Fiber Bundle model with Highly Disordered Breaking Thresholds

TL;DR

This study analyzes a fiber bundle model with highly disordered breaking thresholds following a power law distribution, revealing critical behavior, failure thresholds, and avalanche size distributions through analytical and numerical methods.

Contribution

It introduces a fiber bundle model with thresholds drawn from a power law distribution and characterizes its critical behavior and failure dynamics.

Findings

Critical load approaches an analytical value depending on disorder parameter

Fraction of broken fibers before failure is derived

Avalanche size distribution follows a power law with two regimes

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 $10^{-\beta}$ to $10^{\beta}$. Tuning the value of $\beta$ 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 $\sigma_c(\beta,N)$ for the bundle of size $N$ approaches its asymptotic value $\sigma_c(\beta)$ as $\sigma_c(\beta,N) = \sigma_c(\beta)+AN^{-1/\nu(\beta)}$ where $\sigma_c(\beta)$ has been obtained analytically as $\sigma_c(\beta) = 10^\beta/(2\beta e\ln10)$ for $\beta \geq \beta_u = 1/(2\ln10)$, and for $\beta<\beta_u$ the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to $\sigma_c(\beta) = 10^{-\beta}$; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form $1-1/(2\beta \ln10)$; (iii) the distribution $D(\Delta)$ of the avalanches of size $\Delta$ follows a power law $D(\Delta)\sim \Delta^{-\xi}$ with $\xi = 5/2$ for $\Delta \gg \Delta_c(\beta)$ and $\xi = 3/2$ for $\Delta \ll \Delta_c(\beta)$, where the crossover avalanche size $\Delta_c(\beta) = 2/(1-e10^{-2\beta})^2$.