Size Effect on Nanoscale Phononic Crystal with Pure Differential Model

The nonlocal elastic theory (NLE) has been widely applied to study the nanoscale phonon crystals (PnCs) owing to its simplicity and concision in calculating the size effect. Nevertheless, we find that the widely used integral-differential models (IDMs) can lead to pseudo bandgap and other defects because of a not rigorous mathematical derivation. Consequently, we introduce a pure differential model (PDM) to overcome the defects. The results of PDM agree well with those of the finite element method (FEM) and the first principle. Based on PDM, the size effect on the band structure and transmission is investigated in detail, and it is found that the size effect reduces the center frequency and narrows bandwidth, but enhances the attenuation of the bandgaps. A size effect factor is introduced for the first time to characterize the dense band zones (DBZs) and the cut-off frequencies induced by the NLE. The numerical results show that the size effect factor can fully predict the behavior of the cut-off frequencies and the DBZs. The study may further improve the theories related to nonlocality, such as NLE, nonlocal piezoelectric theories, and nonlocal strain gradient theory. It may provide a theoretical guidance for the design and applications of the novel nanoscale wave devices.