Scaling Shift in Multicracked Fiber Bundles

Bundles of fibers, wires, or filaments are ubiquitous structures in both natural and artificial materials. We investigate the bundle degradation induced by an external damaging action through a theoretical model describing an assembly of parallel fibers, progressively damaged by a random population of cracks. Fibers in our model interact by means of a lateral linear coupling, thus retaining structural integrity even after substantial damage. Monte Carlo simulations of the Young’s modulus degradation for increasing crack density demonstrate a remarkable scaling shift between an exponential and a power-law regime. Analytical solutions of the model confirm this behavior, and provide a thorough understanding of the underlying physics.

Figure 1

Cross-sectional view of the bundle structures. In both the flat (a) and in the circular bundle (b), we considered $M$ fibers with $7\le M\le 19$ (note that $M=7,13,19$ in the circular bundle correspond to exactly one, two, or three shells of neighbors). In (c) the arrangement of $M$ parallel fibers of length $l$ with $N$ breaks at $x_1,\dots ,x_N$ is shown; $E_i$ is the longitudinal (Young) elastic modulus of the $i$th fiber, and $k_{ij}$ is the lateral coupling modulus of the pair $ij$.

Figure 2

Monte Carlo results for $⟨E_{\mathrm{eff}}⟩/E$ versus $N$ in semilogarithmic (main plot) and bilogarithmic (inset) scales. We considered $k{l}^2/E=0.045$ and 13 values of $M$, from 7 to 19.

Figure 3

Monte Carlo results (noisy blue lines) and theoretical interpolation (continuous red lines) for $⟨E_{\mathrm{eff}}⟩/(ME)$ versus $N$ for different values of $k{l}^2/E$ (0.045, 0.08, 0.14, 0.25, 0.45, 0.8, 1.4, 2.5, 4.5, and 8) and $M=19$ (flat bundle).

Figure 4

Numerical determination of the function $\phi (\xi )$ defined in Eq. (8). The black curve (flat bundle) and the red curve (circular bundle) are asymptotically converging to $\phi (\xi )=c/\xi$ with $c=1.45$ (flat bundle) and $c=0.85$ (circular bundle).