From nucleation to percolation: the effect of system size when disorder and stress localization compete

In this section, we have studied the fiber bundle model in the local load sharing limit. In this limit, the stress of a broken fiber is redistributed among the nearest neighbors only. This scenario can be achieved by setting a very high value of $\gamma$.

We start our numerical simulation by observing the characteristics of the patches (or cracks) that are generated within the bundle during the evolution of the model. A certain patch is defined by the combination of an intact fiber and a broken fiber in its neighborhood. If we denote the intact fiber by 1 and broken fiber by 0, then any (1,0) or (0,1) combination on the one-dimensional chain of fibers will be characterized as a patch or crack. The patch density $\rho$ at time $t$ is defined by the number of patches at that time, normalized by the size $L$ of the system. We also note the fraction $B$ of broken fibers at time $t$. $B$ is defined as the number of broken fibers divided by size $L$ of the system. Time, in this case, is represented as the sum of stress increment and redistribution steps (see ref [59] for details). Figure 1 shows the variation of $\rho$ with $B$ for different disorder strength values ranging in between 0.4 and 2.0 for a bundle of size $L=10^{5}$.

Figure 1(a) shows a non-monotonic behavior of $\rho$ when $B$ increases from 0 to 1. Earlier papers [42, 48] have already discussed that such crack density shows a non-monotonic behavior as the model evolves. $B=0$ stands for the initial configuration where all fibers are intact while we obtain $B=1$ when all fibers are broken. The patch density $\rho$ is zero for $B=0$ as no cracks are there in the bundle. On the other hand, when the model is close to the failure point, there will be a single patch, making $\rho=1/L$ when $B$ approaches 1. At an intermediate $B$, $\rho$ reaches a maximum value $\rho_{m}$. Before this maxima, $\rho$ is an increasing function of $B$ as new patches are generated within the bundle. After the maxima, the patches start to coalesce with each other and we observe a lesser and lesser number of cracks with increasing time (hence increasing $B$). Figure 1(a) shows as disorder strength is increased, $\rho_{m}$ shifts to a higher value. We will discuss the variation of $\rho_{m}$ with disorder next. The dotted line in the same figure is the locus of $\rho=B(1-B)$. This dotted line represents failure events random in space for a 1d FBM. This can be understood as follows. If $B$ is the fraction of fibers broken, then the probability of having a broken and an intact fiber will be $B$ and $(1-B)$ respectively. A patch then will be created by placing an intact fiber beside a broken one, the probability of which will be $B(1-B)$ on a 1d lattice. By equating $d\rho/dB$ for this locus to zero we get the maximum $\rho_{m}=0.25$ and the $B$ value to be 0.5 at this maxima. The failure pattern becomes more and more random when $\beta$, the disorder strength, is high enough. On the other hand, when the disorder strength is very low ($\beta=0.4$), we see $\rho=1/L$ independent of the value of $B$. This suggests a pure nucleating failure starting from the very beginning until the global failure.

Figure 1(b) shows the variation of $\rho_{m}$ explicitly when $\beta$ is continuously varied. We observe three different regions.

(I) Nucleation ($\beta\leq 0.4$) $-$ Here $\rho_{m}$ has a value close to $1/L$. This suggests that only a single crack is generated within the bundle in this limit and this crack nucleates to create global failure. Due to the low strength of the disorder, the failure process here is guided by the local stress concentration at the crack tips.

(III) Percolation ($\beta\geq 1.2$) $-$ In this limit the behavior of $B$ vs $\rho$ matches closely with $\rho=B(1-B)$. $\rho_{m}$ has a value close to $0.25$. The failure events are random in space here making it reminiscent of percolation in 1d lattice. The failure process is completely guided by the disorder strength and the local stress concentration is almost non-existing.

(II) Avalanche ($0.4<\beta<1.2$) $-$ in the intermediate disorder strength, there is an interplay between the disorder strength and the local stress concentration. The failure process here starts in a percolating manner but later the local stress concentration takes over making the rest of the failure events nucleating. We will be discussing this spatial correlation in more detail later in this paper.

