Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid

The flexural vibration of viscoelastic carbon nanotubes (CNTs) conveying fluid and embedded in viscous fluid is investigated by the nonlocal Timoshenko beam model. The governing equations are developed by Hamilton's principle, including the effects of structural damping of the CNT, internal moving fluid, external viscous fluid, temperature change and nonlocal parameter. Applying Galerkin’s approach, the resulting equations are transformed into a set of eigenvalue equations. The validity of the present analysis is confirmed by comparing the results with those obtained in literature. The effects of the main parameters on the vibration characteristics of the CNT are also elucidated. Most results presented in the present investigation have been absent from the literature for the vibration and instability of the CNT conveying fluid.     

Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method

Nonlinear free vibration analysis of curved double-walled carbon nanotubes (DWNTs) embedded in an elastic medium is studied in this study. Nonlinearities considered are due to large deflection of carbon nanotubes (geometric nonlinearity) and nonlinear interlayer van der Waals forces between inner and outer tubes. The differential quadrature method (DQM) is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of algebraic equations of motion. The effect of nonlinearities, different end conditions, initial curvature, and stiffness of the surrounding elastic medium, and vibrational modes on the nonlinear free vibration of DWCNTs is studied. Results show that it is possible to detect different vibration modes occurring at a single vibration frequency when CNTs vibrate in the out-of-phase vibration mode. Moreover, it is observed that boundary conditions have significant effect on the nonlinear natural frequencies of the DWCNT including multiple solutions.     

Natural frequencies and mode shapes of initially curved carbon nanotube resonators under electric excitation

Estimating accurately the natural frequencies of electrically actuated carbon nanotubes (CNTs) has been an active research subject over the past few years. Despite the importance of the topic, robust knowledge is still missing in the understanding of the role of various physical parameters affecting the natural frequencies, such as the stretching of doubly clamped CNTs, the DC electrostatic force, and the initial curvature of slack CNTs. In this investigation, we use a 2D nonlinear curved beam model in the form of an arch to simulate the coupled in-plane and out-of-plane motions of a CNT with curvature. We calculate the variation of its natural frequencies and mode shapes with the level of slackness and the DC electrostatic load. Towards this end, we derive a reduced-order model using a multimode Galerkin procedure. We show various scenarios of mode crossing and mode veering as the levels of slackness and DC load are varied. Finally, we tackle the forced vibration problem of a curved CNT when actuated by small DC and AC loads. The results show the transfer of energy among the vibration modes involved in the veering phenomenon.     

Nonlinear vibration of a double-walled carbon nanotube embedded in a polymer matrix

In the current work, the nonlinear vibration of an embedded double-walled carbon nanotube (DWCNT) aroused by nonlinear van der Waals (vdW) interaction forces from both surrounding medium and adjacent tubes is studied. Using both Euler–Bernoulli and Timoshenko beam models, the relation between deflection amplitudes and resonant frequencies of the DWCNT is derived through harmonic balance method. It is found that the nonlinear vdW forces from the surrounding medium result in noncoaxial vibration of the embedded DWCNT. The noncoaxial vibration includes both uni-directional and bi-directional vibration modes. It is found that the surrounding matrix has more prominent effect on the uni-directional vibration in comparison to the bi-directional vibration. The axial load effect on the vibrational behavior of the embedded DWCNT is also discussed. Due to the influence of the surrounding polymer, the prediction on the resonant frequencies of embedded CNTs is quite different from that for free-standing CNTs. A softening behavior for the deflection amplitude-resonant frequency relation is observed for the first time in the bi-directional vibration of the embedded DWCNT, which can only be obtained using the Timoshenko beam theory.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Vibrational analysis of carbon nanotubes using molecular mechanics and artificial neural network

This paper presents the molecular mechanics based finite element modeling of carbon nanotubes (CNTs) and their applications as mass sensors. The beam element with elastic behavior is considered as the bond between the carbon atoms and its properties are obtained using equating continuum and molecular characteristics. The first five natural frequencies of CNTs in cantilever and doubly clamped boundary conditions (BCs) and their corresponding mode shapes are studied in detail. Furthermore, a multilayer perceptron neural network is used to predict the fundamental vibration frequencies of the CNTs with different diameters and lengths. In addition, variations of the natural frequencies of the CNTs with distorted cross sections are investigated. Moreover, the effects of some attached masses with various values on the first three natural frequencies of a considered CNT are studied here.     

