Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory

doi:10.1016/j.physe.2010.12.010

Physica E: Low-dimensional Systems and Nanostructures

Volume 43, Issue 5, March 2011, Pages 1031-1039

https://doi.org/10.1016/j.physe.2010.12.010 Get rights and content

Abstract

Vibration and instability of fluid-conveying double-walled carbon nanotubes (DWNTs) are investigated in this paper based on the modified couple stress theory and the Timoshenko beam theory. The microstructure-dependent Timoshenko beam model, which contains a material length scale parameter and can take the size effect into account, is employed. The Poisson's ratio effect is also included in this model. The surrounding elastic medium is described as the Winkler model characterized by the spring. The higher-order governing equations and boundary conditions are derived by using Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the governing equations, which are then solved to obtain the resonant frequencies of fluid-conveying DWNTs with different boundary conditions. A detailed parametric study is conducted to study the influences of length scale parameter, Poisson's ratio, spring constant, aspect ratio of the DWNTs, velocity of the fluid and end supports on the vibration and flow-induced instability of DWNTs. Results show that the imaginary component of the frequency and the critical flow velocity of the fluid-conveying DWNTs increase with increase in the length scale parameter.

Research highlights

► The imaginary components of the frequencies increase with increase in the length scale parameter. ► The critical flow velocity of the DWNTs increases as the length scale parameter increases. ► DWNTs with large aspect ratio are more likely to cause the divergence instability.

Introduction

Carbon nanotubes (CNTs) have become the most promising materials for nanoelectronics, nanodevices and nanocomposites because of their novel electronic, mechanical and other physical and chemical properties [1]. Experiments and molecular dynamic simulations were considered to be the accurate methods for the mechanical analysis of CNTs. As controlled experiments in nanoscale are difficult and molecular dynamic simulations are computationally expensive, classical continuum mechanics models have been regarded as an effective method and widely employed to study the mechanical properties of CNTs [2], [3], [4], [5], [6], [7]. However, the applicability of the classical continuum models is questionable because at small length scales the materials microstructures (such as lattice space between individual atoms) become increasingly important and the discrete structure of the materials can no longer be homogenized into the continuum. CNTs are of the dimension at nanoscale and their properties are closely related to their microstructure. Therefore, it is significant to consider the size effect in the theoretical and experimental studies of the CNTs. The classical continuum theory is not capable of describing the size effect due to the lack of a material length scale. To overcome this problem, various size-dependent continuum theories (such as couple stress elasticity [8], [9], nonlocal elasticity [10], strain gradient elasticity [11] and surface elasticity [12]), which can capture small scale effects, have been employed to study the mechanical characteristics of CNTs [13], [14], [15], [16], [17], [18].

The nonlocal elasticity theory assumes that the stress at a reference point is a functional of strain field at every point in the body [10]. Therefore, the nonlocal elasticity theory has considered the information related to the internal length scale of nanoscale structures. The couple stress theory is a class of strain gradient theory, which introduces the material length parameter in the constitutive equations to characterize the size effect. In the present study, our focus is placed on using the couple stress theory to study the size effect on CNTs. In conventional mechanics, a force drives a material particle to translate. In the couple stress theory for the linear elastic materials, the applied loads on the materials particle include not only a force to drive the material particle to translate but also a couple to drive it to rotate [8], [9]. By introducing the microstructure-dependent length scale parameter to characterize the effect of the couple stress, the couple stress theory, which takes the rotation gradient into account, can successfully be employed to predict the size effect in micro- or nano-structures. In mechanical analysis of CNTs, the rotation should be considered due to their microstructure-dependent properties. Therefore, the couple stress effect is expected to be relevant for CNTs and can be used to capture the size effect of CNTs.

The classical couple stress theory developed by Toupin [8] and Mindlin and Tiersten [9], which is one class of the higher-order continuum theories, contains two classical and two additional material constants for isotropic elastic materials [19]. The two additional constants are related to the underlying microstructure of the material and are inherently difficult to determine. Yang et al. [20] first proposed the modified couple stress theory by introducing the concept of the representative element and defining the force and couple applied to a single material particle. In this theory, the constitutive equations involved only one additional material length scale parameter besides two classical material constants. After Yang et al.'s [20] pioneering work, the modified couple stress theory has been concentrated on and developed in the past seven years [21], [22]. Based on the modified couple theory, Wang [23] and Xia and Wang [24] developed the non-classical Euler/Timoshenko beam models to analyze the size-dependent vibration and stability characteristics of fluid-conveying microtubes. And further, Xia et al. [25] conducted a nonlinear non-classical beam model and discussed the size effect on the nonlinear bending, nonlinear vibration and postbuckling of microbeams.

Recently, the study of CNTs conveying the fluid has received considerable attention [26], [27], [28] because of the potential applications of CNTs in nanobiological devices and nanomechanical systems such as fluid conveyance and drug delivery. Yoon et al. [29], [30] studied the vibration and stability of the simply supported and cantilever CNTs. They found that the effect of the flow velocity is significant for the CNTs. At the critical flow velocity, the structure instability of the SWCNTs emerges. Based on the Timoshenko beam model, Khosravian and Rafii-Tabar [31], [32] presented the computional modeling of the non-viscous and viscous fluid flow in the multi-walled CNTs. Natsuki et al. [33] used the Euler–Bernoulli beam model to study the vibration in the DWNTs conveying fluid. Wang et al. [34] and Wang and Ni [35] considered the vibration and instability of single- and double-walled carbon nanotubes conveying fluid. Rasekh and Khadem [36] presented the influence of internal moving fluid and compressive axial load on the nonlinear vibration and stability of embedded CNTs. Furthermore, Lee and Chang [37], [38] and Wang [39] used Eringen's nonlocal elasticity theory to analyze the size effect on the fluid-conveying single- and double-walled carbon nanotubes. However, the nonlocal effect of internal fluid was not accounted for in the final equations of motion given in these three Refs. [37], [38], [39]. The improved equations were given by Tounsi et al. [40] and Wang [41].

