Nucleation versus percolation: Scaling criterion for failure in disordered solids

One of the major factors governing the mode of failure in disordered solids is the effective range $R$ over which the stress field is modified following a local rupture event. In a random fiber bundle model, considered as a prototype of disordered solids, we show that the failure mode is nucleation dominated in the large system size limit, as long as $R$ scales slower than $L^{\zeta }$, with $\zeta =2/3$. For a faster increase in $R$, the failure properties are dominated by the mean-field critical point, where the damages are uncorrelated in space. In that limit, the precursory avalanches of all sizes are obtained even in the large system size limit. We expect these results to be valid for systems with finite (normalizable) disorder.

Figure 1

The upper panel shows the variation of the stresses on the fibers in the system with redistribution steps $\tau$. The onset of nucleation can be seen from the tip of the cone, beyond which one broken patch grows and the total number of patch starts decreasing. The onset time gets shifted to a higher value as the stress redistribution range $R$ is increased (a–c) and expected to merge with failure time in the global load-sharing limit. The panel below (d–f) shows the variation of average stress per fiber ($\langle \sigma \rangle$; denoted by dotted lines) and number $\left(n_p\right)$ of patches (denoted by solid lines), scaled by system size, for corresponding $R$ values. The onset of nucleation $\left({\tau }_n\right)$ is the point where the number of patch starts decreasing and the stress per fiber saturates (at the critical value) until the failure point $\left({\tau }_f\right)$ is reached.

Figure 2

The data collapse for the time difference $\mathrm{\Delta }\tau ={\tau }_f-{\tau }_n$ for different system sizes $L$ and for different range $R$ as given by the scaling form [Eq. (1)]. The inverse decay marks the nucleation regime, which stops when $R\sim L^{2/3}$ and global load-sharing region begins. In this regime, $\mathrm{\Delta }\tau$ becomes $R$ independent but depends on $L$ (as can be seen from the inset). The initial collapsed region in the inset confirms the dependence $\mathrm{\Delta }\tau \sim L/R$ in the nucleation regime, and the lines spread out as soon as global mode starts dominating.

Figure 3

(a) The variation of the onset time for nucleation ${\tau }_n$ and failure time ${\tau }_f$ with system size $L$ for two values of $R$. For small system sizes and larger $R$ value, the two times are very close to each other. In this region, mean-field “critical behavior” can be observed. For fixed $R$ value, this “criticality” will not survive in the large system size limit. (b) The avalanche size distribution for fixed $R$ value (100) while the system size is increased. For $L={10}^3$, the ratio $R/L^{2/3}=1$, putting the system in the critical regime where $P\left(S\right)\sim S^{-2.5}$. But for $L={10}^4$ and $L={10}^5$ the ratio becomes $\approx 0.215$ and $\approx 0.046$, respectively. The avalanche size distribution in the last two cases deviates from the above scale-free distribution.

Figure 4

(a) The failure threshold ${\sigma }_c$ goes to zero with $1/lnL$ when $\gamma <{\gamma }_c=4/3$ in the case of power-law load redistribution, showing the validity of the proposed scaling criterion that the effective range must scale faster than $L^{2/3}$ (or $\gamma <4/3$) for the global load-sharing mode. (b) The inverse logarithmic decay is similar to what is observed for uniform load sharing within range $R$ for $R<L^{2/3}<1$.