Free torsional vibration analysis of Nano-spherical particle using Gurtn surface elasticity model

The torsional vibrational characteristics of nano-scale sphere using an exact size-dependent elasticity solution based on Gurtin-Murdoch’s surface elasticity model are studied. In the absence of body forces, the displacement field is governed by the classical Navier’s equation. Helmholtz decomposition is used to separate the dynamic equations of motion into the decoupled vector wave equations. The motion under consideration is assumed to be torsional and vector wave equation exactly is solved and displacement field and stress tensor are obtained. Size-dependent elasticity solution based on Gurtin-Murdoch surface energy model is employed to incorporate the surface stress terms into the pertinent boundary conditions, leading to frequency equations involving spherical Bessel functions. Isotropic aluminum with two different set of surface properties corresponding to the crystallography directions are considered and extensive numerical calculations have been carried out to illustrate the size effect of the nano-sphere on the first and second dimensionless vibrational natural frequencies. The numerical results describe the imperative influence of surface energy and radii ratio on vibrational characteristic frequency of nano-sphere. In particular, the surface energy is much important when inner radius is smaller than 50 nm