Article,
Fiber bundle model with highly disordered breaking thresholds
Physical Review E, (Mar 2, 2015)
DOI: 10.1103/PhysRevE.91.032103
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)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/\nu(\beta)\$ where
\$\sigma\_c(\beta)\$ has been obtained analytically as \$\sigma\_c(\beta) =
10^\beta/(2ełn10)\$ for \$\beta\_u = 1/(2łn10)\$, 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/(2łn10)\$;
(iii) the distribution \$D(\Delta)\$ of the avalanches of size \$\Delta\$ follows a
power law \$D(\Delta)\Delta^-\xi\$ with \$= 5/2\$ for \$\Delta \gg
\Delta\_c(\beta)\$ and \$= 3/2\$ for \$\Delta \Delta\_c(\beta)\$, where the
crossover avalanche size \$\Delta\_c(\beta) = 2/(1-e10^-2\beta)^2\$.
