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We discuss the cooperative failure dynamics in the Fiber Bundle Model where the individual elements or fibers are Hookean springs, having identical spring constant but different breaking strengths. When the bundle is stressed or strained, especially in the equal-load-sharing scheme, the load supported by the failed fiber gets shared equally by the rest of the surviving fibers. This mean-field type statistical feature (absence of fluctuations) in the load-sharing mechanism helped major analytical developments in the study of breaking dynamics in the model and precise comparisons with simulation results. We intend to present a brief review on these developments.
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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 $tau = 1/2$.
We note that the social inequality, represented by the Lorenz function obtained plotting the fraction of wealth possessed by the faction of people (starting from the poorest in an economy), or the plot or function representing the citation numbers against the respective number of papers by a scientist (starting from the highest cited paper in scientometrics), captured by the corresponding inequality indices (namely the Kolkata $k$ and the Hirsch $h$ indices respectively), are given by the fixed points of these nonlinear functions. It has been shown that under extreme competitions (in the markets or in the universities), the $k$ index approaches to an universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (pre-failure) avalanches (obtainable from ultrasonic emissions), given by their nonlinear size distributions in the Fiber Bundle Models (FBM) of non-brittle materials. We will show how a prior knowledge of this terminal and (almost) universal value of the $k$ index (for a range of values of the Weibull modulus characterizing the disorder, and also for uniformly dispersed disorder, in the FBM) for avalanche distributions (as the failure dynamics progresses) can help predicting the point (stress) or time (for uniform increasing rate of stress) for complete failure of the bundle. This observation has also been complemented by noting a similar (but not identical) behavior of the Hirsch index ($h$), redefined for such avalanche statistics.
We investigate how the dimensionality of the embedding space affects the microscopic crackling dynamics and the macroscopic response of heterogeneous materials. Using a fiber bundle model with localized load sharing computer simulations are performed from 1 to 8 dimensions slowly increasing the external load up to failure. Analyzing the constitutive curve, fracture strength and avalanche statistics of bundles we demonstrate that a gradual crossover emerges from the universality class of localized behavior to the mean field class of fracture as the embedding dimension increases. The evolution between the two universality classes is described by an exponential functional form. Simulations revealed that the average temporal profile of crackling avalanches evolves with the dimensionality of the system from a strongly asymmetric shape to a symmetric parabola characteristic for localized stresses and homogeneous stress fields, respectively.
The present work deals with the behavior of fiber bundle model under heterogeneous loading condition. The model is explored both in the mean-field limit as well as with local stress concentration. In the mean field limit, the failure abruptness decreases with increasing order k of heterogeneous loading. In this limit, a brittle to quasi-brittle transition is observed at a particular strength of disorder which changes with k. On the other hand, the model is hardly affected by such heterogeneity in the limit where local stress concentration plays a crucial role. The continuous limit of the heterogeneous loading is also studied and discussed in this paper. Some of the important results related to fiber bundle model are reviewed and their responses to our new scheme of heterogeneous loading are studied in details. Our findings are universal with respect to the nature of the threshold distribution adopted to assign strength to an individual fiber.
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 $Delta_c(beta) = 2/(1-e10^{-2beta})^2$.
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