7.2.1 Nonlocal Beam and Rod Models for Vibration of SWCNTs

The Euler–Bernoulli beam model is usually employed in the analysis of longer one-dimensional beamlike structures, normally with ratio of the length to the thickness over 20 in engineering structures. If the cross-sectional dimensions are not small enough compared to the length of the beam, we need to consider the effects of rotary inertia and shear deformation using Timoshenko beam model [68]. To account for small-scale effects on CNT’s mechanical behaviors, the nonlocal beam and shell models have been employed to overcome the limitation of the applicability of the classical or local continuum modes at small length scales [35,40,41,47,65,69].

The natural frequencies of the local Euler–Bernoulli beam are shown as [68]

${\omega }_n={\left({\beta }_{n}l\right)}^2\sqrt{\frac{EI}{\rho A{l}^4}},\left(n=1,\mathrm{2..}.\right)$ (7.1)

(7.1)

where ${\beta }_{1}l$=1.875 and 4.73 are for the fundamental frequency of a cantilevered beam and a clamped–clamped beam, respectively. l is the length of the beam.

The governing equation for free vibration of the nonlocal Euler–Bernoulli beam [67] is proposed to be:

$EI\frac{{\partial }^{4}u}{\partial x^4}+\rho A\left[\frac{{\partial }^{2}u}{\partial t^2}-{\left(e_{0}a\right)}^2\frac{{\partial }^{4}u}{\partial x^2\partial t^2}\right]=0$ (7.2)

(7.2)

The nonlocal Timoshenko model needs to be employed to account for the small scale effect and the effects of rotary inertia and shear deformation. The governing equations for the vibrations of nonlocal Timoshenko beam are as follows [47]:

$\begin{array}{l}GA\kappa \left(\frac{\partial \phi }{\partial x}+\frac{{\partial }^{2}u}{\partial x^2}\right)+\rho A{\omega }^{2}u=0,\hfill \\ EI\frac{{\partial }^2\phi }{\partial x^2}-GA\kappa \left(\frac{\partial u}{\partial x}+\phi \right)+\rho I{\omega }^2\phi -{\left(e_{0}a\right)}^2\left(\rho A{\omega }^2\frac{\partial u}{\partial x}+\rho I{\omega }^2\frac{{\partial }^2\phi }{\partial x^2}\right)=0\hfill \end{array}$ (7.3)

(7.3)

where $A$ is the beam cross-sectional area. $\rho$= 2.27 g/cm³ is the density of CNT. $\kappa$ is Timoshenko’s shear coefficient which depends on the shape of the cross-section. $\kappa$=0.5 is used for SWCNT [57]. G=E/(2(1+$\nu$)) is the shear modulus. $I=(\pi (D_O^4-D_I^4))/64$ is the area moment of inertia, where $D_O$ and $D_I$ are the outside diameter and inside diameter of SWCNTs, respectively. The wall thickness of an SWCNT is 0.34 nm. The length of the C–C bond is a=0.142 nm. The parameter $e_{0}a$ is the scale coefficient revealing the small scale effect on the responses of structures of nanosize.

Eringen [63] proposed $e_0$ as 0.39 by matching the dispersion curves from the nonlocal theory for a plane waves with the Born–Karman model of lattice dynamics. However, there are no experiments conducted to determine the value of $e_0$ for CNTs. Wang and Hu proposed $e_0=1/\sqrt{12}\approx 0.288$ used in the nonlocal beam model [57]. Zhang et al. [35] estimated $e_0\approx 0.82$ by matching the theoretical buckling strain obtained by the nonlocal thin shell model [66] to those from molecular mechanics simulations given by Sears and Batra [70]. A conservative estimate of the scale coefficient $e_{0}a$<2.0 nm for an SWCNT if the measured wave propagation frequency value for the SWCNT is greater than 10 THz [67]. Hu et al. [41] estimated the values of $e_0$=0.6 and 0.2 for the transverse and torsional wave propagation in CNTs, respectively, by matching the dispersions of CNTs observed from the MD results with the numerical results obtained from the nonlocal shell model. In this study, we will fix a=0.142 nm which is the length of C–C bond and provide an estimation of parameter $e_0$ by matching the fundamental frequency of SWCNTs observed from MD simulations with the numerical results obtained from the nonlocal beam and rod models. The parameter$e_0$ is postulated to be dependent on the diameter and boundary conditions, and independent on the length of SWCNT in this study.

