First-order rigidity transition and multiple stability regimes for random networks with internal stresses

By applying effective medium-style calculations to random spring networks, we demonstrate that internal stresses fundamentally alter the nature of the rigidity transition in disordered materials, changing it from continuous to first order and increasing the mean coordination number z at which rigidity first occurs. Furthermore, we predict the existence of a novel stability regime at low z when the distribution of stresses is asymmetric. Means of verifying these predictions are suggested.