Burst avalanches in solvable models of fibrous materials

TL;DR

This paper reviews models of fiber bundle fracture, analyzing avalanche size distributions under global and local load sharing, revealing conditions for anomalous behavior and explicit calculations for specific threshold distributions.

Contribution

It compares global and local load sharing models in fiber bundles, identifying conditions for anomalous avalanche distributions and providing explicit calculations for certain threshold distributions.

Findings

Global model shows deviations from typical power-law behavior under certain conditions.

Local model lacks universal power-law asymptotics but allows explicit calculations.

Avalanche distributions depend on load sharing rules and threshold distributions.

Abstract

We review limiting models for fracture in bundles of fibers, with statistically distributed thresholds for breakdown of individual fibers. During the breakdown process, avalanches consisting of simultaneous rupture of several fibers occur, and the distribution $D(\Delta)$ of the magnitude $\Delta$ of such avalanches is the central characteristics in our analysis. For a bundle of parallel fibers two limiting models of load sharing are studied and contrasted: the global model in which the load carried by a bursting fiber is equally distributed among the surviving members, and the local model in which the nearest surviving neighbors take up the load. For the global model we investigate in particular the conditions on the threshold distribution which would lead to anomalous behavior, i.e. deviations from the asymptotics $D(\Delta) \sim \Delta^{-5/2}$, known to be the generic behavior. For the local model no universal power-law asymptotics exists, but we show for a particular threshold distribution how the avalanche distribution can nevertheless be explicitly calculated in the large-bundle limit.