The bending moment and shear force of a nonlocal Timoshenko beam are as follows [47]:

$M=EI\frac{d\varphi }{dx}-{\left(e_{0}a\right)}^2\left(\rho A{\omega }^{2}u+\rho I{\omega }^2\frac{d\varphi }{dx}\right)$ (7.4)

(7.4)

$Q=GA\kappa \left(\varphi +\frac{du}{dx}\right)$ (7.5)

(7.5)

The boundary conditions for a cantilevered nonlocal Timoshenko beam:

$M{|}_{x=0}=Q{|}_{x=0}=0,u{|}_{x=l}=\phi {|}_{x=l}=0$ (7.6)

(7.6)

The boundary conditions for a clamped–clamped nonlocal Timoshenko beam:

$u{|}_{x=0,l}=0,\phi {|}_{x=0,l}=0,$ (7.7)

(7.7)

The free vibration solution for CNTs can be expressed as:

$u(x,t)=U(x)e^{i\omega t},\phi (x,t)=\varphi (x)e^{i\omega t},$ (7.8)

(7.8)

where $i=\sqrt{-1}$. The natural frequencies and mode shapes of nonlocal Timoshenko beam can be calculated by the finite difference method.

Next, we will introduce nonlocal rod theory for the analysis of longitudinal and torsional vibrations of SWCNT. The nonlocal constitutive relation is as follows [63]:

$\sigma (x)-{(e_{0}a)}^2\frac{d^2\sigma (x)}{d{x}^2}=E\epsilon (x)$ (7.9)

(7.9)

Based on Eq. (7.9), the governing equation for the longitudinal vibration of the nonlocal rod is given by

$\frac{{\partial }^{2}u}{\partial t^2}-{(e_{0}a)}^2\frac{{\partial }^{4}u}{\partial x^2\partial t^2}=\frac{E}{\rho }\frac{{\partial }^{2}u}{\partial x^2}$ (7.10)

(7.10)

Substituting u=U(x)e^iwt into (7.10) and using the boundary conditions, we can find the natural frequency of longitudinal and torsional vibrations of cantilevered and clamped–clamped nonlocal rods.

The natural frequency of longitudinal vibration of a cantilevered nonlocal rod is derived as

${\omega }_n=\frac{(2n-1)\pi }{2l}\sqrt{\frac{E}{\rho [1+{(e_{0}a)}^2{((2n-1)\pi /(2l))}^2]}},n=(1,2\dots )$ (7.11)

(7.11)

The natural frequency of longitudinal vibration of a clamped–clamped nonlocal rod is derived as

${\omega }_n=\frac{n\pi }{l}\sqrt{\frac{E}{\rho [1+{(e_{0}a)}^2{(n\pi /l)}^2]}},n=(1,2\dots )$ (7.12)

(7.12)

For torsional vibration of nonlocal rod, the Young’s modulus E in the above formulas of natural frequencies should be replaced by shear modulus G. Poisson’s ratio $\nu$ and Young’s modulus E calculated from the MD simulations for the test of axial tension are listed in Table 7.1.

Table 7.1

Young’s Modulus and Poisson’s Ratio of SWCNT [40]

SWCNT	Young’s Modulus (TPa)	Poisson’s Ratio
(5, 5)	E=0.89	$\nu$=0.25
(10, 10),(15, 15)	E=0.83	$\nu$=0.317
(10, 0)	E=0.96	$\nu$=0.21
(19, 0)	E=0.81	$\nu$=0.35
(20, 20)	E=0.84	$\nu$=0.312

MD simulations for SWCNTs are conducted to verify the applicability of the nonlocal beam and nonlocal rod models for free transverse, longitudinal, and torsional vibrations of SWCNTs [71]. The interaction of atoms is described by the second-generation REBO potential energy expression for hydrocarbons [60] and there is no temperature control in the MD simulations. The verlet algorithm in the velocity form with time step 1 fs is used to simulate the vibration of SWCNTs. The shortest and longest SWCNTs including (5, 5), (10, 10), (15, 15), and (20, 20) SWCNTs in the MD simulations are 1.2 and 9.0 nm, respectively.

