Small scale effect on flow-induced instability of double-walled carbon nanotubes
Highlights
► Critical flow velocity of DWCNTs decreases as small scale effect increases. ► Critical flow velocity is strictly related to ratio of length to radius of DWCNTs. ► Natural frequency of DWCNTs decreases as small scale effect increases. ► DWCNTs get through multi-bifurcations in turn as flow velocity increases. ► Small scale effect is crucial on instability in fluid-conveying DWCNTs.
Introduction
Carbon nanotubes (CNTs) discovered by Iijima (1991) have attracted worldwide attention due to their potential use in the fields of chemistry, physics, nano-engineering, electrical engineering, materials science, reinforced composite structures and construction engineering. CNTs can be used as strong, light and high toughness fibers for nanocomposite structures, parts of nano-devices and hydrogen storage (Bachtold et al., 2001, Dresselhaus and Avouris, 2001, Tu and Yang, 2002, Lau and Hui, 2003). Yao and Han (2007) adopted an elastic double-shell model to study the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. Later, Yao and Han (2008) performed nonlinear buckling and postbuckling analysis of single-walled carbon nanotubes subjected to torsional load by elastic shell model. Sun and Liu (2008) used an elastic double-shell model based on continuum mechanics to investigate the dynamic torsional buckling of an embedded double-walled carbon nanotube. 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 (Sheehan and Lieber, 1996, Yakobson et al., 1997). Based on the theory of nonlocal elasticity (Eringen, 1976), 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. (2003). 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 (2003) investigated the infinitesimal column buckling of multi-walled nanotubes (MWNTs) 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. (2004) investigated a nonlocal multi-shell model for the axial buckling of MWNTs 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., 2005a, Zhang et al., 2005b adopted the theory of nonlocal elasticity to investigate free transverse vibrations of double-walled carbon nanotubes. Wang and Hu (2005) 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 (2005) studied wave propagation in carbon nanotubes by nonlocal continuum mechanics. They investigated wave propagation in carbon nanotubes (CNTs) with both Euler–Bernoulli and Timoshenko beam models by considering the nonlocal elasticity.
Recently, fluid flowing inside carbon nanotubes (CNTs) has become an interesting subject to many researchers. This is because CNTs can be used as nanopipes for conveying fluids due to its perfect hollow geometry and excellent mechanical characteristics. The carbon nanotubes (CNTs) promises many new applications in nanobiological devices and nanomechanical systems such as fluid storage, fluid transport and drug delivery (Che and Lakshmil, 1998, Liu et al., 1998, Hummer et al., 2001, Gao and Bando, 2002). Basically, the behavior of the fluid inside CNTs is expected to be quite different from that of the fluid in macro or micro systems at low velocities since CNTs has very small diameter, and hence it offers itself as a challenging and significant research topic for many researchers.
The properties of fluidity, diffusivity, and viscosity, and the dynamics of fluid in a fine pore have been investigated (Bitsanis et al., 1987, Paul and Chandra, 2003). The dynamic properties of hydrogen bonding (Hummer et al., 2001), the effects of wall-fluid interaction (Sokhan et al., 2001), the dependence of fluid behavior on the spatial size of CNTs (Tuzun et al., 1996), 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. (1996) investigated the elastic buckling of MWCNTs under external radial pressure by 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 SWCNTs has been investigated by modeling carbon nanotubes with the Eulerian beam model (Yoon et al., 2006). Besides, Donnell’s shell model for fluid-conveying MWCNTs with the consideration of the van der Waals (vdW) interaction has been presented in the study (Yan et al., 2007).The main goal of the present study 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’s nonlocal elasticity model (Eringen, 1983), 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 iswhere symbols ‘:’ is the inner product with double contraction, C0 is the
Small scale effect on coupled shell model for fluid-conveying DWCNTs
Based on Donnell’s cylindrical shell theory (Amabili et al., 1999), the DWCNTs with vdW interaction between the inner and outer walls are modeled as shown in Fig. 1. The fluid inside the inner tube is assumed to be an ideal incompressible, and the flow is driven by pressure. The small scale effect of the DWCNTs is considered by adopting the theory of nonlocal elasticity described previously. The governing equations of motion of the nonlocal shell model for DWCNTs are expressed as follows
Numerical examples and discussions
The DWCNTs under consideration are simply supported at both ends, as shown in Fig. 1, and water that is driven by pressure goes through the inner tube at a steady velocity. The inner and outer tubes have the same length L. The gap between the two walls is 0.34 nm. The parameters that are used to calculate the vdW interaction coefficient are taken as ε=2.968 mev, a = 1.42Ao and σ=3.407Ao (He et al., 2005, Saito et al., 2001). The other modeling parameters of the tubes are Eh = 360 J/m2, the
Conclusions
Based on Donnell’s shell theory, double-walled carbon nanotubes (DWCNTs) conveying fluid are modeled by considering the vdW interaction between the inner and outer walls, in addition, the small scale effects of the DWCNTs are taken into account by using the theory of nonlocal elasticity. The instability of the DWCNTs that is induced by a pressure-driven steady flow is investigated. The numerical computations imply that as the flow velocity increases, DWCNTs have a destabilizing style to get
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.
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