Out of plane vibration analysis of single-walled carbon nanotubes using stress and strain gradient theories based on Donnell's thin shell theory

Nowadays, micro/nano materials are widely used in various engineering applications such as nanoelectro- mechanical systems, opto-electronics, nuclear engineering, aerospace engineering, energy storage, civil engineering and etc. recently, different non-classical theories such as the couple stress, the nonlocal elasticity and the strain gradient elasticity theories have been developed to consider the size dependency behavior of the structures in small-scales. In this paper, out of plane vibration analysis of single-walled carbon nanotubes were studied using stress gradient theory, strain gradient theory and the classical theory of elasticity based on the assumptions of the Donnell's thin shell theory.. To determining the free vibration parameters of the carbon nanotube, the kinetic and strain (potential) energies were obtained and will be maximized then using Rayleigh's method the natural frequency was obtained. Natural frequency of the Rayleigh and Love modes of out of plane vibration of the single wall carbon nanotubes out are estimated using stress gradient theory, strain gradient theory and the classical theory of elasticity based on the assumptions of the Donnell's thin shell theory. In order to verify the accuracy and reliability of the present study, the results were obtained in this study were compared and validated with available data in the literature. Using numerical data provided, effect of different parameters including length, thickness and radius of the single-walled carbon nanotubes on the natural frequency of the Rayleigh and Love modes are examined and discussed in detail.