The atoms at the end of SWCNT and the middle of SWCNT are moved away from their equilibrium positions in the Y direction as shown in Fig. 7.1A and B, respectively. Fig. 7.1C and D shows that the atoms will be moved in the Z direction to produce the longitudinal vibration of SWCNT. The torsional vibration of SWCNT can be realized through releasing the torsional SWCNT in MD simulation. For example, when SWCNT are stretched from its equilibrium position in the Y direction, the tension acts as the restoring force that brings the SWCNT to its original undeformed position. The deformed SWCNT with different boundary conditions can vibrate transversely at its natural frequency. The displacement d of a cross-section of the SWCNT is calculated as follows: $d=1/n{\sum }_{i=1}^n{y}_i$, where n is the number of atoms at the cross-section of SWCNT. $y_i$ is the displacement of the i atom. Each atom at the cross-section passes through its respective equilibrium position simultaneously which is observed from the MD simulations. Fig. 7.2 shows that the fundamental vibration mode can be excited after a small deflection of cantilevered SWCNT is produced. The transverse deflections of SWCNT during the vibration at different time are presented. The vibrations of the end of the cantilevered (10, 10) SWCNT with different lengths are shown in Fig. 7.3. The periods and fundamental frequencies can be calculated from the MD simulations.

7.2.2 Nonlocal Elastic Beam Models for Flexural Wave Propagation in DWCNTs

In nonlocal elasticity [62], the stress at a reference point x is considered to be a function of the strain field at every point in the body. The basic equations for linear, homogeneous, isotropic, nonlocal elastic solid with zero body force are given as [63]

$\begin{array}{ll}{\sigma }_{ij.j}\hfill & =0,\hfill \\ {\sigma }_{ij}(x)\hfill & =\int \alpha (\left|x-x\prime \right|,\tau )C_{ijkl}{\epsilon }_{kl}(x\prime )dV(x\prime ),\forall x\in V,\hfill \\ {\epsilon }_{ij}\hfill & =\frac{1}{2}(u_{i,j}+u_{j,i}),\hfill \end{array}$ (7.13)

(7.13)

where ${\sigma }_{ij}$ and ${\epsilon }_{ij}$ are stress and strain tensors, respectively, $C_{ijkl}$ is the elastic modulus tensor in classical isotropic elasticity, $u_i$ is the displacement vector, $\alpha (\left|x-x\prime \right|,\tau )$ is the nonlocal modulus or attenuation function which incorporates the nonlocal effects at the reference point x produced by local strain at the source $x\prime$ into the constitutive equations, and $\left|x-x\prime \right|$ is the Euclidean distance. In$\tau =e_{0}a/l,$ e₀ is a constant appropriate to each material, a is an internal characteristic length (e.g., length of C–C bond, lattice parameter, and granular distance), and l is an external characteristic length (e.g., crack length and wavelength).

Integral-partial differential equations of the above linear nonlocal elasticity are reduced to singular partial differential equations of a special class of physically admissible kernel [63]. Hence, Hooke’s law for stress and strain relation in the polar coordinate system is thus expressed by

$\sigma -{(e_{0}a)}^2\frac{{\partial }^2\sigma }{\partial x^2}=E\epsilon ,$ (7.14)

(7.14)

where $\sigma$ is the axial stress and $\epsilon$ is the axial strain. The parameter e₀a is the scale coefficient revealing the small-scale effect on the responses of structures of nanosize. The parameter e₀=0.39 was given by Eringen [63], but there are no experiments conducted to determine the value of e₀ for CNT. Wang and Hu [57] proposed e₀$=1/\sqrt{12}\approx 0.028$ used in the nonlocal beam model. Zhang et al. [35] estimated e₀≈0.82 by matching the theoretical buckling strain obtained by the nonlocal thin shell model [66] to those from molecular mechanics simulations given by Sears and Batra [70]. A conservative estimate of the scale coefficient e₀a<2.0 nm for an SWCNT if the measured wave propagation frequency value for the SWCNT is greater than 10 THz [67]. Hu et al. [41] gave the estimation of e₀ investigated through the comparison between the dispersion relations obtained from the numerical results of governing equations of shell and MD simulation. The value of e₀ which is a coefficient to identify the scale effect should be different for different issues. In this study, we will fix a=0.142 nm and give the estimation of parameter e₀ by matching the dispersions of CNTs observed from the MD results with the numerical results obtained from the nonlocal double-beam model. The parameter a will be assumed to be independent of the length, diameter, and chirality of CNTs for the wave dispersion.

