Non-local free vibration spectra of nanostructures

Frequency spectra of free standing, traction free sphere-and cube-shaped Si, C, and Ge nanoparticles are investigated using both local and non-local elasticity theories. Different particle sizes, internal material lengths, internal-material-length to particle-size ratios, and weighting factors, are considered. Results confirmed that non-local natural frequencies exhibit a non-linear dependence on particle size, indicating that the continuum frequency scale invariance is not valid at the nanoscale. As the l/R ratio decreases, frequencies increase and approach the classical elasticity values. However, for the present non-local formulation, setting l = 0 alone is not sufficient to recover the local elasticity frequencies, the non-local factor must also approach zero. Compared to local elasticity results, nonlocal frequencies are lower, with reductions of up to 44% observed for large non-local weighting factors. In addition, modal shapes and frequency degeneracy are identical for both approaches. Local elasticity formulations provide accurate estimates for the lowest frequencies when the characteristic particle dimension is at least 50 times larger than the internal material length. Finally, polynomial relations are proposed to estimate non-local frequencies from known local elasticity values.