Cellular automaton model of damage

We investigate the role of equilibrium methods and stress transfer range in describing the process of damage. We find that equilibrium approaches are not applicable to the description of damage and the catastrophic failure mechanism if the stress transfer is short ranged. In the long-range limit, equilibrium methods apply only if the healing mechanism associated with ruptured elements is instantaneous. Furthermore we find that the nature of the catastrophic failure depends strongly on the stress transfer range. Long-range transfer systems have a failure mechanism that resembles nucleation. In short-range stress transfer systems, the catastrophic failure is a continuous process that, in some respects, resembles a critical point.

Figure 1

The inverse TM metric and the order parameter, $\varphi =N_{\text{unbroken}}/N$, (shown in the inset) both as a function of time for the TFB with $N={256}^2=65536$ fibers, $\sigma /N={10}^{-5}$, $\kappa =1$, $D=1$, and $T=0.5$. The metric shows that once the order parameter reaches its metastable value, the system becomes effectively ergodic.

Figure 2

The inverse TM metric for the nearest-neighbor $\left(R=1\right)$ (a), $R=30$ (b), and mean field $\left(q=N\right)$ (c) base model simulated on a $d=2$, periodic, square lattice of linear size $L=100,512,768$ for $R=1,30$, “$\infty$,” respectively. The parameters of the model are ${\sigma }^f=2.0$, ${\sigma }^r=1.25\pm 0.25$, and $\alpha =0.2$. The solid line is the inverse TM metric and the vertical dashed line indicates the time at which the first site dies.

Figure 3

The inverse TM metric for the base model when sites are healed after one plate updates. The parameters are given by $R=20$, $L=512$, ${\sigma }^f=2$, ${\sigma }^r=1\pm 0.2$, and $\alpha =0.1$.

Figure 4

The inverse TM metric for the $R=10$ step-down model. The model is simulated on a square lattice of linear size $L=256$ with periodic boundary conditions and with ${\sigma }_f\left(t=0\right)=50$, ${\sigma }_r=0.25\pm 0.25$, and $\alpha =0.05$. We consider a site dead when its failure threshold drops below some predefined value which we take as ${\sigma }_j^f\left(t\right)<{\sigma }_j^r+{10}^{-5}\Delta$ where $\Delta ={\sigma }^f\left(t=0\right)-⟨{\sigma }^r⟩=49.75$.

Figure 5

The number of clusters $\left(n_s\right)$ of size $s$ plotted on a log-log plot for various values of $\varphi =N_{\text{alive}}/N$ where stress is passed to all site, alive or dead. For $\varphi =1$, we reproduce the known results of the OFC model where $n_s\sim s^{-\tau }$ with $\tau =3/2$. As we decrease $\varphi$, $\tau$ begins to increase from 3/2 and the scaling region gets smaller and smaller suggesting the damage present in the system drives it away from the (pseudo) spinodal. The crosses indicate the data while the dashed line is the best fit to the data which gives $s\sim s^{-\tau }$ with $\tau \approx 2.07$.

Figure 6

The number of clusters $\left(n_s\right)$ of size $s$ plotted on a log-log plot for all values of $\varphi$. We find that there does appear to be a scaling law but the exponent is now approximately 2. The crosses indicate the data while the dashed line is the best fit to the scaling regime.

Figure 7

The number of clusters $\left(n_s\right)$ of size $s$ plotted on a log-log plot where stress is passed to all sites for $\varphi =0.8$ and $\varphi =0.5$ (crosses and exes, respectively) and their corresponding undamaged system with ${\alpha }^{\prime }=1-\varphi \left(1-\alpha \right)$ (circles for the system corresponding to $\varphi =0.8$ and boxes for the system corresponding to $\varphi =0.5$).

Figure 8

The number of clusters $\left(n_s\right)$ of size $s$ plotted on a log-log plot for various values of $\varphi$ where stress is passed only to the live sites. For all values of $\varphi$, we get the same scaling behavior as the pure OFC model where $n_s\sim s^{-\tau }$ with $\tau =3/2$.

Figure 9

The number of clusters $\left(n_s\right)$ of size $s$ plotted on a log-log plot where stress is passed only to the live sites for $\varphi =0.8$ (exes) and its corresponding undamaged system with $L^{\prime }=\sqrt{\varphi }L$ (boxes). Both scale as $n_s\sim s^{-\tau }$ with $\tau =3/2$.

Figure 10

(Color online) The catastrophic failure event in the long-range base model. The event begins in a dense region of dead sites (a) which, locally, overwhelm the system (b) and grow outward (c) in the shape of the interaction, ultimately failing the entire lattice (d) in a single time step.

Figure 11

(Color online) Two typical nearest-neighbor base model lattices at the time of catastrophic failure (percolating cluster). Figures (a) and (b) show all of the clusters at the time the spanning cluster appears. Figures (c) and (d) isolate the percolating cluster from figures (a) and (b), respectively.