Dynamic critical behavior of failure and plastic deformation in the random fiber bundle model

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.