Brittle-to-ductile transition in a fiber bundle with strong heterogeneity

We analyze the failure process of a two-component system with widely different fracture strength in the framework of a fiber bundle model with localized load sharing. A fraction $0\le \alpha \le 1$ of the bundle is strong and it is represented by unbreakable fibers, while fibers of the weak component have randomly distributed failure strength. Computer simulations revealed that there exists a critical composition ${\alpha }_c$ which separates two qualitatively different behaviors: Below the critical point, the failure of the bundle is brittle, characterized by an abrupt damage growth within the breakable part of the system. Above ${\alpha }_c$, however, the macroscopic response becomes ductile, providing stability during the entire breaking process. The transition occurs at an astonishingly low fraction of strong fibers which can have importance for applications. We show that in the ductile phase, the size distribution of breaking bursts has a power law functional form with an exponent $\mu =2$ followed by an exponential cutoff. In the brittle phase, the power law also prevails but with a higher exponent $\mu =\frac{9}{2}$. The transition between the two phases shows analogies to continuous phase transitions. Analyzing the microstructure of the damage, it was found that at the beginning of the fracture process cracks nucleate randomly, while later on growth and coalescence of cracks dominate, which give rise to power law distributed crack sizes.

Figure 1

The constitutive behavior of the two-component bundle for ELS (black continuous lines) and LLS (dashed-dotted lines with different colors). The transition from brittle to ductile behavior can clearly be observed as $\alpha$ increases from bottom to top.

Figure 2

Avalanche size distributions $P\left(\Delta \right)$ at different values of $\alpha$ below and above ${\alpha }_c$. Straight lines with slope $\frac{9}{2}$ and $2.0$ are drawn to guide the eye. The value of $\alpha$ increases from bottom to top in the range $\Delta <10$.

Figure 3

Scaling plot of avalanche size distributions above ${\alpha }_c$ using the average size of the largest avalanche ${\overline{\Delta }}_{\max }$ as scaling variable. An excellent collapse is achieved with the exponents $\xi =1.24$ and $\beta =2.5$. The bold line represents the fit with Eq. (4), where $\mu =1.97$ was obtained. The original distributions $P\left(\Delta \right)$ are presented in the inset where $\alpha$ increases from right to left.

Figure 4

Average size of the largest avalanche ${\overline{\Delta }}_{\max }$ as a function of $\alpha$. The position of the sharp peak ${\alpha }_c$ coincides with the critical value of $\alpha$ we identified based on the constitutive curves. Inset: the same quantity plotted as a function of the distance from the critical point $\alpha -{\alpha }_c$ for $\alpha >{\alpha }_c$. A straight line of slope $1.95$ is drawn to guide the eye.

Figure 5

Snapshots of a damaging bundle of size $L=50$ at four values of $\varepsilon$ during the loading process with $\alpha =0.07$. Blue (gray) and yellow (light gray) colors represent intact weak and strong fibers, respectively, while red (dark gray) stands for the broken ones.

Figure 6

Average cluster size $\langle S_{av}\rangle$ (a) and average number of clusters $\langle n_c\rangle$ (b) as function of $\varepsilon$ for several compositions $\alpha$ of the system. For clarity, the vertical straight lines indicate the position ${\varepsilon }_m^{nc}$ of the maximum of $\langle n_c\rangle$ for a few $\alpha$ values: $0.0,0.13,0.27,0.44$.

Figure 7

The maximum value of the average cluster size ${\langle S_{av}\rangle }^{\max }$ averaged over a large number of samples at a given $\alpha$. Three distinct regimes can clearly be distinguished, which are separated by ${\alpha }_c$ and ${\alpha }^{*}$. The inset presents the same quantity in the range $\alpha >{\alpha }^{*}$ as a function of $p_c-p_w$, where $p_w=1-\alpha$. The straight line has slope $2.32$.

Figure 8

(a) Average number of cracks as a function of $\varepsilon$ rescaled along the horizontal axis with $p_w=1-\alpha$. High quality data collapse can be observed up to the maximum. (b) Cluster size distributions determined at ${\varepsilon }_m^S$ for several compositions $\alpha$.

Figure 9

Snapshots of the cluster structure in systems of size $L=100$ taken at the deformation ${\varepsilon }_m^S$ where the average cluster size $\langle S_{av}\rangle$ has a maximum for different values of the control parameter $\alpha$: (a) 0, (b) $0.08$, (c) $0.35$, (d) $0.4$. The assignment of colors is the same as in Fig. 5. When the system only contains weak fibers (a) all the clusters are small compared to the system size. Above the critical point ${\alpha }_c$, growth and merging result in large cluster sizes which are only limited by the system size and by the underlying lattice structure of the weak and strong components.