Figure 2(a) shows how the maximum patch density $\rho_{m}$ responds to the size of the bundle. The results are repeated for three different $\beta$ values 0.3, 0.7, and 1.2, in order to cover all three regions$-$ nucleation, avalanche, and percolation, mentioned above. When disorder strength $\beta$ is low (0.3), $\rho_{m}$ decreases with $L$ in a scale-free manner with exponent $-1$. This suggests that the maximum number of cracks observed in the bundle decreases with increasing size and the model goes towards nucleation more and more as the model approaches the thermodynamic limit. On the other hand, for $\beta=1.2$, $\rho_{m}$ is independent of $L$ and saturates at a value close to 0.25. As mentioned above, the failure process is percolating here and remains the same irrespective of the size of the bundle. In the intermediate disorder, where the disorder strength and local stress concentration compete with each other, we observe

where the exponent $\xi$ is a function of $\beta$.

Figure 2(b) shows the variation of exponent $\xi$ with the strength of disorder $\beta$. $\xi$ remains at 1 for low $\beta$ where the failure is nucleating, gradually decreases in region avalanche, and becomes constant at 0 in the limit of percolation. The nature of patch density remains the same in the percolation region ($\xi=0$) only. In both avalanche and nucleation, fewer patches grow as the size of the bundle is increased, suggesting that the effect of the local stress concentration becomes more prominent here as the model goes towards the thermodynamic limit.

Here, we will discuss a dynamic parameter that helps us to understand the onset of the nucleation process with time more clearly. As explained earlier, time $t$ here is analogous to the total number of redistribution plus stress increment steps prior to the global failure. We start by breaking the weakest fiber, say $i$, at time $t=0$ by the first stress increment. Let us assume further $n_{1}$ fibers break at the next time step ($t=1$) upon redistributing the stress carried by the weakest fiber. We consider the distance $\Delta r$ between these two consecutive events to be the minimum of distances between fiber $i$ and other $n_{1}$ fibers that break after redistribution. Here, $\Delta r$ is not the exact lattice distance as only intact fibers are considered while calculating it. The distance across a broken patch is considered to be 1 independent of the size of the patch. This is due to the LLS scheme that we have adopted. Whenever a fiber at a notch breaks and the redistributed stress breaks the fiber at the other notch, the failure is still nucleating, no matter how large this patch is. Next, we square this distance and average it over $10^{4}$ realizations to get $\langle\Delta r^{2}\rangle$ at time $t=0$. Next, we move our reference frame to the fiber among those $n_{1}$ fibers that had the minimum distance from fiber $i$. Let’s denote this new fiber as $j$. If further $n_{2}$ fibers break in the next redistribution, $\langle\Delta r^{2}\rangle$ at $t=1$ will be calculated by the same procedure: find $\Delta r$ from the minimum of distances between fiber $j$ and those $n_{2}$ fibers, square it and average over $10^{4}$ realizations. Such a parameter was explored earlier by Stormo et. al [58] in the context of the soft clamp model to point out the onset of localization. Figure 3 shows this variation of $\langle\Delta r^{2}\rangle$ with time $t$ for $\beta=0.3$, 0.7 and 1.2. For all three disorder strength values, $\langle\Delta r^{2}\rangle$ starts from a high value at low $t$ and then decreases towards 1 when $t$ is high. A high value of $\langle\Delta r^{2}\rangle$ suggests the fibers that break consecutively are far from each other. This is a spatially uncorrelated failure. On the other hand, when $\langle\Delta r^{2}\rangle=1$, the consecutive failures take place from the neighboring fibers only. This behavior stands for pure nucleation. For $\beta=0.3$, $\langle\Delta r^{2}\rangle$ becomes 1 very fast and stays there independent of $t$ until the bundle reaches global failure. For $\beta=1.2$, we observe the opposite behavior where $\langle\Delta r^{2}\rangle$ stays at a high value for a long time and falls to 1 just before global failure. The former behavior is nucleating (I) while the latter one is percolating (III). The visualization of the failure process for both (I) and (III) is shown below figure 3. The x-axis of each plot is time and the y-axis is fiber index. The color gradient is over the local stress profile. The yellow color stands for the failed fibers. For (I), we see a single crack growing in a nucleating manner from the very beginning. For (III), on the other hand, there are no nucleating yellow-colored fibers and the rupture events are spatially uncorrelated. For the avalanche (II) behavior, there is a spatially uncorrelated failure in the beginning as well as nucleation close to global failure. Figure 3 shows the point for $\beta=0.7$ where $\langle\Delta r^{2}\rangle$ becomes 1 indicating onset of localized (nucleating) failure events.

