Flow-induced instability of double-walled carbon nanotubes based on nonlocal elasticity theory

doi:10.1016/j.physe.2011.03.015

Physica E: Low-dimensional Systems and Nanostructures

Volume 43, Issue 8, June 2011, Pages 1419-1426

https://doi.org/10.1016/j.physe.2011.03.015 Get rights and content

Abstract

Instability occurs in double-walled carbon nanotubes when a fluid flows through them. This is investigated using an elastic shell model based on Donnell's shell theory. The dynamic governing equations of double-walled carbon nanotubes are derived on the basis of nonlocal elasticity theory, and the van der Waals interaction between the inner and outer walls is considered. Instability induced by a pressure-driven steady flow is studied. The numerical computations reveal that as the flow velocity increases, double-walled carbon nanotubes have a destabilizing style to get through multi-bifurcations of the first (pitchfork) and second (Hamiltonian Hopf) bifurcations in turn. It can be concluded that the critical flow velocity of the flow-induced instability is closely correlated to the ratio of the length to the radius of double-walled carbon nanotubes, the pressure of the fluid and the small size effects.

Highlights

► The critical flow velocity of the DWCNTs is correlated to the small size effects. ► The nonlocal natural frequencies of DWCNTs decrease as compared to local ones. ► As the flow velocity increases, DWNCTs get through multi-bifurcations in turn.

Introduction

Since the landmark paper by Iijima [1], carbon nanotubes (CNTs) have attracted worldwide attention due to their potential use in the fields of chemistry, physics, nano-engineering, electrical engineering and materials science. CNTs can be used as strong, light and high toughness fibers for composite structures, parts of nano-devices and for hydrogen storage [2], [3], [4], [5]. Although classical or local continuum models, such as beam and shell models, are practical in analyzing CNTs for large systems, however, size effects often become remarkable at nanometer scales, therefore, the modeling of size-dependent phenomena has become a topic of interest [6], [7]. Based on the theory of nonlocal elasticity [8], the scale effect was clarified in elasticity by assuming the stress to be a functional of the strain field at every point in the body. In this sense, the internal size scale could be considered simply as a material parameter in the constitutive equations. The application of nonlocal elasticity models in nanomaterials was proposed by Peddieson et al. [9]. They applied the nonlocal elasticity to formulate a nonlocal version of Euler–Bernoulli beam model, and concluded that nonlocal continuum mechanics could potentially play an important role in nanotechnology applications. Further applications of the nonlocal continuum mechanics have been utilized in investigating the mechanical behavior of CNTs. Sudak [10] investigated the infinitesimal column buckling of multi-walled carbon nanotubes (MWCNTs) combining not only van der Waals forces but also the effects of small length scales. His results demonstrated that as the small length scale gets larger in magnitude the critical axial strain gets smaller compared to the results with classical continuum mechanics. Zhang et al. [11] investigated a nonlocal multi-shell model for the axial buckling of MWCNTs under axial compression. Their results indicated that both the buckling mode and the length of tubes have contributions to the influence of the small scale on the axial buckling strain. Zhang et al. [12], [13] adopted the theory of nonlocal elasticity to investigate free transverse vibrations of double-walled carbon nanotubes. Wang and Hu [14] investigated flexural wave propagation in single-walled carbon nanotubes, their study focuses on the wave dispersion by considering a model of traditional Timoshenko beam in conjunction with the theory of nonlocal elasticity. Wang [15] studied wave propagation in carbon nanotubes by nonlocal continuum mechanics. They investigated wave propagation in CNTs with both Euler–Bernoulli and Timoshenko beam models by considering the nonlocal elasticity.

Recently, fluid flow inside CNTs has become an interesting subject. The properties of fluidity, diffusivity, and viscosity, and the dynamics of fluid in a fine pore have been investigated [16], [17]. The dynamic properties of hydrogen bonding [18], the effects of wall–fluid interaction [19], the dependence of fluid behavior on the spatial size of CNTs [20], and other issues have been extensively investigated in the nanoflow and microflow fields. The instability problems of fluid-filled CNTs are of central interest in the field. Wang et al. [21] investigated the elastic buckling of multi-walled carbon nanotubes (MWCNTs) under external radial pressure using a multi-walled shell model, and the results demonstrated that the multi-walled shell model is in good agreement with the experiment. Recently, the flow-induced instability of single-walled carbon nanotubes (SWCNTs) has been investigated by modeling carbon nanotubes with the Eulerian beam model [22]. Besides, Donnell's shell model for fluid-conveying MWCNTs with the consideration of the van der Waals interaction has been presented in Ref. [23]. Wang [24] developed a nonlocal double-elastic beam model to perform the vibration analysis of double-walled carbon nanotubes conveying fluid. Lee and Chang [25] investigated the vibrations of fluid-conveying double-walled carbon nanotubes by adopting a nonlocal double-elastic beam model. Wang [26] initiated a theoretical analysis of wave propagation of fluid-conveying single-walled carbon nanotubes based on strain gradient elasticity theory with consideration of both inertia and strain gradients, in which two small-scale parameters were accounted for. Wang [27] developed a new theoretical model, based on the modified couple stress theory, for the vibration analysis of fluid-conveying microtubes by introducing one internal material length scale parameter.

