Nonlocal effect in carbon nanotube resonators: A comprehensive review

Abstract

This article provides a comprehensive review about the use of nonlocal theory in modeling of the vibrations of the carbon nanotube resonators. It is fully described how the nonlocal effect has been exploited by researchers to mathematically model vibrations of carbon nanotube resonators. Fusion of different classical beam theories with nonlocal theory is discussed. Also, the article presents the combination of nonlocal models with different physical phenomena which have influence in the vibrations of carbon nanotube resonators.

Introduction

Discovery of carbon nanotubes (CNTs) by Iijima ¹ in 1991 was an important step toward developing different scientific fields. In 1991, in the revolutionary article by Iijima, he reported a new type of finite carbon structure including needle-like tubes. His significant article in Nature originally introduces extraordinary materials with exceptional properties which fascinate researchers to exploit them in a wide range of applications from mechanical engineering to medicine. From the early stage of CNTs discovery, interest about better understanding of their properties was highly stimulated, and accordingly, researchers had too many speculations and ideas about these extraordinary materials. As a result, scientists have been prompted to scrutinize different features of CNTs considering different points of view. In addition, they broached different techniques for synthesis of CNTs. Table 1 briefly describes few of the principal and revolutionary articles about CNTs, and the first steps in the road of the CNTs exploration.

Table 1.

Principal works toward exploring of CNTs.

Principal contribution
Discovery of CNTs 1
Growth model for CNTs 2
Electronic structure 3
The synthesis of graphitic nanotubes in gram quantities 4
Nanocapillarity of Tubules 5
Reporting striking variations in electronic transport of CNTs 6
Synthesis of single-shell tubes with diameter of 1 nm 7
Raman spectra of CNTs 8
Effect of van der Waals forces on CNTs 9
Interlayer spacing 10
Effect of magnetic field on CNTs 11
Energy bands of CNTs 12
Optical properties of CNTs 13
Electrical resistance of CNTs 14
Field emission source of CNTs 15
Thermal properties of CNTs 16
Geometry analysis of CNTs 17
Structural flexibility of CNTs 18

CNTs: carbon nanotubes.

The aforementioned fundamental works reveal peculiar features of CNTs. Providing an extensive review about fundamental works in CNTs is beyond the scope of this review article. This review article aims to provide information about the utilization of nonlocal effect in vibration analysis of CNTs. In the next section, a few principle works for vibration modeling of carbon nanotube resonators (CNTRs) are presented, and then, we only focus on the nonlocal theory and its application in CNTRs.

Vibration theory for CNTRs modeling

Extraordinary mechanical properties of the CNTRs have prompted vibration researchers to explore their vibrational features. CNTRs have been experimentally and theoretically studied by many researchers for the purpose of having better understanding from the peculiar behavior of CNTRs. For example, the pioneering effort of Treacy et al. ¹⁹ results in finding exceptionally high Young’s modulus of the CNTs. Another primary endeavor for vibration exploring of the CNTs is an article by Sumpter and Noid. ²⁰ They showed the significance of the nonlinear resonance to positional stability in CNTs. The fundamental resonance of arc-produced multi-walled CNTs induced by an electric field in transmission electron microscopy (TEM) was initially measured by Poncharal et al. ²¹ By the beginning of 21st century, trends of publication in the field of vibration analysis of CNTRs drastically increase. During this time, a group of researchers at University of Alberta was working on the theoretical modeling of CNTRs. In one of their principal work, they formulated coaxial vibrations of the multi-walled CNTRs. Their proposed model is mathematically described by the following equations ²²

$$\begin{array}{l}{C}_1[w_2-w_1]=E{I}_1\frac{d^4{w}_1}{d{x}^4}+\rho A_1\frac{d^2{w}_1}{d{t}^2},\\ C_2[w_3-w_2-]-C_1[w_2-w_1]=E{I}_2\frac{d^4{w}_2}{d{t}^4}+\rho A_2\frac{d^2{w}_2}{d{t}^2},\\ ⋮\\ C_{N-1}[w_N-w_{N-1}]=E{I}_N\frac{d^4{w}_2}{d{x}^4}+\rho A_N\frac{d^2{w}_N}{d{t}^2}\end{array}$$

(1)

where x represents the axial coordinate; t is the time; $w_k(x,t)(k=1,2,\dots ,N)$ shows the deflection of the kth tube; and $A_k$ and $I_k$ indicate the cross-sectional area and the moment of inertia of the kth tube, respectively. The subscripts 1, 2,…, N show the quantities of the innermost tube, its adjacent tube, and the outermost tube, respectively. E, $\rho$ , and $C_k$ indicate the modulus of elasticity, density of CNTs, and interaction coefficients, respectively. ²² After developing equation (1), they proposed novel analytical models for CNTRs considering different phenomena such as terahertz wave propagation, ²³ sound wave propagation, ²⁴ and instability of fluid-conveying CNTs. ²⁵ During 2000–2005, different mathematical models were proposed for vibrations analysis of CNTs. Table 2 describes the proposed mathematical models by researchers.

Table 2.

Vibrations models for CNTRs: early stage of developments.

Developed mathematical model
Mahan 26 $-{\omega }^{2}u(r,t)=c_t^2{\nabla }^{2}u(r,t)+\Lambda \nabla (\nabla ·u(r,t))$ where $\Lambda =c_l^2-c_t^2$ . $c_t,c_l$ and $u(r,t)$ show the transverse and longitudinal velocities of sounds, and vector of displacement, respectively
Sound wave propagation in CNTs 24 $\begin{array}{c}\multicolumn{1}{c}{C_1[w_2-w_1]=E{I}_1\frac{d^4{w}_1}{d{t}^4}+\rho A_1\frac{d^2{w}_1}{d{t}^2},}\\ \multicolumn{1}{c}{C_2[w_3-w_2-]-C_1[w_2-w_1]=}\\ \multicolumn{1}{c}{E{I}_2\frac{d^4{w}_2}{d{t}^4}+\rho A_2\frac{d^2{w}_2}{d{t}^2},}\\ \multicolumn{1}{c}{C_{N-1}[w_N-w_{N-1}]=E{I}_N\frac{d^4{w}_2}{d{t}^4}}\\ \multicolumn{1}{c}{+\rho A_N\frac{d^2{w}_N}{d{t}^2}}\end{array}$
Thermally induced vibrations of CNTs 27 $\frac{d{\zeta }^2}{d{t}^2}=\frac{\mathrm{YI}}{\rho A}\frac{d{\zeta }^4}{d{x}^4}$ , ${\omega }_i=\frac{{\beta }_i^2}{L^2}\sqrt{\frac{\mathrm{YI}}{\pi d{\rho }_{2d}}}$
Equivalent space frame-like structures and intertube $[M]\stackrel{··}{y}+[K]y=0$ where M and K are the equivalent mass, and stiffness of CNTRs 28
Timoshenko beam theory for CNTRs $-\mathrm{KG}{A}_1(\frac{\partial {\varphi }_1}{\partial x}-\frac{{\partial }^2{Y}_1}{\partial x^2})+p=\rho A_1\frac{{\partial }^2{Y_1}^2}{\partial x^2}$ ; $-E{I}_1\left(\frac{{\partial }^2{\varphi }_1}{\partial x^2}\right)-\mathrm{KG}{A}_1({\varphi }_1-\frac{{\partial }^2{Y_1}^2}{\partial x^2})=\rho I_1\frac{{\partial }^2{\varphi }^2}{\partial t^2}$ $-\mathrm{KG}{A}_2(\frac{\partial {\varphi }_2}{\partial x}-\frac{{\partial }^2{Y}_2}{\partial x^2})-p=\rho A_2\frac{{\partial }^2{Y_2}^2}{\partial x^2}$ ; $-E{I}_2\left(\frac{{\partial }^2{\varphi }_2}{\partial x^2}\right)-\mathrm{KG}{A}_2({\varphi }_1-\frac{{\partial }^2{Y}_1}{\partial x^2})=\rho I_2\frac{{\partial }^2{\varphi }^2}{\partial t^2}$ where $p=c(Y_2-Y_1)$ 29
Wave propagation of CNTRs 30 $L_A^1{w}_1+L_B^1[{\eta }_1(w_2-w_1)]=0$ , $L_A^1{w}_2+L_B^1[-{\eta }_1(w_2-w_1)]=0,\eta =\mathrm{cK}$ $L_A^1$ and $L_B^1$ are differential operators
Instability in suspended CNTRs 31 $m_{\mathrm{eff}}\stackrel{··}{x}+\gamma \stackrel{·}{x}+\mathrm{kx}=\frac{F_0{(C_{2}V-q)}^2}{1-\alpha x^2}$
Compressive axial load on vibrations of CNTRs 32 $C[w_2-w_1]+{\sigma }_x^0{A}_1{w″}_1=E{I}_1{w}_1^{\mathrm{IV}}+\rho A_1{\stackrel{··}{w}}_1$ $C[w_1-w_2]+{\sigma }_x^0{A}_2{w″}_1=E{I}_2{w}_2^{\mathrm{IV}}+\rho A_2{\stackrel{··}{w}}_2$
Fluid-conveying CNTRs 25 $\begin{array}{c}EI\frac{{\partial }^{4}w}{\partial x^4}+[M{V}^2+P^{\ast }-T]\frac{{\partial }^{2}w}{\partial x^2}+2MV\frac{{\partial }^{2}w}{\partial x\partial t}\\ =(M+m)\frac{{\partial }^{2}w}{\partial t^2}+\mathrm{kw}=0\end{array}$
Parametric excitation of CNTs 33 $\rho A\frac{{\partial }^{2}y}{\partial t^2}+P\frac{{\partial }^{2}y}{\partial x^2}+E_b{I}_G\frac{{\partial }^{4}y}{\partial x^4}$ where $P=P_0+P(t)$ $P_0={\alpha }^2\beta e^2\left[\frac{V_{\mathrm{dc}}^2+V_{\mathrm{ac}}^2}{2}\right]$ $P(t)={\alpha }^2\beta e^2[2V_{\mathrm{dc}}{V}_{\mathrm{ac}}\cos (\omega t)+\left(\frac{V_{\mathrm{ac}}^2}{2}\right)\cos (2\omega t)]$

CNTs: carbon nanotubes; CNTRs: carbon nanotube resonators.

To develop a mathematical model for the nonlinear vibrations of CNTRs, Liu et al. ³⁴ derived the governing equation and obtained effect of rippling mode on resonance of CNTRs. Vibrations of suspended nanotube oscillator were studied by Ustunel et al., ³⁵ considering nonlinearity. It was shown that the frequencies of such systems are extremely sensitive to the slack. All of the above-mentioned proposed models are based on the local continuum theories. However, all of these models cannot accurately predict CNTRs vibrations due to the ignorance of the small length scales of the material micro-structures such as lattice spacing between individual atoms. This is indeed the major shortcoming of the continuum theory for vibration analysis of CNTRs. The weakness of continuum theory instigated researchers to develop a new way for mathematically modeling of CNTRs. This leads to a fundamental work by Peddieson et al. ³⁶ They first used the nonlocal continuum theory for developing a new version of Euler–Bernoulli beam model which overcomes shortcoming of the continuum theory for modeling of the nanostructures. ³⁶ Their pioneering work attracts attention of the researchers to use the nonlocal continuum theory for modeling of CNTRs considering different physical phenomena. This article aims to provide an extensive review regarding the use of nonlocal theory in the vibration analysis of CNTRs. We briefly illustrate the nonlocal theory, and then, the focus of the article will be on the implication of nonlocal theory in CNTRs.

