Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods

Abstract

Because of future promising exploration of nanotechnology, focus is being put in the miniaturization of mechanical and electromechanical devices. Attention is sought toward the development of nanodevices and nanomachines. The length scales associated with nanostructures like are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size, etc. This makes a physically consistent classical continuum model formulation very challenging. So this work presents Eringen's nonlocal elasticity theory, that has been incorporated into classical torsional rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different torsional wave behaviors of nanorods from those of macroscopic rods. Nonlocal torsional rod model is developed for nanorods. Explicit expressions are derived for torsional wavenumbers and wave speeds of nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in torsional wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite or wave speed tends to zero. Next, the Spectral Finite Element formulation of nanorods is performed. The exact frequency dependent shape functions and the dynamic stiffness matrix for the nanorod are obtained as a function of nonlocal scale parameter. It has been found that the nonlocal small scale has significant effect on the exact shape functions and the elements of the dynamic stiffness matrix. These effects are also captured in the present work. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.