Elasticity of Floppy and Stiff Random Networks

We study the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination (average number of springs per node) relative to that of a marginally rigid network deltaz: a floppy network has deltaz<0, while a stiff network has deltaz>0. Under the influence of an externally applied load, we observe that the response of both floppy and stiff networks is controlled by the critical point corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the heterogeneity of the response, and the network stiffening as a function of deltaz and derive these theoretically, thus allowing us to predict aspects of the mechanical response of glasses and fibrous networks.