Transverse vibration analysis of double-walled carbon nanotubes in an elastic medium under temperature gradients and electrical fields based on nonlocal Reddy beam theory

Abstract

This work studies the mechanical behavior of double-walled carbon nanotubes (DWCNTs) under temperature gradients and electrical fields based on the nonlocal reddy beam model, focusing on the free vibration of DWCNTs. The DWCNTs include the interaction of van der Waals (vdW) forces between the outer and inner tubes, and they are supposed to be embedded in a Winkler-Pasternak foundation. Eringen’s nonlocal elasticity theory is taken into account to consider the small-scale effects. The governing equations are achieved according to Hamilton's principle and then discretized and solved using the generalized differential quadrature method. The impact of various parameters, such as nonlocal parameters, length and ratio of thickness-to-radius, Winkler, Pasternak coefficients and piezoelectric effects and radius to length ratio, has been studied. The outcomes show that with increasing outer radius of the nanotube, the first dimensionless frequency decreases. In addition, the results illustrate that when the length of the nanotube and the Winkler spring constant increase, the initial frequencies are increased. Moreover, the desirable value for ratios of L/r2 of DWCNTs in order to obtain the minimum and maximum dimensionless natural frequencies for Winkler spring model Kw is approximately 0.7 and 6.3, respectively, whereas this value for the shear constant of Pasternak Gp is about 1.25 and 4.75.