Scaling forms for relaxation times of the fiber bundle model

Roy, Chandreyee; Kundu, Sumanta; Manna, S. S.

doi:10.1103/physreve.87.062137

Cited by 22 publications

(23 citation statements)

References 17 publications

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“…As an effect the strength of the system will decrease. For α = 1 the behavior of this model is same as the mean field version of the Fiber Bundle model [12,13,14], the elements break one by one with increasing sequence of their breaking thresholds.…”

Section: Discussionmentioning

confidence: 99%

A simple discrete-element-model of Brazilian test

2016

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We present a statistical model which is able to capture some interesting features exhibited in the Brazilian test of rock samples. The model is based on elements which break irreversibly when the force experienced by the elements exceed their own load capacity. If an element breaks the load capacity of the neighboring elements are decreased by a certain amount, assuming weakening effect around the defected zone. From the model we numerically investigate the stress-strain behavior, the strength of the system, how it scales with the system size and also it's fluctuation for both uniform and Weibull distribution of breaking thresholds in the system. To check the validity of our statistical model we perform few Brazilian tests on Sandstone and Chalk samples. The stress-strain curve from model results agree qualitatively well with the lab-test data. Also, the damage profile right at the point when the stress-strain curve reaches it's maximum is seen to mimic the crack patterns observed in our Brazilian test experiments.

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Section: Discussionmentioning

confidence: 99%

A simple discrete-element-model of Brazilian test

2016

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“…It is known that at the critical point, the relaxation time scales as τ ∼ N α , with α = 1/3 for b = 0 [25,29]. But for b > 0 the power law scaling seems to be lost, with τ becoming almost independent of N for b > 1 (see Fig.…”

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Maximizing the Strength of Fiber Bundles under Uniform Loading

Biswas

Sen

2015

Phys. Rev. Lett.

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The collective strength of a system of fibers, each having a failure threshold drawn randomly from a distribution, indicates the maximum load carrying capacity of different disordered systems ranging from disordered solids, power-grid networks, to traffic in a parallel system of roads. In many of the cases where the redistribution of load following a local failure can be controlled, it is a natural requirement to find the most efficient redistribution scheme, i.e., following which system can carry the maximum load. We address the question here and find that the answer depends on the mode of loading. We analytically find the maximum strength and corresponding redistribution schemes for sudden and quasi static loading. The associated phase transition from partial to total failure by increasing the load has been studied. The universality class is found to be dependent on the redistribution mechanism.The response of an ensemble of elements having random failure thresholds plays a crucial role in the breakdown properties of heterogeneous systems [1]. Such systems can be, for example, disordered solids under stress, power-grid networks carrying currents, roads carrying car traffic, or redundant computer circuitry etc. The failure properties of these systems are strongly dependent upon the load redistribution mechanism following a local breakdown. While in some cases (e.g. solids under stress) such mechanisms are properties inherent to the system, in many other cases (e.g. power-grids [2], traffic controls [3]) it is a matter of design that can be optimized so as to achieve the most robust configuration (see e.g. [4]). Such robustness properties of networks connecting elements with varying failure thresholds under targeted or random attacks have received substantial attention [5,6].In this Letter, we address the question of maximizing the strength of a disordered system by finding the most effective redistribution scheme analytically using the fiber bundle model as a generic example. This model was introduced [7], indeed, to estimate the strength of cotton in textile engineering. Since then, it has been extensively studied in the context of distribution of failure strength and failure time of disordered materials under tensile loading or twist by viewing fibers as elements of the disordered solids having a finite failure threshold (or even a finite lifetime dependent of loading) [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] (see [23] for a review).Conventionally, the fiber bundle model is viewed as a set of parallel fibers, having failure thresholds randomly drawn from a distribution (say, uniform in [0 : 1]), clamped between two horizontal plates. When the bottom plate is loaded, some of the weak fibers break, and their load is redistributed among the surviving fibers, which may in turn break or survive depending on their failure thresholds. The load transfer rule, generally a function of distance from the broken fiber, plays a vital role (see e.g., [24,25]). While substantial attention has been concentra...

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“…We now investigate the dependence of the finite size correction exponent ν(β) on the cut-off parameter β. We recall that in the case of a uniform breaking threshold distribution, the plot of σ c (N )−σ c as a function of N −1/ν gives an excellent straight line with σ c = 1/4 and ν = 3/2 [24][25][26][27]. Similarly, for our model of highly disordered FBM, the plot of σ c (β, N ) − σ c (β) against N −1/ν(β) is carried out for different values of β.…”

Section: Highly Disordered Fiber Bundlesmentioning

confidence: 99%

Fiber bundle model with highly disordered breaking thresholds

2015

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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 .

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Scaling forms for relaxation times of the fiber bundle model

Cited by 22 publications

References 17 publications

A simple discrete-element-model of Brazilian test

A simple discrete-element-model of Brazilian test

Maximizing the Strength of Fiber Bundles under Uniform Loading

Fiber bundle model with highly disordered breaking thresholds

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