This paper is concerned with the use of the nonlocal Timoshenko beam model for free vibration analysis of single-walled carbon nanotubes (CNTs) including the thermal effect. Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. The elastic Timoshenko beam model is reformulated using the nonlocal differential constitutive relations of Eringen (1972 Int. J. Eng. Sci. 10 1–16). The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the traditional Timoshenko beam model but also by the nonlocal elasticity characterizing the microstructure of CNTs in a wide frequency range up to terahertz. Numerical results are presented using the nonlocal beam theory to bring out the effect of both the nonlocal parameter and the temperature change on the properties of transverse vibrations of CNTs. The exact nonlocal Timoshenko beam solution presented here should be useful to engineers who are designing microelectromechanical and nanoelectromechanical devices.
The effect of small size on wave propagation in double-walled carbon nanotubes ͑DWCNTs͒ under temperature field is investigated using the Euler-Bernoulli beam model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal thermal elastic theory. The effects of temperature change and van der Waals forces between the inner and outer nanotubes are taken into account. Results show the significance of the small-scale effect on wave propagation in DWCNTs and that some properties of transverse vibrations of DWCNTs are dependent on the change in temperature. The results demonstrate the great potential of the proposed nonlocal beam theory in studying wave propagation in CNTs including thermal effects and also indicate the limitations of local continuum mechanics in analysis of small-scale effects. The work should be useful in the design and application of nanoelectronics and nanoelectromechanical devices.
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