The nonlocal continuum theory was first introduced by Eringen in 1972. The Eringen nonlocal elasticity theory illustrates that the stress state at a reference point x in the body is considered to be reliant on both the strain state at x and also on the strain states at all other points of the body. ³⁷ We can observe similar concept in both atomic theory of lattice dynamics and also phonon dispersion. ³⁷ To provide a mathematical view from nonlocal continuum theory, we write its formulation for homogeneous and isotropic elastic solids as follows ³⁷

$${\sigma }_{kl,k}+\rho (f_1-{\stackrel{··}{u}}_l)=0$$

(2)

$${\sigma }_{kl}(x)={\displaystyle \underset{V}{\int }\alpha (x\mathrm{,}{x}^{\prime }){\tau }_{kl}(x^{\prime })dV(x^{\prime })}$$

(3)

$${\tau }_{\mathrm{kl}}(x\prime )=\lambda {\epsilon }_{\mathrm{mm}}(x\prime ){\delta }_{\mathrm{kl}}+2\mu {\epsilon }_{\mathrm{kl}}(x\prime )$$

(4)

$${\tau }_{\mathrm{kl}}(x\prime )=\lambda {\epsilon }_{\mathrm{mm}}(x\prime ){\delta }_{\mathrm{kl}}+2\mu {\epsilon }_{\mathrm{kl}}(x\prime )$$

(5)

$${\epsilon }_{\mathrm{kl}}(x\prime )=\frac{1}{2}(\frac{\partial u_k(x\prime )}{\partial {x\prime }_l}+\frac{\partial u_l(x\prime )}{\partial {x\prime }_k})$$

(6)

where $f_l$ , $\rho$ , ${\sigma }_{\mathrm{kl}}$ , and $u_l$ show body force density, mass density, nonlocal stress tensor, and the displacement of a reference point x, respectively. ${\tau }_{\mathrm{kl}}(x\prime )$ and ${\epsilon }_{\mathrm{kl}}(x\prime )$ indicate classical stress tensor and the strain tensor, correspondingly. $\lambda$ , $\mu$ and $\alpha (x,x\prime )$ represent lame constants and the kernel function, respectively. Using the above-mentioned theory, Peddieson et al. formulated vibrations of CNTRs based on the nonlocal theory in conjunction with Euler–Bernoulli beam theory. They considered a beam model for the CNTRs and developed its governing equation with combining of nonlocal and Euler–Bernoulli beam theories. Figure 1 represents the considered model in their article.

Figure 1.

(a) Considered beam model and (b) a differential element of the beam.

V and M show the shear force and the bending moment, respectively. The prime represents differentiation with respect to x. Based on Figure 1(b), we can easily write the relation between the applied force, shear force as well as bending moment, as Peddieson et al. ³⁶ reported in their article. Accordingly, one could have

$$V\prime +F=0$$

(7)

and

$$M\prime +V=0$$

(8)

Combining equations (7) and (8) results in

$$M″=F$$

(9)

Implementing the Hook’s law based on equation (2), we reach

$$\sigma -(e_{0}a{)}^2\sigma ″=E\epsilon$$

(10)

where $\sigma$ , $\epsilon$ , and E are axial stress, axial strain, and Young’s modulus, respectively. a and $e_{0}a$ represent the internal characteristic length and material constant, respectively.

A small deflection is supposed for the considered beam; thus, the following relation is derived between strain and the curvature as follows

$$\epsilon =-\mathrm{yw}″$$

(11)

where w shows the displacement of the beam. Equation (11) is combined with equation (10), and it results in

$$\sigma -(e_{0}a{)}^2\sigma ″=-\mathrm{Eyw}″$$

(12)

The moment resultant of the stress distribution of the considered beam is obtained by the following equation

$$M=\int \underset{A}{\int }\sigma \mathrm{ydA}$$

(13)

After a few mathematical procedures, the following equation is obtained

$$M=\mathrm{EIw}″+(e_{0}a{)}^{2}M″$$

(14)

Peddieson et al. ³⁶ called equation (14) as nonlocal Bernoulli/Euler beam deflection equation. Substituting equation (14) into equations (2) and (3) results in

$$V=-(\mathrm{EIw}″″+(e_{0}a{)}^{2}F″)$$

(15)

and

$$w^{\mathrm{IV}}=\frac{(p-{(e_{0}a)}^{2}p″)}{\mathrm{EI}}$$

(16)

The above formula is called nonlocal Euler–Bernoulli deflection equation. The above-mentioned equations illustrate the basis of utilization of nonlocal theory for CNTRs. Next section provides detail about the nonlocal Euler–Bernoulli beam theory and the implications of this method for CNTRs modeling.

It must be noted that appropriate selection of the nonlocal parameter plays a critical role for having an accurate analysis from vibrations of CNTs. Accordingly, it is highly necessary to use the calibrated nonlocal parameter in the modeling and the numerical analysis. Interesting and technical information related to the calibration of the nonlocal parameter can be found in Ansari et al., ³⁸ Ansari and Rouhi, ³⁹ Duanna et al., ⁴⁰ and Ansari and Sahmani. ⁴¹

Nonlocal Euler–Bernoulli beam theory

The focus of this part will be on the nonlocal Euler–Bernoulli beam theory and its implementation in modeling of the CNTRs. One of the primary endeavors to utilize nonlocal Euler–Bernoulli theory is an article by Zhang et al. ³⁷ Their developed mathematical model considers vibrations of double-walled carbon nanotubes (DWCNTs). The developed formulation has the following form

$$\begin{array}{l}C[w_2-w_1]=E{I}_1\frac{\partial w_1^4}{\partial x^4}+\rho A_1\frac{\partial w_1^2}{\partial t^2}-e_0{a}^2\\ \left[\rho A_1\frac{\partial w_1^4}{\partial x^2\partial t^2}-C\frac{{\partial }^2}{\partial x^2}(w_2-w_1)\right]\end{array}$$

(17)

and

$$\begin{array}{l}-C[w_2-w_1]=E{I}_2\frac{\partial w_2^4}{\partial x^4}+\rho A_2\frac{\partial w_1^2}{\partial t^2}-e_0{a}^2\\ \left[\rho A_2\frac{\partial w_1^4}{\partial x^2\partial t^2}-C\frac{{\partial }^2}{\partial x^2}(w_2-w_1)\right]\end{array}$$

(18)

where C is the intertube interaction coefficient per unit length between two tubes and is obtained using equation (19)

$$C=\frac{320(2R_1)\mathrm{erg}/\mathrm{c}{\mathrm{m}}^2}{0.16d^2}$$

(19)

Wang has played an important role for mathematical modeling of CNTs using the nonlocal theory. In one of his article, he utilized nonlocal Euler–Bernoulli beam theory to study CNTs characteristics. ⁴² In his pioneering article, he obtained the ratio of phase velocity between the nonlocal and the local Euler–Bernoulli beam theory as equation (20) ⁴²

$$\frac{v_{\mathrm{nE}}}{v_{\mathrm{lE}}}=\frac{1}{\sqrt{1+{(e_{0}a)}^2{k}^2}}$$

(20)

where

$$v_{\mathrm{lE}}=k\sqrt{\frac{\mathrm{EI}}{\rho A}}$$

(21)

His paper shows that by increasing the small-scale parameter, the velocity ratio decreases. In addition, the reduction of the wave length results in decrement in the velocity ratio. Figure 2 shows the effect of the above-mentioned parameters based on Wang. ⁴²

Figure 2.

Small effect on wave velocity by Euler–Bernoulli model. ⁴²

In another article, Wang and Varadan show the influence of the small length scale on the frequency ratio. Based on their paper, ratio of the frequency of the CNTRs considering nonlocal and local theory is obtained by the following equation

$$\frac{{\omega }_n}{{\omega \prime }_n}=\frac{1}{\sqrt{1+(n^2\pi /L^2){(e_{0}a)}^2}}$$

(22)

where $\omega {\prime }_n=(n^2\pi /L^2)\sqrt{(\mathrm{EI}/\rho A)}$ shows the resonant frequency of CNTRs using Euler–Bernoulli beam theory. Figure 3 shows the effect of small length scale on the aforesaid ratio based on Wang and Varadan. ⁴³ As evidenced in Figure 3, increasing in the small length scale results in the reduction of the frequency ratio. In addition, the higher mode that we consider the less ratio between frequencies is obtained.

Figure 3.

Effect of the small length scale on the frequency for $L=15\mathrm{nm}$ . ⁴³

An interesting paper by Reddy ⁴⁴ provides detailed formulation of the nonlocal higher-order theories. Reddy’s paper provides useful information about various beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories. The paper presents a combination of classical beam theories with nonlocal differential constitutive relations of Eringen. For the nonlocal Euler–Bernoulli theory, his paper proposes the following formulation

$$\begin{array}{l}\frac{{\partial }^2}{\partial x^2}\left(-EI\frac{\partial w^2}{\partial x^2}\right)+\mu \left[\frac{\partial }{\partial x}\left(N\frac{\partial w}{\partial x}\right)-q+m_0\frac{{\partial }^{2}w}{\partial t^2}-m_2\frac{{\partial }^{2}w}{\partial t^2}\right]\\ +q-\frac{\partial }{\partial x}\left(N\frac{\partial w}{\partial x}\right)=m_0\frac{{\partial }^{2}w}{\partial t^2}-m_2\frac{{\partial }^{2}w}{\partial t^2\partial x^2}\end{array}$$

(23)

where w, E, I, N, and $\mu$ show transverse displacement, modulus of elasticity, moment of inertia, the applied axial compressive force, and small length scale, respectively. Table 3 represents a few of developed models based on the nonlocal Euler–Bernoulli beam theory. The first row of Table 3 represents a model by Heireche et al. ⁴⁵ in which they investigated sound wave propagation in CNTRs considering initial axial stress based on the nonlocal theory. One of the main results of their paper shows the ratio of the nonlocal to the local velocity considering initial stress as represented in equation (24)

$$\frac{v_{\mathrm{nE}}^2}{v_{\mathrm{lE}}^2}=\frac{1}{1+{(e_{0}a)}^2{(n\pi /l)}^2}-\frac{\delta N_x}{\mathrm{EI}{(n\pi /l)}^2}$$

(24)

where $\delta =1$ shows the wave propagation in a beam with initial compressive stress, $\delta =-1$ expresses the wave propagation in a beam with initial tensile stress, and $\delta =0$ represents that there is no initial stress. $N_x={\sigma }_x^{0}A$ in which ${\sigma }_x^0$ is the initial stress. In sections 6–9, plenty of models are presented based on nonlocal Euler–Bernoulli beam theory considering different physical phenomena. ⁴⁶

Table 3.

