Abstract
We derive an asymptotic series expansion for the burst size distribution in the equal load sharing fiber bundle model, a predominant model for breakdown in disordered media. Earlier calculations give expressions with correct asymptotic behavior for large bursts, but low accuracy for small bursts, up to an order of magnitude off. The approximations from the expansion we present here give relative errors of at most several percent when compared with exact results or simulation results for large systems. We also solve the burst size distribution exactly for the Weibull threshold distributions.
Highlights
The fiber bundle model [1,2,3,4] is a prime example of what Bouchaud calls a metaphorical model [5]; that is, a model which reveals mechanisms that lie hidden beneath layers of complexity in realistic models and are completely lost in phenomenological models
In the equal load sharing fiber bundle model, we have found an analytic solution of the burst size distribution for Weibull threshold distributions: Equation (14)
This is a significant improvement over previous results, which describe the asymptotic behavior as the burst size diverges
Summary
The fiber bundle model [1,2,3,4] is a prime example of what Bouchaud calls a metaphorical model [5]; that is, a model which reveals mechanisms that lie hidden beneath layers of complexity in realistic models and are completely lost in phenomenological models. Even in its simplest form, the equal load sharing (ELS) model, the number of papers written on the fiber bundle model may be counted in the thousands. By following the development of the crossover burst size—that is, the burst size that constitutes the watershed between bursts following the 5/2 law and the bursts following the 3/2 law—it is possible to quantitatively measure of how far the bundle is from collapse This approach has the problem that it requires knowledge of large (and rare) bursts, which have poor statistics. We provide in this article a method to analytically calculate the burst distribution accurately for small bursts
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