Finally, we have constructed the phase diagram of disorder strength $\beta$ and system size $L$ to show all three failure processes. In figure 4, $1/\beta$ is plotted against $1/L$. This is done in this way so that the origin (0,0) of this plot corresponds to $L\rightarrow\infty$ and $\beta\rightarrow\infty$, an infinite disorder in the thermodynamic limit. As discussed earlier, if the disorder strength is increased, we start with nucleation, go through an avalanche, and finally reach percolation behavior. The spatial rupturing events are shown in the corresponding phases. Now, if the disorder strength is kept constant and the size of the bundle is increased, a percolating behavior moves towards avalanche and an avalanche behavior moves towards nucleation. Due to weak dependence of parameters like $\rho_{m}$ on $L$ (see figure 2), it will not be possible to see (as the system size has to be very high) this change from percolation to avalanche if we are well inside the percolation region. To see this change at relatively lower system sizes, it is required to keep the disorder strength at a value so that the model is closer to the percolation-avalanche interface. The opposite happens if the system size is decreased instead of increasing. This suggests we observe only nucleating failure in the thermodynamic limit unless the disorder is infinitely high. This effect of disorder was explored earlier in the context of random fuse network by Shekhawat et al. [34] and Moreira et al. [35]. We observe that the fiber bundle model in one dimension follows the same trend.

In this section, the model is studied with a continuous variation in both $\gamma$ when $\beta$. We start our numerical simulation by observing the spatial correlation through the rupture events with increasing time as the bundle fails.

Figure 5 shows such correlation for $\beta$ values ranging in between 0.3 and 1.2 and $\gamma$ values within $0$ and $3$. For each small figure, the x-axis shows the time $t$ and the y-axis shows the fiber index. From left to right, the figures are plotted for increasing values of $\gamma$ keeping the disorder strength $\beta$ constant. On the other hand, from bottom to the top the figures are plotted with increasing $\beta$ and keeping $\gamma$ constant. We observe the following behavior:

For low $\beta$ and high $\gamma$, the fibers break in a nucleating manner. Due to the low stress release range the stress concentration plays a crucial role and dominates the failure process. Moreover, due to the low value of disorder strength, the probability that the fibers break with redistribution (without any increment in external stress) from the neighborhood is high. On the other hand, for high $\beta$ and low $\gamma$, the fluctuation between threshold strength as well as the stress release range is high. As a result, we observe rupture events random in space and through increment in external stress.

Now, keeping the $\beta$ fixed at a low value, as we decrease $\gamma$, the model slowly goes towards the mean-field limit. In this limit, the stress of the broken fibers are redistributed among all surviving fibers. This increases the chance that whenever one fiber breaks, the next rupture event may take place from somewhere which is not the neighborhood of the broken fiber. The failure process, in this case gradually deviates from the nucleating behavior as $\gamma$ decreases.

Instead, if we keep $\gamma$ fixed at a high value and increase $\beta$, the fluctuation among fiber strengths increases. Here, the stress of the broken fiber is redistributed in the neighborhood (as $\gamma$ is high) but due to this increase in fluctuation we will find more more strong fibers in this neighborhood that will finally arrest the growth of a crack - a phenomena known as lattice trapping or intrinsic crack resistance [19, 20, 21, 22, 23, 24]. This forces the growth of a different crack from a different place.

As discussed in figure 5, a single crack or a number of cracks are observed in the bundle depending on what the values of disorder strength $\beta$ and the stress release range $\gamma$ are. In figure 6, we have studied how the maximum number of cracks ($\rho_{m}$) varies as we vary both $\beta$ and $\gamma$. At first, we observe the variation of $\rho_{m}$ with $\gamma$ for a constant value of $\beta$. The study is then repeated for $\beta$ values ranging in between $0.2$ and $0.8$.

For low $\gamma$, $\rho_{m}$ has a higher value and decreases as $\gamma$ increases and crosses the critical value $\gamma_{c}$ [42]. The results can be discussed in three parts. At an intermediate disorder ($0.3\leq\beta\leq 0.5$), $\rho_{m}$ saturates close to 0.25 for low $\gamma$ and decreases to $1/L$ when $\gamma$ is high. A $\rho_{m}$ value close to $1/L$ suggests there is only one crack that propagates throughout the system. On the other hand, as already discussed in the manuscript, $\rho_{m}$ close to 0.25 suggests a failure process random in space. In this limit, all three regions are accessible with a variation in $\gamma$.

