Creeplike behavior in athermal threshold dynamics: Effects of disorder and stress

We study the dynamical aspects of a statistical-mechanical model for fracture of heterogeneous media: the fiber bundle model with various interaction ranges. Although the model does not include any nontrivial elementary processes such as nonlinear rheology or stochasticity, the system exhibits creeplike behaviors under a constant load being slightly above the critical value. These creeplike behaviors comprise three stages: primary, secondary, and tertiary. In the primary and tertiary stages, the strain rate exhibits power-law behaviors with time, which are well described by the Omori-Utsu and the inverse Omori laws, respectively, although the exponents are larger than those typically observed in experiments. A characteristic time that defines the onset of power-law behavior in the Omori-Utsu law is found to decrease with the strength of disorder in the system. The analytical solution, which agrees with the above numerical results, is obtained for the mean-field limit. Beyond the mean-field limit, the exponent for the Omori-Utsu law tends to be even larger but decreases with the disorder in the system. Increasing the spatial range of interactions, this exponent is found to be independent of disorder and to converge to the mean-field value. In contrast, the inverse Omori law remains independent of the spatial range of interaction and the disorder strength.

Figure 1

Variation of the function $\mathrm{\Phi }\left(f\right)$ with increasing force per fiber, $f$. The time evolution is given by $f_{i+1}=\mathrm{\Phi }\left(f_i\right)$ and $f_0$ is the initial condition for $f$. Note that $\mathrm{\Phi }\left(0\right)=f_0$. The plots are shown for $f_0<f_0^{*}$, $f_0=f_0^{*}$, and $f_0>f_0^{*}$ (from the bottom to the top), respectively.

Figure 2

(a) Behavior of strain, showing the primary, secondary, and tertiary stages in the time evolution normalized by failure time $t_f$. [(b), (c)] Variation of the strain rate with time respectively in the primary and tertiary stages, along with the comparison with the analytical findings (lines with no points). (d) Variation of $c$ value with $\delta$, the strength of disorder, in the primary stage.

Figure 3

Time evolution of the model with Weibull threshold distribution. (a) Omori-like behavior in the primary stage with a continuous variation of disorder. (b) Variation of $c$ value with disorder at different loading conditions. (c) Inverse-Omori-like behavior close to the failure point.

Figure 4

Omori-like behavior at different loading conditions. [(a), (b)] $\dot{\epsilon }$ vs $t$ for two different disorder values $\beta =2.0$ and $\beta =1.0$. (c) Variation of $c$ value with $\mathrm{\Delta }f$ for $0.5<\beta <2.0$.

Figure 5

Variation of $c$ value when both $\mathrm{\Delta }f$ and $\beta$ are continuously varying parameters. Results are shown for (a) uniform and (b) Weibull distributions.

Figure 6

Strain rate vs time in the (a) primary and (b) tertiary stages of the creep process. The threshold distribution considered here is $p\left(f\right)=f^{-\alpha }$ within the window $[1,{10}^3]$.

Figure 7

Schematic diagram of fiber bundle model, showing the effect of memory during the rupture events. The dashed line shows a ruptured fiber. (a) In the case of the conventional model, the local stresses on the crack tips are uneven. (b) After memory is removed from the model, both the tips carries the same stress.

Figure 8

(a) Time evolution of strain rates in the conventional algorithm and the memory-free algorithm. Each curve is obtained at the critical stress, which depends on the algorithm. (b) System-size effect on the critical stress ($f_c$) for each algorithm. In both cases, it is described by $f_c=a-m/logL$, where log denotes the natural logarithm. For the conventional algorithm, $m=1.0$ and $a=0$. The coefficient $m$ decreases to 0.5 for the memoryless algorithm, where the critical stress is higher. The constant $a$ also shifts to a nonzero value ($\approx 0.1)$.

Figure 9

Variation of strain rate with time (in the primary stage) for disorder $\delta$, ranging in between 0.2 and 0.5, while a critical stress is applied on it. The study is repeated for $\rho =0.93$, 0.46, and 0.05.

Figure 10

Plot for $\dot{\epsilon }$ vs $t$ in the primary stage at three different disorder values ($\delta =0.3$, 0.4, and 0.5). The stress is kept constant at the critical value while $\rho$ is continuously varied.

Figure 11

Variation of the $c$ value and and the exponent $p$ with increasing stress localization. The model approaches the mean-field limit toward $\rho =1$, and $p$ approaches its mean-field value 1.8 gradually with increasing $\rho$.