The governing equation for DWCNT’s wave propagation by considering the vdW effect via the nonlocal Euler–Bernoulli beam model is given as [35]

$\begin{array}{c}E{I}_1\frac{{\partial }^4{u}_1}{\partial x^4}+\rho A_1\frac{{\partial }^2{u}_1}{\partial t^2}-{(e_{0}a)}^2(\rho A_1\frac{{\partial }^4{u}_1}{\partial x^2\partial t^2}-\frac{{\partial }^2{p}_1}{\partial x^2})=p_1,\\ E{I}_2\frac{{\partial }^4{u}_1}{\partial x^4}+\rho A_2\frac{{\partial }^2{u}_1}{\partial t^2}-{(e_{0}a)}^2(\rho A_2\frac{{\partial }^4{u}_1}{\partial x^2\partial t^2}-\frac{{\partial }^2{p}_2}{\partial x^2})=p_2.\end{array}$ (7.15)

(7.15)

The wave propagation solution for CNTs can be expressed as

$u_i(x,t)=U_i{e}^{i(kx-\omega t)}$ (7.16)

(7.16)

where U_i is the amplitude of the wave motion, k is the wave number, and $\omega$ is the frequency of the wave motion.

The analytic solution can be obtained from the Euler–Bernoulli beam model,

$\begin{array}{l}{\upsilon }_1=\sqrt{\frac{-B+\sqrt{B^2-4AC}}{2A}},\\ {\upsilon }_2=\sqrt{\frac{-B-\sqrt{B^2-4AC}}{2A}}\end{array}$ (7.17)

(7.17)

where

$\begin{array}{ll}A\hfill & ={\rho }^2{A}_1{A}_2{k}^4{\left[1+{(e_{0}a)}^2{k}^2\right]}^2,\hfill \\ B\hfill & =-\rho k^2\left[1+{(e_{0}a)}^2{k}^2\right]\left[\left(A_1{I}_2+A_2{I}_1\right)E{k}^4+{\left(e_{0}a\right)}^2{k}^2\left(A_1{c}_{21}-A_2{c}_{12}\right)-A_1{c}_{21}-A_2{c}_{12}\right],\hfill \\ C\hfill & =E{I}_1{I}_2{k}^8-E{I}_1{k}^4{c}_{21}\left[1-{(e_{0}a)}^2{k}^2\right]-E{I}_2{k}^4{c}_{12}\left[1+{(e_{0}a)}^2{k}^2\right]\hfill \end{array}$ (7.18)

(7.18)

The governing equation for DWCNT’s wave propagation by considering the vdW effect via the nonlocal Timoshenko beam model is given as [43]

$\begin{array}{l}G{A}_1\kappa \left(\frac{\partial {\phi }_1}{\partial x}-\frac{{\partial }^2{u}_1}{\partial x^2}\right)+\rho A_1\frac{{\partial }^2{u}_1}{\partial t^2}=p_1,\hfill \\ G{A}_1\kappa \left[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2}\right]\left(\frac{\partial u_1}{\partial x}-{\phi }_1\right)+E{I}_1\frac{{\partial }^2{\phi }_1}{\partial x^2}-\rho I_1\frac{{\partial }^2}{\partial t^2}\left[{\phi }_1-{\left(e_{0}a\right)}^2\frac{{\partial }^2{\phi }_1}{\partial x^2}\right]=0,\hfill \\ G{A}_2\kappa \left(\frac{\partial {\phi }_2}{\partial x}-\frac{{\partial }^2{u}_2}{\partial x^2}\right)+\rho A_2\frac{{\partial }^2{u}_2}{\partial t^2}=p_2,\hfill \\ G{A}_2\kappa \left[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2}\right]\left(\frac{\partial u_2}{\partial x}-{\phi }_1\right)+E{I}_2\frac{{\partial }^2{\phi }_2}{\partial x^2}-\rho I_2\frac{{\partial }^2}{\partial t^2}\left[{\phi }_2-{(e_{0}a)}^2\frac{{\partial }^2{\phi }_2}{\partial x^2}\right]=0.\hfill \end{array}$ (7.19)

(7.19)

where p_k (k=1, 2) is the vdW force between the adjacent tubes. The (inward positive) net radial pressure p_k on the tube k is given as follows:

