Newman W I, Phoenix S L
Departments of Earth and Space Sciences, Physics and Astronomy, and Mathematics, University of California, Los Angeles, California 90095-1567, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Feb;63(2 Pt 1):021507. doi: 10.1103/PhysRevE.63.021507. Epub 2001 Jan 24.
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 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 rho that yield drastically different behavior as N increases. For 1/2< or =rho< or =1, ELS and LLS have remarkably similar behavior (they have identical lifetime distributions at rho=1) with approximate Gaussian bundle lifetime statistics and a finite limiting mean. For 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<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 rho=0.
纤维束模型中，纤维的寿命取决于其负载历史且具有随机性，是解释异质材料中随时间变化的失效现象的有用工具。这类模型有助于阐明各种现象，如结构材料中的疲劳以及地球物理环境中的地震。针对具有不同几何形状和纤维负载分担机制的纤维束，已经开发了各种渐近理论和近似理论，但由于较大纤维束的计算需求过高，数值验证受到了阻碍。为了在大尺寸尺度上获得深入理解，人们重新关注一维理想化纤维束模型。这类简化模型通常假设幸存者之间要么是等负载分担（ELS），要么是局部负载分担（LLS），即失效纤维将其负载重新分配给其两侧最近的幸存纤维。随着纤维束尺寸(N)的增加，这类模型通常可以精确求解或渐近求解，但仍能捕捉实际材料失效的本质。目前的工作聚焦于局部负载分担情况下的一维纤维束。与之前的工作一样，纤维的失效率遵循负载水平的幂律，击穿指数为(\rho)。固定负载下的幸存纤维具有独立且呈指数分布的剩余寿命。我们开发了新的渐近理论和新的计算算法，极大地增加了在大量重复计算中能够处理的纤维束尺寸（例如，在数千次实现中有一百万个纤维）。特别是，我们开发了一种算法，该算法采用了计算机科学家熟知但物理学家相对陌生的几个概念和方法，在不损失精度的情况下显著提高了计算速度。我们考虑了不同的(\rho)取值范围，随着(N)的增加，这些取值范围会产生截然不同的行为。对于(1/2\leqslant\rho\leqslant1)，等负载分担和局部负载分担具有非常相似的行为（在(\rho = 1)时它们具有相同的寿命分布），具有近似高斯的纤维束寿命统计特性和有限的极限均值。对于(\rho>1)，这种高斯行为也适用于等负载分担，而局部负载分担行为则急剧发散，从临界簇形成导出的特征元素来看，呈现出脆性、最弱体积行为。对于(0<\rho<1/2)，等负载分担和局部负载分担的行为再次相似，但纤维束寿命由少数长寿纤维主导，并呈现出最强链、极值分布的特征，这在(\rho = 0)时完全适用。
