Avalanche dynamics in hierarchical fiber bundles

Heterogeneous materials are often organized in a hierarchical manner, where a basic unit is repeated over multiple scales. The structure then acquires a self-similar pattern. Examples of such structure are found in various biological and synthetic materials. The hierarchical structure can have significant consequences for the failure strength and the mechanical response of such systems. Here we consider a fiber bundle model with hierarchical structure and study the avalanche dynamics exhibited by the model during the approach to failure. We show that the failure strength of the model generally decreases in a hierarchical structure, as opposed to the situation where no such hierarchy exists. However, we also report a special arrangement of the hierarchy for which the failure threshold could be substantially above that of a nonhierarchical reference structure.

Figure 1

Schematic illustration of a hierarchical fiber bundle.

Figure 2

The critical load of a two-level hierarchical fiber bundle model for fixed system size as a function of the level-one unit size. In the two limiting cases, when the unit size is 1 or equal to the full system size, the critical load is trivially $1/4$, which is the result for an ELS fiber bundle with uniform threshold distribution. In all intermediate cases, the critical load is lower due to confinement-enhanced damage accumulation.

Figure 3

Distribution of loads on elementary fibers in two-level hierarchical fiber bundles at the point of failure, with system size $N=10000$ fixed; distributions are averaged over an ensemble of fiber bundle realizations. The width of the distributions is largest for intermediate $N_1,N_2$ values, where the system is weak.

Figure 4

Distribution of loads on elementary fibers in two-level hierarchical fiber bundles at the point of failure for fixed $N_1$ value ($N_1=2$) and various $N_2$ values. The peak positions remain unchanged at approximately $1/3$ and $2/3$ (see text).

Figure 5

Size distribution of avalanches for the same systems as in Fig. 3. Large avalanches that span multiple units show a power-law decay with exponent value $-5/2$, and smaller avalanches which are confined in one unit show an exponentially truncated power-law decay. The crossover occurs for avalanche sizes approximately equal to the unit size $N_1$ and is marked by a dip in the distribution.

Figure 6

The horizontal line shows the analytical estimate ${\sigma }_c\approx 0.1898$ for the critical load of a three-level hierarchical fiber bundle model where $N_1=2$, $N_2=2$, and $N_3$ is large. The points, joined by a curve, are the simulation results for various values of $N_3$. For large values, the critical load approaches the analytical estimate according to the finite-size scaling relation ${\sigma }_c\left(N_3\right)={\sigma }_c\left(\infty \right)+A{N}_3^{-1/\nu }$. This is verified in the inset plot, and $\nu =3/2$, which is the finite-size scaling exponent of the ELS fiber bundle model [17], gives a reasonable fit.

Figure 7

Distribution of loads on elementary fibers in three-level hierarchical fiber bundle models at the point of failure, system size $N=10000$ fixed, for different values of $N_1,N_2$ and $N_3=N/\left(N_1{N}_2\right)$. The effects of stress confinement are evident for intermediate values of $N_1,N_2$, where the stress distribution is widest.

Figure 8

Size distribution of avalanches for the same systems as in Fig. 7. Large avalanches that span multiple units show a power-law decay with exponent value $-3/2$, and smaller avalanches which are confined in units of lower hierarchy show an exponential decay.

Figure 9

Size distributions of avalanches for a hierarchical structure with two fibers in each unit and multiple levels of hierarchy; the peaks at large avalanche sizes correspond to system-spanning avalanches at the point of failure.

Figure 10

The critical load ${\sigma }_c$, as a function of the partitioning parameter ${\sigma }_l$ in Eq. (14). The maximum strength is obtained for large $m$ values when the groups are well partitioned into almost equal strengths and is substantially higher than the single hierarchy limit $1/4$, which is in turn higher than all other hierarchical arrangements.