APPROXIMATIONS FOR THE STRENGTH DISTRIBUTION AND SIZE EFFECT IN AN IDEALIZED LATTICE MODEL OF MATERIAL BREAKDOWN

VARIOUS random network models have been developed recently to explain certain features of fracture development in materials, including the character of 'cracks', the form of the strength distribution, the size effect, and the connection to percolation theory. Applications include fibrous composites, random fuse networks, superconducting networks, dielectrics and elastic lattices. Because of extreme analytical difficulties researchers have relied on Monte Carlo simulation to validate various scaling hypotheses and approximations. Since only small network sizes are presently accessible, features which eventually emerge at the largest scale may not be uncovered. To shed light on this issue we consider a simple, idealized model where elements have strength zero or one with probabilities alpha or 1 - alpha, respectively. The load of a failed element is redistributed equally onto the nearest surviving neighbors, and open boundary conditions are considered for simplicity of calculation. Various exact, asymptotic and numerical results are obtained including a careful evaluation of any errors. Features of the results are in conflict with some of those in the literature for more complex stress redistribution situations. For alpha close to one, the mean strength of the network is dominated by small-scale and boundary effects which may persist up to relatively large network sizes (1000 x 1000) before large-scale effects ultimately dominate.