Effective-medium theory of percolation on central-force elastic networks. II. Further results

The effective-medium theory developed in a previous paper for elastic networks with a fraction p of the bonds present is extended to networks which have central forces of arbitrary range. The results are illustrated by studying a square lattice with a fraction p1 of nearest-neighbor bonds present and a fraction p2 of next-nearest-neighbor bonds present. We show that effective-medium theory gives an excellent description of the elastic properties of the networks. An argument using constraints is used to show that the network loses its elastic properties when p1+p2<1 and that the number of zero-frequency modes depends only on p1+p2. We construct flow diagrams to show that a line of fixed points exists when p1+p2=1, along which the ratio of elastic constants attains a universal value that depends on p1 but not on the spring constants. The simulations show no significant deviations from the effective-medium results.