Acoustic radiation from a fluid-filled, subsurface vascular tube with internal turbulent flow due to a constriction

The vibration of a thin-walled cylindrical, compliant viscoelastic tube with internal turbulent flow due to an axisymmetric constriction is studied theoretically and experimentally. Vibration of the tube is considered with internal fluid coupling only, and with coupling to internal-flowing fluid and external stagnant fluid or external tissue-like viscoelastic material. The theoretical analysis includes the adaptation of a model for turbulence in the internal fluid and its vibratory excitation of and interaction with the tube wall and surrounding viscoelastic medium. Analytical predictions are compared with experimental measurements conducted on a flow model system using laser Doppler vibrometry to measure tube vibration and the vibration of the surrounding viscoelastic medium. Fluid pressure within the tube was measured with miniature hydrophones. Discrepancies between theory and experiment, as well as the coupled nature of the fluid-structure interaction, are described. This study is relevant to and may lead to further insight into the patency and mechanisms of vascular failure, as well as diagnostic techniques utilizing noninvasive acoustic measurements.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

Vibrations of carbon nanotube-reinforced composites

This work deals with a study of the vibrational properties of carbon nanotube-reinforced composites by employing an equivalent continuum model based on the Eshelby-Mori-Tanaka approach. The theory allows the calculation of the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (carbon nanotubes). The devised computational approach is shown to yield predictions in good agreement with the experimentally obtained elastic moduli of composites reinforced with uniformly aligned single-walled carbon nanotubes (CNTs). The primary contribution of the present work deals with the global elastic modal properties of nano-structured composite plates. The investigated composite plates are made of a purely isotropic elastic hosting matrix of three different types (epoxy, rubber, and concrete) with embedded single-walled CNTs. The computations are carried out via a finite element (FE) discretization of the composite plates. The effects of the CNT alignment and volume fraction are studied in depth to assess how the modal properties are influenced both globally and locally. As a major outcome, the lowest natural frequencies of CNT-reinforced rubber composites are shown to increase up to 500 percent.     

Viscoelastic flow through a wavy curved channel

Time-dependent flow of a viscoelastic material through a wavy curved channel is studied. The flow traversing the wavy curved domain is modelled in the curvilinear coordinates. The viscoelastic fluid is described by the fluid relaxation and retardation times, and their variations are considered in relation to the flow characteristic time scale. The wavy walls of the curved channels are in sinusoidal form, with arbitrary phase difference. Using domain perturbation technique, the analytical solution of the model is obtained, including the velocity and the volumetric flow rate. The analytical solution is explained as a function of the flow domain structure, the viscoelastic fluid properties, and the flow generating force. Flow enhancement due to the flow domain is discussed; we have shown that the flow augmentation depends on the properties of the viscoelastic fluid. Furthermore, comparisons have been made with Newtonian fluid flow, and the prominent variations in the flow behaviours have been reported.     

Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle

In this study, an analytical method of the small scale parameter on the vibration of single-walled Boron Nitride nanotube (SWBNNT) under a moving nanoparticle is presented. SWBNNT is embedded in bundle of carbon nanotubes (CNTs) which is simulated as Pasternak foundation. Using Euler–Bernoulli beam (EBB) model, Hamilton's principle and nonlocal piezoelasticity theory, the higher order governing equation is derived. The effects of electric field, elastic medium, slenderness ratio and small scale parameter are investigated on the vibration behavior of SWBNNT under a moving nanoparticle. Results indicate the importance of using surrounding elastic medium in decrease of normalized dynamic deflection. Indeed, the normalized dynamic deflection decreases with the increase of the elastic medium stiffness values. The electric field has significant role on the nondimensional fundamental frequencies, as a smart controller. The results of this work is hoped to be of use in design and manufacturing of smart nano-electro-mechanical devices in advanced medical applications such as drug delivery systems with great applications in biomechanics.     

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.     

Elastic response of carbon nanotube forests to aerodynamic stresses

The ability to determine static and (hydro)dynamic properties of carbon nanotubes (CNTs) is crucial for many applications. While their static properties (e.g., solubility and wettability) are fairly well understood, their mechanical responses (e.g., deflection under shear) to ambient fluid flow are to a large extent unknown. We analyze the elastic response of single-walled CNT forests, attached to the bottom wall of a channel, to the aerodynamic loading exerted by both laminar and turbulent flows. Our analysis yields analytical expressions for velocity distributions, the drag coefficient, and bending profiles of individual CNTs. This enables us to determine flexural rigidity of CNTs in wind-tunnel experiments. The model predictions agree with laboratory experiments for a large range of channel velocities.     

A note on the spheroidal modes vibration of an elastic sphere in linear viscoelastic fluid

Vibration characteristics of elastic nanostructures embedded in fluid medium have been used for biological and mechanical sensing, and also to investigate materials mechanical properties. An analytical approach based on the exact theory has been developed in this paper, to establish a new accurate and simple generalized frequency equation to predict spheroidal vibration of an elastic nanosphere, in a compressible viscoelastic fluid using linear Maxwell fluid model. To demonstrate the accuracy of the present approach, a comparison is made with the published theoretical results in the literature in some particular cases, which shows a very good agreement. Thus, the obtained frequency equation can be very useful to interpret the experimental measurements of vibrational dynamics of nanospheres and can serve as benchmark solution in design of liquid sensors.     