The purpose of the present paper is to investigate the small scale effects on the flow-induced instability of double-walled carbon nanotubes (DWCNTs) using Donnell's shell model.

Section snippets

Nonlocal elasticity theory

Based on Eringen nonlocal elasticity model [28], the stress at a reference point x in a body is considered as a function of strains of all the points in the near region. The above assumption is in agreement with the atomic theory of lattice dynamics and experimental observations on phonon dispersion.

Consider a homogeneous and isotropic elastic solid, the constitutive equation is $σ (x) = C_{0} : \int_{V} α (| x' - x |, τ) ε (x') d V (x'),$ where symbols ‘:’ is the inner product with double contraction, C₀ is the elastic

Small scale effect on coupled shell model for DWCNT conveying fluid

In Fig. 1, DWCNTs with van der Waals interaction between the inner and outer walls are modeled using Donnell's cylindrical shell theory [29]. The fluid inside the inner tube is assumed to be an ideal incompressible, and the flow is driven by pressure. The small scale effect is considered by using the nonlocal elasticity theory stated before.

The governing equations of motion of the model are written as follows [30]: $D \nabla^{4} w_{1} + ρ_{t} h \frac{\partial^{2}}{\partial t^{2}} (1 - {(e_{0} a)}^{2} \nabla^{2}) w_{1} = (1 - {(e_{0} a)}^{2} \nabla^{2}) (c_{12} (w_{1} - w_{2})) - (1 - {(e_{0} a)}^{2} \nabla^{2}) p + \frac{1}{R_{1}} \frac{\partial^{2} F_{1}}{\partial x^{2}},$ $D \nabla^{4}$

Numerical computations and discussions

Two quantitative comparisons are performed to prove the validity of the present approach. In quantitative comparisons, first of all, the numerical results based on the present study are compared with those obtained from Ref. [24]. All the input data must be exactly the same for comparison, therefore, the gap between the two walls is taken as h=0.34 nm, Young's modulus of the DWCNT is E=1 TPa, and the mass density is ρ_t=2300 kg/m³ for CNTs and pf=1000 kg/m³ for the fluid. The inner radius and outer

Conclusions

Fluid-conveying double-walled carbon nanotubes (DWCNTs) are modeled based on Donnell's shell theory by taking into account the van der Waals interaction between the inner and outer walls, in conjunction with the consideration of the small scale effects by nonlocal elasticity theory. The instability that is induced by a pressure-driven steady flow is studied. The numerical computations indicate that as the flow velocity increases, DWNCTs have a way to get through multi-bifurcations of the first

Acknowledgment

This research was partially supported by the National Science Council in Taiwan through Grant NSC-96-2221-E-327-018-MY2. The authors are grateful for this support.

References (33)

M.S. Dresselhaus et al.
Top. Appl. Phys.
(2001)
J. Peddieson et al.
Int. J. Eng. Sci.
(2003)
Y.Q. Zhang et al.
Phys. Lett. A
(2005)
S. Paul et al.
Chem. Phys. Lett.
(2003)
J. Yoon et al.
Int. J. Solids Struct.
(2006)
L. Wang
Comput. Mater. Sci.
(2009)
L. Wang
Comput. Mater. Sci.
(2010)
L. Wang
J. Fluids Struct.
(2010)
M. Amabili et al.
J. Sound Vib.
(1999)
H.S. Shen et al.
Composite Struct.
(2010)

X.Q. He et al.

J. Mech. Phys. Solids

(2005)

R. Saito et al.

Chem. Phys. Lett.

(2001)

S. Iijima

Nature

(1991)

A. Bachtold et al.

Science

(2001)

Z.C. Tu et al.

Phys. Rev. B

(2002)

K.T. Lau et al.

