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A new microscopic derivation of the elastic constants of amorphous solids ispresented within the framework of nonaffine lattice dynamics, which makes useof a perturbative form of the low-frequency eigenvectors of the dynamicalmatrix 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 theeffect of internal stresses. The theory shows that the Maxwell rigiditycriterion $z=2d$ is violated with internal stress. In particular, $\mu \sim(z-2df)$ where $f<1$ if the bonds are, on average, stretched, and the solid isthus rigid below the Maxwell isostatic limit, while $f>1$ if the bonds are, onaverage, compressed. The coefficient $f$ is derived in analytical form anddepends only on $d$ and on the average particle displacement from theinteraction energy minimum.

Theory of elastic constants of athermal amorphous solids with internal stresses