Competition of strength and stress disorder in creep rupture

Based on a fiber bundle model of subcritical fracture with localized load sharing, we show that the interplay of threshold disorder and the inhomogeneous stress field gives rise to a rich dynamics with intriguing aspects. In the model, fibers fail either due to immediate breaking or to a slow damage process. When the disorder is strong, a large amount of damage occurs, which is randomly diffused over the system; however, for weak disorder, a single growing crack is formed, which proceeds in a large number of localized bursts. The microstructure of cracks is characterized by a power-law size distribution, which is analogous to percolation in the regime of diffusive damage; however, it becomes significantly steeper when a single crack dominates. Simulations showed that the size distribution of breaking bursts and of the waiting times in between have a power-law functional form with a load-dependent cutoff. The burst size exponent proved to be independent of the damage process; however, it strongly depends on the external load with a minimum value of 1.75. The waiting time distribution is sensitive to the details of the damage process with an exponent decreasing from 2.0 to 1.4 as bursts get more and more localized to an advancing crack front.

Figure 1

Disorder distribution of the damage thresholds $f\left(c_{th}\right)$. By varying the width $W$ of the distribution, we can control the amount of disorder: the red (gray) curve represents a lower disorder than the black one. In the specific calculations, $C$ had the value $C=1$, so that $W$ was varied in the interval $0⩽W⩽1$.

Figure 2

Phase diagram of the system. The curve corresponding to Eq. (7) separates the regimes of single crack growth and diffusive damage.

Figure 3

Cluster structure of broken fibers formed during the time evolution of a system of size $L=201$ for the same load, ${\sigma }_0/{\sigma }_c=0.084$, and amount of disorder, $W/C=0.2$, at different values of the $\gamma$ exponent: (a) 1.0, (b) 2.0, (c) 5.0, and (d) 8.0. Green color indicates the fibers that broke due to damage accumulation, while different colored spots represent broken fibers of bursts triggered by damage sequences. Periodic boundary conditions were used in both directions. (a) At low values of $\gamma$, most of the fibers break due to damage. (b) By increasing $\gamma$, damage sequences get spatially localized, which can trigger larger and larger bursts. (c), (d) At high $\gamma$, only a small amount of dispersed damage occurs, and practically a single large cluster is formed, which is a growing crack in the system.

Figure 4

Average value of characteristic quantities of the bundle as a function of the external load, ${\sigma }_0/{\sigma }_c$, for three values of $\gamma$ with the amount of disorder $W/C=0.2$. In all of the subfigures, the same legend is used, which is presented in (c). (a) Average lifetime $\langle t_f\rangle$ of the bundle. Power-law behavior is obtained, showing the validity of the Basquin law. The slope of the straight lines is equal to the corresponding value of the damage accumulation exponent $\gamma$. (b) Average burst size. (c) Average size of the largest cluster of broken fibers. (d) Average cluster size. In (c), the thick line represents the fit obtained with Eq. (10).

Figure 5

Spatial structure of bursts in a system of size $L=401$ with the parameter values $\gamma =8$, $W/C=0.2$, and ${\sigma }_0/{\sigma }_c=0.05$. Light green color indicates fibers broken due to damage accumulation. Spots of randomly selected colors represent bursts, i.e., simultaneously broken fibers, so that the extension of colored spots gives the burst size $\Delta$. The black circle on the left middle of the picture indicates the starting point of the crack and the white arrows show the direction of propagation (note the periodic boundary conditions in both directions).

Figure 6

Burst size distributions for three values of $\gamma$ varying the load ${\sigma }_0$ in a broad range. It can be observed that at the load ${\sigma }_0/{\sigma }_c=0.11$, where the average burst size has a maximum, the exponent $\xi$ of the distribution $P\left(\Delta \right)$ has a minimum $\xi =1.75$.

Figure 7

Cluster size distributions for three $\gamma$ values at the peak load of the average cluster size $\langle S_{av}\rangle$. The exponent $\tau =2.05$ obtained for $\gamma =1$ is consistent with site percolation on a square lattice in two dimensions.

Figure 8

Distribution of waiting times $P\left(T\right)$ in a system of size $L=401$. A power-law functional form can be observed over eight orders of magnitude. The exponent of the power law $z$ is independent of the load ${\sigma }_0$; however, it changes from 1.4 to 2.0 as $\gamma$ is increased.