Nonlocal Euler–Bernoulli beam models.

Developed mathematical model
Effect of initial axial stress 45 $\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}+\delta N_x\frac{{\partial }^{2}w}{\partial x^2}-(e_{0}a{)}^2(\rho A\frac{{\partial }^{4}w}{\partial x^2\partial t^2}+\delta N_x\frac{{\partial }^{4}w}{\partial x^4})=0$ where $\delta$ can be −1, 0, or 1, and $N_x=A{\sigma }_x^0$ . ${\sigma }_x^0$ shows the initially axial stress
Considering surrounding medium 47 $\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}+\mathrm{kw}-(e_{0}a{)}^2(k\frac{{\partial }^{2}w}{\partial x^2}+\rho A\frac{{\partial }^{4}w}{\partial x^2\partial t^2})$ where k represents the stiffness of surrounding medium
Considering external force 48 $\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}-(e_{0}a{)}^2(\rho A\frac{{\partial }^{4}w}{\partial x^2\partial t^2}-\frac{{\partial }^{2}q}{\partial x^2})=q$ where q shows the distributed transverse force along the x-axis of CNTRs
Axial vibration of carbon nanotube heterojunctions 49 $E_i{A}_i\frac{{\partial }^2{u}_i}{\partial x^2}=(1-(e_{0}a{)}_i^2\frac{{\partial }^2}{\partial x^2})m_i\frac{{\partial }^2{u}_i}{\partial x^2},i=1,2,3,\dots ,n$ where n denotes the number of elements in the CNTRs
Forced vibration of an elastically connected double-carbon nanotubes 50 $\mathrm{EI}\frac{{\partial }^4{w}_1}{\partial x^4}+k(w_1-w_2)-k(e_{0}a{)}^2({w″}_1-{w″}_2)+\rho A{\stackrel{··}{w}}_1-\rho A(e_{0}a{)}^2{\stackrel{··}{w}″}_1=p-(e_{0}a{)}^{2}p″$ $\mathrm{EI}\frac{{\partial }^4{w}_2}{\partial x^4}+k(w_2-w_1)-k(e_{0}a{)}^2({w″}_2-{w″}_1)+\rho A{\stackrel{··}{w}}_2-\rho A(e_{0}a{)}^2{\stackrel{··}{w}″}_2=0$ where $p=P(t)\delta (x-x_p)$ and $x_p$ is position of the applied force on CNTRs
Torsional wave propagation in CNTRs 51 $\frac{\mathrm{Eh}}{1+2\nu }(1+\frac{h^2}{12a^2})\frac{d^{2}v}{d{x}^2}=(1-{(e_{0}a)}^2\frac{d^2}{d{x}^2})(\rho h\stackrel{··}{v})$ where v is the circumferential displacement
Two coupled carbon nanotubes conveying lagged moving nanoparticles 52 $\mathrm{EI}\frac{{\partial }^4{w}_1}{\partial x^4}+k(w_1-w_2)-k(e_{0}a{)}^2({w″}_1-{w″}_2)+\rho A{\stackrel{··}{w}}_1-\rho A(e_{0}a{)}^2{\stackrel{··}{w}″}_1=p_1-(e_{0}a{)}^2{p″}_1$ $\mathrm{EI}\frac{{\partial }^4{w}_2}{\partial x^4}+k(w_2-w_1)-k(e_{0}a{)}^2({w″}_2-{w″}_1)+\rho A{\stackrel{··}{w}}_2-\rho A(e_{0}a{)}^2{\stackrel{··}{w}″}_2=p_2-(e_{0}a{)}^2{p″}_2$ where $p_i=P_i(t)\delta (x-x_{p_i})$ and $x_{p_i}$ is the position of the applied force on CNTRs
Pre-stressed carbon nanotubes undergoing rotation 53 $\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}-(e_{0}a{)}^2\rho A\frac{{\partial }^{4}w}{\partial t^2\partial x^2}+N_x\frac{{\partial }^{2}w}{\partial x^2}}\\ \multicolumn{1}{c}{-(e_{0}a{)}^2{N}_x\frac{{\partial }^{4}w}{\partial x^4}-\rho A{\Omega }^2[(-r-x)\frac{\partial w}{\partial x}+(\mathrm{rL}+\frac{L^2}{2}-\mathrm{rx}-\frac{x^2}{2})\frac{{\partial }^{2}w}{\partial x^2}]}\\ \multicolumn{1}{c}{+(e_{0}a{)}^2\rho A{\Omega }^2[-3\frac{{\partial }^{2}w}{\partial x^2}+3(-r-x)\frac{{\partial }^{3}w}{\partial x^3}+(\mathrm{rL}+\frac{L^2}{2}-\mathrm{rx}-\frac{x^2}{2})\frac{{\partial }^{4}w}{\partial x^4}]=0}\end{array}$ where r, L, and $\Omega$ represent hub radius, length, and angular velocity of CNTRs, respectively
CNTs under moving harmonic load 54 $\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A(1-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial t^2})=P(x,t)-(e_{0}a{)}^2\frac{{\partial }^{2}P(x,t)}{\partial x^2}$ where $P(x,t)=P(t)\delta (x-x_p)$ and $P(t)=P_0\sin (\Omega t)$

CNTs: carbon nanotubes; CNTRs: carbon nanotube resonators.

Nonlocal Timoshenko beam theory

This section provides specific focus on the utilization of nonlocal Timoshenko beam theory for the vibration modeling of the CNTRs. One of the earliest attempts for vibration analysis of CNTRs using nonlocal Timoshenko theory can be found in Wang and Varadan ⁴³ in which the following model was developed for CNTRs

$$\mathrm{GAk}(\frac{\partial \varphi }{\partial x}-\frac{{\partial }^{2}u}{\partial x^2})-\rho A\frac{{\partial }^{2}u}{\partial t^2}=0$$

(25)

$$GAk\left(1-{(e_{0}a)}^2\frac{{\partial }^2}{{\partial }^{2}x}\right)\left(\frac{\partial u}{\partial x}-\varphi \right)+EI\frac{{\partial }^2\varphi }{\partial x^2}-\rho I\frac{{\partial }^2}{\partial t^2}\left(\varphi -{(e_{0}a)}^2\frac{{\partial }^{2}u}{\partial x^2}\right)=0$$

(26)

where $\varphi$ shows the slope of the cross section due to the bending, $G=(E/(1+2\nu ))$ and $k=10/9$ for circular cross section. Indeed, equations (25) and (26) are obtained using equation (10) and the following equation

$${\tau }_{\mathrm{xy}}-(e_{0}a{)}^2{\tau ″}_{\mathrm{xy}}=G\gamma$$

(27)

and it is apparent that the strains $\epsilon$ and $\gamma$ for Timoshenko beam are obtained through the following equations ⁵⁵

$$\epsilon =y\frac{\partial \varphi }{\partial x}$$

(28)

and

$$\gamma =\frac{\partial u}{\partial x}-\varphi$$

(29)

Lu et al. ⁵⁵ have considered axial loads in their model and obtained the following governing equation for their considered system as

$$c_s^2\left(\frac{\partial \varphi }{\partial x}-\frac{{\partial }^{2}u}{\partial x^2}\right)+\frac{1}{\rho A}\left(p-e_0{a}^2\frac{{\partial }^{2}p}{\partial x^2}\right)-\left(\frac{{\partial }^{2}u}{\partial t^2}-{(e_{0}a)}^2\frac{{\partial }^{2}u}{\partial t^2}\right)=0$$

(30)

$$c_s^2(\frac{\partial u}{\partial x}-\varphi )+c_1^2{r}_0^2\frac{{\partial }^2\varphi }{\partial x^2}-r_0^2\frac{{\partial }^2}{\partial t^2}(\varphi -{(e_{0}a)}^2\frac{{\partial }^2\varphi }{\partial x^2})=0$$

(31)

where $c_s=k\prime G/\rho$ , $r_0=\sqrt{I/A}$ , and $c_1=\sqrt{E/\rho }$ . Accordingly, natural frequency of the considered system is obtained by the following equation ⁵⁵

$$\frac{w_{\mathrm{nT}}}{w_{\mathrm{lT}}}=\frac{1}{\sqrt{1+{(e_{0}a)}^2{k}^2}}$$

(32)

where ${\omega }_{l}T$ are the positive real frequencies based on the local Timoshenko beam theories. For the phase velocity, they obtained the following formulation

$$\frac{v_{\mathrm{nT}}}{v_{\mathrm{lT}}}=\frac{1}{\sqrt{1+{(e_{0}a)}^2{k}^2}}$$

(33)

Equations (34) and (35) represent ${\omega }_{\mathrm{lT}}$ and $n_{\mathrm{lT}}$ , respectively

$${\omega }_{\mathrm{lT}}={(\frac{1}{2}(c_1^2+c_s^2)k^2+\frac{c_s^2}{r_0^2}\pm \sqrt{\frac{1}{4}{((c_1^2+c_s^2)k^2+\frac{c_s^2}{r_0^2})}^2-c_s^2{c}_l^2})}^{\frac{1}{2}}$$

(34)

and Lu et al. ⁵⁵

$$v_{\mathrm{lT}}={(\frac{1}{2}(c_1^2+c_s^2+\frac{c_s^2}{r_0^2{k}^2})\pm \sqrt{\frac{1}{4}{((c_1^2+c_s^2)+\frac{c_s^2}{r_0^2{k}^2})}^2-c_s^2{c}_l^2})}^{\frac{1}{2}}$$

(35)

As another example in which the natural frequency of the CNTRs was obtained based on nonlocal Timoshenko beam theory, we can refer to a paper by Lee and Chang ⁵⁶ in which they obtained the following equations as the natural frequency of CNTRs

$${\omega }_n=\sqrt{\frac{(\mathrm{EI}{\lambda }_n^4/g_n)+\beta ((\mathrm{EI}{\lambda }_n/\mathrm{kGA})+1)}{(\rho A+m)((\mathrm{EI}{\lambda }_n^4/\mathrm{kGA})+1)+\rho I((\beta g_n/\mathrm{kGA})+{\lambda }_n^2)}}$$

(36)

where ${\lambda }_n=(n\pi )/L$ , $g_n=1+\mu {\lambda }_n^2$ , $\beta$ is the Winkler coefficient, and $\mu =\mathrm{ea}$ . In a recently published article, Boumia et al. ⁵⁷ have implemented Timoshenko beam theory to investigate chiral SWCNTRs. The results of their paper indicate significant dependence of the natural frequencies on the chirality of single-walled carbon, and also the small-scale parameter. Figure 4 shows the effect of the mode number as well as small length parameter on the $\beta =(({\omega }_{N}L)/{\omega }_L)$ for zigzag (14, 0) SWCNTRs. Similar to Figure 4, effect of the mode number and also small length parameter on $\beta$ is shown in Figure 5. Effect of the aspect ratio and chirality of CNT are presented in Figure 6 based on Boumia et al. ⁵⁷

Figure 4.

