Scaling and localization in fracture of disordered central-force spring lattices: comparison with random damage percolation

We analyze statistical and scaling properties of the fracture of two-dimensional (2D) central-force spring lattices with strong disorder by means of computer simulation. We run fracture simulations for two types of boundary conditions and compare the results both with the simulation of random damage percolation on the same lattices and with the analytical scaling relations of percolation theory. We investigate the scaling behavior of the macroscopic failure thresholds, the main features of the developing microscopic cluster statistics and damage pattern, and the roughness scaling of the final crack. Our observations show that simulated fracture has three clearly distinguished regimes. The initial phase displays short-range localization of damage, but it is soon replaced by a regime where damage develops in a uniform manner, qualitatively as in random percolation. Already before the maximum-stress point macroscopic localization and anisotropy come into play, resulting in final crack formation. The data of the second, uniform-damage regime can be fitted consistent with the scaling laws of random percolation. Beyond this regime a clear difference is observed with percolation theory and with earlier results from fuse-network models. Nevertheless, the final-crack roughness is found to scale accurately over at least three decades, with a roughness exponent consistent with limited available data for 2D systems and marginally consistent with the value for 2D percolation in a gradient.