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 $p(b)\sim b^{-1}$ in the range $10^{-β}$ to $10^β$. 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 $σ_c(β,N)$ for the bundle of size $N$ approaches its asymptotic value $σ_c(β)$ as $σ_c(β,N) = σ_c(β)+AN^{-1/ν(β)}$ where $σ_c(β)$ has been obtained analytically as $σ_c(β) = 10^β/(2βe\ln10)$ for $β\geq β_u = 1/(2\ln10)$, and for $β<β_u$ 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 $1-1/(2β\ln10)$; (iii) the distribution $D(Δ)$ of the avalanches of size $Δ$ follows a power law $D(Δ)\sim Δ^{-ξ}$ with $ξ= 5/2$ for $Δ\gg Δ_c(β)$ and $ξ= 3/2$ for $Δ\ll Δ_c(β)$, where the crossover avalanche size $Δ_c(β) = 2/(1-e10^{-2β})^2$.