  Viscoelasticity has been described since the time of Maxwell as an
interpolation of purely viscous and purely elastic response, but its
microscopic atomic-level mechanism in solids has remained elusive. We studied
three model disordered solids: a random lattice, the bond-depleted fcc lattice,
and the fcc lattice with vacancies. Within the harmonic approximation for
central-force lattices, we applied sum-rules for viscoelastic response derived
on the basis of non-affine atomic motions. The latter motions are a direct
result of local structural disorder, and in particular, of the lack of
inversion-symmetry in disordered lattices. By defining a suitable quantitative
and general atomic-level measure of nonaffinity and inversion-symmetry, we show
that the viscoelastic responses of all three systems collapse onto a master
curve upon normalizing by the overall strength of inversion-symmetry breaking
in each system. Close to the isostatic point for central-force lattices,
power-law creep $G(t)\sim t^{-1/2}$ emerges as a consequence of the interplay
between soft vibrational modes and non-affine dynamics, and various analytical
scalings, supported by numerical calculations, are predicted by the theory.
