Study on size-dependent vibration and stability of DWCNTs subjected to moving nanoparticles and embedded on two-parameter foundations

Highlights

• Dynamic instability of double-walled carbon nanotubes surrounded by elastic medium.

• DWCNT is modeled as two Euler–Bernoulli beams interacting between them.

• Partial differential equations of motion are reduced to couple ordinary differential equations.

• Considering the vdW effects, increasing the amplitude of the static axial tensile force.

Abstract

Parametric resonance is an important phenomenon that may be evinced in applying carbon nanotubes for the delivery of nanoparticles. This paper aims to investigate dynamics instability of double-walled carbon nanotubes (DWCNTs) surrounded by elastic medium and excited by a sequence of moving nanoparticles. The DWCNT is modeled as two Euler-Bernoulli beams interacting between them through van der Waals (vdW) forces. Based on Eringen's nonlocal elastic theory to consider the small-scale effects, the governing equations are derived by using Hamilton's principle. All inertial terms of the moving nanoparticles are taken into account. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the nanotube is considered. By utilization of the Galerkin method, the partial differential equations (PDEs) of motion are reduced to couple ordinary differential equations with time-varying coefficients describing a parametrically excited nanosystem. Then, an incremental harmonic balance (IHB) method is implemented to calculate the instability regions of the DWCNT. The results show that considering the vdW effects, increasing the amplitude of the static axial tensile force, reducing the amplitude of axial oscillating force, and increasing the stiffness of the elastic medium improve stability of the system. A comparison between the results with those reported in the literature is performed to verify the precision of the presented analyses.

Introduction

Nearly after three decades of carbon nanotube (CNT) discovery by Iijima (1991), researchers have not stopped investigating for identification of their new properties to use them in nanostructural applications. This is due to the unique properties of CNTs, such as especial physical, mechanical, chemical, thermal and electronic characteristics. Among too many proposed industrial applications for CNTs, applying them as Hydrogen storages, fluid conveyance, and nanoparticle delivery systems are attended thoroughly notable because of their great mechanical properties besides their unique molecular structures, that is, cylindrical shape with inner hollow space (Ajayan and Zhou, 2001).

The issue of drug delivery can be understood by considering the interior hollow space of CNTs as the container, and the fullerenes, e.g. C60, C70, C80, and C84 as the nanoparticles. Accordingly, numerous researchers have studied different aspects of applying CNTs as drugs delivery systems. For example, Rezapour and Araghi (2019b) analyzed dynamics of a viscoelastic single-walled carbon nanotube (SWCNT) in nanoparticle delivery and showed that the interaction and friction force effects between nanoparticle and SWCNTs reveals significant influence on dynamic response of the system. In another study, Rezapour and Araghi (2019a) concerned dynamic behavior of CNTs delivering a nanoparticle with constant velocity. In his study, Kiani (2014) employed the nonlocal Rayleigh beam theory to study nonlinear vibrations of SWCNTs as nanoparticle delivery systems. The results showed that a nonlinear analysis is necessary especially for large amounts of the mass and velocity of the moving nanoparticle. Another study by Lee and Chang (2010) studied dynamic behavior of SWCNTs for nanoparticle delivery. Their numerical results show that increasing the non-local parameter decreases the dynamic displacement of SWCNT while increasing the velocity of nanoparticle increases the maximum displacement.

When applications like drug delivery systems or systems conveying fluids are considered, investigating vibrational response of CNTs as the containers is unavoidable. It is because of the nanostructure-nanoparticle interaction and the small-scale effect. Therefore, dynamic analysis and the study of induced dynamic instability of a CNT due to passage of nanoparticles or fluids through it become important and are of great academic as well as practical concern. Molecular dynamics (MD) simulation, hybrid atomistic-continuum mechanics, experimental research and continuum mechanics (Farokhi and Ghayesh, 2017, 2018; Ghayesh and Farajpour, 2018; Kazemirad et al., 2013; Farokhi et al., 2013; Ghayesh et al., 2013c; Ghayesh, 2018b,c ; Ghayesh and Farokhi, 2015) are four common methods which are used to study mechanical behavior of CNTs. The first two methods are especially complex and time-consuming and just may be used for systems with small number of atoms. In addition, performing exact and reliable experiments at nanoscale is difficult and expensive. However, because of dependency of nanostructures mechanical behavior on the length scale, application of classical continuum mechanics may lead to erroneous results. Consequently, in recent years significant investigation has been done to explore dynamic behavior of CNTs by employing several non-classical continuum theories including the nonlocal elasticity (Eringen, 1983), the coupled stress (Yang et al., 2002), the surface stress (Gurtin and Murdoch, 1978) and the strain gradient (Lam et al., 2003) theories, where the scale effect is considered in analyses. Some investigations done to find vibrational characteristics of CNTs using the non-classical continuum theories are as the following.the study by Lü et al. (2015) investigated the transverse vibration of simply supported DWCNTs both conveying moving nanoparticles. Effect of some system parameters was analyzed on the tubes dynamics. Their results show that the maximum transverse deflections of both coupled tubes can be reduced because of the time lag. In another study, Hashemi and Khaniki (2018) examined nonlocal continuum model of simply supported Euler–Bernoulli nanobeams under a moving nanoparticle using Eringen's nonlocal theory. Beam layers were coupled by Winkler elastic medium. Their results show that small-scale parameter has an important role on dynamic response of nanobeams under moving nanoparticles. The work of Kiani and Roshan (2019) investigated transverse vibrations of doubly parallel nanotubes acted by doubly lagged-moving nanoparticles applying the nonlocal Rayleigh and higher-order beam models. They considered the nonlocal inertial force as well as the lag of moving nanoparticles. The effect of nonlocality, shear deformation, lag effect, and kinematic properties of the moving nanoparticles on the dynamic deflections of the tubes was studied. Furthermore, Karličić et al. (2017) using the nonlocal continuum theory studied the nonlinear vibrations of SWCNTs influenced by a time-varying axial load and a longitudinal magnetic field. Using the method of multiple scales the amplitude-frequency relationship was derived. They approximated an analytical expression for nonlinear frequency. They presented instability regions for the linear vibration of the system. It was shown that the magnetic field, the nonlocal parameter, and stiffness coefficient of the viscoelastic medium have important effects on the vibration and instability behavior of the nanobeam. In their study, Pourseifi et al. (2015) evaluated active vibration control of nanotubes under action of a moving nanoscale particle. The effects of the moving nanoparticle velocity, small scale effect parameter and slenderness ratio of nanotube on the dynamic deflection were investigated. They showed the efficiency of the control algorithm in suppressing the vibrations of the nanostructure. Hołubowski et al. (2019) based on non-local elasticity theory studied dynamics of SWCNTs under distributed random loads. They examined the influence of load standard deviation and non-local parameters on dynamic response and showed that random load perturbations should not be neglected in dynamic analyses. Based on different beam theories and using the nonlocal continuum theory of Kiani and Wang (2012) studied the interaction of a moving nanoparticle with a single-walled carbon nanotube. Examining forced vibration of a simply supported SWCNT excited by a moving harmonic load, Şimşek (2010) investigated the effects of aspect ratio, nonlocal parameter, and velocity and the excitation frequency of the moving load on the dynamic response of SWCNTs.