Influence of small-scale effect $(e_{0}a)$ and the mode number (N) on $\beta$ for zigzag (14, 0) SWCNTRs. ⁵⁷

Figure 5.

Influence of the small-scale effect $(e_{0}a)$ and the mode number (N) on $\beta$ for chiral (16, 8) SWCNTRs. ⁵⁷

Figure 6.

Effect of aspect ratio and chirality of carbon nanotube on the values of ratio $\beta$ in the fundamental mode and the scale coefficient ( $e_{0}a=2\mathrm{nm}$ ). ⁵⁷

In Table 4, we provide different models for CNTRs based on the nonlocal Timoshenko beam theory.

Table 4.

Nonlocal Timoshenko beam models.

Developed mathematical model
Timoshenko model by Hu et al. 58 for DWCNTRs $\begin{array}{c}\multicolumn{1}{c}{G{A}_{1}k(\frac{\partial {\varphi }_1}{\partial x}-\frac{{\partial }^2{u}_1}{\partial x^2})+\rho A_1\frac{{\partial }^2{u}_1}{\partial x^2}=p_1}\\ \multicolumn{1}{c}{G{A}_{1}k[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2}](\frac{\partial u_1}{\partial x}-{\varphi }_1)+E{I}_1\frac{\partial {{\varphi }_1}^2}{\partial x^2}-\rho I_1\frac{{\partial }^2}{\partial t^2}({\varphi }_1-{(e_{0}a)}^2\frac{\partial {{\varphi }_1}^2}{\partial x^2})=0}\end{array}$ and $\begin{array}{c}\multicolumn{1}{c}{G{A}_{2}k(\frac{\partial {\varphi }_2}{\partial x}-\frac{{\partial }^2{u}_2}{\partial x^2})+\rho A_1\frac{{\partial }^2{u}_2}{\partial x^2}=p_2}\\ \multicolumn{1}{c}{G{A}_{2}k[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2}](\frac{\partial u_2}{\partial x}-{\varphi }_1)+E{I}_2\frac{\partial {{\varphi }_2}^2}{\partial x^2}-\rho I_2\frac{{\partial }^2}{\partial t^2}({\varphi }_2-{(e_{0}a)}^2\frac{\partial {{\varphi }_2}^2}{\partial x^2})=0}\end{array}$ where $p_1$ and $p_2$ represents interaction force between CNTs
Modified Timoshenko beam theory 56 $E^{*}{I}^{*}\frac{{\partial }^2\varphi }{\partial x^2}+(1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2})[\mathrm{kGA}(\frac{\partial Y}{\partial x}-\varphi )-\rho I\frac{{\partial }^2\varphi }{\partial x^2}]=0$ $\frac{\partial }{\partial x}kGA\left(\frac{\partial Y}{\partial x}-\varphi \right)+H\frac{{\partial }^{2}Y}{\partial x^2}=\rho A\frac{{\partial }^{2}Y}{\partial t^2}$ where k, $E^{*}{I}^{*}$ , and H are shear factor, the effective flexural rigidity, and the constant parameter which is obtained by the residual surface tension and the shape of cross section, respectively. It is critical to know that $H=4\tau (R_i+R_o)$ and $E^{*}{I}^{*}=\mathrm{EI}+E^s(R_i^3+R_o^3)$ where $\tau$ and $E^s$ are the surface elasticity modulus and residual surface tension per length on the CNTRs, respectively
Timoshenko beam model for CNTRs by Ansari et al. 59 $(\mathrm{EI}+4\tau T)\frac{{\partial }^{4}w}{\partial x^4}+\rho \pi t{R}_1\frac{{\partial }^{2}w}{\partial x^2}-4\tau R_1\frac{{\partial }^{2}w}{\partial x^2}-\rho (\pi \mathrm{tT}+\frac{\pi t{\lambda }^R}{4\lambda }+\frac{\pi t{\lambda }^R}{E})\frac{{\partial }^{4}w}{\partial x^2\partial x^2}+\frac{{\rho }^2\pi t{\lambda }^R}{4\pi }\frac{{\partial }^{4}w}{\partial t^4}=0$ where $T=({\lambda }^R/4)+({\lambda }^{\mathrm{ER}}/t)$ , ${\lambda }^R=\lambda R_1{R}_2$ , ${\lambda }^{\mathrm{ER}}=\lambda E^{\lambda }{R}_3$ , $E^{\lambda }=E^s/E$ , $\lambda =E/\mathrm{kG}$ , $R_j=R_o^j+R_i^j$ , and $t=R_o-R_i$
Fractional Timoshenko beam model 59 $\rho A(1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2})\frac{{\partial }^{2}w}{\partial t^2}-\mathrm{kGA}(1+{\tau }_d\frac{{\partial }^{\alpha }}{\partial t^{\alpha }})(\frac{\partial \theta }{\partial x}+\frac{{\partial }^{2}w}{\partial x^2})=[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial {\theta }^2}]p$ $\rho A(1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2})\frac{{\partial }^2\theta }{\partial x^2}+\mathrm{kGA}(1+{\tau }_d\frac{{\partial }^{\alpha }}{\partial t^{\alpha }})(\theta +\frac{\partial w}{\partial x})-\mathrm{EI}(1+{\tau }_d\frac{{\partial }^{\alpha }}{\partial t^{\alpha }})\frac{{\partial }^2\theta }{\partial x^2}=[1-{(e_{0}a)}^2\frac{{\partial }^2}{\partial x^2}]m$ where m and p indicate external loads. $\alpha$ represents the fractional derivative

CNTs: carbon nanotubes; CNTRs: carbon nanotube resonators; DWCNTRs: double-walled carbon nanotube resonators.

Nonlocal Rayleigh beam theory

Rayleigh beam theory is another way for modeling CNTRs. This section provides an illustration about nonlocal Rayleigh beam theory. In comparison with Timoshenko and Euler–Bernoulli beam theories, nonlocal Rayleigh beam theory is rarely used for modeling of CNTRs. The main difference between Euler–Bernoulli and Rayleigh beam theories is in a term which is related to the rotational inertia of cross section, it is considered in the Rayleigh beam theory. Equation (37) represents the nonlocal Rayleigh beam model which has been modeled by Mustapha and Zhong ⁶⁰

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2}{\partial x^2}\left(\mathrm{EI}\frac{{\partial }^{2}w}{\partial x^2}\right)+(e_{0}a{)}^2\frac{{\partial }^2}{\partial x^2}}\\ \multicolumn{1}{c}{[-\frac{\partial }{\partial x}\left(N\frac{\partial w}{\partial x}\right)-K_{f}w+k_s\frac{{\partial }^{2}w}{\partial x^2}-\rho A\frac{{\partial }^{2}w}{\partial t^2}+\rho I\frac{{\partial }^{4}w}{\partial x^2\partial t^2}]}\\ \multicolumn{1}{c}{+\frac{\partial }{\partial x}\left(N\frac{\partial w}{\partial x}\right)+k_{f}w-k_s\frac{\partial w^4}{\partial x^2}=\rho A\frac{{\partial }^{2}w}{\partial t^2}+\rho I\frac{{\partial }^{4}w}{\partial x^2\partial t^2}}\end{array}$$

(37)

In their model, they considered Winkler and Pasternak type foundations. $k_f$ and $k_s$ show the Winkler and Pasternak coefficients, respectively. Comparing with Euler–Bernoulli beam theory, we observe two new terms including $\rho I({\partial }^2/\partial x^2)[(({\partial }^{4}w)/(\partial x^2\partial t^2))]$ and $\rho I(({\partial }^{4}w)/(\partial x^2\partial t^2))$ . As another example of using nonlocal Rayleigh beam theory, we can refer to an article by K. Kiani in which longitudinal and transverse vibrations of a single-walled carbon nanotube (SWCNT) subjected to a moving nanoparticle were studied. Kiani ⁶¹ has used Rayleigh beam theory for modeling CNTRs considering different physical effects. The following equation represents the mathematical model relevant to the transverse vibration

$$\begin{array}{l}\rho (A\stackrel{··}{w}-I{\stackrel{··}{w}}_{xx})-{(e_{0}a)}^2\rho (A{\stackrel{··}{w}}_{xx}-I{\stackrel{··}{w}}_{xxxx})+EI{w}_{xxxx}\\ =M\left[\left(g-\frac{D^{2}w}{D{t}^2}\right)\delta (x-x_M)-{(e_{0}a)}^2{\left(\left(g-\frac{D^{2}w}{D{t}^2}\right)\delta (x-x_M)\right)}_{xx}\right]H(L-x_M)\end{array}$$

(38)

where $x_M=\mathrm{vt}$ . Free transverse vibration of an elastically supported stocky DWCNT embedded in an elastic matrix under initial axial force was studied by Kiani using nonlocal Rayleigh beam theory. Results of his paper reveal that an increase in the lateral stiffness as well as rotational stiffness of the surrounding matrix would result in the increase in the predicted first dimensionless frequencies of the DWCNT for all boundary conditions. ⁶² In another paper, Kiani ⁶³ has studied longitudinal, transverse, and torsional vibrations of axially moving SWCNTs using nonlocal Rayleigh beam theory. Results of his paper show divergence and flutter modes for longitudinal, transverse, and torsional motions of the moving SWCNT. Pirmohammadi et al. developed mathematical model for the vibrations of CNTRs considering harmonic load using nonlocal Rayleigh beam model. Their developed model has the following mathematical formulation ⁶⁴

$$\begin{array}{l}\rho (A\stackrel{··}{w}-I{\stackrel{··}{w}}_{xx})-{(e_{0}a)}^2\rho (A{\stackrel{··}{w}}_{xx}-I{\stackrel{··}{w}}_{xxxx})+EI{w}_{xxxx}\\ =F_0\sin (\Omega t)\left[\delta (x-x_f)-{(e_{0}a)}^2\frac{{\partial }^2\delta (x-x_f)}{\partial x^2}\right]\end{array}$$

(39)

The following equation was obtained as the natural frequency of the considered system for the case of free vibrations ⁶⁴

$$w_{\mathrm{nR}}=\sqrt{\frac{\mathrm{EI}{(n\pi /L)}^4}{\rho A(1+(I/A){(n\pi /L)}^2)(1+{((n{e}_{0}a\pi )/L)}^2)}}$$

(40)

Pourseifi et al. ⁶⁵ actively controlled the vibrations of CNTRs under the action of a moving nanoscale particle using nonlocal Rayleigh beam theory. They considered the following form for the applied force

$$F(x,t)=\mathrm{mg}\delta (x-x_f)+\sum _{i=1}^p{u}_i(t)\delta (x-x_i)$$

(41)

$u_i(t)$ is the required control force, and $i=1,2,\dots ,p$ . Accordingly, they developed the following mathematical model for CNTRs

$$\begin{array}{c}\multicolumn{1}{c}{\rho (A\stackrel{··}{w}-I{\stackrel{··}{w}}_{\mathrm{xx}})-(e_{0}a{)}^2\rho (A{\stackrel{··}{w}}_{\mathrm{xx}}-I{\stackrel{··}{w}}_{\mathrm{xxxx}})}\\ \multicolumn{1}{c}{+\mathrm{EI}{w}_{\mathrm{xxxx}}=\mathrm{mg}[\delta (x-x_f)-{(e_{0}a)}^2\frac{{\partial }^2\delta (x-x_f)}{\partial x^2}]}\\ \multicolumn{1}{c}{+\sum _{i=1}^p{u}_i(t)[\delta (x-x_i)-{(e_{0}a)}^2\frac{{\partial }^2\delta (x-x_i)}{\partial x^2}]}\end{array}$$

(42)

Based on the above equation, their paper presents a linear classical optimal control algorithm with a time varying gain matrix and displacement–velocity feedback to suppress vibrations of CNTRs. ⁶⁵ The nonlocal Rayleigh beam theory has been used in modeling of CNTRs under magnetic field and also fluid-conveying CNTRs. In addition to the above-mentioned works, a few researchers focused on the axial vibration analysis of CNTs. For example, M Aydogdu ⁶⁶ studied axial vibrations of CNTs using nonlocal theory. Results of his paper show that the axial vibration frequencies of SWCNT embedded in an elastic medium highly over estimated by the classical rod model. In another paper, M Aydogdu ⁶⁷ studied axial vibrations of CNTs considering van der Waals (vdW) effect and nonlocal parameters.

