The fiber bundle model describes a collection of elastic fibers under load. The fibers fail sucessively and for each failure, the load distribution among the surviving fibers changes. Even though very simple, this model captures the essentials of failure processes in a large number of materials and settings. We present here a review of the fiber bundle model with different load redistribution mechanisms from the point of view of statistics and statistical physics rather than materials science, with a focus on concepts such as criticality, universality and fluctuations. We discuss the fiber bundle model as a tool for understanding phenomena such as creep, and fatigue, how it is used to describe the behavior of fiber reinforced composites as well as modelling e.g. network failure, traffic jams and earthquake dynamics.

The statistics of damage avalanches during a failure process typically follows a power law. When these avalanches are recorded only near the point at which the system fails catastrophically, one finds that the power law has an exponent which is different from that one finds if the recording of events starts away from the vicinity of catastrophic failure. We demonstrate this analytically for bundles of many fibers, with statistically distributed breakdown thresholds for the individual fibers and where the load is uniformly distributed among the surviving fibers. In this case the distribution D(Delta) of the avalanches (Delta) follows the power law Delta-xi with xi=3/2 near catastrophic failure and xi=5/2 away from it. We also study numerically square networks of electrical fuses and find xi=2.0 near catastrophic failure and xi=3.0 away from it. We propose that this crossover in xi may be used as a signal of imminent failure.

The random fiber bundle (RFB) model, with the strength of the fibers distributed uniformly within a finite interval, is studied under the assumption of global load sharing among all unbroken fibers of the bundle. At any fixed value of the applied stress sigma (load per fiber initially present in the bundle), the fraction U(t)(sigma) of fibers that remain unbroken at successive time steps t is shown to follow simple recurrence relations. The model is found to have stable fixed point U*, filled (sigma) for applied stress in the range 0 < or = sigma < or = sigma(c), beyond which total failure of the bundle takes place discontinuously [abruptly from U*, filled (sigma(c)) to 0]. The dynamic critical behavior near this sigma(c) has been studied for this model analyzing the recurrence relations. We also investigated the finite size scaling behavior near sigma(c). At the critical point sigma = sigma(c), one finds strict power law decay (with time t) of the fraction of unbroken fibers U(t)(sigma(c)) (as t--> infinity). The avalanche size distribution for this mean-field dynamics of failure at sigma < sigma(c) has been studied. The elastic response of the RFB model has also been studied analytically for a specific probability distribution of fiber strengths, where the bundle shows plastic behavior before complete failure, following an initial linear response.

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