart, see Figure 1.2. By removing rivets, no mechanism was present that could lead to crack arrest. A growing crack in a car wind shield is effectively stopped by drilling a hole in front of it. The high stress at the crack tip, which drives the crack forward, is lowered as it is spread over the surface of the drilled hole when the crack reaches it. In the same way, rivets would stop growing cracks in the hull.

Figure 1.2 The Schenectady after it broke into two on January 16, 1943, in dock in Portland, Oregon. The ship had just been finished and was being outfitted. The failure was sudden and unexpected.

This contradicts the important role played by rivets in the AA flight 243 incident where rivets were the cause of the failure. Here, the lack of rivets was the reason for the failure.

Are there fundamental and general principles at work here that can explain the difference between the two incidents? The answer is yes. But, to be able to understand these principles, we need to simplify the problem. We need models.

It is here that the fiber bundle models enter. They are models that simplify the problem of failure to the point where the very powerful methods of theoretical physics, statistics, and mathematics may be fully explored. They help us understand the subtle interplay between forces and strength that control the failure process. They help us understand what is generally present in all failure processes and what is specific for a given failure process.

1.1.1 What Are Models Good For?

Since the use of models is sometimes viewed with some skepticism by the engineering community, we elaborate some more on what precisely is a model.

Fundamental sciences, and physics in particular, approach Nature in a hierarchical way [3]: more general questions are posed and answered before more specific questions. We may illustrate this by the following example: in the 1920s, general quantum mechanics was developed. In the 1930s, a general theory of metals was developed. This allowed for studying specific metals, but it also opened up for the search for a class of materials that were between metals and insulators: semiconductors. In the 1940s, this resulted in the construction of the first transistor-and the electronics age was born. One may only speculate how long it would have taken to construct the transistor if this path from the more general to the more specific had not been followed. How long would it take before someone accidentally stumbled across semiconductors?

This hierarchical approach lies behind the extensive use of models in theoretical physics. The fiber bundle model is a good example of the use of physical models to study the physical phenomena of interest with the minimum of ingredients needed: these models are stripped of any irrelevant contents. In fact, the models, and the approach of physics to science, are related to Occam's dictum: Numquam ponenda est pluralitas sine necessitate [plurality must never be posited without necessity] [4].1

Still, the fiber bundle models have proved to be very effective in practical applications such as fiber-reinforced composites. In this context, the models have a history that goes back to the 1920s [6], and they constitute today an elaborate toolbox for studying such materials, rendering computer studies orders of magnitudes more efficient than brute force methods. Since the late 1980s [7], these models have received increasing attention in the physics community due to their deceivingly simple appearance coupled with an extraordinary richness of behaviors. As these models are just at the edge of what is possible analytically and typically not being very challenging from a numerical point of view so that extremely good statistics on large systems are available, they are perfect as model systems for studying failure processes as a part of theoretical physics.

1.2 Fracture and Failure: A Short Summary

Fracture and material stability have for practical reasons interested humanity ever since we started using tools: our pottery should be able to withstand handling, our huts should be able to withstand normal weather. As science took on the form we know today during the Renaissance, Leonardo da Vinci studied 500 years ago experimentally the strength of wires-fiber bundles-as a function of their length [8]. Systematic strength studies, but on beams, were also pursued by Galileo Galilei 100 years later, as was done by Edme Mariotte (of gas law fame), who pressurized vessels until they burst in connection with the construction of a fountain at Versailles. For some reason, mainstream physics moved away from fracture and breakdown problems in the nineteenth century, and it is only during the last 20 years that fracture problems have been studied within physics proper. The reason for this is most probably the advent of computers as a research tool, rendering problems that were beyond the reach of systematic theoretical study now accessible.

If we were to single out the most important modern contribution from the physics community with respect to fracture phenomena, it must be the focus on fluctuations rather than averages. What good is the knowledge of the average behavior of a system when faced with a single sample and this being liable to breakdown given the right fluctuation? This book, being written by physicists, reflects this point of view, and hence, fluctuations play an important role throughout it.

1.3 The Fiber Bundle Model in Statistics

Even though we may trace the study of fiber bundles to Leonardo da Vinci, their modern story starts with the already mentioned work by Peirce [6]. In 1945, Daniels published a seminal review-cum-research article on fiber bundles, which still today must be regarded as essential reading in the field [9]. In this paper, the fiber bundle...