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 $10^{-\beta }$ 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 ${\sigma }_c(\beta ,N)={\sigma }_c(\beta )+A{N}^{-1/\nu (\beta )}$ where ${\sigma }_c(\beta )$ has been obtained analytically as ${\sigma }_c(\beta )=10^{\beta }/(2\beta e\ln 10)$ for $\beta \ge {\beta }_u=1/(2\ln 10)$, 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 $1-1/(2\beta \ln 10)$; (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 ${\Delta }_c(\beta )=2/(1-e10^{-2\beta }{)}^2$.

7 pages, 7 figures

References in corpus (4)

• Universality class of fiber bundles with strong heterogeneities
• Brittle to Ductile Transition in a Fiber Bundle with Strong Heterogeneity
• Scaling forms for Relaxation Times of the Fiber Bundle model
• A Fiber Bundle Model of Traffic Jams

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• Inequality of avalanche sizes in models of fracture
• Fiber bundle model under heterogeneous loading
• Failure process of fiber bundles with random misalignment
• Stability in fiber bundle model : Existence of strong links and the effect of disorder
• Impact induced transition from damage to perforation