$p_1=c_{12}(u_1-u_2)\mathrm{and}{p}_2=c_{21}(u_2-u_1),$ (7.20)

(7.20)

where c_ij is the vdW interaction coefficient, which is estimated as [24]

$c_{ij}=-\left[\frac{1001\pi \epsilon {\sigma }^{12}}{3a^4}{E}_{ij}^{13}-\frac{1120\pi \epsilon {\sigma }^6}{9a^4}{E}_{ij}^{17}\right]R_j$ (7.21)

(7.21)

and

$E_{ij}^m={(R_j+R_i)}^{-m}{\int }_0^{\pi /2}\frac{d\theta }{{\left[1-K_{ij}{\cos }^2\theta \right]}^{m/2}},$ (7.22)

(7.22)

in which

$K_{ij}=\frac{4R_j{R}_i}{{(R_j+R_i)}^2}.$ (7.23)

(7.23)

Nonlocal beam theory can conform to the local theory if the scale coefficient e₀ is set to zero.

In order to check the applicability of the dispersion relations by the nonlocal double-beam model, MD simulation for DWCNTs is also presented. The lengths of the DWCNTs are 19.7 nm. The interaction of atoms in each tube is described by the second-generation REBO potential energy expression for hydrocarbons [60], while the vdW force between inner and outer walls is described by the Lennard–Jones potential [72]. Poisson’s ratio and Young’s modulus calculated from the MD simulations for the test of axial tension are listed in Table 7.1.

7.5.1 Nonlocal Elastic Beam Models for Flexural Wave Propagation

The phase velocity and wavenumber can be determined from the flexural vibration of two arbitrary sections of a CNT that is simulated using MD. The transverse displacement of some section of an armchair–armchair (10, 10)@(15, 15) DWCNT can be obtained by computing the average displacements of the atoms in the section. The harmonic deflection is achieved by shifting the edge atoms at one end of the CNT shown in Fig. 7.11. For example, the atoms that are located at the end of the armchair–armchair (10, 10)@(15, 15) DWCNT denoted by 0 are assumed to be subjected to a harmonic deflection of period T=500 fs, as shown in Fig. 7.12. The corresponding angular frequency is $\omega$=2/T $\approx$ 1.26$\times$10¹³ rad/s while the other end is kept free. Fig. 7.12B and C shows the flexural vibrations of 1 at x₁=2.46 nm and 2 at x₂=4.92 nm, respectively, of the DWCNT that is simulated using MD. If the transient deflection of the first two periods is neglected, then the propagation duration $\mathrm{\Delta }$t of the wave from 1 to 2 can be estimated as

$\mathrm{\Delta }t\approx \frac{(t_{23}-t_{13})+(t_{24}-t_{14})+\cdots +(t_{2n}-t_{1n})}{n-2}$ (7.31)

(7.31)

The phase velocity and wave number are expressed as follows, respectively,

$c=\frac{x_2-x_1}{\mathrm{\Delta }t},k=\frac{2\pi }{\lambda }=\frac{\omega T}{\lambda }=\frac{\omega }{c}.$ (7.32)

(7.32)

One section of the inner tube and outer tube at period T=500 fs is presented in Fig. 7.13. The inner tube and outer tube are observed to vibrate coaxially in the figure.

Fig. 7.14 illustrates the dispersion relations between the phase velocity c and the wave number k of the flexural wave in the (10, 10)@(15, 15) DWCNT. The flexural wave dispersion relation derived from the above governing equation for nonlocal double Euler–Bernoulli and Timoshenko beam models are showed in Fig. 7.14. The classical (or local) beam theories cannot reveal the small effect on flexural wave dispersion. The scale coefficients, e₀=6.5 and a=0.142 nm, are used in the nonlocal elastic double Timoshenko beam model. When the wave number is approximately smaller than 1$\times$10⁹ 1/m, the phase velocities given by local or nonlocal beam models are close to each other and they can all predict the MD simulations well. When the wave number is larger than 1.5$\times$10⁹ 1/m, the phase velocity decreases, and it is found that only the nonlocal elastic double Timoshenko beam model can predict the flexural wave dispersion of the DWCNT. Figs. 7.15 and 7.16 show the dispersion relation of flexural wave in (5, 5)@(10, 10) DWCNT and (10, 0)@(19, 0)DWCNT. It is clearly seen that only the nonlocal elastic double Timoshenko beam model can predict the flexural wave dispersion of the DWCNTs when the wave numbers are larger than 2.4$\times$10⁹ 1/m and 2.44$\times$10⁹ 1/m for (5, 5)@(10, 10) DWCNT and (10, 0)@(19, 0) DWCNT, respectively. The nonlocal Euler beam model fails to describe wave propagation in the DWCNTs. The values of the scale coefficient e₀=4.3 and 6.1 are used in the nonlocal double Timoshenko beam model for the two DWCNTs. In addition, it is found that there are little differences among nonlocal models with different scale parameters e₀ when the wave number is small. The good agreement between the dispersion relation induced from the nonlocal elastic double Timoshenko beam model and the MD simulations verifies the effectiveness of this model.

