Burst avalanches in solvable models of fibrous materials

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 $D(\ensuremath{\Delta})$ of the magnitude $\ensuremath{\Delta}$ of such avalanches is the central characteristic 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 $D(\ensuremath{\Delta})\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}5/2}$, 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.