Time-dependent fiber bundles with local load sharing - art. no. 021507

Fiber bundle models, where fibers have random lifetimes depending on their load histories, are useful tools in explaining time-dependent failure in he terogeneous materials. Such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings. Va rious asymptotic and approximate theories have been developed for bundles w ith various geometries and fiber load-sharing mechanisms, but numerical ver ification has been hampered by severe computational demands in larger bundl es. To gain insight at large size scales, interest has returned to idealize d fiber bundle models in 1D. Such simplified models typically assume either equal load sharing (ELS) among survivors, or local load sharing (LLS) wher e a failed fiber redistributes its load onto its two nearest flanking survi vors. Such models can often be solved exactly or asymptotically in increasi ng bundle size, N, yet still capture the essence of failure in real materia ls. 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 brea kdown exponent rho. Surviving fibers under fixed loads have remaining lifet imes that are independent and exponentially distributed. We develop both ne w asymptotic theories and new computational algorithms that greatly increas e the bundle sizes that can be treated in large replications (e.g., one mil lion fibers in thousands of realizations). In particular we develop an algo rithm that adapts several concepts and methods that are well-known among co mputer scientists, but relatively unknown among physicists, to dramatically increase the computational speed with no attendant loss of accuracy. We co nsider various regimes of rho that yield drastically different behavior as N increases. For 1/2 less than or equal to rho less than or equal to1, ELS and LLS have remarkably similar behavior (they have identical lifetime dist ributions at rho =1) with approximate Gaussian bundle lifetime statistics a nd a finite limiting mean. For rho >1 this Gaussian behavior also applies t o ELS, whereas LLS behavior diverges sharply showing brittle, weakest volum e behavior in terms of characteristic elements derived from critical cluste r formation. For 0< <rho><1/2, ELS and LLS again behave similarly, but the bundle lifetimes are dominated by a few long-lived fibers, and show charact eristics of strongest link, extreme value distributions, which apply exactl y to <rho>=0.