A fascinating topic in the strength of materials is the effect of size on the strength of a system. Typically, doubling the size of a system increases the likelihood of a serious flaw, which weakens the system and prevents the system from doubling its load.

We investigated the breakdown properties of a random resistor-fuse network in which each network element behaves as a linear resistor for a current less than a one amp threshold. Above that threshold, the element irreversibly breaks to become an insulator. Our first contribution was to describe the strength of a crack--a contiguous group of breaks--as it varies with the crack size. Our second contribution was to quantify the likelihood of a more serious flaw as the lattice size increases.

This paper is limited to two-dimensional rectangular lattices in which each element is initially broken (or missing) with a probability of nearly zero. As a result a typical lattice will contain a number of flaws consisting of broken and unbroken elements arranged in some sort of pattern. We showed how the current flow around an arbitrary flaw can be calculated using the current flow around a single dipole current source in a lattice.

Our primary result was that the maximum current enhancement around a crack has an asymptotic upperbound which is proportional to the square root of the length of the crack. To do so, we framed the problem as the expected stopping time of a symmetric random walk with infinite variance, and examined the Green function of the stopped random walk.

Finally, we examined the probabilities of various flaw patterns. We showed that mixed patterns of intact and broken arcs are most likely to lead to system failure under simple spatial correlation conditions, and, using the Chen-Stein method, that the number and location of critical flaws are asymptotically distributed as a Poisson random variable.