  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.
