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We present a theory of viscoelasticity of amorphous media, which takes intoaccount the effects of confinement along one of three spatial dimensions. Theframework is based on the nonaffine extension of lattice dynamics to amorphoussystems, or nonaffine response theory. The size effects due to the confinementare taken into account via the nonaffine part of the shear storage modulus$G'$. The nonaffine contribution is written as a sum over modes in $k$-space.With a rigorous argument based on the analysis of the $k$-space integral overmodes, it is shown that the confinement size $L$ in one spatial dimension, e.g.the $z$ axis, leads to a infrared cut-off for the modes contributing to thenonaffine (softening) correction to the modulus that scales as $L^{-3}$.Corrections for finite sample size $D$ in the two perpendicular dimensionsscale as $\sim (L/D)^4$, and are negligible for $L \ll D$. For liquids it ispredicted that $G'\sim L^{-3}$ in agreement with a previous more approximateanalysis, whereas for amorphous materials $G' \sim G'_{bulk} + \beta L^{-3}$.For the case of liquids, four different experimental systems are shown to bevery well described by the $L^{-3}$ law.

Universal $L^{-3}$ finite-size effects in the viscoelasticity of confined amorphous systems