Modeling of nonlinearity

Due to the small scale of CNTRs, a small perturbation may lead to strong nonlinearity in the dynamical behavior of the CNTRs. Thus, it is critical to have a precise mathematical model for vibration analysis of the CNTRs. Generally, there could be five main potential reasons for emerging nonlinearity in the vibrations of CNTRs: (a) nonlinearity due to the material properties, (b) large amplitude vibrations and geometrical nonlinearity, (c) nonlinear elastic medium, (d) nonlinear electromagnetic and electrostatic force, and (e) intermolecular interaction such as vdW force. This section presents mathematical models of CNTRs in which we can observe nonlinearity due to the first three above-mentioned potential reasons. One of the earliest modeling of nonlinearity in the vibration of CNTRs was formulated by Liu et al. ³⁴ Based on their mathematical model, they scrutinized the effect of a rippling mode on the resonances of CNTRs. Their developed mathematical model has the following form

$$w″″+2\mu \stackrel{·}{w}+\stackrel{··}{w}=N+F$$

(43)

where

$$N=3{\alpha }_3{\left(\frac{h}{L}\right)}^2[2w″{(w‴)}^2+{(w″)}^{2}w″″]+\gamma [(w″{)}^3+6w^{\prime }w″w‴+{(w^{\prime })}^{2}w″″]$$

(44)

and F is the driving force. After a few mathematical procedures, they reach well-known Duffing equation. ⁶⁸ Wang et al. ⁶⁹ proposed a nonlinear model for CNTRs to study their bending modulus with considering the rippling deformation. Based on their proposed mathematical model and nonlinear analysis, the effective bending modulus ratio of the CNTRs has the following relation with D/L (Figure 7)

$$\frac{E_{\mathrm{eff}}}{E}={(1-[3876.44{\left(\frac{D}{L}\right)}^2+0.0186]{\left(\frac{G_1^{*}}{{\mu }^{*}}\right)}^2)}^2$$

(45)

where $(G_1^{\ast }/{\mu }^{\ast })$ shows the ratio of the excitation load to the damping coefficient. In a pioneering paper, Askari et al. ⁷⁰ proposed a nonlinear model for the vibrations of CNTRs based on Euler–Bernoulli beam theory. Their mathematical model is described by the following equations

$$\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}=\left[\frac{\mathrm{EA}}{2L}\underset{0}{\overset{2L}{\int }}\frac{\partial w}{\partial x}\mathrm{dx}\right]\frac{{\partial }^{2}w}{\partial x^2}+p$$

(46)

where $p=-\mathrm{kw}$ represent linear Winkler foundation. The integral terms using the Galerkin approach for equation (46), the following nonlinear differential equation only represents its time-dependent part

$$\frac{d^{2}u}{d{t}^2}+(\frac{{\pi }^4\mathrm{EI}}{l^4\rho A}+\frac{k}{\rho A})u+\left(\frac{{\pi }^{4}E}{4l^4\rho }\right)u^3=0$$

(47)

Equation (47) was then solved by Askari et al. ⁷⁰ using homotopy analysis and variational iteration method and obtained the following equation as the natural frequency of the considered system ⁷⁰

$$\omega =\sqrt{(\frac{{\pi }^4\mathrm{EI}}{l^4\rho A}+\frac{k}{\rho A})+\frac{3}{4}\left(\frac{{\pi }^{4}E}{4l^4\rho }\right)a^2}$$

(48)

Figure 7.

The corresponding relation between $E_{\mathrm{eff}}/E$ and $D/L$ . ⁶⁹

Figure 8 shows the effect considering nonlinearity in the vibrations of CNTs and the linear stiffness. As depicted from Figure 8, having an elastic medium increases the natural frequency of the vibrations of CNTs. Accordingly, it helps the stability of their vibrations especially when CNTs convey fluid. A lot of papers have been published so far about the effect of the elastic medium on the vibrations of CNTs. A few important examples include Mustapha and Zhong, ⁶⁰ M Aydogdu, ⁶⁶ Fu et al., ⁷¹ Murmu and Pradhan, ⁷² and Zhen and Fang. ⁷³

Figure 8.

Variations of frequency ratio with respect to the initial amplitudes. ⁷¹

Ke et al. ⁷⁴ used nonlocal Timoshenko beam model to investigate the influence of nonlinearity and have a better insight into its dynamical behavior. One of the crucial results of their paper shows that at a given vibration amplitude, an increase in nonlocal parameter leads to smaller linear and nonlinear frequencies but a higher nonlinear frequency ratio. In addition, it was illustrated that the nonlocal parameter almost has no influence in the nonlinear mode shape for the hinged–hinged DWNTs. Their model is presented in section pertinent to vdW force. An elastic continuum approach for modeling the nonlinear vibration of DWCNTs under harmonic excitation was developed by Hawwa and Al-Qahtani. ⁷⁵ They developed the model based on the Euler–Bernoulli beam theory. The source of nonlinearity in their model is vdW force. Nonlinear free vibration of DWCNTs based on the nonlocal elasticity theory was studied by Fang et al. ⁷⁶ They considered two sources of nonlinearity in their model including vdW force and also geometrical nonlinearity. An important result of the paper is relevant to simultaneously considering two sources of nonlinearity. Based on the conclusions of the paper, the amplitudes of the noncoaxial vibration increase with the increase in the noncoaxial frequency. Inversely, by only considering the nonlinear vdW forces, the amplitudes of the noncoaxial vibration decrease with the increase in frequency, and the amplitudes are larger than those considering both the geometric nonlinearity and the nonlinear vdW forces for frequency less than a certain value. Wang and Li ⁷⁷ studied damping effect in the vibrations of CNTRs using a nonlinear nonlocal Euler–Bernoulli beam theory. Their developed mathematical model is as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}=\left[\frac{\mathrm{EA}}{2L}\underset{0}{\overset{2L}{\int }}\frac{{\partial }^{2}w}{\partial x^2}\mathrm{dx}\right]}\\ \multicolumn{1}{c}{\frac{{\partial }^2}{\partial x^2}(e_0{a}^2\frac{{\partial }^{2}w}{\partial x^2}-w)=\rho A\frac{{\partial }^2}{\partial x^2}(e_0{a}^2\frac{{\partial }^{2}w}{\partial x^2}-w)}\\ \multicolumn{1}{c}{+k_w(e_0{a}^2\frac{{\partial }^{2}w}{\partial x^2}-w)+c\frac{\partial }{\partial t}(e_0{a}^2\frac{{\partial }^{2}w}{\partial x^2}-w)=0}\end{array}$$

(49)

Employing the Galerkin procedure, they obtained Damped-Duffing equation as follows ⁷⁷

$$\stackrel{··}{u}+{\omega }_n^{2}u+\mu \stackrel{·}{u}+\alpha u^3=0$$

(50)

where ${\omega }_n$ , $\mu$ , and $\alpha$ show the linear natural frequency, damping coefficient, and nonlinear coefficient, respectively. Nonlinear modeling was also used for delivering of nanoparticles. In 2014, Kiani formulated the governing equations of the vibrations of CNTRs containing a moving nanoparticle considering nonlinearity. Using the nonlocal Rayleigh beam theory, results of his paper show that increasing in the velocity and the mass weight of the moving nanoparticle increases the discrepancies between the predicted results by the linear and nonlinear analyses. Furthermore, his paper shows that decreasing the slenderness ratio of CNTs results in increasing the discrepancies between the predictions of nonlinear and linear modeling. The above two important points of Kiani’s ⁷⁸ article indicate the significance of nonlinear modeling for the vibrations of CNTRs. One of the potential reasons of having nonlinearity in the vibrations of CNTRs can be a nonlinear foundation. Askari mathematically modeled vibrations of CNTRs considering nonlinear Winkler foundation. He developed CNTRs governing equation based on the nonlocal Euler–Bernoulli beam theory as presented in equation (51). As evidenced by the equation, the model includes nonlinear Winkler foundation, damping term, Pasternak foundation as well as external force ⁷⁹

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}+\mu \frac{{\partial }^{2}w}{\partial t^2}-k_p\frac{{\partial }^{2}w}{\partial x^2}+k_{1}w+k_3{w}^3}\\ \multicolumn{1}{c}{-e_0{a}^2\left(\frac{{\partial }^{2}w}{\partial x^2}\right)(\rho A\frac{{\partial }^{2}w}{\partial t^2}+\mu \frac{{\partial }^{2}w}{\partial t^2}-k_p\frac{{\partial }^{2}w}{\partial x^2}+k_{1}w+k_3{w}^3)}\\ \multicolumn{1}{c}{=F(x,t)(1-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2})}\end{array}$$

(51)

Using the Galerkin procedure, the principal mode vibration equation was obtained based on the above-mentioned mathematical model

$$\stackrel{··}{u}+{\omega }_n^{2}u+\mu \stackrel{·}{u}+\alpha u^3=f\cos (\Omega t)$$

(52)

Interested readers can find the mathematical formulation of $\alpha$ , ${\omega }_n$ , $\mu$ in Askari. ⁷⁹ . Also, analysis of the second mode of vibrations of CNTRs can be found in Askari. ⁷⁹ Considering two modes of vibrations based on equation (51), the following set of coupled nonlinear differential equations is obtained

$${\stackrel{··}{u}}_1+{\omega }_1^2{u}_1+\mu {\stackrel{·}{u}}_1+{\alpha }_1{u}_1^3+{\alpha }_2{u}_1{u}_2^2=f_1\cos (\Omega t)$$

(53)

$${\stackrel{··}{u}}_1+{\omega }_2^2{u}_2+\mu {\stackrel{·}{u}}_1+{\alpha }_3{u}_2^3+{\alpha }_4{u}_2{u}_1^2=f_2\cos (\Omega t)$$

(54)

Accordingly, the primary and secondary resonances of CNTRs can be found in Askari. ⁷⁹ Figure 9 shows the effect of the chirality on the primary resonance of the vibrations of CNTRs. Figure 10 shows the effect of surface effect on the vibrations of nanoresonators considering Armchair (20, 20) CNTRs. In Figure 10, $\eta$ shows the ratio of nonlinear frequency to the linear frequency. As shown in Figure 10, increasing the initial amplitude of oscillation results in increasing in the considered ratio. In addition, we observe that for a given initial condition, increasing the small length parameter results in decreasing in the ratio of nonlinear frequency to the linear frequency. Figure 11 shows the effect of applied force position on the amplitude of oscillations. As depicted in Figure 11, the amplitude of oscillation increases as the position of the applied force is going to be closer to the middle of the CNTRs. It means that if there is a moving particle on top of the CNTR, motion toward its middle results in increasing in its displacement. A few of recent studies concerning nonlinearity in the vibrations of CNTs include research works by Ansari et al., ⁸⁰ P Ribeiro, ⁸¹ and Ansari et al. ⁸² In next sections, it is shown how the nonlinearity can be influential on the dynamical behavior of CNTRs considering different parameters such geometrical imperfection, thermal effect, and fluid–structure interaction (FSI).