As we mentioned above, Xia and Wang [24] developed a non-classical Timoshenko beam model to analyze the size-dependent vibration and stability characteristics of fluid-conveying microtubes. In this paper, we extend their model to study the vibration and instability of fluid-conveying DWNTs with consideration of the van der Waals (vdW) interaction forces between the inner and outer tubes. The microstructure-dependent Timoshenko beam model, which contains a material length scale parameter and can take the size effect into account, is employed. The Poisson's ratio effect is also incorporated in the model. The surrounding elastic medium is described as the Winkler model characterized by the spring. The higher-order governing equations and boundary conditions are derived by using Hamilton's principle. The DQ method is employed to discretize the governing equations, which are then solved to obtain the resonant frequencies of fluid-conveying DWNTs with different boundary conditions. The influences of length scale parameter, Poisson's ratio, spring constant, aspect ratio of the DWNTs, velocity of the fluid and end supports on the vibration and flow-induced instability of DWNTs are discussed in detail.

Section snippets

The modified couple stress theory

Yang et al. [20] first proposed the modified couple stress theory, in which the couple stress tensor is symmetric and only one internal material length scale parameter is involved. In this theory, the strain energy density is considered as the function of both strain tensor (conjugated to stress tensor) and curvature tensor (conjugated to couple stress tensor). Then, the strain energy Π_S in an isotropic linear elastic materials occupying region Λ can be written as $Π_{S} = \frac{1}{2} \int_{Λ} (σ : ε + m : χ) d Λ,$ where ε is

Vibration and stability of fluid-conveying DWNTs

Fig. 1 shows the DWNTs modeled as a Timoshenko beam with length L, inner radius r₁, outer radius r₂ and equal thickness h for each tube embedded in an elastic medium. The surrounding medium is described by the Winkler foundation model with spring constant k. Based on Timoshenko beam theory and one-dimension beam theory, the displacement field can be expressed as follows [21]: $\tilde{U} (x, y, z, t) = z Ψ (x, t), \tilde{V} (x, y, z, t) = 0, \tilde{W} (x, y, z, t) = W (x, t),$ where $\tilde{U}$ , $\tilde{V}$ and $\tilde{W}$ are the x-, y- and z-components of the

Differential quadrature method

The DQ method is used to solve the governing Eqs. (27), (28), (29), (30) and the associated boundary conditions to determine the vibration and instability of the fluid-conveying DWNTs. The main idea of the DQ method is that the derivative of a function at a sample point can be approximated as a weighted linear summation of the function value at all of the sample points in the domain. The functions w_i and ψ_i (i=1,2) and their kth derivatives with respect to x can be approximated as follows [48]: ${$

Results and discussion

Table 1 lists the dimensionless fundamental frequencies ( $ω = Ω L \sqrt{M_{0} / E A_{0}}$ ) of fluid-conveying DWNTs (l/h=3, L/r₂=20, v₀=0.0168, k=10⁸ N/m²) with varying total numbers of nodes N in the DQ method. The DWNTs are modeled as size-dependent Timoshenko beams with H–H and C–C boundary conditions. The parameters used in this example are taken with r₁=0.77 nm, outer radius r₂=1.11 nm, Young's modulus E=1 TPa, Poisson's ratio v=0.25, the thickness of each tube h=0.34 nm, shear correction factor K_s1=0.6255 and K_s2

Conclusions

This paper investigates the vibration and instability of the embedded fluid-conveying DWNTs based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains both size effect and Poisson's ratio effect. The higher-order governing equations and boundary conditions are derived by using Hamilton's principle. The DQ method is employed to discretize the governing equations, which are then solved to obtain the resonant frequencies of DWNTs with

Acknowledgements

The work described in this paper was supported by National Natural Science Foundation of China (no. 11002019), Ph.D. Programs Foundation of Ministry of Education of China (no. 20100009120018) and Fundamental Research Funds for the Central Universities (no. 2009JBM073).

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Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory

Abstract

Research highlights

Introduction

Section snippets

The modified couple stress theory

Vibration and stability of fluid-conveying DWNTs

Differential quadrature method

Results and discussion

Conclusions

Acknowledgements

Compos. Sci. Technol.

Compos. Sci. Technol.

J. Sound Vib.

Int. J. Solids Struct.

Int. J. Solids Struct.

Comp. Mater. Sci.

Int. J. Eng. Sci.

Int. J. Solids Struct.

Comp. Mater. Sci.

Physica E

Comp. Mater. Sci.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.

J. Fluid Struct.

Compos. Sci. Technol.

Int. J. Solids Struct.

Comp. Mater. Sci.

Comp. Mater. Sci.

Comp. Mater. Sci.

J. Mater. Res.

J. Appl. Phys.

Phys. Lett. A

J. Eng. Mech.

Arch. Ration. Mech. Anal.