The paper presents the study on the flexural wave propagation in a single-walled carbon nanotube through the use of the continuum mechanics and the molecular dynamics simulation based on the Terroff-Brenner potential. The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the model of a traditional Timoshenko beam, but also by the nonlocal elasticity characterizing the microstructure of carbon nanotube in a wide frequency range up to THz. For this purpose, the paper starts with the dynamic equation of a generalized Timoshenko beam made of the nonlocal elastic material, and then gives the dispersion relations of the flexural wave in the nonlocal elastic Timoshenko beam, the traditional Timoshenko beam and the Euler beam, respectively. Afterwards, it presents the molecular dynamics simulations for the flexural wave propagation in an armchair (5,5) and an armchair (10,10) single-walled carbon nanotubes for a wide range of wave numbers. The simulation results show that the Euler beam holds for describing the dispersion of flexural waves in the two single-walled carbon nanotubes only when the wave number is small. The Timoshenko beam provides a better prediction for the dispersion of flexural waves in the two single-walled carbon nanotubes when the wave number becomes a little bit large. Only the nonlocal elastic Timoshenko beam is able to predict the decrease of phase velocity when the wave number is so large that the microstructure of carbon nanotubes has a significant influence on the flexural wave dispersion.

Mechanical metamaterials can be used to control elastic waves, but it is challenging to obtain multiple or ultrawide bandgaps. A one-dimensional simple periodic system with multi-resonator unit cells can achieve multiple locally resonant bandgaps. A unit cell that comprises multiple cells is called a hybrid unit cell. Two different metamaterials with hybrid unit cells are proposed to achieve a wider coupled bandgap. The first type of metamaterial with a hybrid unit cell comprises two simple cells that have different bandgaps connected by a spring. A new Bragg bandgap appears near the locally resonant bandgaps. By adjusting the spring stiffness, these two types of bandgaps can be coupled to achieve an ultrawide coupled bandgap in a lower frequency range. The second type of metamaterial with a hybrid unit cell comprises two different sub-periodic systems. The bandgaps can be combined to achieve a wider bandgap. With this hybrid periodic system, a wider bandgap can be achieved by designing sub-periodic systems with different bandgaps. In addition, the transmission of a finite periodic system is calculated by the transfer-coefficient method, and the results show that elastic waves can be suppressed in wider frequency range in hybrid periodic systems. This paper provides new ways to design metamaterials with wider and lower bandgaps, which can be used for wide and low-frequency vibration isolation in engineering applications.