Breakdown of two-phase random resistor networks

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Abstract
We describe the failure of two-component square- and cubic-lattice random resistor networks. The model behavior is dependent on the ratio of the conductances of the two components g, the ratio of the (brittle) failure thresholds of the two components i, the volume fraction p, and the sample size L. For much of the parameter space, the average strength of the networks shows a rather weak size effect, and a scaling argument suggests that this size effect is logarithmic. As usual, near the percolation points, there can be algebraic scaling provided i and g are very large or small. Near the limits p=0 and p=1, there is a logarithmic (‘‘dilute-limit’’) singularity in average strength. The ability to absorb damage is very strongly dependent on the model parameters. When one phase is more conducting and weaker than the other, and the strong phase is connected, the damage is usually extensive. Basically most of the weak bonds fail prior to the failure of the whole network. In the other regions of parameter space, damage is not extensive, but it does sometimes scale in a nontrivial way with the sample size.

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