Criticality in Fiber Bundle Model

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 $\delta \ge {\delta }_c$, $P_b=0$. At ${\delta }_c,P_b\sim L^{-\eta }$, with $\eta \approx 1\mathrm{/}3$, where $L$ is the size of the system. As $\delta \rightarrow \delta_c$,therelaxationtime$\tau$ diverges obeying the finite size scaling law: $\tau \sim L^{\beta }(\mathrm{\mid }\delta -{\delta }_c\mathrm{\mid }{L}^{\alpha })$ with $\alpha, \beta = 0.33 \pm 0.05$\mathrm{.}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.