Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory

In this paper, a gradient elasticity shell formulation is presented for free vibration analysis of single-walled carbon nanotube placed on Winkler/Pasternak foundation. The proposed formulation is based on the combined strain-inertia gradient elasticity. The combined strain-inertia gradient elasticity provides an extension to the classical equations of elasticity with additional higher-order spatial derivatives of strains and two material length scale parameters related to the inertia and strain gradients, which enable formulation to investigate the size effect on the dynamic behavior of nanotubes. The effects of the length scale parameters, aspect ratio of single-walled carbon nanotube and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. The natural frequencies obtained from the proposed shell formulation show the effects of size-dependent properties. It can be concluded that a continuum model enriched with higher-order inertia terms has been proposed as alternative to the continuum description obtained with classical elasticity theory.