Brittle-to-quasibrittle transition in bundles of nonlinear elastic fibers

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 $s=\mathcal{G}\left(x\right)$ till its breaking threshold. In particular, four different functional forms have been studied: $\mathcal{G}\left(x\right)=e^{\alpha x}, 1+x^{\alpha }, x^{\alpha }$, and $x{e}^{\alpha x}$ where $\alpha$ 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 ${\alpha }_c$ 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 ${\sigma }_c\left(\alpha \right)$ for the global failure of the bundle depends explicitly on $\alpha$ 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.

Figure 1

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) Plot of the critical value of ${\sigma }_c(\alpha ,N)-{\sigma }_c\left(\alpha \right)$ with respect to $N^{-0.665}$ with $\alpha =2$ for $N=253$ to 31 623. The least square fit line has the form ${\sigma }_c(\alpha ,N)-1.3591=2.0733N^{-0.665}$. Therefore, ${\sigma }_c\left(\alpha \right)=1.3591$ which is very close to its analytical value $e/2$ in Eq. (7) for $\alpha =2$. (b) Plot of the variation of ${\sigma }_c\left(\alpha \right)$ against $\alpha$. The solid line represents the analytically obtained expression given by Eq. (7), and the circles represent the numerical data, which match well the analytical values.

Figure 2

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) Plot of the probability of brittle bundles $P(\alpha ,N)$ against $\alpha$ for $N=2^8$ (black), $2^{10}$ (red), $2^{14}$ (blue), and $2^{26}$ (magenta) ($N$ increases from right to left). (b) Plot of ${\alpha }_c\left(N\right)-{\alpha }_c$, with ${\alpha }_c=1$ against $N^{-0.333}$. The least square fitted straight line misses the origin very closely.

Figure 3

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) Plot of the average avalanche size ${\mathrm{\Delta }}_m(\alpha ,N)$ against $\alpha$ for $N=2^{16}$ (black), $2^{18}$ (red), and $2^{20}$ (blue) with $N$ increasing from bottom to top. The values of $\alpha$ for the maximum value of ${\mathrm{\Delta }}_m(\alpha ,N)$ defined in Eq. (15) are $1.030,1.021$, and 1.011, respectively, leading to ${\alpha }_c=1$ as $N\to \infty$. (b) Finite size scaling of the average avalanche size ${\mathrm{\Delta }}_m(\alpha ,N)N^{-0.666}$ against $(\alpha -{\alpha }_c)N^{0.333}$ using the data in (a) exhibits an excellent data collapse.

Figure 4

$\mathcal{G}\left(x\right)=e^{\alpha x}$: Plot of the fraction of broken fibers $f_b(\alpha ,N)$ before the last avalanche against $\alpha$ for $N=2^8$ (black), $2^{10}$ (red), and $2^{14}$ (blue) with $N$ increasing from left to right. The plot with filled circles represents the analytical form given in Eq. (16).

Figure 5

$\mathcal{G}\left(x\right)=e^{\alpha x}$: Subtracting a tunable parameter $a$ from $\mathcal{G}\left(x\right)$ we calculate the critical value ${\alpha }_c\left(a\right)$ of the transition point as a function of $a$. For 11 different equispaced values of $a$, the values of ${\alpha }_c$ are estimated numerically and plotted. These points fit very well with Eq. (20), which describes the boundary between the brittle and the quasibrittle regimes.

Figure 6

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) Plot of the average relaxation time $\langle T(\alpha ,N)\rangle$ against $\mathrm{\Delta }\alpha =\alpha -{\alpha }_c^{\lambda }$. The brittle and the quasibrittle phases correspond to the negative and positive values of $\mathrm{\Delta }\alpha$. The bundle sizes are $N=2^{10}$, (black) $2^{17}$ (blue), and $2^{24}$ (red) ($N$ increases from bottom to top). (b) Variations of the maximal relaxation times $\langle T_b^m(\alpha ,N)\rangle$ (upper curve) and the $\langle T_{qb}^m(\alpha ,N)\rangle$ (lower curve) in the brittle and quasibrittle regions, respectively. The solid lines are the fitted functional forms given in the text.

Figure 7

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) A finite-size scaling of the avalanche size distribution using $D(\mathrm{\Delta },N)N^{\eta }$ against $\mathrm{\Delta }·N^{-\zeta }$ for $\alpha =1$ and $N=2^{12}$ (black), $2^{18}$ (red), and $2^{24}$ (blue). A good collapse of data is obtained with $\eta =1.0$ and $\zeta =0.66$, implying $\tau =\eta /\zeta \approx 1.5$. (b) For a specific bundle size $N=2^{20}$, the avalanche size distribution $D(\mathrm{\Delta },N)$ has been plotted against $\mathrm{\Delta }$ for $\alpha =1.05$ (black), 1.1 (red), 1.2 (blue), 1.4 (green), and 2.0 (magenta) (from top to bottom). As $\alpha$ approaches unity, the crossover avalanche size gradually diverges.

Figure 8

$\mathcal{G}\left(x\right)=e^{\alpha x}$: (a) Plot of the critical width ${\delta }_c\left(\alpha \right)$ against $\alpha$ using open circles. The corresponding analytical form in Eq. (23) has been shown in black solid line. The matching is quite good. (b) For $\alpha ={\alpha }_c=1$ and for ${\delta }_c=1/2$, the probability of occurrence of a brittle bundle has been plotted against $N^{-0.333}$, and a very good straight line fit has been obtained. On extrapolation, this line passes very close to the origin.

Figure 9

Case I: Probability $P(\alpha ,N)$ of finding an arbitrary fiber bundle of size $N$ in the brittle phase for a specific value of $\alpha$. The bundle sizes are $N=2^{10}$ (black), $2^{12}$ (red), $2^{14}$ (green), and $2^{16}$ (blue), and the distribution becomes increasingly sharper with increasing $N$ at the critical value of ${\alpha }_c=1$.

Figure 10

Case I: (a) Numerically obtained values of the asymptotic critical load ${\sigma }_c\left(\alpha \right)$ has been plotted against $\alpha$. It gradually decreases to a value of ${\sigma }_c\left(\alpha \right)=1$. (b) Fraction $f_b(\alpha ,N)$ of broken fibers before the last avalanche has been plotted against $\alpha$ for the bundle sizes $N=2^8$ (black), $2^{10}$ (red), $2^{12}$ (green), and $2^{14}$ (blue), and the curves become increasingly sharper with increasing $N$ at the critical value of ${\alpha }_c=1$.