Thermal vibration analysis of carbon nanotubes embedded in two-parameter elastic foundation based on nonlocal Timoshenko's beam theory

This paper presents a new model to consider the thermal effects, Pasternak’s shear foundation, transverse shear deformation and rotary inertia on vibration analysis of a single-walled carbon nanotube. Nonlocal elasticity theory is implemented to investigate the small-size effect on thermal vibration response of an embedded carbon nanotube. Based on Hamilton’s principle, the governing equations are derived and then solved analytically, in order to determine the nonlocal natural frequencies. Results show that unlike the Pasternak foundation, the influence of Winkler’s constant on nonlocal frequency is negligible for low temperature changes. Moreover, the nonlocal frequencies are always smaller as compared to their local counterparts. In addition, in high shear modulus along with an increase in aspect ratio, the nonlocal frequency decreases.