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 $p_1$ of nearest-neighbor bonds present and a fraction $p_2$ 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 $p_1$+$p_2$<1 and that the number of zero-frequency modes depends only on $p_1$+$p_2$. We construct flow diagrams to show that a line of fixed points exists when $p_1$+$p_2$=1, along which the ratio of elastic constants attains a universal value that depends on $p_1$ but not on the spring constants. The simulations show no significant deviations from the effective-medium results.

• Received 21 January 1985

DOI:https://doi.org/10.1103/PhysRevB.31.7276