Free axial vibration analysis of functionally graded nanorods using surface elasticity theory

In the present paper, free axial vibration behavior of functionally graded nanorods is studied using the surface elasticity theory. For modelling of free axial vibration of nanorods, the Simple theory of rods is implemented. Besides using the Simple theory of rods, the surface elasticity theory is used for considering the surface energy parameters in the governing equations and boundary conditions. The surface energy parameters are the surface elasticity, the surface density, and the surface residual stress. The surface and bulk material properties of nanorod are considered to vary in the length direction according to the power law distribution. Then, the governing equation of motion and boundary conditions of nanorod are derived using the Hamilton’s principle. Due to considering the surface energy parameters, the obtained governing equation of motion becomes non-homogeneous. But in none of the previous researches, for example investigation of free transverse vibration of nanobeams and free torsional vibration of nanorods in presence of the surface energy, the surface energy parameters do not cause the non-homogeneity of the governing equation or the boundary conditions. To extract the natural frequencies of the nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using Galerkin method. In order to have a comprehensive research, effects of various parameters like the length and radius of nanorod on axial frequencies of functionally graded nanorod is investigated.