Surface and crack effects on longitudinal vibration of thick nanorods with various boundary conditions

Considering the effect of surface parameters and crack on the longitudinal vibration of nanorods is the aim of this study. The nanorod is assumed to be thick, and it is modeled based on Bishop theory of rods. In thick nanorods, the effects of the inertia of lateral motions and the shear stiffness are considered in the kinetic energy and the strain energy of the nanorod, respectively. In the present study, the crack is modeled as a linear spring in which its stiffness is related to the crack severity. The surface parameters considered are the surface Lamé constants, the surface density, and the surface stress. Considering all surface parameters causes inhomogeneity of governing equations and boundary conditions. After converting inhomogeneous equations to homogeneous ones, the resulted equations are solved numerically. All possible boundary conditions are introduced but only natural frequencies of nanorod with fixed-fixed and fixed-free are reported. In addition to considering the effect of type of boundary conditions on the natural frequencies, the effect of geometrical parameters like radius and length, and mode number is also considered. The results are presented for both thin and thick nanorods to compare the effect of various parameters especially surface parameters on longitudinal frequencies.