Based on the number of consisting rolled graphene sheets, CNTs are divided into single-, double- and multi-walled carbon nanotubes. While the outer layer of MWCNTs may keep the inner tubes away from chemical interactions with outside environs, each of the nested tubes interacts with the adjacent nanotubes through the vdW interlayer forces. These forces influence vibrational characteristics of CNTs according to the nonlocal interaction of carbon atoms of different layers on each other. The effect of vdW interaction on the vibration characteristics of MWCNTs was studied by modeling it as a radius-dependent function (He et al., 2006). The natural frequencies were calculated for MWCNTs with various number of tubes and radii. It was reported that the vdW interaction plays a significant role on the vibration of MWCNTs with small radii. Wang et al. (2012) investigates rigorous vdW interaction effect on vibration characteristics of MWCNTs under a transverse magnetic field. Their results showed that the rigorous vdW force considerably influences the frequency of MWCNTs. Also, Budarapu et al. (2014) estimated the natural frequencies of MWCNTs embedded in an elastic medium by modeling the interaction between adjacent nanotubes through the vdW forces. Nonlinear vibration of DWCNTs embedded in an elastic medium was investigated by Ansari and Hemmatnezhad (2012). They showed that the governing equations of layers were coupled due to the vdW interlayer forces. Tylikowski (2008) investigated the effect of vdW interaction force on dynamic stability of CNTs under a time-dependent axial load. Fu et al. (2009) investigated the nonlinear dynamic instability of DWNT taking into account the effects of vdW forces. Their results show that when the vdW forces are sufficiently strong, the DWNT can be assumed as a single column. Ke and Wang (2011) studied vibration and stability of fluid-conveying DWNTs and obtained the resonance frequencies of the system. Their results show that the critical flow velocity of the fluid-conveying DWNTs increases with an increase in the length scale parameter. Lei et al. (2012) studied the vibrational frequencies of DWCNTs, as latent materials for drug carriers, using the nonlocal Timoshenko beam model. Their results show that the vibrational frequency is significantly influenced by the aspect ratio, vibration mode and the nonlocal parameter.

Investigating dynamics of CNTs under periodic excitations caused by time-dependent axial loads (Tylikowski, 2008), magnetic and electrical fields (Wang et al., 2012), fluid flows (Ke and Wang, 2011) and repetitive delivery of drugs (Lee and Chang, 2010) is of great interest. In these cases, it is expected that inter-layer radial displacements of MWNTs would come to play an important role. A review on the open literature shows that no comprehensive investigation has been done on the instability of MWCNTs excited by a series of moving nanoparticles until now. Addressing the necessity to bridge to this technical gap, the present study is dedicated to examine dynamic instability of DWCNTs loaded by successive nanoparticles through drug delivery process. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the nanotube, which has been ignored in most of previous studies, is considered. To this end, two consisting layers of a DWCNT are simulated using Euler–Bernoulli beams based on the nonlocal continuum theory. An elastic layer introducing the vdW interaction force between two adjacent tubes connects the beams. In addition, using a confined spring connecting the nanoparticle to the innermost CNT the vdW effect is taken into account. After applying the Hamilton's principle to find the nonlocal partial differential equations of the motion, the Galerkin procedure is used to discretize the unknown fields in the spatial domain. Then, the incremental harmonic balance method is applied to explore stability characteristics of the system and the effects of nonlocal parameter, the elastic medium stiffness and the vdW interaction forces on the system stability are investigated, comprehensively.