Percolation of elastic networks under tension

We examine the elastic properties of random networks of Hooke springs under a tension supplied by a frame. As an illustration of the general ideas we use the triangular net with a fraction p of the nearest-neighbor bonds present. When there is no tension, the transition takes place at the threshold for rigidity percolation pcen(2/3. When the tension is very large (or equivalently the natural length of the springs is zero) the transition takes place at the familiar connectivity percolation pc(1/3. As the tension is varied, the phase boundary evolves smoothly and continuously between these two limits. We give a general discussion of the elasticity of random networks under tension and present extensive computer simulations for all the relevant quantities on the bond depleted triangular network. The antisymmetric (nonrotationally invariant) terms in the elastic moduli tensor are equal to the tension. In general, the Cauchy relations are not obeyed in these networks. Although these networks are nonlinear, we show that a harmonic approximation can be made that leads to a very good effective-medium theory of the phase boundary.