At low $\beta$ ($<0.3$), we observe that $\rho_{m}$ goes to $1/L$ for high $\gamma$ but does not approach $0.25$ even if $\gamma$ is very low. In this limit, we do not see a percolation like a random failure. This is due to very low disorder strength, where the bundle breaks very abruptly before it can reach the real maximum value of $\rho_{m}$ ($\approx 0.25$) at low $\gamma$. On the other hand, for high $\beta$ ($>0.5$), $\rho_{m}$ reaches 0.25 easily at low $\gamma$ but do not reach $1/L$ for high $\gamma$. In this case, $\rho_{m}$ does not reach $1/L$ even at high $\gamma$ due to the intrinsic crack resistance caused by the high fluctuation in threshold strength, which arrests an propagating crack in the process. As a result, a nucleating failure is not observed here.

Figure 7(a) shows the system size effect of $\rho_{m}$ for $\beta=0.6$ and for wide range of stress relaxation $\gamma$ ($0\leq\gamma\leq 3$). We observe $\rho_{m}$ to decrease in a scale-free manner with system size $L$,

where $\zeta$ is an increasing function of $\gamma$. At low $\gamma$, $\rho_{m}$ is almost independent of $L$ and saturates around 0.25. When $\gamma$ is high, $\rho_{m}$ responds to the change in $L$ very sharply and decreases as $L$ increases.

The variation of the exponent $\zeta$ is shown in figure 7(b). $\zeta$ has a value close to $0$ independent of disorder strength $\beta$ when $\gamma$ low. At such a low value of $\gamma$, the model is in the mean field limit and changing the system size does not change the dynamics of the model. As $\gamma$ increases, the model slowly deviates from the mean-field limit and the effect of local stress concentration becomes more and more prominent. In this limit, $\zeta$ starts to increase slowly. When $\gamma$ crosses a certain value that depends on $\beta$, $\zeta$ finally reaches 1. A Higher value of $\beta$ will require a higher $\gamma$ value in order to obtain $\zeta=1$. Finally, when $\beta$ is very high, $\zeta$ remains close to $0$ independent of the stress release range $\gamma$. Here, the failure process is random in space, independent of both $\beta$ and $\gamma$.

Figure 8 shows above mentioned three regions from the study of the maximum number of cracks ($\rho_{m}$) when both disorder strength $\beta$ and the stress release range $\gamma$ are varied simultaneously. The color gradient in figure 8 is on $\rho_{m}$, with a maximum value of 0.25 (lightest color) and minimum value of $1/L$ (darkest color) which is $10^{-5}$ as the size of our bundle is $10^{5}$. We observe $\rho_{m}$ to have the lowest value for lowest possible $\beta$ and highest possible $\gamma$.

This is due to the fact that fluctuation among local strength values are minimum here and at the same time stress release range is also minimum making the local stress concentration most prominent. At this situation, as we decrease $\gamma$, we go towards percolation (light color) through avalanche region. The same behavior is observed (I $\rightarrow$ II $\rightarrow$ III) if we increase $\beta$ keeping $\gamma$ fix. The figure also shows the existence of $\gamma_{c}$ [42, 43], the value of $\gamma$ below which the model enters the mean-field limit. For $\gamma<\gamma_{c}$, we have almost an uniform gradient of light color suggesting $\rho_{m}$ is close to $0.25$ here independent of $\beta$. We observe a very small change in the color gradient if we increase $\gamma$ at a high $\beta$.

Figure 10 shows different regions $-$ nucleation, avalanche and percolation, with their unique nature of crack propagation during the failure process when the disorder strength ($\beta$), stress release range ($\gamma$), and system size ($L$) are continuously varied. Figure 10(a) shows the plane separating the region nucleation from avalanche. The plane seems to diverge for high $\beta$. This is due to the fact that, at high beta the fluctuation among fiber strength values will be high and the $\gamma$ value will also have to be very high to make the local stress concentration prominent enough to create nucleation. At the same time, since an increasing $L$ has already been seen to favor nucleating failure, we achieve such nucleation at relatively lower $\gamma$ value at higher $L$ when $\beta$ is kept fixed. Figure 10(b), on the other hand, shows the plane between avalanche and percolation. We observe the same effect of $L$ here $-$ as $L$ increases, the transition from percolation to avalanche takes place at a lower value of $\gamma$. At the same time, as $\beta$ increases, we have to go to a higher $\gamma$ value to enter the avalanche region from percolation. The sudden upward curvature of nucleation $-$ avalanche plane at high $\beta$ suggests that if the disorder strength is extremely high, we might not get a nucleation region. Similarly, the sudden downward curvature of avalanche $-$ percolation plane at low $\beta$ suggests that if the disorder strength is extremely low, we might not get a percolation region.