Free torsional voting analysis of nanospheres thanks to the Gurtin surface elasticity model

Abstract

The analytical solution of torsion-free vibrations of spherical nanoparticles using the sensitive three-dimensional elasticity theory with the Gortin-Murdoch model for import has been studied for its surface effects. To obtain the equations of motion equations and Helmholtz wave equations using the isolation material environment is written for navir are converted to vector equations, and then torsion for the movement, the assumptions used, SPHERE, spherical vector wave equations in the coordinate system of the stress tensor and the displacement field is solved correctly assume that S has been removed. Below, using the Gortin-Murdoch theory, the effects of surface energy, which in some way represents the nanoscale for the sphere, the problem boundary conditions are introduced. Finally, the frequency characteristic equation is deduced by applying boundary conditions. Various crystallographic aspects of taking the surface of the nano-sphere from aluminum and two types of nano-sphere have been investigated in various numerical examples to influence the surface energy and especially the size of the inner radius, the normal torsion frequencies of the system are shown. It was observed for an aluminum nanosphere with a size of less than 50 Nm. The effects of surface energy on the natural frequency are noticeable.