Fig. 7.17 provides the vibration of the (10, 10)@(15, 15) DWCNT at the period T=330 fs and the corresponding angular frequency is $\omega$=2/T≈1.9$\times$10¹³ rad/s. From the figure, it shows that when the frequency is larger than 1.9$\times$10¹³ rad/s, a noncoaxial vibration of the DWCNT will occur. The average phase difference between the inner tube and the outer tube can be estimated from Fig. 7.17 as $\overline{t}=1/n{\sum }_{i=1}^n(t_{0i}-t_{1i})\approx 165.3\mathrm{fs}$. In order to check the effect of the vdW force on the noncoaxial vibration of the DWCNT, MD simulations for flexural wave in SWCNTs are conducted. Fig. 7.18 indicates that the (10, 10) and (15, 15) SWCNTs have different phases at the period T=330 fs without considering the vdW forces between them. The average phase difference between the (10, 10) and (15, 15) SWCNTs can also be estimated from Fig. 7.18 $\overline{s}=1/n{\sum }_{i=1}^n(s_{0i}-s_{1i})\approx 153.3\mathrm{fs}$. The other average phase differences at different periods are shown in Table 7.5. The phase difference of the vibration becomes increasingly obvious with the decrease in the period. The phase difference increases from 11.8 to 165.3 fs as the period decreases from 600 to 330 fs for DWCNTs. Similarly, the phase difference increases from 12.5 to 153.3 fs as the period decreases from 600 to 330 fs for SWCNTs. The values of $\overline{s}$ and $\overline{t}$ are very close at the periods T=500 fs and T=600 fs, at which the phase velocity also starts to decrease, as indicated in Fig. 7.14. It is concluded that the different small-scale effects of SWCNTs with different diameters have a significant effect on the noncoaxial flexural vibration of DWCNTs at high frequency. The vdW forces make no significant contribution to the phase differences, as the noncoaxial vibration of DWCNTs would occur without consideration of the vdW forces between the adjacent tubes. The continuum mechanics model is inapplicable in the analysis of flexural wave in DWCNT when the noncoaxial vibration of DWCNTs happens at high frequency, especially when the angular frequency is larger than 1.9$\times$10¹³ rad/s.

Table 7.5

Average Phase Differences of the Vibration of the Section Which Is 2.46 nm Ahead of the End of the DWCNT (10, 10)@(15, 15) and the SWCNTs (10, 10) and (15, 15) [40]

Period (fs)	DWCNT(10, 10)@(15, 15) (fs)	SWCNTs (10, 10) and (15, 15) (fs)	$\left|\overline{\mathit{s}}\mathbf{-}\overline{\mathit{t}}\right|$ (fs)
T=330	$\overline{t}$=165.3	$\overline{s}$=153.3	12
T=400	$\overline{t}$=91	$\overline{s}$=82	9
T=500	$\overline{t}$=27	$\overline{s}$=28	1
T=600	$\overline{t}$=11.8	$\overline{s}$=12.5	0.7

In summary, the study of flexural wave dispersion in DWCNTs in a wide range of wavenumbers by the nonlocal elastic double-beam theory and MD simulations is presented. The results indicate that only the nonlocal elastic double Timoshenko beam model is able to predict the small effect on flexural wave dispersion when the wavenumber becomes so large that the CNT microstructure plays an important role in the flexural wave. These small-scale effects have a significant effect on the noncoaxial flexural vibration of DWCNTs at high frequency. The vdW interaction has little effect on the noncoaxial flexural vibration of DWCNT, and the nonlocal elastic Timoshenko beam theory is inapplicable in modeling the noncoaxial wave propagation in CNTs.

