Fiber Bundle model with Highly Disordered Breaking Thresholds

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 $sigma_c(beta,N) = sigma_c(beta)+AN^{-1/ u(beta)}$ where $sigma_c(beta)$ has been obtained analytically as $sigma_c(beta) = 10^beta/(2beta eln10)$ for $beta geq beta_u = 1/(2ln10)$, 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 $1-1/(2beta ln10)$; (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 $Delta_c(beta) = 2/(1-e10^{-2beta})^2$.

Scaling forms for Relaxation Times of the Fiber Bundle model

Using extensive numerical analysis of the Fiber Bundle Model with Equal Load Sharing dynamics we studied the finite-size scaling forms of the relaxation times against the deviations of applied load per fiber from the critical point. Our most crucial result is we have not found any $ln (N)$ dependence of the average relaxation time $langle T(sigma,N) rangle$ in the precritical state. The other results are: (i) The critical load $sigma_c(N)$ for the bundle of size $N$ approaches its asymptotic value $sigma_c(infty)$ as $sigma_c(N) = sigma_c(infty) + AN^{-1/ u}$. (ii) Right at the critical point the average relaxation time $langle T(sigma_c(N),N) rangle$ scales with the bundle size $N$ as: $langle T(sigma_c(N),N) rangle sim N^{eta}$ and this behavior remains valid within a small window of size $|Delta sigma| sim N^{-zeta}$ around the critical point. (iii) When $1/N < |Delta sigma| < 100N^{-zeta}$ the finite-size scaling takes the form: $langle T(sigma,N) rangle / N^{eta} sim {cal G}[{sigma_c(N)-sigma}N^{zeta}]$ so that in the limit of $N to infty$ one has $langle T(sigma) rangle sim (sigma - sigma_c)^{-tau}$. The high precision of our numerical estimates led us to verify that $ u = 3/2$, conjecture that $eta = 1/3$, $zeta = 2/3$ and therefore $tau = 1/2$.