Random network models have recently been developed in the physics literature to explain the strength and size effect in heterogeneous materials. Applications have included the breakdown of random fuse networks, dielectric breakdown and brittle fracture. Unfortunately, conventional scaling approaches of statistical mechanics have yielded incorrect predictions, and new approaches have been proposed which build on field enhancement occurring near the tips of critical, random clusters together with the statistical theory of extremes. New distributions and size scalings for strength have been proposed and supported through Monte Carlo simulation. Here we consider an idealized, one-dimensional model for the failure of such networks where elements of constant strength may be initially present or absent at random. Our idealized rule for local stress redistribution near breaks reflects features we find in a discrete mechanics model that has limiting forms consistent with continuum theories for cracks. We obtain rigorous asymptotic results for the strength distribution and size effect with constants and exponents that are known. The validity of various analytical approximations in the literature is then discussed.

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