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
z: a floppy network has
z < 0, while a stiff network has
z > 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
z and derive these theoretically, thus allowing us to predict aspects of the
mechanical response of glasses and fibrous networks.