7.5.2 Nonlocal Shell Model for Elastic Wave Propagation

It is quite straightforward to determine the phase velocity and wavenumber from the transverse vibration of atoms in two arbitrary sections of a CNT that was simulated using MD. The harmonic deflection of some atoms was achieved by shifting the atoms at one end of the nanotube while the other end was kept free, as shown in Fig. 7.5A. For example, some atoms that are located at the end of the armchair (15, 15) SWCNT denoted by Section 0 were assumed to be subject to the harmonic deflection of period $T=500\mathrm{fs}$, as shown in Fig. 7.19A. The corresponding angular frequency is$\omega =2\mathrm{\pi }/T\approx 1.26\times {10}^{13}\text{rad}/\mathrm{s}$. Fig. 7.19B and C show the transverse vibrations of the atom in Section 1 at x₁=2.46 nm and Section 2 at x₂=4.92 nm, respectively. If the transient deflection of the first two periods is neglected, then the propagation duration $\mathrm{\Delta }t$ of the wave from Section 1 to Section 2 can be estimated as

$\mathrm{\Delta }t\approx \frac{(t_{23}-t_{13})+(t_{24}-t_{14})+\cdots +(t_{2n}-t{}_{1n})}{n-2}$ (7.33)

(7.33)

The phase velocity and wavenumber are, respectively,

$c=\frac{x_2-x_1}{\mathrm{\Delta }t},k=\frac{2\pi }{\lambda }=\frac{\omega T}{\lambda }=\frac{\omega }{c}$ (7.34)

(7.34)

When the atoms at the end of CNT are subjected to the harmonic excitation indicated in Fig. 7.5B, the torsional wave will propagate in SWCNTs as shown in Fig. 7.20. The torsional wave velocity can be calculated through the method for calculating transverse wave velocity.

Fig. 7.21 illustrates the dispersion relations between the phase velocity $c$ and the wavenumber k of the transverse wave in the armchair (15, 15) SWCNT and zigzag (20, 0) SWCNT. Fig. 7.21 shows the nonlocal shell model can predict the MD results better than the local shell model can when the wave lengths are less than $2.36\times {10}^{-9}\mathrm{m}$ and $1.88\times {10}^{-9}\mathrm{m}$ for (15, 15) SWCNT and (20, 0) SWCNT, respectively. The value of the parameter $e_0$=0.6 is used for transverse wave in armchair (15, 15) and zigzag (20, 0) CNT.

Fig. 7.22 shows that the nonlocal elastic cylindrical double-shell model can predict the dispersion relation of the transverse wave better than local double-shell model when the wavenumber is large enough. This is because the interlayer van der Waals interaction of the DWCNT and the small-scale effect on the dispersion of the transverse wave of the SWCNT are considered. The phase velocity decreases when the wavenumber is approximately larger than $2.65\times {10}^9$ 1/m, or the wavelength is approximately less than $2.35\times {10}^{-9}\mathrm{m}$. The nonlocal elastic double-shell model can predict the wave dispersion better than local shell model when the wavelength is approximately smaller than $2.35\times {10}^{-9}\mathrm{m}$. Fig. 7.23 indicates that the nonlocal shell model can predict dispersion relation obtained from the MD simulation of torsional wave in (10, 10) SWCNT and (15, 15) SWCNT better than local shell model can when the wavelength is smaller than $0.95\times {10}^{-9}\mathrm{m}$ and $0.9\times {10}^{-9}\mathrm{m}$, respectively. The good agreement between the nonlocal elastic cylindrical shell models results and the MD simulations verifies the effectiveness of this model. The value of the parameter $e_0$=0.6 is adopted for (10, 10)@(15, 15) DWCNTs to predict the dispersion of a transverse wave in CNTs, and $e_0$=0.2 and 0.23 are used for a torsional wave in SWCNT (10, 10) and SWCNT (15, 15), respectively. The estimates of $e_0$ are investigated through the comparison between the dispersion relations obtained from numerical results of equations and MD simulation. The value of $e_0$ which is a coefficient to identify the scale effect should be different for different issues, such as dispersions of different elastic waves in SWCNT and DWCNT. The scale coefficient is dependent on SWCNTs and DWCNTs for transverse wave, and CNTs for torsional wave. When the value of $e_0$ becomes smaller, the scale effect will be weaker. The torsional wave motion is only related to the displacement in tangential direction, while the transverse wave motion is involved in the displacements in longitudinal, radial, and tangential direction. The carbon atoms more complicated vibration in the transverse wave propagation in CNT leads to the stronger scale effect on the dispersion of the transverse wave. This is the main reason why the vale of $e_0$=0.2 for the torsional wave in SWCNT is much smaller than the values of $e_0$=0.6 used for transverse wave in CNTs. The structure of CNTs, such as chirality, diameter, and layers, may also be the factors to affect the value of $e_0$. But for the same problems, such as transverse wave or torsional wave, in CNT, these structure differences lead to a relatively small change of the value of $e_0$ compared to the changes of values for different problems.

