▎ 摘　　要

The thermo-mechanical vibration of a single-layer graphene sheet (SLGP) and a single-walled carbon nanotube (SWCNT) on a substrate is studied by using a nonlocal elastic plate model and two nonlocal elastic beam models (including Timoshenko-beam model and Euler-beam model) with quantum effects, respectively. The effect of the van der Waals (vdW) interactions between the SLGP (or the SWCNT) and the substrate on the vibration is obtained. Checking against our molecular dynamics simulations shows that the present models are reasonable. In particular, the radial vibration of the SWCNT on the substrate with quantum effects is further derived through the continuum shell model due to the different vdW forces on each point of the SWCNT circumference. The present models show that the normalized transverse frequency decreases and the normalized radial frequency increases with increasing SWCNT radius, respectively. The radial amplitude of the SWCNT (or the amplitude of the SLGP) nonlinearly increases and the transverse amplitude of the SWCNT linearly increases with increasing temperature for a given distance, respectively. The obtained analytical solution should be of great importance for understanding the thermo-mechanical vibration of nanoelectronic devices on a substrate. Published by AIP Publishing.