Local load sharing fiber bundles with a lower cutoff of strength disorder

We study the failure properties of fiber bundles with a finite lower cutoff of the strength disorder varying the range of interaction between the limiting cases of completely global and completely local load sharing. Computer simulations revealed that at any range of load redistribution there exists a critical cutoff strength where the macroscopic response of the bundle becomes perfectly brittle, i.e., linearly elastic behavior is obtained up to global failure, which occurs catastrophically after the breaking of a small number of fibers. As an extension of recent mean field studies [Phys. Rev. Lett. 95, 125501 (2005)], we demonstrate that approaching the critical cutoff, the size distribution of bursts of breaking fibers shows a crossover to a universal power law form with an exponent $3∕2$ independent of the range of interaction.

Figure 1

(a) Failure stress ${\sigma }_c$ and strain ${\epsilon }_c$ of the fiber bundle model at zero cutoff ${\epsilon }_L=0$ compared to the critical cutoff ${\epsilon }_L^c$ as a function of $\gamma$. (b) Constitutive curves for GLS $\gamma =0$ (lines) compared to the case of $\gamma =5$ (triangles) at different values of ${\epsilon }_L$.

Figure 2

(Color online) Characteristic quantities of the failure process of FBM varying the effective range of load sharing $\gamma$ and the cutoff value of failure strength ${\epsilon }_L$: (a) critical deformation ${\epsilon }_c$, (b) critical stress ${\sigma }_c$, (c) $\mid {\epsilon }_c-{\sigma }_c\mid$, (d) the average size of the largest avalanche $⟨{\Delta }_{\max }⟩$ as a function of ${\epsilon }_L$ for various values of $\gamma$.

Figure 3

(Color online) Distribution of burst sizes $D\left(\Delta \right)$ varying ${\epsilon }_L$ at different values of $\gamma$: (a) 2.0, (b) 2.5, (c) 3.0, (d) 6.0. Crossover behavior of $D\left(\Delta \right)$ can be observed as ${\epsilon }_L$ approaches the critical cutoff value ${\epsilon }_L^c\left(\gamma \right)$.

Figure 4

(Color online) The average burst size $⟨\Delta ⟩$ as a function of ${\epsilon }_L$ for different values of $\gamma$. In the short range regime $\gamma >2$, the average avalanche size $⟨\Delta ⟩$ has a clear maximum that coincides with the critical cutoff strength ${\epsilon }_L^c$. Inset, crossover avalanche size ${\Delta }_c$ and GLS analytical solution [13].