Time-dependent fiber bundles with local load sharing. II. General Weibull fibers

Fiber bundle models (FBMs) are useful tools in understanding failure processes in a variety of material systems. While the fibers and load sharing assumptions are easily described, FBM analysis is typically difficult. Monte Carlo methods are also hampered by the severe computational demands of large bundle sizes, which overwhelm just as behavior relevant to real materials starts to emerge. For large size scales, interest continues in idealized FBMs that assume either equal load sharing (ELS) or local load sharing (LLS) among fibers, rules that reflect features of real load redistribution in elastic lattices. The present work focuses on a one-dimensional bundle of $N$ fibers under LLS where life consumption in a fiber follows a power law in its load, with exponent $\ensuremath{\rho}$, and integrated over time. This life consumption function is further embodied in a functional form resulting in a Weibull distribution for lifetime under constant fiber stress and with Weibull exponent, $\ensuremath{\beta}$. Thus the failure rate of a fiber depends on its past load history, except for $\ensuremath{\beta}=1$. We develop asymptotic results validated by Monte Carlo simulation using a computational algorithm developed in our previous work [Phys. Rev. E 63, 021507 (2001)] that greatly increases the size, $N$, of treatable bundles (e.g., ${10}^{6}$ fibers in ${10}^{3}$ realizations). In particular, our algorithm is $\mathcal{O}(N\text{ }\text{ln}\text{ }N)$ in contrast with former algorithms which were $\mathcal{O}({N}^{2})$ making this investigation possible. Regimes are found for $(\ensuremath{\beta},\ensuremath{\rho})$ pairs that yield contrasting behavior for large $N$. For $\ensuremath{\rho}g1$ and large $N$, brittle weakest volume behavior emerges in terms of characteristic elements (groupings of fibers) derived from critical cluster formation, and the lifetime eventually goes to zero as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, unlike ELS, which yields a finite limiting mean. For $1/2\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$, however, LLS has remarkably similar behavior to ELS (appearing to be virtually identical for $\ensuremath{\rho}=1$) with an asymptotic Gaussian lifetime distribution and a finite limiting mean for large $N$. The coefficient of variation follows a power law in increasing $N$ but, except for $\ensuremath{\rho}=1$, the value of the negative exponent is clearly less than 1/2 unlike in ELS bundles where the exponent remains 1/2 for $1/2l\ensuremath{\rho}\ensuremath{\le}1$. For sufficiently small values $0l\ensuremath{\rho}⪡1$, a transition occurs, depending on $\ensuremath{\beta}$, whereby LLS bundle lifetimes become dominated by a few long-lived fibers. Thus the bundle lifetime appears to approximately follow an extreme-value distribution for the longest lived of a parallel group of independent elements, which applies exactly to $\ensuremath{\rho}=0$. The lower the value of $\ensuremath{\beta}$, the higher the transition value of $\ensuremath{\rho}$, below which such extreme-value behavior occurs. No evidence was found for limiting Gaussian behavior for $\ensuremath{\rho}g1$ but with $0l\ensuremath{\beta}(\ensuremath{\rho}+1)l1$, as might be conjectured from quasistatic bundle models where $\ensuremath{\beta}(\ensuremath{\rho}+1)$ mimics the Weibull exponent for fiber strength.