Longitudinal vibrations of functionally graded material Nano-rod based on nonlocal strain gradient theory

This paper investigates the free longitudinal vibrations of functionally graded nanorods with constant cross-section on the basis of nonlocal strain gradient theory. Functionally graded nanostructures can be defined as nonhomogeneous composites which are obtained by combining two different materials with different elasticity moduli and mass densities. Due to the special properties of the functionally graded materials (FGM), the changes of physical properties (such as the elasticity modulus and mass density along the axis of functionally graded nanorods) are expressed by an exponential relation in terms of the location and constant ratios of the material properties. In this paper, numerical results related to these changes are shown. Galerkin solution technique is utilized to obtain an approximate solution to the free longitudinal vibration of FGM nanorods with clamped–clamped boundary conditions. A parametric study is carried out to show the influences of nonlocal parameters on the fundamental natural frequencies such as location-dependent nonlocal parameter and material length scale parameter as well as the physical parameters such as the elasticity modulus and mass density of different material composition.