Composites B

(2003)

Cited by (12)

Nonlinear vibration analysis of an embedded branched nanofluid-conveying carbon nanotube: Influence of downstream angle, temperature change and two dimensional external magnetic field
2020, Nano Materials Science
Citation Excerpt :
Interesting models have also been developed to investigate the physical properties of CNTs [12–17]. By employing continuum theories, like elasticity in nonlocal form [18], exact nonlocal elasticity [19], modified strain gradient elasticity, couple stress, surface elasticity theory [20] and electro-thermal mechanical model [21], the instability analysis associated with CNTs have been studied. Lee and Chang [22] analyzed the influence of the velocity of flow in the SWCNTs on the frequency of vibration and Eigen vectors.
In this study, non-linear thermal-mechanical stability and vibration analyses of different end-shaped single-walled carbon nanotube conveying viscous nano-magnetic fluid embedded in non-linear visco-elastic foundation under the influence of magnetic fields are presented. The development of the equation of motion was based on Euler-Bernoulli theory, Hamilton principle and nonlocal elasticity theory. The results of the analytical solutions using Galerkin decomposition differential transform method (GDDTM) were validated with existing experimental results. From the parametric studies, it was shown that decreasing the temperature difference as well as increasing the downstream angle decreased the system's stability for pre-bifurcation analysis but increased stability of the system for post bifurcation analysis. Also, the results obtained from the dynamic behaviour of the system indicated that the magnetic effect had an attenuating impact of about 45% on the system's response at any mode and for any boundary condition considered. It is hoped that this work will enhance the design and optimization of nano-devices with I, V, Y, L, K and T-shaped junctions under the influence of thermal-magneto-mechanical flow induced vibration.
Nonlinear vibration of single-walled carbon nanotubes with nonlinear damping and random material properties under magnetic field
2017, Composites Part B: Engineering
This paper copes with the statistical dynamic behaviors of nonlinear vibration of the single-walled carbon nanotubes (SWCNTs) under longitudinal magnetic field by considering the effects of the geometric nonlinearity and nonlinear damping. Both the Young's modulus of elasticity and mass density of the SWCNTs are considered as stochastic with respect to the position to actually characterize the random material properties of the SWCNTs. Based on the theory of nonlocal elasticity, the small scale effects of the nonlinear vibration of the SWCNTs are investigated. By using the Hamilton's principle, the nonlinear governing equations of the single-walled carbon nanotubes subjected to longitudinal magnetic field are derived. The Monte Carlo Simulation, Galerkin's method and the multiple scale method are adopted to solve the nonlinear governing equation and to calculate the statistical response of the SWCNTs. Some statistical dynamic responses of the SWCNTs such as the mean values and standard deviations of the midpoint deflections are computed, the effects of the small scale coefficients, magnetic field, nonlinear damping and the elastic stiffness of matrix on the statistical dynamic responses of the SWCNTs are investigated and discussed.
Free vibration of wavy single-walled fluid-conveying carbon nanotubes in multi-physics fields
2015, Applied Mathematical Modelling
Citation Excerpt :
Thus, it is very useful to study the mechanical behavior of fluid-conveying CNTs, particularly because the fluid flow inside CNTs will apparently affect their vibration. In recent years, fluid-conveying CNTs have been investigated widely [3–11] to develop theoretical and experimental methods to study their dynamic behavior. However, improvements are still required in areas such as theoretical modeling.
Fluid-conveying carbon nanotubes (CNTs) have attracted much attention and they are used in nano-electromechanical systems (NEMS). In this study, we investigate the free vibration of embedded single-walled fluid-conveying carbon nanotubes in magnetic and temperature fields. The CNTs are modeled as wavy Timoshenko beams. Based on the nonlocal beam theory, the governing equations of motion are derived using Hamilton’s principle. These equations are solved using the Galerkin approach, thereby obtaining a set of ordinary differential equations from the partial differential equations of motion. Numerical examples are analyzed to determine the differences between the proposed model and some previously reported models, as well as the effects of the nonlocal parameter, the fluid velocity and density, the temperature and magnetic field flux change, and the surrounding elastic medium on the dynamic behavior of wavy CNTs. The numerical results validate the proposed analytical model, thereby allowing us to conclude that this method may facilitate the application of fluid-conveying CNTs as NEMS devices.
Nonlocal vibration of Y-shaped CNT conveying nano-magnetic viscous fluid under magnetic field
2015, Ain Shams Engineering Journal
Citation Excerpt :
