Criticality in Fiber Bundle Model

TL;DR

This paper uncovers a new critical behavior in fiber bundle models, showing how failure probability, relaxation time, and avalanche sizes scale near a critical threshold, revealing a transition from brittle to quasi-brittle failure modes.

Contribution

It identifies a novel critical point in fiber bundle models with specific scaling laws for failure probability, relaxation time, and avalanche distribution, linking to brittle to quasi-brittle transition.

Findings

Failure probability scales as P_b ~ L^{-rac{1}{3}} at criticality.

Relaxation time diverges with system size as τ ~ L^{0.33} near the critical point.

Avalanche sizes follow a power-law distribution with exponent κ ≈ 0.50 at criticality.

Abstract

We report a novel critical behavior in the breakdown of an equal load sharing fiber bundle model at a dispersion $\delta_c$ of the breaking threshold of the fibers. For $\delta < \delta_c$, there is a finite probability $P_b$, that rupturing of the weakest fiber leads to the failure of the entire system. For $\delta \geq \delta_c$, $P_b = 0$. At $\delta_c, P_b \sim L^{-\eta}$, with $\eta \approx 1/3$, where $L$ is the size of the system. As $\delta \rightarrow \delta_c$, the relaxation time $\tau$ diverges obeying the finite size scaling law: $\tau \sim L^{\beta}(|\delta-\delta_c| L^{\alpha})$ with $\alpha, \beta = 0.33 \pm 0.05$. At $\delta_c$, the system fails, at the critical load, in avalanches (of rupturing fibers) of all sizes $s$ following the distribution $P(s) \sim s^{-\kappa}$, with $\kappa = 0.50 \pm 0.01$. We relate this critical behavior to brittle to quasi-brittle transition.