The Elastic Behavior of Entropic "Fisherman's Net"

A new formalism is used for a Monte Carlo determination of the elastic constants of a two-dimensional net of fixed connectivity. The net is composed of point-like atoms each of which is tethered to six neighbors by a bond limiting the distance between them to a certain maximal separation, but having zero energy at all smaller lengths. We measure the elastic constants for many values of the ratio $\gamma$ between the maximal and actual extensions of the net. When the net is very stretched ($\gamma\sim 1$), a simple transformation maps the system into a triangular hard disks solid, and we show that the elastic properties of both systems, coincide. We also show that the crossover to a Gaussian elastic behavior, expected for the non-stressed net, occurs when the net is more loose ($\gamma\sim 3$).