Their results showed that for the DWBNNTs, the nonlocal theory predictions for the natural frequency are lower than that of the classical theory. Chang and Liu [19] investigated the instability of double-walled carbon nanotubes (DWCNTs) conveying fluids based on nonlocal elasticity theory. Their results show that the critical flow is closely correlated to the ratio of the length to the radius of carbon nanotubes, the pressure of the fluid and the small size effects.
This study deals with the vibration and stability analysis of a Y-shaped single-walled carbon nanotube (SWCNT) embedded in visco-Pasternak foundation and conveying nano-magnetic viscous fluid (NMF) based on nonlocal elasticity theory and Euler–Bernoulli beam model. The fluid is two-phases due to the existence of magnetic nanoparticles which its volume fraction is much little in comparison with the base fluid where the influence of 2D magnetic field is taken into account. Also, Knudsen number is used to correct the velocity profile of fluid. The Galerkin method is applied to solve the equation of motion which is obtained by employing Hamilton’s principle. The detail parametric study is conducted, focusing on the combined effects of carbon nanotube and Y-shaped junction fitted at the downstream end, fluid velocity, Knudsen number and elastic medium. The results indicate that increasing the angle between centerline of the CNT and the downstream elbows decreases stability of system.
Stochastic FEM on nonlinear vibration of fluid-loaded double-walled carbon nanotubes subjected to a moving load based on nonlocal elasticity theory
2013, Composites Part B: Engineering
Citation Excerpt :
Moreover, the nonlocal elasticity theory was incorporated into the elastic beam model to study the small scale effect on the dynamics of SWCNT conveying fluid [20]. Chang and Liu [21,22] studied small scale effects on the flow-induced instability of double-walled carbon nanotubes (DWCNTs) by using the nonlocal elasticity theory. More recently, Chang [23,24] investigated the thermal–mechanical vibration and instability of fluid-conveying single-walled carbon nanotubes (SWCNTs) based on nonlocal elasticity theory.
This paper adopts stochastic FEM to study the statistical dynamic behaviors of nonlinear vibration of the fluid-conveying double-walled carbon nanotubes (DWCNTs) under a moving load by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The Young’s modulus of elasticity of the DWCNTs is considered as stochastic with respect to the position to actually characterize the random material properties of the DWCNTs. Besides, the small scale effects of the nonlinear vibration of the DWCNTs are studied by using the theory of nonlocal elasticity. Based on the Hamilton’s principle, the nonlinear governing equations of the fluid-conveying double-walled carbon nanotubes under a moving load are formulated. The stochastic finite element method along with the perturbation technique is adopted to study the statistical dynamic response of the DWCNTs. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the non-dimensional dynamic deflections are computed and checked by the Monte Carlo Simulation, meanwhile the effects of the nonlocal parameter, aspect ratio and the flow velocity on the statistical dynamic response of the DWCNTs are investigated. It can be concluded that the nonlocal solutions of the dynamic deflections get larger with the increase of the nonlocal parameters due to the small scale effect, and as the flow velocity increases, the maxima non-dimensional dynamic deflections of the DWCNTs get larger.
Vibrational properties of two and three junctioned carbon nanotubes
2012, Computational Materials Science
Citation Excerpt :
It is clear that most of these applications, like many other researches in nanotechnology, are potential and they may be not applied in the near future but they should be studied at least theoretically and the related ideas ought to be developed. Ordinary and junctioned CNTs may be used in the fluid transfers and the transferred flow may result in flutter instability or vibration [19–21]. Hence, recognizing the vibrational behaviors of these nanostructures is very important.
This paper deals with the vibrational properties of the two and three junctioned carbon nanotubes (CNTs) with different geometries and boundary conditions. The well-known molecular mechanics method is applied to the modal analysis of the mentioned nanostructures and the presented results cover the natural frequencies and their corresponding mode shapes. The results of the two junctioned CNTs with different diameters and lengths and with cantilever from right side, left side and doubly clamped boundary conditions are presented in detail. Moreover, the natural frequencies of the three junctioned CNTs with different lengths and diameters and various boundary conditions are discussed.

View all citing articles on Scopus

View full text

Flow-induced instability of double-walled carbon nanotubes based on nonlocal elasticity theory

Abstract

Highlights

Introduction

Section snippets

Nonlocal elasticity theory

Small scale effect on coupled shell model for DWCNT conveying fluid

Numerical computations and discussions

Conclusions

Acknowledgment

Top. Appl. Phys.

Int. J. Eng. Sci.

Phys. Lett. A

Chem. Phys. Lett.

Int. J. Solids Struct.

Comput. Mater. Sci.

Comput. Mater. Sci.

J. Fluids Struct.

J. Sound Vib.

Composite Struct.

J. Mech. Phys. Solids

Chem. Phys. Lett.

Nature

Science

Phys. Rev. B

Composites B