System-size-dependent avalanche statistics in the limit of high disorder

Abstract: We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasi-static fracture of heterogeneous materials. We consider a fiber bundle model where the strength of single fibers is sampled from a power law distribution over a finite range, so that the amount of materials' disorder can be controlled by varying the power law exponent and the upper cutoff of fibers' strength. Analytical calculations and computer simulations, performed in the limit of equal load sharing, revealed that depending on the disorder parameters the mechanical response of the bundle is either perfectly brittle where the first fiber breaking triggers a catastrophic avalanche, or it is quasi-brittle where macroscopic failure is preceded by a sequence of bursts. In the quasi-brittle phase, the statistics of avalanche sizes is found to show a high degree of complexity. In particular, we demonstrate that the functional form of the size distribution of bursts depends on the system size: for large upper cutoffs of fibers' strength, in small systems the sequence of bursts has a high degree of stationarity characterized by a power law size distribution with a universal exponent. However, for sufficiently large bundles the breaking process accelerates towards the critical point of failure which gives rise to a crossover between two power laws. The transition between the two regimes occurs at a characteristic system size which depends on the disorder parameters.