Time-dependent fiber bundles with local load sharing

Abstract

Fiber bundle models, where fibers have random lifetimes depending on their load histories, are useful tools in explaining time-dependent failure in heterogeneous materials. Such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings. Various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load-sharing mechanisms, but numerical verification has been hampered by severe computational demands in larger bundles. To gain insight at large size scales, interest has returned to idealized fiber bundle models in 1D. Such simplified models typically assume either equal load sharing (ELS) among survivors, or local load sharing (LLS) where a failed fiber redistributes its load onto its two nearest flanking survivors. Such models can often be solved exactly or asymptotically in increasing bundle size, N, yet still capture the essence of failure in real materials. The present work focuses on 1D bundles under LLS. As in previous works, a fiber has failure rate following a power law in its load level with breakdown exponent $\ensuremath{\rho}.$ Surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed. We develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications (e.g., one million fibers in thousands of realizations). In particular we develop an algorithm that adapts several concepts and methods that are well-known among computer scientists, but relatively unknown among physicists, to dramatically increase the computational speed with no attendant loss of accuracy. We consider various regimes of $\ensuremath{\rho}$ that yield drastically different behavior as N increases. For $1/2<~\ensuremath{\rho}<~1,$ ELS and LLS have remarkably similar behavior (they have identical lifetime distributions at $\ensuremath{\rho}=1)$ with approximate Gaussian bundle lifetime statistics and a finite limiting mean. For $\ensuremath{\rho}>1$ this Gaussian behavior also applies to ELS, whereas LLS behavior diverges sharply showing brittle, weakest volume behavior in terms of characteristic elements derived from critical cluster formation. For $0<\ensuremath{\rho}<1/2,$ ELS and LLS again behave similarly, but the bundle lifetimes are dominated by a few long-lived fibers, and show characteristics of strongest link, extreme value distributions, which apply exactly to $\ensuremath{\rho}=0.$