System-size-dependent avalanche statistics in the limit of high disorder

We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasistatic fracture of heterogeneous materials. We consider a fiber bundle model where the strength of single fibers is sampled from a power-law distribution over a finite range, so that the amount of materials' disorder can be controlled by varying the power-law exponent and the upper cutoff of fibers' strength. Analytical calculations and computer simulations, performed in the limit of equal load sharing, revealed that depending on the disorder parameters the mechanical response of the bundle is either perfectly brittle where the first fiber breaking triggers a catastrophic avalanche, or it is quasibrittle where macroscopic failure is preceded by a sequence of bursts. In the quasibrittle phase, the statistics of avalanche sizes is found to show a high degree of complexity. In particular, we demonstrate that the functional form of the size distribution of bursts depends on the system size: for large upper cutoffs of fibers' strength, in small systems the sequence of bursts has a high degree of stationarity characterized by a power-law size distribution with a universal exponent. However, for sufficiently large bundles the breaking process accelerates towards the critical point of failure, which gives rise to a crossover between two power laws. The transition between the two regimes occurs at a characteristic system size which depends on the disorder parameters.

Figure 1

Stress-strain curves $\sigma \left(\varepsilon \right)$ of the bundle (a) for a fixed value of the upper cutoff ${\varepsilon }_{max}=40$ varying the exponent $\mu$, and $\left(b\right)$ for a fixed $\mu =0.7$ exponent varying the upper cutoff ${\varepsilon }_{max}$ by means of the multiplication factor $\lambda$, where ${\varepsilon }_{max}=\lambda {\varepsilon }_{max}^c$. Approaching the phase boundary, in both cases the system becomes more and more brittle; i.e., the maximum of $\sigma \left(\varepsilon \right)$ is preceded by a smaller and smaller amount of fiber breakings. For comparison, the curve corresponding to the case of an infinite upper cutoff ${\varepsilon }_{max}\to \infty$ is also presented.

Figure 2

Phase diagram of the system. The phase boundary separating the brittle and quasibrittle macroscopic response is given by Eq. (6). Under stress controlled loading, in the brittle phase the bundle suffers immediate abrupt failure at the breaking of the weakest fiber, while in the quasibrittle phase failure is preceded by a sequence of breaking bursts. For $\mu \ge 1$ the bundle is always in the brittle phase. The horizontal and vertical dashed lines indicate the parameter sets for which avalanche size distributions were determined by computer simulations.

Figure 3

Series of bursts in a small system of $N={10}^5$ fibers at the exponent $\mu =0.8$ for two different values of the upper cutoff (a) $\lambda =100$ and (b) $\lambda =+\infty$. The size of bursts $\mathrm{\Delta }$ is presented as a function of the order number $i$ of events. The yellow lines represent the moving average of burst sizes $\mathrm{\Delta }$ averaging over 25 consecutive data points.

Figure 4

The average number of breaking fibers $a\left(\varepsilon \right)$ (9) triggered by the failure of one fiber due to the increase of the external load for several values of the disorder exponent $\mu$. The cutoff strength ${\varepsilon }_{max}$ is fixed to ${\varepsilon }_{max}/{\varepsilon }_{min}=1000$. All curves are presented from ${\varepsilon }_{min}$ to the corresponding value of ${\varepsilon }_c(\mu ,{\varepsilon }_{max})$. For $\mu \to 0$ the critical point converges to ${\varepsilon }_c={\varepsilon }_{max}/e$.

Figure 5

Inset: Size distribution of bursts $p\left(\mathrm{\Delta }\right)$ for a bundle of $N={10}^6$ fibers at several values of the disorder exponent $\mu$ when the cutoff strength of fibers is infinite ${\varepsilon }_{max}=+\infty$. Main panel: Data collapse of the curves of the inset obtained by rescaling with a power of the distance from the critical point ${\mu }_c=1$. Along the horizontal axis the scaling exponent is $\nu =2$ in agreement with Eq. (14), while along the vertical axis the product $\nu \tau$ is used with $\tau =3/2$. The straight line represents a power law of exponent $3/2$.

Figure 6

Burst size distributions $p\left(\mathrm{\Delta }\right)$ in a bundle of size $N={10}^7$ at a fixed upper cutoff ${\varepsilon }_{max}=10$ varying the value of the exponent $\mu$. The two straight lines represent power laws of exponent $3/2$ and $5/2$.

Figure 7

Crossover burst size ${\mathrm{\Delta }}_0$ as a function of the disorder exponent $\mu$ for the cutoff strength ${\varepsilon }_{max}=10$. The arrow indicates the position of the corresponding critical point ${\mu }_c$. The inset presents ${\mathrm{\Delta }}_0$ as a function of the distance from the critical point ${\mu }_c\left({\varepsilon }_{max}\right)-\mu$ for two upper cutoffs on a double logarithmic plot.

Figure 8

Burst size distributions $p\left(\mathrm{\Delta }\right)$ for a fixed value of the exponent $\mu =0.85$ varying the upper cutoff $\lambda$. The bundle is composed of ${10}^7$ fibers. The two straight lines represent power laws of exponent $3/2$ and $5/2$. The case of an infinite upper cutoff $\lambda =+\infty$ is also included for comparison.

Figure 9

Size distributions $p\left(\mathrm{\Delta }\right)$ of the first $L$ bursts of a bundle of $N={10}^7$ fibers varying $L$ in a broad range at the fixed disorder parameters $\mu =0.8$ and $\lambda =500$. The complete distribution of the entire failure process is also presented together with the distribution of the last events just preceding global failure. The total number of events is about $1.08\times {10}^6$. The two straight lines represent power laws of exponents $3/2$ and $5/2$.

Figure 10

Size distribution of bursts $p\left(\mathrm{\Delta }\right)$ for different system sizes $N$ at fixed values of the disorder parameters $\mu =0.8$ and $\lambda =500$. For small system sizes $p\left(\mathrm{\Delta }\right)$ agrees with the corresponding distribution of the infinite cutoff $\lambda =+\infty$. Above a characteristic system size a second power-law regime gradually develops for large bursts.

Figure 11

Average size of the catastrophic avalanche $〈{\mathrm{\Delta }}_c〉$ as a function of the system size $N$ for several values of the upper cutoff $\lambda$ of fibers strength at a fixed exponent $\mu =0.8$.

Figure 12

The same data as in Fig. 11 are presented in such a way that along the horizontal axis the system size $N$ is rescaled with ${\lambda }^{\mu }$. High-quality data collapse is obtained. The straight line represents a power law of exponent 1.