Figure 9.

Effect of the chirality on the primary resonance of CNTRs.

Figure 10.

Effect of the surface effect on the ratio of nonlinear frequency to the linear frequency.

Figure 11.

Effect of position of applied force on the primary resonance of the CNTs. ⁷⁹

Thermal effect

Study of the temperature variation influence in the vibrations of CNTRs is an interesting topic for researchers, and many different theoretical, numerical, and experimental methods have been developed so far to scrutinize the thermal effect in CNTRs. For example, Cao et al. ⁸³ used molecular dynamics simulation to study the natural thermal vibration behaviors of SWCNTs. One of the earliest mathematical models in which thermal effect was considered can be found in a paper by Hsu et al. ⁸⁴ They used Timoshenko beam theory and developed the following equation for their system considering the thermal effect

$$EI\frac{{\partial }^{4}w}{\partial x^4}+\frac{F_T}{\rho A}\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial t^2}-\frac{I}{A}\left(1+\frac{E}{KG}\right)\frac{{\partial }^{4}w}{\partial x^4}+\frac{I}{A}\frac{\rho }{KG}\frac{{\partial }^{4}w}{\partial t^4}=0$$

(55)

where $F_T$ shows the axial force due to the temperature, and it is calculated by the following equation

$$F_T=\Phi \mathrm{TEA}$$

(56)

where $\Phi$ denotes the thermal expansion coefficient of the nanotube. Apparently, we can have an especial case from equation (55) which is the Euler–Bernoulli beam theory. Based on the paper by Hsu et al., ⁸⁴ we have the following equation as the natural frequency of the system

$$\omega =\frac{\pi \sqrt{{\pi }^2-\delta }}{L^2}\sqrt{\frac{\mathrm{EI}}{\rho A}}$$

(57)

where

$$\delta =\frac{4\Phi T}{{(D/L)}^2+{(t_c/L)}^2}$$

(58)

Figure 12 shows the variation of the natural frequency with considering temperature effect. The considered temperature is 800 K. We can see the effect of chirality on the natural frequency of the CNTRs vibrations. Euler–Bernoulli beam model was developed for the vibration and instability analysis of fluid-conveying SWCNTs considering the thermal effect by Wang et al. ⁸⁵ Results of their work indicate the dependence of natural frequencies on the flow velocity as well as temperature change. One of the earliest studies, in which nonlocal Timoshenko beam theory was used for modeling CNTR considering thermal effect, is a paper by Benzair et al. ⁸⁶ In their mathematical model, they have added the following term to the nonlocal Timoshenko beam equation which is pertinent to the thermal effect

$$N_t\frac{{\partial }^2}{\partial x^2}(w-\frac{{\partial }^{2}w}{\partial x^2})$$

(59)

where $N_t$ is the thermal effect coefficient and is completely same as $F_T$ in equation (57). Figure 13 shows the effect of the mode number and the temperature on the $\chi =({\omega }_{\mathrm{NL}}/{\omega }_{\mathrm{NL}}^0)$ in which ${\omega }_{\mathrm{NL}}^0$ is the nonlocal natural frequency of the CNTRs without consideration of the temperature. As can be seen in Figure 13, increasing the temperature results in increasing the $\chi$ for a given mode number. Figure 14 shows the effect of mode number on $\chi$ when a high temperature is considered. As evidenced in Figure 14, increasing mode number results in increasing $\chi$ . In addition, Figure 14 represents that the considered ratio decreases by increasing the temperature for a given mode number. The thermal effect was then considered by Tounsi et al. ⁸⁷ in modeling of DWCNTRs based on Timoshenko beam theories. In their model, they have added two terms related to the thermal effect to the corresponding nonlocal Timoshenko beam model for DWCNTRs. The terms related to the first and the second walls of CNTRs are as follows

$$E{A}_1\alpha T\frac{{\partial }^{2}w}{\partial x^2}+E{A}_1\alpha T(\mathrm{ea}{)}^2\frac{{\partial }^{4}w}{\partial x^4}$$

(60)

and

$$E{A}_2\alpha T\frac{{\partial }^{2}w}{\partial x^2}+E{A}_2\alpha T(\mathrm{ea}{)}^2\frac{{\partial }^{4}w}{\partial x^4}$$

(61)

Figure 12.

Effect of the aspect ratio and chirality on the linear frequency of CNTRs considering temperature. ⁸⁴

Figure 13.

Effect of the mode number and temperature on $\chi$ for the vibrations of CNTRs in low or temperature room. ⁸⁶

Figure 14.

Effect of the mode number and temperature on $\chi$ for the vibrations of CNTRs in high temperature. ⁸⁶

Another interesting model considering temperature was developed by Murmu and Pradhan. ⁷² They obtained the following governing equation for their system based on the nonlocal Euler–Bernoulli beam theory considering thermal effect as well as Winkler foundation

$$[EI+N_{temp}{(e_{0}a)}^2]\frac{{\partial }^{4}w}{\partial x^4}-m{(e_{0}a)}^2\frac{{\partial }^{4}w}{\partial x^2\partial t^2}-[k{(e_{0}a)}^2+N_{temp}]\frac{{\partial }^{2}w}{\partial x^2}+kw+m\frac{{\partial }^{2}w}{\partial t^2}=0$$

(62)

where $N_{\mathrm{temp}}=-((\mathrm{EA})/(1-2\nu )){\alpha }_{x}T$ . Results of their paper show that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. A few other models can be found in the literature for vibration modeling of CNTRs considering thermal effects. Mustapha and Zhong, ⁶⁰ Askari et al., ⁸⁸ Ansari and Ramezannezhad, ⁸⁹ Maachou et al., ⁹⁰ Zidour et al., ⁹¹ Chen et al., ⁹² and Chang ⁹³ have investigated thermal effect on the vibrations of CNTRs. Table 5 presents the mathematical models of CNTRs considering thermal effect which have been developed by the above-mentioned researchers.

Table 5.

Thermal effects in vibrations of CNTRs.

Developed mathematical model
Nonlocal Timoshenko model considering large oscillations and thermal effects 89 $\rho I(1-(e_{0}a{)}^2{\nabla }^2)\frac{{\partial }^2\varphi }{\partial x^2}-\mathrm{EI}\frac{{\partial }^2\varphi }{\partial x^2}+\mathrm{kGA}(\varphi +\frac{\partial \varphi }{\partial x})=0$ $\rho A(1-(e_{0}a{)}^2{\nabla }^2)\frac{{\partial }^{2}w}{\partial x^2}-\mathrm{kGA}(\frac{\partial \varphi }{\partial x}+\frac{{\partial }^2(w)}{\partial x^2})-[\frac{\mathrm{EA}}{2L}\underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^2\mathrm{dx}+N_t]A(1-(e_{0}a{)}^2{\nabla }^2)\frac{{\partial }^{2}w}{\partial x^2}=(1-(e_{0}a{)}^2{\nabla }^2)P(x,t)$
Euler–Bernoulli beam model considering large oscillations and thermal effects 94 $\rho A\frac{{\partial }^{2}w}{\partial t^2}+\mathrm{EI}\frac{{\partial }^{4}w}{\partial t^4}+\mathrm{kw}=\frac{\mathrm{EA}}{2L}[\underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^2\mathrm{dx}+N_t]\frac{{\partial }^{2}w}{\partial x^2}$
Nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes 90 $k_{55}\frac{\partial }{\partial x}(\varphi +\frac{\partial w}{\partial x})-N_t\frac{{\partial }^2}{\partial x^2}-N_t\frac{{\partial }^2}{\partial x^2}(w-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2})=\rho A\frac{{\partial }^2}{\partial t^2}(w-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2})$ $D_{11}\frac{{\partial }^2\varphi }{\partial x^2}-c_{11}{F}_{11}(\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^{3}w}{\partial x^3})-k_{55}(\varphi +\frac{\partial w}{\partial x})=\rho I\frac{{\partial }^2}{\partial t^2}(\varphi -\mu \frac{{\partial }^2\varphi }{\partial x^2})$
Nonlocal Euler–Bernoulli beam theory considering forced vibrations and thermal effects 88 $\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+(\rho A\frac{{\partial }^{2}w}{\partial t^2}+k_{1}w+k_2{w}^3+\mu \frac{\partial w}{\partial t}-\frac{\mathrm{EA}}{2L}[\underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^2\mathrm{dx}+N_t]\frac{{\partial }^{2}w}{\partial x^2}-F(x,t))}\\ \multicolumn{1}{c}{-(e_{0}a{)}^2\frac{{\partial }^w}{\partial x^2}(\rho A\frac{{\partial }^{2}w}{\partial t^2}+k_{1}w+k_2{w}^3+\mu \frac{\partial w}{\partial t}-\frac{\mathrm{EA}}{2L}[\underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^2\mathrm{dx}+N_t]\frac{{\partial }^{2}w}{\partial x^2}-F(x,t))=0}\end{array}$

Fluid-conveying CNTRs

Vibration analysis of fluid-conveying CNTRs has attracted great attention. Many researchers have focused on them to study the effect of fluid on their vibrations. The main reason of the high attention of researchers to fluid-conveying CNTRs is because of its applications in nanofluidic devices. In this section, we briefly go over different proposed nonlocal-based mathematical models for vibrations of CNTRs considering fluid effect. The topic of this section can be comprehensively considered in another review article with special focus on vibrations of fluid-conveying CNTRs. One of the first analytical models for the vibration analysis of the fluid-conveying CNTRs was developed by Yoon et al. ²⁵ Results of their paper show the dependence of resonant frequencies on the flow velocity. Furthermore, they showed that the internal moving fluid could substantially affect resonant frequencies especially for suspended longer CNTs. Their developed mathematical model can be found in Table 2. In another work, Yoon et al. ⁹⁵ studied flow-induced flutter instability for a cantilever CNTRs. In their work, they showed that the internal moving fluid substantially affects vibrational frequencies and the decaying rate of amplitude especially for longer cantilever CNTs. Reddy et al. ⁹⁶ considered the following model for the vibrations of CNTRs

$$(m_f+m_c)\frac{{\partial }^{2}u}{\partial t^2}+2m_{f}v\frac{{\partial }^{2}u}{\partial t\partial x}+m_f{v}^2\frac{{\partial }^{2}u}{\partial x^2}+\mathrm{YI}\frac{{\partial }^{4}u}{\partial x^4}=0$$

