Emergent patterns of localized damage as a precursor to catastrophic failure in a random fuse network

We study the failure of disordered materials by numerical simulations of the random fuse model. We identify emergent patterns of localized damage prior to catastrophic failure by statistically averaging the density of damage around the eventual failure nucleation point. The resulting pattern depends on fracture density and obeys the same scaling relations as would be expected for the stress field generated by a critical crack nucleating in a finite, disorder-free effective medium of varying size. The growth of this critical crack absorbs preexisting clusters according to a well-defined scaling relation. Unfortunately, in single model runs such precursory signals are not obvious. Our results imply that reliable and accurate prediction of failure in time-independent, microscopically brittle random materials in a real case is inherently problematic, and degrades with system size.

Figure 1

Sketch of a fuse network with applied global current $I$ and voltage $U$. Initially deleted bonds are indicated as thick red (dark) lines while the corresponding cracks, i.e., the barriers to the current, are indicated as dashed blue lines. In the simulation the applied current is increased, resulting in sequential breaking of the thick yellow (light) bonds until one crack (the critical crack) splits the network in two parts.

Figure 2

Damage density $\rho$ (in units of damage per fuse) around nucleation sites in systems of size $L=512$ with dilution $f=0.1$. (a) Single realization with binary color code (dark blue $=$ intact, white $=$ broken). (b–d) Damage density averaged over ensembles with increasing number $N$ of realizations, showing the emergence of a continuous damage distribution. The scale in (d) is reduced to better show details of the crack phantom pattern.

Figure 3

Upper row: Theoretical damage density $\rho$ (in units damage per fuse) calculated from Eq. (2) using the same scale as in Fig. 2 for $f=0.1$, $L=$ (a) 128, (b) 512. Panel (b) is a best fit to (d) in Fig. 2. Lower row: normalized density $(\rho -f)/({\rho }_{\max }-f)$ for $L=256$, $f=$ (c) 0.15, (d) 0.3.

Figure 4

Dependence of phantom size $\xi$ (a, c) (in units of fuses) and failure stress ${\sigma }_c$ (b, d) (in units $i_c/L$) on dilution fraction $f$ (a, b) (dimensionless) and system size $L$ (c, d) (in units of fuses); (a, b) system size $L=256$, dilution fraction $0.05\le f\le 0.3$; (c, d) dilution fraction $f=0.1$, system size $32\le L\le 512$. Dotted line in (c), $\xi \propto \log \left(L\right)$; full lines in (b) and (d), Predictions of failure stresses according to Eqs. (3) and (5), respectively, where $K_{\infty }$ is replaced by $K_{\infty }/\kappa (a/L)$ with $\kappa (a/L)$ according to Eq. (5).

Figure 5

Relative fraction $f\left(s\right)$ of absorbed clusters of size $s$ directly before the peak current, which join the final system-wide crack. In the inset, the curves have been rescaled by Eq. (7) with $L_0=512$. The quantities $s$, $\tilde{s}$, $L$, and $L_0$ are given in units of fuses, whereas $f$ is a dimensionless quantity.