  A new microscopic derivation of the elastic constants of amorphous solids is
presented within the framework of nonaffine lattice dynamics, which makes use
of a perturbative form of the low-frequency eigenvectors of the dynamical
matrix introduced in [V. Mazzacurati, G. Ruocco, M. Sampoli EPL 34, 681
(1996)]. The theory correctly recovers the shear modulus at jamming, $\mu \sim
(z-2d)$, including prefactors in quantitative agreement with simulations.
Furthermore, this framework allows us, for the first time, to include the
effect of internal stresses. The theory shows that the Maxwell rigidity
criterion $z=2d$ is violated with internal stress. In particular, $\mu \sim
(z-2df)$ where $f<1$ if the bonds are, on average, stretched, and the solid is
thus rigid below the Maxwell isostatic limit, while $f>1$ if the bonds are, on
average, compressed. The coefficient $f$ is derived in analytical form and
depends only on $d$ and on the average particle displacement from the
interaction energy minimum.