(63)

where $m_f$ , $m_c$ , and v represent fluid mass, CNTRs mass, and also the velocity of the fluid, respectively. Their paper presents a technique which is capable of measuring the mass flow rate of fluid. One year later, Lee and Chang ⁹⁷ used nonlocal Euler–Bernoulli beam theory to model transverse vibration of fluid-conveying CNTRs. The governing equation of their developed mathematical model is as follows

$$EI\frac{{\partial }^{4}w}{\partial x^4}+m_f{v}^2\frac{{\partial }^{2}w}{\partial x^2}+2m_{f}v\frac{{\partial }^{2}w}{\partial x\partial t}+\frac{{\partial }^2}{\partial t^2}\left[(m_c+m_f)w-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2}\right]=0$$

(64)

In their work, they showed that obtaining the natural frequency of CNTRs is particularly useful to estimate the mass flow rate of fluid. One of the earliest attempts for the vibration analysis of fluid-conveying DWCNTs can be found in a paper by Wang et al. ⁹⁸ Using the Euler–Bernoulli beam theory, they obtained the following model for the vibrations of their objective system

$$c[w_2-w_1]=E{I}_1\frac{{\partial }^4{w}_1}{\partial x^4}+M{U}^2\frac{{\partial }^2{w}_1}{\partial x^2}+2MU\frac{{\partial }^2{w}_1}{\partial x\partial t}+(M+\rho A_1)\frac{\partial w_1^2}{\partial t^2}$$

(65)

$$-k{w}_2-c_1[w_2-w_1]=E{I}_2\frac{{\partial }^4{w}_2}{\partial x^4}+\rho A_2\frac{{\partial }^2{w}_2}{\partial t^2}$$

(66)

Results of their paper indicate that the effect on the critical dimensionless flow velocity of the slenderness ratio is significant in the fluid-conveying DWNTs, in contrast to the SWCNTs conveying fluid. In addition, their paper shows that as the spring constant k is increased, the critical dimensionless flow velocity becomes higher, both for the first and second modes. ⁹⁸ There are plenty of papers in which authors used local theory for vibrations of the fluid-conveying CNTRs. It is actually beyond the scope of this review article to focus on them. Model presented in equations (65) and (66) was then modified by Wang ⁹⁹ using nonlocal theory to investigate the effect of small length on its vibrations. The mathematical formulation of his model is as follows

$$\begin{array}{c}\multicolumn{1}{c}{c[w_2-w_1]=E{I}_1\frac{{\partial }^4{w}_1}{\partial x^4}+M{U}^2\frac{{\partial }^2{w}_1}{\partial x^2}+2\mathrm{MU}\frac{{\partial }^2{w}_1}{\partial x\partial t}}\\ \multicolumn{1}{c}{+(M+\rho A_1)\frac{{\partial }^2{w}_1}{\partial t^2}-(e_{0}a{)}^2}\\ \multicolumn{1}{c}{[\rho A_1\frac{{\partial }^{4}w}{\partial t^2\partial x^2}-c_1(\frac{{\partial }^2{w}_2}{\partial x_2}-\frac{{\partial }^2{w}_1}{\partial x^2})]}\end{array}$$

(67)

and

$$-c[w_2-w_1]=E{I}_2\frac{{\partial }^4{w}_2}{\partial x^4}+\rho A_2\frac{{\partial }^2{w}_2}{\partial t^2}-{(e_{0}a)}^2\left[\rho A_2\frac{{\partial }^4{w}_2}{\partial t^2\partial x^2}-c_1\left(\frac{{\partial }^2{w}_2}{\partial x_2}-\frac{{\partial }^2{w}_1}{\partial x^2}\right)\right]$$

(68)

One of the critical results of Wang’s paper is related to the effect of the small length scale on the critical velocity. The paper indicates that the small length scale is not influential on the critical flow velocity. It means that the local theories can provide a good estimation from the critical flow velocity. As depicted in the above equation, the term related to the fluid effect is completely similar to the local theory. In 2009, Tounsi et al. have shown that the above model is incomplete and modified it using nonlocal Euler–Bernoulli beam theory as follows ¹⁰⁰

$$\begin{array}{c}\multicolumn{1}{c}{c[w_2-w_1]=E{I}_1\frac{{\partial }^4{w}_1}{\partial x^4}+M{U}^2\frac{{\partial }^2{w}_1}{\partial x^2}+2\mathrm{MU}\frac{{\partial }^2{w}_1}{\partial x\partial t}}\\ \multicolumn{1}{c}{+(M+\rho A_1)\frac{{\partial }^2{w}_1}{\partial t^2}-(e_{0}a{)}^2}\\ \multicolumn{1}{c}{[\rho A_1\frac{{\partial }^{4}w}{\partial t^2\partial x^2}2\mathrm{MU}\frac{{\partial }^4{w}_1}{\partial x^3\partial t}+M{U}^2\frac{{\partial }^4{w}_1}{\partial x^4}-c_1(\frac{{\partial }^2{w}_2}{\partial x_2}-\frac{{\partial }^2{w}_1}{\partial x^2})]}\end{array}$$

(69)

and

$$-c[w_2-w_1]=E{I}_2\frac{{\partial }^4{w}_2}{\partial x^4}+\rho A_2\frac{{\partial }^2{w}_2}{\partial t^2}-{(e_{0}a)}^2\left[\rho A_2\frac{{\partial }^4{w}_2}{\partial t^2\partial x^2}-c_1\left(\frac{{\partial }^2{w}_2}{\partial x_2}-\frac{{\partial }^2{w}_1}{\partial x^2}\right)\right]$$

(70)

In their paper, they reported that previously published papers missing the fluid terms concerning the nonlocal effects. Table 6 presents a few mathematical models for vibrations of the CNTRs considering fluid interaction. In this part of the article, we focus on a model which has been recently developed by Askari ⁷⁹ for vibrations of fluid-conveying CNTRs

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}+f_{f_e}+\mu \frac{{\partial }^{2}w}{\partial t^2}-k_p\frac{{\partial }^{2}w}{\partial x^2}}\\ \multicolumn{1}{c}{+k_{1}w+k_3{w}^3-e_0{a}^2\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}\\ \multicolumn{1}{c}{(\rho A\frac{{\partial }^{2}w}{\partial t^2}+f_{f_e}+\mu \frac{{\partial }^{2}w}{\partial t^2}-k_p\frac{{\partial }^{2}w}{\partial x^2}+k_{1}w+k_3{w}^3)}\\ \multicolumn{1}{c}{=F(x,t)(1-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2})}\end{array}$$

(71)

where

$$f_{f_e}=m_f(2v\frac{{\partial }^{2}w}{\partial x\partial t}+v^2\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial t^2})$$

(72)

Table 6.

Fluid effect in the vibrational models of CNTRs.

