Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field

 Vietnam Journal of Mechanics, VAST, Vol.43, No. 1 (2021), pp. 55 – 77
 DOI: https://doi.org/10.15625/0866-7136/15467
 NONLINEAR VIBRATION OF NONLOCAL STRAIN
 GRADIENT NANOTUBES UNDER LONGITUDINAL
 MAGNETIC FIELD
 N. D. Anh1,2, D. V. Hieu3,∗
 1Institute of Mechanics, VAST, Hanoi, Vietnam
 2VNU University of Engineering and Technology, Hanoi, Vietnam
 3Thai Nguyen University of Technology, Vietnam
 ∗E-mail: hieudv@tnut.edu.vn
 Received: 20 September 2020 / Published online: 10 January 2021
 Abstract. The nonlinear free vibration of embedded nanotubes under longitudinal mag-
 netic field is studied in this paper. The governing equation for the nanotube is formu-
 lated by employing Euler–Bernoulli beam model and the nonlocal strain gradient theory.
 The analytical expression of the nonlinear frequency of the nanotube is obtained by us-
 ing Galerkin method and the equivalent linearization method with the weighted averag-
 ing value. The accuracy of the obtained solution has been verified by comparison with
 the published solutions and the exact solution. The influences of the nonlocal parame-
 ter, material length scale parameter, aspect ratio, diameter ratio, Winkler parameter and
 longitudinal magnetic field on the nonlinear vibration responses of the nanotubes with
 pinned-pinned and clamped-clamped boundary conditions are investigated and and dis-
 cussed.
 Keywords: nonlinear vibration, carbon nanotube, nonlocal strain gradient, magnetic field,
 Galerkin method, equivalent linearization, weighted averaging.
 1. INTRODUCTION
 First discovered in 1991 by Iijima [1], carbon nanotubes (CNTs) show many advan-
tages compared to conventional steel tubes. With theirs advantages and small sizes,
CNTs have many applications such as nanoactuator [2], nano-electromechanical systems
(NEMS) [3,4], nano-devices for electronics [5,6], nano-medicine [7], or nano-devices for
conveying fluid and gas storage [8,9].
 Researching the mechanical behavior of CNTs is an extremely important problem
and attracts many interested scientists. Unlike macro-sized tubes, the size-dependent ef-
fect plays an important role in the response of CNTs. Some elasticity theories have been
proposed to study the size-dependent effect on the mechanical response of micro-/nano-
structures such as Eringen’s nonlocal elasticity theory [10, 11], the strain gradient theory
 © 2021 Vietnam Academy of Science and Technology
56 N. D. Anh, D. V. Hieu
(SGT) [12, 13], and the modified couple stress theory (MCST) [14]. To date, many works
related to the static and dynamic responses of CNTs have been published using these
higher-order elasticity theories. Nonlinear free vibration responses of the single-walled
carbon nanotubes (SWCNTs) were reported by Yang et al. [15] using the Timoshenko
beam theory and Eringen’s nonlocal elasticity theory. The effects of nonlocal parameter,
length and radius of the SWCNTs with different boundary conditions on the nonlinear
free vibration behaviors of SWCNTs were examined in this work. Narendar et al. [16]
studied the wave propagation problem in the SWCNTs under longitudinal magnetic field
based on Eringen’s nonlocal elasticity theory and the Euler-Bernoulli beam theory. The
wave method was employed by Zhang et al. [17] to analyze the nonlinear free vibration of
the fluid-conveying SWCNTs based on Eringen’s nonlocal elasticity theory. Based on the
Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory, Wang and Li [18]
investigated the nonlinear free vibration of the SWCNTs with the viscous damping effect.
Zhen and Fang [19] used the Lindstedt–Poincare´ method to investigate the nonlinear vi-
bration of the fluid-conveying SWCTNs under harmonic excitation. Nonlinear vibration
of the embedded SWCNTs was reported by Valipour et al. [20] using Eringen’s nonlocal
elasticity theory and the parameterized perturbation method. Goughari et al. [21] stud-
ied effects of magnetic-fluid flow on the instability of the fluid-conveying SWCNTs under
the magnetic field. Free vibration of the SWCNTs resting on the exponentially varying
elastic foundation was examined by Chakraverty and Jena using Eringen’s nonlocal elas-
ticity theory [2]. Based on the MCST and the Euler-Bernoulli beam theory, Wang [22]
investigated the size-dependent vibration responses of the fluid-conveying microtubes.
Flexural size-dependent vibrations of the microtubes conveying fluid was carried out by
Wang et al. [23] utilizing the MCST. Based on the MCST, Tang et al. [24] developed a non-
linear model to study the size-dependent vibration of the curved microtubes conveying
fluid. Xia and Wang [25] studied vibration and stability behaviors of the microscale pipes
conveying fluid based on Timoshenko beam theory and the MCST. Based on the second
SGT, vibration and stability behaviors of the SWCNTs conveying fluid were presented by
Ghazavi et al. [26].
 In 2015, the nonlocal parameter and the material length sca ... lutions. 
thepinned accuracy and of the clamped-clamped obtained approximate boundary solutions.conditions. The comparison of the obtained
 approximate solutions with the exact ones and the published ones shows the accuracy of
 the obtained approximate solutions.
 Effects of the nonlocal parameter ea/L, the material length scale parameter l/L, the
 aspect ratio L/D, the diameter ratio d/D, the elastic foundation KW and the magnetic
 field H on the nonlinear vibration responses of the nanotubes are examined. It can be
 concluded that:
 - The nonlocal parameter leads to a decrease in the nonlinear frequencies of the nan-
 otubes; while the material length scale parameter, the aspect ratio, the diameter ratio, the
 elastic foundation and the magnetic field lead to an increase in the nonlinear frequencies
 of the nanotubes.
 - When l < ea, the frequency ratios of the nanotube increase as the nonlocal parame-
 ter ea/L increases. However, when l > ea, the frequency ratios of the nanotube decrease
 as the nonlocal parameter ea/L increases. The frequency ratios of the NSGT nanotubes
 are always smaller than the ones of the nonlocal nanotubes (c = l/ea = 0). When c < 1,
 the frequency ratios of the nanotubes increase as the nonlocal and length scale parame-
 ters increase; while c > 1, the frequency ratios of the nanotubes decrease as the nonlocal
 and length scale parameters increase.
74 N. D. Anh, D. V. Hieu
 - The diameter ratio, the elastic foundation and the magnetic field lead to a decrease
in the frequency ratios of the nanotubes; while the aspect ratio leads to an increase in the
frequency ratios of the nanotubes.
 ACKNOWLEDGMENT
 This work was supported by Vietnam National Foundation for Science and Technol-
ogy Development (NAFOSTED) under grant number 107.02-2020.03.
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