Effective force constant for a central-force random network.

Abstract

Previous numerical results for the effective force constant of a two-dimensional nearest-neighbor central-force random network with a broad distribution of force constants are reevaluated by applying a well-known percolation argument developed by Ambegaokar, Halperin, and Langer [Phys. Rev. B 4, 2612 (1971)]. This argument, which was constructed for networks of conductors, is shown to give excellent agreement with the numerical results for the elastic properties. Single-bond effective-medium theory is analytically shown to agree closely with the prediction of this argument in the limit ${\ensuremath{\alpha}}_{2}$\ensuremath{\gg}${\ensuremath{\alpha}}_{1}$, where the network's force constants are evenly distributed between ${\ensuremath{\alpha}}_{1}$ and ${\ensuremath{\alpha}}_{2}$. However, the physical picture that underlies the Ambegaokar-Halperin-Langer argument is not correct for this model, because the backbone in rigidity percolation is quite different from that in ordinary connectivity percolation.