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

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 $\kappa$-Weibull distribution. The upper tail of the $\kappa$-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the $\kappa$-Weibull distribution decreases linearly after a waiting time ${\tau }_c\propto n^{1/m}$, 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 $\kappa$ 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 $\kappa$-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Figure 1

$\kappa$-Weibull pdfs for $x_s=10$, different values of $\kappa$, and (a) $m=0.7$ and (b) $m=3$.

Figure 2

Schematic illustrating the definition of a return interval ${\tau }_j$ as the time between two consecutive events with magnitudes $M_j$ and $M_{j+k}$ that exceed the threshold $M_c$, i.e., the events occurring at times $t_j$ and $t_{j+k}$.

Figure 3

Median ratio ${\tau }_{\mathrm{med};n}/{\tau }_{\mathrm{med};\infty }$ for the $\kappa$-Weibull distribution versus the Weibull modulus $m$ and the system size $n$.

Figure 4

Log-log plot of hazard rates $h\left(\tau \right)$ versus $\tau$ for different models. (K) $\kappa$-Weibull hazard rate based on (15) for $n=100$ with $m=0.5$ (magenta solid line and diamonds), $m=1$ (black solid line), $m=2$ (purple solid line with squares), and $m=5$ (cyan line with circles). (W) Weibull hazard rate obtained from (20) with $m=0.5$ (magenta dashed line with diamonds), $m=2$ (purple dashed line with squares), and $m=5$ (cyan dashed line with circles). (Exponential) Exponential hazard rate obtained from (20) with $m=1$ (black dash-dot line).

Figure 5

${\Phi }_n\left(\tau \right)$ versus $ln(\tau /{\tau }_s)$ for the Weibull distribution with $m=0.7$ (green solid line), the $\kappa$ Weibull with $m=0.7$ and $n=1$ (blue dashed line), $n=10$ (red dashed and dotted line), $n=100$ (black dotted line), and the gamma distribution with $\alpha =0.7$ (magenta dashed line with small circles).

Figure 6

${\Phi }_n\left(\tau \right)$ versus $ln(\tau /{\tau }_s)$ for the Weibull distribution with $m=3$ (green solid line); the $\kappa$ Weibull with $m=3$ and $n=1$ (blue dashed line), $n=10$ (red dashed and dotted line), and $n=100$ (black dotted line); and the gamma distribution with $\alpha =3$ (magenta dashed line with small circles).

Figure 7

$\Phi \left(\tau \right)$ versus $ln\left(\tau \right)$ for earthquake return intervals of the Cretan earthquake sequence (CES). The return intervals refer to 628 events with magnitudes exceeding 2.3 ($M_L$). The magnitude of completeness is $\approx$2.2 ($M_L$). The maximum likelihood estimates of the $\kappa$-Weibull parameters are ${\widehat{\tau }}_0\approx 3.19\times {10}^4$ (s), $\widehat{m}\approx 0.78$, $\widehat{\kappa }\approx 0.33$. The estimate of $\Phi \left(\tau \right)$ using the empirical (data-based) cdf is shown with the solid blue line. The optimal $\kappa$-Weibull fit is shown with the red dashed line, whereas the optimal Weibull fit is shown with the green dashed and dotted line. The left vertical axis measures $\Phi \left(\tau \right)$, whereas the right vertical axis marks the corresponding cdf values.

Figure 8

Sample quantiles versus the corresponding quantiles of the optimal $\kappa$-Weibull distribution for the Cretan earthquake sequence return intervals (CES ERI)—$\Phi \left(\tau \right)$ of the data shown in Fig. 7. The $\kappa$-Weibull quantiles are obtained from (16) using the maximum likelihood estimates of the $\kappa$-Weibull parameters (see the caption of Fig. 7).

Figure 9

Schematic of a fiber bundle of initial length $L$ elongated by $x$ due to the loading force $F$. Broken fibers do not contribute to the strength of the bundle.

Figure 10

Evolution of avalanche events in time for a bundle of ${10}^7$ fibers with Weibull-distributed strength thresholds ($m=5,x_s=1$). The most energetic avalanche $({log}_{10}{E}_f\approx 7.31)$ is generated when the bundle fails (during the last event of the breaking sequence) at time $t_f\approx 0.725$.

Figure 11

Quantile-quantile plots of FBM return intervals (dimensionless) versus the best-fit $\kappa$-Weibull distribution. Data are based on a bundle of $5\times {10}^7$ fibers with strength thresholds drawn from the Weibull distribution with $m=5$, $x_s=1$, unit elastic constant, and unit strain rate.