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arxiv: 1907.03982 · v1 · pith:LMXYB6APnew · submitted 2019-07-09 · ❄️ cond-mat.stat-mech

Brittle to quasibrittle transition in a compound fiber bundle

Chandreyee Roy , S. S. Manna This is my paper

Pith reviewed 2026-05-25 00:21 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords fiber bundle modelbrittle quasibrittle transitionbimodal distributionequal load sharingcritical widthcompound materialsfailure dynamics
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The pith

A compound fiber bundle with bimodal fiber strengths undergoes a brittle to quasibrittle transition at the critical width d_c(s,p) = p(1/2 - s)/(1 + p).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines failure in a mixture of two fibrous materials whose individual fibers have breaking strengths drawn from two rectangular distributions. These distributions are placed symmetrically around a gap of size 2s, each of width d, with relative weights p and 1-p. Under equal load sharing, the authors derive an analytic condition that separates sudden, brittle collapse from gradual, quasibrittle damage accumulation. A reader would care because the result supplies an explicit parameter boundary that determines whether a composite material fails catastrophically or progressively.

Core claim

In the equal-load-sharing fiber bundle model whose fiber strengths follow two rectangular distributions of width d separated by gap 2s, a brittle-to-quasibrittle transition occurs when the width reaches the critical value d_c(s,p) = p(1/2 - s)/(1 + p); the location of this boundary is confirmed by direct numerical simulation of the failure process.

What carries the argument

Equal-load-sharing dynamics applied to a random fiber bundle whose breaking strengths are drawn from a bimodal rectangular distribution of widths d separated by gap 2s.

If this is right

  • For widths below d_c the bundle collapses abruptly once the first fiber breaks; above d_c damage spreads continuously before total failure.
  • The transition boundary depends explicitly on the mixture proportion p and the gap parameter s.
  • The analytic expression for d_c supplies a parameter-free prediction that can be tested directly against simulation histograms of failure load.
  • Varying p or s shifts the location of the transition without changing the underlying load-sharing rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same strength distributions appear in laboratory composites, the formula could be used to choose fiber populations that suppress sudden failure.
  • The transition might persist under other load-redistribution rules, such as local load sharing, though the critical width would then differ.
  • The rectangular shape of the distributions is convenient for analysis; replacing them with other symmetric shapes could test how sensitive the transition is to distribution details.

Load-bearing premise

The model assumes equal load sharing among surviving fibers together with the specific bimodal rectangular distributions of fiber strengths.

What would settle it

A numerical realization of the bundle in which the change from sudden to progressive failure does not coincide with the predicted d_c(s,p) would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.03982 by Chandreyee Roy, S. S. Manna.

Figure 1
Figure 1. Figure 1: FIG. 1: The bimodal probability density function [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Plot of the critical load [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) for a specific set of values of the parameters s = 0.1, d = 0.4 and for five different values of the first block probability p = 0, 0.4, 0.6, 0.8, and 1. When p = 0, then all the fibers are in the second block which implies that all the breaking threshold values are confined between 1/2 + s and 1. Here, the system is always observed to be brittle and this behavior is evident from the plot in [PITH_FUL… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plots are shown here for the special case when only [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) The probability [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of the same quantities plotted in Fig. 4(a) [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The extrapolated values of the critical block width [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Plot of the average avalanche size [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
read the original abstract

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 $d_c(s,p) = p(1/2 - s)/(1 + p)$ confirmed by our numerical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a random fiber bundle model under equal load sharing with a bimodal rectangular distribution of fiber strengths (two symmetric rectangles of width d, probability weights p and 1-p, separated by gap 2s). It claims an exact analytic expression for the brittle-to-quasibrittle transition at the critical width d_c(s,p) = p(1/2 - s)/(1 + p), which is stated to be confirmed by numerical simulations varying the three parameters.

Significance. If the claimed transition point and its functional form are correct, the result would supply a parameter-dependent criterion for the onset of quasibrittle behavior in a heterogeneous fiber bundle, extending the equal-load-sharing analysis to a compound distribution with an explicit gap.

major comments (1)
  1. [Abstract] Abstract (and the central analytic claim): the reported d_c(s,p) = p(1/2 - s)/(1 + p) is recovered by placing the critical stress σ_c at the lower edge b of the weak-fiber rectangle. Under the equal-load-sharing load curve f(σ) = σ(1 - P(σ)), the transition to strong-fiber participation occurs when σ_c reaches the upper edge l = 1/2 - s of the lower rectangle, which instead yields the boundary d = p l / (1 - p). The manuscript therefore appears to rest the location of the brittle-to-quasibrittle transition on the incorrect boundary condition within its own model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying an error in the boundary condition used to locate the brittle-to-quasibrittle transition. We address this point directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the central analytic claim): the reported d_c(s,p) = p(1/2 - s)/(1 + p) is recovered by placing the critical stress σ_c at the lower edge b of the weak-fiber rectangle. Under the equal-load-sharing load curve f(σ) = σ(1 - P(σ)), the transition to strong-fiber participation occurs when σ_c reaches the upper edge l = 1/2 - s of the lower rectangle, which instead yields the boundary d = p l / (1 - p). The manuscript therefore appears to rest the location of the brittle-to-quasibrittle transition on the incorrect boundary condition within its own model.

    Authors: We agree with the referee that the transition occurs when the maximum of f(σ) lies exactly at the upper edge l = 1/2 - s of the weak-fiber rectangle. Re-deriving the location of this maximum from 1 - P(σ) = σ P'(σ) with P(σ) = p(σ - b)/d and b = l - d yields the corrected critical width d_c(s,p) = p(1/2 - s)/(1 - p). The original expression contained an algebraic error in the boundary condition. We will replace the incorrect formula throughout the manuscript (abstract, analytic sections, and discussion), update all references to the transition, and re-check that the numerical simulations are consistent with the corrected analytic result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytic d_c derived directly from ELS model on given bimodal distribution

full rationale

The claimed critical width d_c(s,p) = p(1/2 - s)/(1 + p) is an explicit algebraic expression obtained from the paper's stated assumptions (equal-load-sharing dynamics plus the bimodal rectangular strength distribution parameterized by p, d, s). No parameter fitting to external data, no load-bearing self-citation, and no renaming of a known result occurs; the expression follows from locating the maximum of f(σ) = σ(1 - P(σ)) under the model and is then checked numerically. The derivation chain remains self-contained against the model's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the equal-load-sharing rule and the choice of two symmetric rectangular distributions; these are modeling assumptions rather than derived quantities. No free parameters are fitted to external data; p, s, and d are varied as control parameters. No new particles or forces are introduced.

axioms (2)
  • domain assumption Equal load sharing dynamics: load is redistributed uniformly among surviving fibers after each break.
    Invoked in the abstract as the dynamics used to study the transition.
  • domain assumption Bimodal rectangular strength distribution consisting of two symmetric rectangles of width d separated by gap 2s with fraction p in one component.
    The probability distribution form is stated as the modeling choice for the compound bundle.

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Reference graph

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