Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects

Abstract

We establish the physics and understanding of nonlocal nanoscale wave propagation in carbon nanotubes (CNTs) based on nonlocal elastic stress field theory. This is done by developing an analytical nonlocal nanotube model based on the variational principle for wave propagation in CNTs. Specifically, we successfully derive benchmark governing equations of motion for analyzing wave propagation based on an analytical nonlocal shear deformable model. The physical insights of the analytical nonlocal stress model are presented through examples. Analytical solutions with significant observation of wave propagation have been predicted and the prediction compares favorably with molecular dynamic simulations. Qualitative comparisons with other non-nonlocal approaches, including the strain gradients model, the couple stress model and experiments, justify the stiffness enhancement conclusion as predicted by the new nonlocal stress model. New dispersion and spectrum relations derived using this analytical nonlocal model bring an important focus onto the critical wavenumber: stiffness of CNTs and wave propagation are enhanced below the critical wavenumber, while beyond that a sharp decrease in wave propagation is observed. The physics of nanoscale wave propagation in nanotubes are further illustrated by relating the nanoscale and the phase velocity ratio.

Higher order nonlocal strain gradient approach for wave characteristics of carbon nanorod

“…The nonlocal stress depends not only on the strain at that location but also on the strain at all other points within the domain in a diminishing influence away from the reference location. The nonlocal elastic field theory for homogeneous and isotropic solids is described using the following basic equations [11,12,14]:…”

“…It is interesting to observe that the second order strain gradient model (11) and sixth order strain gradient model (12) gives the critical wave number as…”

“…This theory is based on c Vilnius University, 2014 the atomic theory of lattice dynamics and on phonon dispersions. Recently several authors [10][11][12][13][14][15] worked on the wave propagation in carbon nanotubes using strain gradient models. The effect of nanoscale on the frequency, phase velocity and group velocities and several other properties have been studied in certain special strain gradient models.…”

Nonlocal modeling of a Carbon Nanotube actuated by an electrostatic force

“…Numerous studies investigated the linear free and vibration and wave propagation of CNTs [6][7][8][9]. Thongyothee et al [6] investigated the free vibration problem of CNTs including the effect of nonlocal elasticity to study the effect of their chirality and various boundary conditions.…”

“…They accounted for the van der Waals forces between the inner and outer nanotubes. Lim and Yang [8] discussed the physics a e-mail: houakad@kfupm.edu.sa and understanding of nonlocal nanoscale wave propagation in CNTs based on nonlocal elastic stress field theory. In this regards, they developed an analytical nonlocal shear deformable nanobeam model based on the variational principle for wave propagation in CNTs.…”