Variability in the power-law distributions of rupture events

Abstract

Rupture events, as the propagation of cracks or the sliding along faults, associated with the deformation of brittle materials are observed to obey power-law distributions. This is verified at scales ranging from laboratory samples to the Earth's crust, for various materials and under various loading modes. Besides the claim that this is a universal characteristic of the deformation of heterogeneous media, spatial and temporal variations are observed in the exponent and tail-shape. These have considerable implications for the ability and the reliability of forecasting large events from smaller ones. There is a growing interest in identifying the factors responsible for these variations. In this present work, we first present observations at various scales (laboratory tests, field experiments, landslides, mining induced seismicity, crustal Earthquakes) showing that substantial variations exist in both the slope and the tailshape of the rupture event size distribution. This review allows us to identify potential explanations for these variations (incorrect statistical methods, heterogeneity, stress, brittle/ductile transition, finite size effects, proximity to the failure). A possible link with the critical point theory is also drawn showing that it is able to explain a part of the observed variations considering the distance to the critical point. Using numerical simulations of progressive failure we investigate the role of mechanical properties on the powerlaw distributions. The results of simulations agree with the critical point theory for various macroscopic behaviors ranging from ductility to brittleness providing a unified framework for the understanding of power-law variability observed in rupture phenomena.

Evidence of frost-cracking inferred from acoustic emissions in a high-alpine rock-wall

“…This scaling relationship is the signature of the rock micro-fracturing activity. Beside the claim of the possible universality of the b-value, this parameter has been proposed to be dependent on various parameters, in particular on the strength heterogeneity of the material, the applied stress and the proximity of failure (see Amitrano, 2012, and references therein for an extended discussion).…”

Scalable statistics of correlated random variables and extremes applied to deep borehole porosities

“…Of special concern are the statistics of extremes, which have received much attention among hydrologists (Katz et al, 2002) and others concerned with a wide range of phenomena including snow avalanches on mountain slopes (Ancey, 2012); rupture events associated with the propagation of cracks or sliding along faults in brittle materials including rock failure, landslides and earthquakes (Amitrano, 2012;Lei, 2012;Main and Naylor, 2012) as well as volcanic eruptions, landslides, wildfires and floods (Sachs et al, 2012;Schoenberg and Patel, 2012;Süveges and Davison, 2012); demographic and financial crises (Akaev et al, 2012;Janczura and Weron, 2012); neuronal avalanches and coherence potentials in the mammalian cerebral cortex (de Arcangelis, 2012;Plenz, 2012); citations of scientific papers (Golosovsky and Solomon, 2012); and distributions of city sizes (Pisarenko and Sornette, 2012). Extreme values cluster around heavy tails of data frequency distributions which are often modeled as stretched exponential, lognormal or power functions.…”

Damage cluster distributions in numerical concrete at the mesoscale

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