  We present a mathematical description of amorphous solid deformation and
plasticity by extending the concept of instantaneous normal modes (INMs) to
deformed systems, which allows us to retain the effect of strain on the
vibrational density of states (VDOS). Starting from the nonaffine lattice
dynamics (NALD) description of elasticity and viscoelasticity of glasses, we
formulate the linear response theory up to large deformations by considering
the strain-dependent tangent modulus at finite values of shear strain. The
(nonaffine) tangent shear modulus is computed from the vibrational density of
states (VDOS) of affinely strained configurations at varying strain values. The
affine strain, found analytically on the static (undeformed) snapshot of the
glass, leads to configurations that are rich of soft low-energy modes as well
as unstable modes (negative eigenvalues) that are otherwise completely "washed
out" and lost, if one lets the system relax after strain. This procedure is
fully consistent with the structure of NALD. The INM spectrum of deformed
states allows for the analytical prediction of the stress-strain curve of a
model glass, including the prediction of the yield point. Good parameter-free
quantitative agreement is shown between the prediction and simulations of
athermal quasi-static shear of a coarse-grained polymer glass.