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.     

Nonlinear vibration of viscoelastic embedded-DWCNTs integrated with piezoelectric layers-conveying viscous fluid considering surface effects

This article studies the nonlinear vibration of viscoelastic embedded nano-sandwich structures containing of a double walled carbon nanotube (DWCNT) integrated with two piezoelectric Zinc oxide (ZnO) layers. DWCNT and ZnO layers are subjected to magnetic and electric fields, respectively. This system is conveying viscous fluid and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. Visco–Pasternak model with three parameters of the Winkler modulus, shear modulus, and damp coefficient is used for simulation of viscoelastic medium. The nano-structure is simulated as an orthotropic Timoshenko beam (TB) and the effects of small scale, structural damping and surface stress are considered based on Eringen's, Kelvin-voigt and Gurtin–Murdoch theories. Energy method and Hamilton's principle are employed to derive motion equations which are then solved using differential quadrature method (DQM). The detailed parametric study is conducted, focusing on the combined effects of small scale effect, fluid velocity, thickness of piezoelectric layer, boundary condition, surface effects, van der Waals (vdW) force on the frequency and critical velocity of nano-structure. Results indicate that the frequency and critical velocity increases with assume of surface effects.     

Energy harvesting from transverse galloping

Some elastic bluff bodies under the action of a fluid flow can experience transverse galloping and lose stability if the flow velocity exceeds a critical value. For flow velocities higher than this critical value, there is an energy transfer from the flow to the body and the body develops an oscillatory motion. Usually, it is considered as an undesirable effect for civil or marine structures but here we will show that if the vibration is substantial, it can be used to extract useful energy from the surrounding flow. This paper explores analytically the potential use of transverse galloping in order to obtain energy. To this end, transverse galloping is described by a one-degree-of-freedom model where fluid forces obey the quasi-steady hypothesis. The influence of cross-section geometry and mechanical properties in the energy conversion factor is investigated.     

The thermal effect on vibration and instability of carbon nanotubes conveying fluid

Based on the theory of thermal elasticity mechanics, an elastic Bernoulli–Euler beam model is developed for vibration and instability analysis of fluid-conveying single-walled carbon nanotubes (SWNTs) considering the thermal effect. Results are demonstrated for the dependence of natural frequencies on the flow velocity as well as temperature change. The influence of temperature change on the critical flow velocity at which buckling instability occurs is investigated. It is concluded that the effect of temperature change on the instability of SWNTs conveying fluid is significant.     

Longitudinal waves in carbon nanotubes in the presence of transverse magnetic field and elastic medium

In the present paper, the coupling effect of transverse magnetic field and elastic medium on the longitudinal wave propagation along a carbon nanotube (CNT) is studied. Based on the nonlocal elasticity theory and Hamilton's principle, a unified nonlocal rod theory which takes into account the effects of small size scale, lateral inertia and radial deformation is proposed. The existing rod theories including the classic rod theory, the Rayleigh-Love theory and Rayleigh-Bishop theory for macro solids can be treated as the special cases of the present model. A two-parameter foundation model (Pasternak-type model) is used to represent the elastic medium. The influence of transverse magnetic field, Pasternak-type elastic medium and small size scale on the longitudinal wave propagation behavior of the CNT is investigated in detail. It is shown that the influences of lateral inertia and radial deformation cannot be neglected in analyzing the longitudinal wave propagation characteristics of the CNT. The results also show that the elastic medium and the transverse magnetic field will also affect the longitudinal wave dispersion behavior of the CNT significantly. The results obtained in this paper are helpful for understanding the mechanical behaviors of nanostructures embedded in an elastic medium.     

Compressive characteristics of single walled carbon nanotube with water interactions investigated by using molecular dynamics simulation

The elastic properties of single walled carbon nanotube (SWCNT) with surrounding water interactions are studied using molecular dynamics simulation technique. The compressive loading characteristic of carbon nanotubes (CNTs) in a fluidic medium such as water is critical for its role in determining the lifetime and stability of CNT based nano-fluidic devices. In this paper, we conducted a comprehensive analysis on the effect of geometry, chirality and density of encapsulated water on the elastic properties of SWCNT. Our studies show that defect density and distribution can strongly impact the compressive resistance of SWCNTs in water. Further studies were conducted on capped SWCNTs with varying densities of encapsulated water, which is necessary to understand the strength of CNT as a potential drug carrier. The results obtained from this paper will help determining the potential applications of CNTs in the field of nano-electromechanical systems (NEMS) such as nano-biological and nano-fluidic devices.