Size scaling of failure strength with fat-tailed disorder in a fiber bundle model

We investigate the size scaling of the macroscopic fracture strength of heterogeneous materials when microscopic disorder is controlled by fat-tailed distributions. We consider a fiber bundle model where the strength of single fibers is described by a power law distribution over a finite range. Tuning the amount of disorder by varying the power law exponent and the upper cutoff of fibers' strength, in the limit of equal load sharing an astonishing size effect is revealed: For small system sizes the bundle strength increases with the number of fibers, and the usual decreasing size effect of heterogeneous materials is restored only beyond a characteristic size. We show analytically that the extreme order statistics of fibers' strength is responsible for this peculiar behavior. Analyzing the results of computer simulations we deduce a scaling form which describes the dependence of the macroscopic strength of fiber bundles on the parameters of microscopic disorder over the entire range of system sizes.

Figure 1

(a) The macroscopic response $\sigma \left(\varepsilon \right)$ of the system for the same upper cutoff ${\varepsilon }_{\max }=50$ varying the value of the exponent $\mu$ ($\mu$ increases from top to bottom). (b) Constitutive curves for a fixed $\mu =0.7$ exponent varying the upper cutoff ${\varepsilon }_{\max }$ with the multiplication factor $k$ ($k$ increases from bottom to top). Approaching the phase boundary in both cases the system becomes more and more brittle. The curve on the top of (b) corresponds to the case ${\varepsilon }_{\max }\to \infty$. The dashed lines represent the full analytical curves of Eq. (6) while the colored dots show the results of stress-controlled simulations.

Figure 2

Phase diagram of the system. The phase boundary separating the brittle and quasibrittle macroscopic response is given by Eq. (9). Note that for $\mu \ge 1$ the bundle is always in the brittle phase.

Figure 3

(a) The average value of the critical strain $〈{\varepsilon }_c〉$ as a function of the bundle size $N$ for several values of the upper cutoff ${\varepsilon }_{\max }$ of the strength of single fibers. The horizontal lines represent the corresponding asymptotic strength obtained from Eq. (7). The value of the exponent $\mu$ is fixed to $\mu =0.8$. The upper cutoff ${\varepsilon }_{\max }$ is parametrized by $k$ such that the legend is the same as in panel (b). The red bold line gives the analytic curve of Eq. (15). (b) Scaling plot of the data presented in panel (a). After rescaling with the asymptotic strength ${\varepsilon }_c\left(\infty \right)$ we subtracted 1 from the result in order to demonstrate the asymptotic power law behavior. The straight line represents a power law of exponent $-2/3$.

Figure 4

Inset: The average fracture stress $〈{\sigma }_c〉\left(N\right)$ as a function of the number of fibers $N$ for the same values of the upper cutoff as in Fig. 3 using also the same legend. The bold line represents the curve of Eq. (17). Main panel: Rescaling the two axis of the inset the curves obtained at different cutoff values can be collapsed on a master curve.