Non-Gaussian Nature of Fracture and the Survival of Fat-Tail Exponents

Abstract

We study the fluctuations of the global velocity V(l)(t), computed at various length scales l, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity--and extreme event distributions--by measuring only at a global scale.

The Burst‐Like Behavior of Aseismic Slip on a Rough Fault: The Creeping Section of the Haiyuan Fault, China

“…The presence of a long tail of large creep velocities has already been observed in several studies in material physics, focusing on the avalanche-like propagation of a front line into a heterogeneous media, such as mode I fractures (Måløy et al, 2006;Tallakstad et al, 2013) or imbibition fronts into a porous medium (Planet et al, 2009). In these studies, nonGaussian PDFs of velocity fluctuations, comparable to that in Figure 3c, have been related to the cascading of bursts and long-range interactions along the front.…”

“…In these studies, nonGaussian PDFs of velocity fluctuations, comparable to that in Figure 3c, have been related to the cascading of bursts and long-range interactions along the front. These distributions can be approximated using either generalized Gumbel distributions (Planet et al, 2009) or stable Lévy distributions (Tallakstad et al, 2013). Unfortunately, the noise in our data is too large to decipher between these two kinds of distributions.…”

Triggering Processes in Rock Fracture

“…Pore collapse and grain fragmentation are the main mechanisms associated with compaction band formation [63,64], and compactiontype events also dominate in shear banding [63] as well as in fracture zone formation (the polarity and focal mechanism analysis in Table S2 [37] gives 68% compaction-type events for sample Wgrn07 and even higher values for trigger and triggered events). Hence, our work proves conclusively that the occurrence of these empirical laws extends well beyond purely frictional sliding events, with potential applications in other types of crackling noise as well [65][66][67][68][69][70].…”

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

“…The non-Markovian generalisation of that model [7] contains a fractional equation which has a form different from Eq. (14) and the predicted motion is always subdiffusive. The density has a finite value at the origin in contrast to the our approach: p(x, t), Eq.…”

“…However, presence of the Lévy flights does not need to imply infinite fluctuations because any physical system is finite and, if one introduces a truncation of the distribution tail, the diffusion properties are well-determined. It has been recently demonstrated that cracking of heterogeneous materials reveals a slowly falling power-law tail of the local velocity distribution of the crack front [14] but, despite that, the authors were able to determine the variance due to the finiteness of the system. Systems with memory can be conveniently handled in terms of a Langevin equation by introducing an auxiliary operational time.…”