Fig. 7.24 indicates that the obvious phase difference of inner and outer tube of the (10, 10)@(15, 15) DWCNT will occur at the period T=300 fs. The corresponding angular frequency is $\omega =2\pi /\mathit{T}\approx 3.14\times {10}^{13}$ rad/s. The average phase difference between the inner tube and the outer tube can be estimated from Fig. 7.8 as $\overline{t}=\frac{1}{n}{\sum }_{i=1}^n(t_{0i}-t_{1i})\approx 103.2\mathrm{fs}$. Fig. 7.25 indicates that the SWCNTs (10, 10) and (15, 15) have different phases at the period T=300 fs without considering the van der Waals forces between them. The average phase difference between the SWCNTs (10, 10) and (15, 15) can also be estimated from Fig. 7.25: $\overline{s}=\frac{1}{n}{\sum }_{i=1}^n(s_{0i}-s_{1i})\approx 101.2$ fs. The other average phase differences at different periods are shown in Table 7.6. The phase difference of the vibration becomes increasingly obvious with the decrease of the period. The phase difference increases from 43.3 to 103.2 fs as the period decreases from 700 to 300 fs for DWCNTs. Similarly, the phase difference increases from 40 to 101.2 fs as the period decreases from 700 to 300 fs for SWCNTs. The van der Waals forces make no significant contribution to the phase differences of the transverse wave in the inner and outer tube, as the phase differences would occur without consideration of the van der Waals forces between the adjacent tubes. This indicates that the small-scale effects can change the phase when the frequency is extremely high.

Table 7.6

The Average Phase Differences of the Vibrations of Atom at the Section, 2.46 nm Ahead of the End of the DWCNT (10, 10)@(15, 15) and the SWCNTs (10, 10) and (15, 15) [73]

Period (fs)	DWCNT (10, 10)@(15, 15) (fs)	SWCNTs (10, 10) and (15, 15) (fs)	|$\overline{\mathit{s}}\mathit{-}\overline{\mathit{t}}$| (fs)
T=300	$\overline{t}=$103.2	$\overline{s}=$101.2	2
T=400	$\overline{t}=$80.3	$\overline{s}=$74.1	6.2
T=500	$\overline{t}=$59.8	$\overline{s}=$38.8	21
T=600	$\overline{t}=$54	$\overline{s}=$29	25
T=700	$\overline{t}=$43.3	$\overline{s}=$40	3.3

In summary, the study of transverse and torsional wave dispersion in SWCNTs and DWCNTs on the basis of the nonlocal elastic cylindrical shell theory and the use of MD simulations for SWCNT and an armchair–armchair (10, 10)@(15, 15) DWCNT in a wide range of wavenumbers are presented. The value of parameter $e_0$ is estimated based on the MD result to predict the dispersion of a transverse wave in CNTs through using the nonlocal shell models. The results of this study indicate that the nonlocal elastic cylindrical shell model is able to offer a better prediction for transverse and torsional wave dispersion in CNTs than the local elastic shell model when the wavenumber is large enough. The nonlocal shell models are required when the wavelengths are approximately less than $2.36\times {10}^{-9}\mathrm{m}$ and $0.95\times {10}^{-9}\mathrm{m}$ for a transverse wave in armchair (15, 15) SWCNTs and a torsional wave in armchair (10, 10) SWCNTs, respectively. These small-scale effects have a significant effect on the phase difference of a transverse wave in the inner and outer tubes of DWCNTs at high frequency.