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) similar to 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/upsilon(beta)), where sigma(c) (beta) has been obtained analytically as sigma(c) (beta) = 10(beta)/(2 beta e ln 10) for beta >= 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) similar to Delta(-xi). with xi= 5/2 for Delta >> Delta(c) (beta) and xi = 3/2 for Delta << Delta(c) (beta), where the crossover avalanche size Delta(c) (beta) = 2/(1 - e10(-2 beta))(2).