Competing surface and geometric effects in the size-dependent vibrational response of hollow silicon nanobeams

Classical continuum theories neglect surface-dominated effects that govern the mechanical response of nanostructures at high surface-to-volume ratios (SVR). Using molecular dynamics simulations, this work examines the size-dependent free transverse vibration of solid and hollow silicon nanobeams and shows that hollow geometries exhibit a non-monotonic dependence of natural frequency on cavity size, in contrast to the monotonic surface-induced softening displayed by solid nanobeams. For small cavities, geometric stiffening from the redistribution of material away from the neutral axis outweighs surface softening, increasing the frequency despite a rising SVR; as the cavity enlarges and the walls thin, the negative surface elastic constants of silicon dominate, producing a pronounced frequency drop. This competition is shown to be quantitatively consistent with an extension of the Miller–Shenoy surface-elasticity framework to hollow cross-sections, and is distilled into a minimal two-term dimensionless scaling law that balances a geometric stiffening term against a surface-softening term governed by the ratio of an intrinsic material length scale to the beam's outer dimension. The scaling law predicts the cavity ratio at which the stiffening-to-softening transition occurs, and the molecular dynamics results show this transition at a smaller cavity ratio than the continuum prediction, pointing to an additional, cross-section-dependent atomistic contribution to surface softening not captured by the linear continuum model. These results clarify the competing mechanisms governing the vibrational response of hollow nanobeams and provide a predictive, extensible framework connecting molecular-scale surface elasticity to continuum-level nanoscale design.