Editorial: The Fiber Bundle

A long time has passed since Fredrick Thomas Peirce, an Australian physicist working for the British Cotton Industry Research Association, published the paper Tensile tests for cotton yarns: “the weakest link” theorems on the strength of long and of composite specimens in 1926 [1]. In it, he describes a simplified model for failure of yarn where the variations of the strength in the yarn competes with increased load it carries when there is a failure. Here it is in modern language: Imagine a set of fibers of equal length clamped between two parallel stiff plates. The fibers all have the same elastic constant, so that when the plates are moved apart, all the fibers carry the same load, which is the total load on the plates divided by the number of fibers. Assume each fiber has its own strength, that is the maximum load it can carry before it fails irreversibly. As the fibers fail one by one, the load on the remaining fibers increases as there are fewer and fewer left to share the total load. Such a model seems overly simple. It cannot possibly contribute to our understanding of fracture phenomena! The mathematical statistician Henry Daniels did not think so. He did his PhD on the strength of fiber bundles and in 1945 he published a seminal paper on his work that moves the model beyond the strength of yarn [2], reshaping it into a general model for failure processes. Up to the point when he published his paper, we find six papers referring to the Peirce paper [1].1 In the decade that follows 1945, there are 35 references to the Peirce paper, but the titles have now changed. We find e.g., “The Fracture of Metals” [3].2 It is in the seventies that the fiber bundle model really gains traction within the mechanics community [4]. Together with the statistics community, the subtleties of the model is gradually uncovered, while at the same time variants of it is being proposed, e.g., the Local Load Sharing Model [5].3 The statistical physics community, those with a background in critical phenomena, had in the mid eighties began to discover that fracture in fact is a very interesting problem with properties that sometimes created a déjà vu feeling among them [6]. This resulted in the creating of the Fuse Network Model [7]. This was network of fuses, e.g. a square lattice, where each link would be a fuse with a trip current drawn from some statistical distribution. What would happen when the current running through this network was increased? A lot of things would happen: there would be avalanches of fuses blowing simultaneously, there would be localization, i.e., the fuses that blew would be concentrated in some region, there would be instabilities where suddenly a crack of blown fuses would form zipping through the network. The problem with the fuse model, however, was that it was almost impossible to approach it analytically—it was a model for the computer, and the computer only. The book by Herrmann and Roux from 1990 (and reprinted in 2014) covers the status of the field at that time [6]. In 1989 Sornette published a paper entitled Elasticity and Failure of a Set of Elements Loaded in Parallel [8]. This was the entrance of the fiber bundle model into the statistical physics community. It was the perfect model in some sense. It was almost as rich as the fuse network model, but it was at the same time simple enough to be analytically tractable to a considerable degree. In some sense, the fiber bundle model is gaining a status in the physics of fracture community similar to that of the Ising model in the critical phenomena community. Two reviews of the fiber bundle model have appeared in this community [9, 10]. There are hundreds of papers, if not more, that have appeared over the last years in this field. The aim of this Research Topic has been to display some of the richness of ideas and uses that the fiber bundle model has produced. The eleven papers it contains does indeed do that.