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

Abstract

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 s (load per fiber initially present in the bundle), the fraction U_t(σ) 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^⋆(σ) for applied stress in the range 0>∼σ >∼σ_c, beyond which total failure of the bundle takes place discontinuously [abruptly from U^⋆(σ_c) to 0]. The dynamic critical behavior near this sc has been studied for this model analyzing the recurrence relations. We also investigated the finite size scaling behavior near σ_c. At the critical point σ=σ_c, one finds strict power law decay (with time t) of the fraction of unbroken fibers U_t(σ_c) (as t^→∞). The avalanche size distribution for this mean-field dynamics of failure at σ<σ_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.