First-passage percolation, semi-directed Bernoulli percolation and failure in brittle materials

We present a two-dimensional, quasistatic model of Fracture in disordered brittle materials that contains elements of first-passage percolation, i.e., we use a minimum-energy-consumption criterion For the fracture path. The first-passage model is employed in conjunction with a "semi-directed" Bernoulli percolation model, for which we calculate critical properties such as the correlation length exponent v(sdir) and the percolation threshold p(c)(sdir). Among other results, our numerics suggest that v(sdir) is exactly 3/2, which lies between the corresponding known values in the literature for usual and directed Bernoulli percolation. We also iind that the well-known scaling relation between the "wandering" and energy fluctuation exponents breaks down in the vicinity of the threshold for semi-directed percolation. For a restricted class of materials, we study the dependence of the fracture energy (toughness) on the width of the distribution of the specific Fracture energy and find that it is quadratic in the width for small widths for two different random fields, suggesting that this dependence may be universal.