Fiber bundle model with highly disordered breaking thresholds

Abstract

We present a study of the fiber bundle model using equal load-sharing dynamics where the breaking thresholds of the fibers are drawn randomly from a power-law distribution of the form p(b)∼b-1 in the range 10-β to 10β. Tuning the value of β continuously over a wide range, the critical behavior of the fiber bundle has been studied both analytically as well as numerically. Our results are: (i) The critical load σc(β,N) for the bundle of size N approaches its asymptotic value σc(β) as σc(β,N)=σc(β)+AN-1/ν(β), where σc(β) has been obtained analytically as σc(β)=10β/(2βeln10) for β≥βu=1/(2ln10), and for β<βu the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to σ_{c}(β)=10-β; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form 1-1/(2βln10); (iii) the distribution D(Δ) of the avalanches of size Δ follows a power-law D(Δ)∼Δ-ξ with ξ=5/2 for Δ≫Δc(β) and ξ=3/2 for Δ≪Δc(β), where the crossover avalanche size Δc(β)=2/(1-e10-2β)2.