Aftershock sequences and seismic-like organization of acoustic events produced by a single propagating crack

Abstract

Brittle fractures of inhomogeneous materials like rocks, concrete, or ceramics are of two types: Nominally brittle and driven by the propagation of a single dominant crack or quasi-brittle and resulting from the accumulation of many microcracks. The latter goes along with acoustic noise, whose analysis has revealed that events form aftershock sequences obeying characteristic laws reminiscent of those in seismology. Yet, their origin lacks explanation. Here we show that such a statistical organization is not only specific to the multi-cracking situations of quasi-brittle failure and seismology, but also rules the acoustic events produced by a propagating crack. This simpler situation has permitted us to relate these laws to the overall scale-free distribution of inter-event time and energy and to uncover their selection by the crack speed. These results provide a comprehensive picture of how acoustic events are organized upon material failure in the most fundamental of fracture states: single propagating cracks.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks

“…A more general explanation has been proposed in [54][55][56]: Power-law distributed inter-event waiting time simply arise when a finite detection threshold is applied to separate the events from the background noise. This argument, together with the power-law distributed sizes and waiting times, naturally yield aftershock sequences and seismic laws [38,48], and that an experimentally finite driving rate naturally implies the use of a finite detection threshold, may provide an explanation of the seismic-like temporal organization widely reported in damage and fracture problems. Still, the specific conditions leading to this organization remains to clarify.…”

“…Figure4 presents the mean number of AS, N AS , as a function of the size S th prescribed for the triggering M S. In between two cutoffs, N AS goes as a power-law with S th as expected from the productivity law. Following [38], we checked that the N AS v.s. S M S curve remains unchanged after:…”

“…This demonstrates that the productivity law is a simple consequence of the size distribution. The relation between the two can be rationalized using the argument provided in [38,48]: The total number of events with a size larger than the prescribed value S M S gives, by definition, the total number of M S of size S M S , and hence the total number of AS sequences. The total number of events with a size smaller that S M S gives the total number to be labeled AS in the catalog.…”

“…Finally, we looked at the rate of AS produced by a M S of size S M S and its evolution as a function of the time elapsed since M S: R AS (t−t M S |E M S ). To compute these curves, we adopted the procedure developed in [38]: For each simulation, all sequences triggered by M S of size falling within a prescribed interval are sorted out; subsequently the AS events are binned over t − t M S and the so-obtained curves are finally averaged. Figure 7 shows the resulting curves in a typical simulation.…”

“…Another illustrative example is found in seismology; earthquakes get organized into aftershock (AS) sequences which obeys characteristic laws [39]: Productivity law [40,41] stating that the number of produced aftershocks goes as a power-law with the mainshock (M S) energy; Båth's law [42] stipulating that the ratio between the M S energy and that of its largest AS is independent of the M S magnitude; and Omori-Utsu law [43][44][45] telling that the production rate of AS decays al-gebraically with the elapsed time since M S. These laws, referred to as the fundamental laws of seismology, are central in the implementation of probabilistic forecasting models of earthquakes [46]. They are not specific to seismology, but were also reported, at the lab scale, in the acoustic emission associated with the damaging of different materials loaded under compression [7,8], in the global dynamics of a sheared granular material [47] and in the simpler situation of a single tensile crack slowly driven in artificial rocks [38]. In the latter case, it has been possible to show that the fundamental laws of seismology are direct consequences of the individual scale free statistics of both the event sizes and inter-event waiting times [38,48]; productivity and Båth's law [42] for AS sequences result from the power-law distribution of sizes and Omori-Utsu law results from the power-law distribution of waiting time.…”