Finite-size effects on return interval distributions for weakest-link-scaling systems

Abstract

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Weakest-Link Scaling and Extreme Events in Finite-Sized Systems

“…We have focused on various statistical mechanisms leading to deviations from the Weibull distribution, which is commonly used in weakest-link scaling systems. Our motivation stems from empirical distributions that deviate from the Weibull in physical systems expected to follow weakest-link scaling, e.g., [7,15,27,28,36]. Our results include a modified gamma distribution, the κ-Weibull distribution and weakest-link scaling distributions with non-homogeneous links that have random (possibly correlated) parameters.…”

“…For N → ∞, the κ-Weibull tends to the Weibull model, and the power-law decay at x 1 is replaced by the exponential Weibull dependence. We have argued in [27] that the κ-Weibull distribution can be observed in the case of earthquake recurrence times if the monitored area comprises a finite number of faults. We have also linked the number of effective units in this system to the inverse of κ.…”

“…Whereas the Weibull distribution has proven quite successful in the analysis of extreme value data, in certain cases, deviations from the Weibull scaling have been noticed in the tails of empirical distributions [27,28]. As shown above, such deviations could occur due to fluctuations of the link hazard rate parameters.…”

“…The term κ-Weibull is due to the presence of the κ-exponential. This distribution has been used in economics to model the distribution of income [32,34] and in statistical physics as a model of earthquake recurrence times [27]. Other potential applications include the fracture strength of quasi-brittle materials, which exhibit deviations from Weibull scaling [7,28], and reliability models that involve the Pareto distribution.…”

“…We expect that the variability of link parameters is a plausible scenario in heterogeneous physical or engineered systems. For example, a theoretical framework has been proposed that links the statistics of earthquake recurrence times to fracture strength in the Earth's crust [15,27]. Since the crust is a heterogeneous material decorated with flaws (faults) of widely varying sizes and orientations, it is reasonable to expect that the fracture strength parameters exhibit local random fluctuations in addition to systematic changes with depth [25].…”