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 in the precritical state. The other results are as follows: (i) The critical load σ(c)(N) for the bundle of size N approaches its asymptotic value σ(c)(∞) as σ(c)(N)=σ(c)(∞)+AN(-1/ν). (ii) Right at the critical point the average relaxation time scales with the bundle size N as ~N(η) and this behavior remains valid within a small window of size |Δσ|~N(-ζ) around the critical point. (iii) When 1/N<|Δσ|<100N(-ζ) the finite-size scaling takes the form /N(η)~G[{σ(c)(N)-σ}N(ζ)] so in the limit of N→∞ one has ~(σ-σ(c))(-τ). The high precision of our numerical estimates led us to verify that ν=3/2, conjecture that η=1/3, ζ=2/3, and, therefore, τ=1/2.

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 −β to 10 β . Tuning the value of β 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 σc(β, N ) for the bundle of size N approaches its asymptotic value σc(β) as σc(β, N ) = σc(β)+AN −1/ν(β) where σc(β) has been obtained analytically as σc(β) = 10 β /(2βe ln 10) for β ≥ βu = 1/(2 ln 10), and for β < βu the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to σc(β) = 10 −β ; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form 1 − 1/(2β ln 10); (iii) the distribution D(∆) of the avalanches of size ∆ follows a power law D(∆) ∼ ∆ −ξ with ξ = 5/2 for ∆ ≫ ∆c(β) and ξ = 3/2 for ∆ ≪ ∆c(β), where the crossover avalanche size ∆c(β) = 2/(1−e10 −2β ) 2 .

Desert roses are gypsum crystals that consist of intersecting disks. We determine their geometrical structure using computer assisted tomography. By mapping the geometrical structure onto a graph, the topology of the desert rose is analyzed and compared to a model based on diffusion limited aggregation. By comparing the topology, we find that the model gets a number of the features of the real desert rose right, whereas others do not fit so well.

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to be consisting of two symmetrically placed rectangular probability distributions of strengths p and 1 − p, each of width d, and separated by a gap 2s. Different properties of the transition have been studied varying these three parameters and using the well known equal load sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width dc(s, p) = p(1/2 − s)/(1 + p) confirmed by our numerical results.

Properties of the fiber bundle model have been studied using equal load-sharing dynamics where each fiber obeys a nonlinear stress (s)-strain (x) characteristic function s=G(x) till its breaking threshold. In particular, four different functional forms have been studied: G(x)=e^{αx}, 1+x^{α}, x^{α}, and xe^{αx} where α is a continuously tunable parameter of the model in all cases. Analytical studies, supported by extensive numerical calculations of this model, exhibit a brittle to quasibrittle phase transition at a critical value of α_{c} only in the first two cases. This transition is characterized by the weak power law modulated logarithmic (brittle) and logarithmic (quasibrittle) dependence of the relaxation time on the two sides of the critical point. Moreover, the critical load σ_{c}(α) for the global failure of the bundle depends explicitly on α in all cases. In addition, four more cases have also been studied, where either the nonlinear functional form or the probability distribution of breaking thresholds has been suitably modified. In all these cases similar brittle to quasibrittle transitions have been observed.