Developed mathematical model
Nonlocal Euler–Bernoulli model considering curvature and fluid effects 101 $\begin{array}{c}\multicolumn{1}{c}{E{I}_1\frac{{\partial }^4{w}_i}{\partial x^4}+[M{U}^2+{\Pi }_0]\frac{{\partial }^2{w}_i}{\partial x^2}+2\mathrm{MU}\frac{{\partial }^2{w}_i}{\partial x\partial t}}\\ \multicolumn{1}{c}{+(M+m)\frac{{\partial }^2{w}_i}{\partial t^2}-(e_{0}a{)}^2[m\frac{{\partial }^4{w}_i}{\partial t^2\partial x^2}+[M{U}^2+{\Pi }_0]\frac{{\partial }^4{w}_i}{\partial x^4}2\mathrm{MU}\frac{{\partial }^4{w}_i}{\partial x^3\partial t}]=f_i(x,t),\mathrm{where}i=1,2}\end{array}$
Nonlocal Euler–Bernoulli beam model considering thermal and fluid effects 73 $\mathrm{EI}\frac{{\partial }^{4}w}{\partial x^4}+m_c\frac{{\partial }^{2}w}{\partial t^2}+\mathrm{kw}-\frac{\partial }{\partial x}\left(N_T\frac{\partial w}{\partial x}\right)+f_{f_e}-(e_{0}a{)}^2(m_c\frac{{\partial }^{4}w}{\partial x^2\partial t^2}+\frac{{\partial }^2}{\partial x^2}[\mathrm{kw}-\frac{\partial }{\partial x}\left(N_T\frac{\partial w}{\partial x}\right)+f_{f_e}])$ where equation (72) defines $f_{f_e}$ . Soltani et al. 102 supposed viscoelastic foundation and added following term $c(\partial w/\partial t)$ which is relevant to damping
Nonlocal Euler–Bernoulli beam theory considering surface tension and fluid effects 103 $\begin{array}{c}\multicolumn{1}{c}{(\mathrm{EI}+\hslash )\frac{{\partial }^{4}w}{\partial x^4}+[M{U}^2-{\Pi }_0]\frac{{\partial }^{2}w}{\partial x^2}+2\mathrm{MU}\frac{{\partial }^{2}w}{\partial x\partial t}}\\ \multicolumn{1}{c}{+(M+m)\frac{{\partial }^{2}w}{\partial t^2}-(e_{0}a{)}^2[m\frac{{\partial }^4{w}_i}{\partial t^2\partial x^2}+[M{U}^2-{\Pi }_0]\frac{{\partial }^4{w}_i}{\partial x^4}2\mathrm{MU}\frac{{\partial }^4{w}_i}{\partial x^3\partial t}]=0}\end{array}$ where $\hslash$ shows the additional flexural rigidity due to the inner and outer surface layers and ${\Pi }_0$ is relevant to the residual surface tension
Non-uniform fluid-conveying CNTRs 104 $\frac{{\partial }^2}{\partial x^2}\left(\mathrm{EI}\frac{{\partial }^{2}w}{\partial x^2}\right)-\frac{{\rho }_w{A}_i}{2\pi }\frac{d{A}_i}{\mathrm{dx}}(\frac{{\partial }^{3}w}{\partial x^2\partial t}+2U_i\frac{{\partial }^{3}w}{\partial x^2\partial t}+U_i^2\frac{{\partial }^{3}w}{\partial x^3})$ $\begin{array}{c}\multicolumn{1}{c}{\rho A_i(\frac{{\partial }^{2}w}{\partial t^2}+2U_i\frac{{\partial }^{2}w}{\partial x\partial t}+U_i^2\frac{{\partial }^{2}w}{\partial x^2})+\mathrm{kw}+c\frac{\partial w}{\partial t}+m\frac{{\partial }^{2}w}{\partial t^2}}\\ \multicolumn{1}{c}{-(e_{0}a{)}^2[{\rho }_w{A}_i(\frac{{\partial }^{4}w}{\partial x^2\partial t^2}+2U_i\frac{{\partial }^{4}w}{\partial x^3\partial t}+U_i^2\frac{{\partial }^{4}w}{\partial x^2\partial t^2}+4U_i\frac{d{U}_i}{\mathrm{dx}}\frac{{\partial }^{3}w}{\partial x^3}+4U_i\frac{d{U}_i}{\mathrm{dx}}\frac{{\partial }^{3}w}{\partial x^2\partial t})+k\frac{{\partial }^{2}w}{\partial x^2}+c\frac{{\partial }^{3}w}{\partial x^2\partial t}+m\frac{{\partial }^{4}w}{\partial x^2\partial t^2}]}\\ \multicolumn{1}{c}{-(e_{0}a{)}^2[2{\rho }_w\frac{d{A}_i}{\mathrm{dx}}(\frac{{\partial }^{3}w}{\partial x\partial t^2}+2U_i\frac{{\partial }^{3}w}{\partial x^2\partial t}+U_i^2\frac{{\partial }^{3}w}{\partial x^3})-\frac{{\rho }_w{A}_i}{2\pi }\frac{d{A}_i}{\mathrm{dX}}(\frac{{\partial }^{3}w}{\partial x^3\partial t^2}+2U_i\frac{{\partial }^{5}w}{\partial x^4\partial t}+U_i^2\frac{{\partial }^{5}w}{\partial x^5})]}\end{array}$ where $I(x)=I_0(1-\alpha x{)}^{\Phi +2}$ , $A_i(x)=A_{i}0(1-\alpha x{)}^{\Phi }$ , and $U_i(x)=U_{i0}(1-\alpha x{)}^{-\Phi }$ where $\alpha$ and $\Phi$ represent the taper ratio and non-uniform parameter for the non-uniform CNTRs, respectively.
CNTRs conveying pulsating and viscous fluid 105 $\begin{array}{c}\multicolumn{1}{c}{(1+c\frac{\partial }{\partial t})\mathrm{EI}\frac{{\partial }^{4}y}{\partial x^4}+[M{U}^2-N+\mathrm{PA}(1-2\nu )+M\frac{\partial U}{\partial t}(L-x)]\frac{{\partial }^{2}y}{\partial x^2}+(M+m)\frac{{\partial }^{2}y}{\partial t^2}+2\mathrm{MU}\frac{{\partial }^{2}y}{\partial x\partial t}}\\ \multicolumn{1}{c}{-\mathrm{rA}(U\frac{{\partial }^{3}y}{\partial x^3}+\frac{{\partial }^{3}y}{\partial x^2\partial t})-(e_{0}a{)}^2[(M+m)\frac{{\partial }^{4}y}{\partial x^2\partial t^2}+M{U}^2\frac{{\partial }^{4}w}{\partial x^4}+2\mathrm{MU}\frac{{\partial }^{4}y}{\partial x^3\partial t}]=0}\end{array}$
Wavy fluid-conveying CNTRs considering thermal and electromagnetic field effects using nonlocal Timoshenko beam theory 106 $-(m_c+m_f)\frac{{\partial }^{2}w}{\partial t^2}-2m_{f}V\frac{{\partial }^{2}w}{\partial x\partial t}+\beta \mathrm{GA}(\frac{{\partial }^{2}w}{\partial x^2}+\frac{\partial \varphi }{\partial x})$ $N_x\frac{d^2{w}_0}{d{x}^2}-k_w+(k_g+N_t+\eta {\mathrm{AH}}_x^2-m_f{V}^2)\frac{{\partial }^{2}w}{\partial x^2}$ $-(e_{0}a{)}^2\frac{{\partial }^2}{\partial x^2}[-(m_c+m_f)\frac{{\partial }^{2}w}{\partial t^2}-2m_{f}V\frac{{\partial }^{2}w}{\partial x\partial t}+N_x\frac{d^2{w}_0}{d{x}^2}-k_w+(k_g+N_t+\eta {\mathrm{AH}}_x^2-m_f{V}^2)\frac{{\partial }^{2}w}{\partial x^2}]=0$ $\mathrm{EI}\frac{{\partial }^2\varphi }{\partial x^2}-\beta \mathrm{GA}(\frac{\partial w}{\partial x}+\varphi )+(e_{0}a{)}^2(J_c+J_f)\frac{{\partial }^4\varphi }{\partial x^2\partial t^2}=(J_c+J_f)\frac{{\partial }^2\varphi }{\partial t^2}$
FSI governing equation of fluid-conveying CNTRs Dimensionless form 107 $-\tau \frac{{\partial }^{6}w}{\partial x^6}+\frac{{\partial }^{4}w}{\partial x^4}+u^2\frac{{\partial }^{2}w}{\partial x^2}-\Psi \frac{{\partial }^{2}w}{\partial x^2}+2\mu \sqrt{\beta }\frac{{\partial }^{2}w}{\partial x\partial t}+\frac{{\partial }^{2}w}{\partial t^2}=0$ $\frac{{\partial }^{3}w}{\partial x^3}-{\tau }^2\frac{{\partial }^{5}w}{\partial x^5}+u^2\frac{\partial w}{\partial x}+u\sqrt{\beta }\frac{\partial w}{\partial t}=0$ or $w=0$ (at $x=0,1$ ) $\frac{{\partial }^{2}w}{\partial x^2}-{\tau }^2\frac{{\partial }^{4}w}{\partial x^4}=0$ or $\frac{\partial w}{\partial x}=0$ at $(x=0,1)$ $\frac{{\partial }^{3}w}{\partial x^3}+3{\tau }^2\frac{{\partial }^{5}w}{\partial x^5}=0$ or $\frac{{\partial }^{2}w}{\partial x^2}=0$ at $(x=0,1)$ where $\tau$ , u, $\Psi$ , and $\beta$ are pertinent to nonlocal parameter, dimensionless flow velocity, longitudinal magnetic field, and mass ratio, respectively

CNTRs: carbon nanotube resonators; FSI: fluid–structure interaction.

In Figure 15, we can see the effect of the type of the fluid on the nonlinear behavior of the CNTRs based on the model developed by Askari. ⁷⁹ As depicted from the figure, increasing the fluid density results in decreasing the amplitude of the oscillations. In Figure 16, the effect of fluid velocity is presented on the linear natural frequency of the CNTRs. As evidenced in Figure 16, there is a fluid velocity in which the natural frequency of the CNTRs is zero. This velocity is called critical fluid velocity. The highest critical fluid velocity occurs when the boundary conditions of CNTRs are clamped–clamped (CC). Combining equations (59) and (71), we can have both thermal and fluid velocity effects. Figure 17 represents the effect of both temperature and fluid velocity for Armchair (20, 20) CNTRs. It shows that increasing the temperature results in an increase in the critical fluid velocity.

Figure 15.

Effect of the fluid type on the frequency behavior of CNTRs. ⁷⁹

Figure 16.

Effect of the fluid velocity and boundary conditions on the frequency behavior of CNTRs (Armchair (20, 20)) ⁷⁹ (CC: clamped–clamped, CS: clamped–simply, SS: simply–simply).

Figure 17.

Effect of the fluid velocity and boundary conditions on the frequency behavior of CNTRs (Armchair (20, 20)). ⁷⁹

Electromagnetic field effect

Electromagnetic field effect on the vibrations of CNTs was investigated by many researchers in the field of vibration analysis of nanostructures. The Maxwell’s equations can easily describe the distribution of the electrical and magnetic fields within the nanostructures. The main four parts of Maxwell’s equations can be illustrated by the following equation

$$\nabla \times H=\stackrel{·}{D}+J$$

(73)

$$\nabla \times E=-\stackrel{·}{B}$$

(74)

$$\nabla ·D={\rho }_b$$

(75)

$$\nabla ·B=0$$

(76)

where E, H, J, B, and D are the electric field intensity, magnetic field, current density vector, magnetic field density, and the displacement current density, respectively. Formulating of the Lorentz force is the main part for the vibration analysis of CNTs considering electromagnetic field effect. The Lorentz force is formulated in terms of the displacement field of the CNTs and also the strength of the magnetic field as below

$$f_m=\eta \nabla \times \nabla \times (u\times H_0)\times H_0$$

(77)

$\mathrm{where}\nabla$ , $\eta$ , $H_0$ , and u show gradient operator, permeability of CNTs, magnetic field, and the displacement field of CNTs, respectively. Recently, a few papers have been published which show the effect of the electromagnetic field on the vibrations of CNTs. For example, in Kiani, ¹⁰⁸ the transverse waves within embedded DWCNTs which are subjected to an axial magnetic field were investigated. Results of the paper show that flexural frequencies increase the longitudinal magnetic field. An interesting work by Zhang et al. ¹⁰⁹ has analyzed vibrations of a horn-shaped SWCNT embedded in a viscoelastic medium under longitudinal magnetic field. They showed that nonlocal effect significantly decreases by increasing the longitudinal magnetic field $H_x$ . In addition, the rigidity of the system increases by increasing the longitudinal magnetic field. Figure 18 schematically shows their model. Hosseini and Sadeghi-Goughari ¹⁰⁷ analyzed vibrations of CNTs considering the effect of both longitudinal magnetic field $H_x$ and fluid interaction with CNTs. Results of their paper show that the effect of the longitudinal magnetic field is insignificant on the vibration behavior of CNTs when $0<H_x<{10}^7\mathrm{A}/\mathrm{m}$ . The vibration properties of CNTs abruptly change when the longitudinal magnetic field goes to the range of ${10}^7<H_x<{10}^8\mathrm{A}/\mathrm{m}$ . They obtained the following equation as Lorentz force for the vibrations of their system in Z-direction

$$f_m=\eta H_x^2\frac{{\partial }^{2}W}{\partial x^2}$$

(78)

where W is the lateral motion of CNTs in ZX-plane. Karlicic et al. ¹¹⁰ investigated the free transverse vibration analysis of nanosystem composed of multiple single-walled carbon nanotubes (MSWCNT) embedded in a polymer matrix and under the influence of an axial magnetic field. Vibration of a double viscoelastic CNTs conveying viscous fluid coupled by visco-Pasternak medium under longitudinal magnetic field was studied by Arani et al. ¹¹¹ They showed that increasing the magnetic field significantly increases the stability of the system. Accordingly, the magnetic field can be used to control stability of CNTs.

Figure 18.

Schematic of horn-shaped CNTs under longitudinal magnetic field. ¹⁰⁹

vdW force

The focus of this part will be on the vdW force and mathematical modeling of CNTRs considering the above-mentioned force. The topic of this part includes plenty of research works and it has the potential to be investigated in a separate review article. Accordingly, we just illustrate one example based on the nonlocal continuum theory. Originally, Ruoff et al. ⁹ reported the effect of the vdW force on CNTs. The result of their paper shows that the vdW force between two adjacent nanotubes can deform them substantially. This result has stimulated researchers to study the vdW forces on the properties of CNTs, especially their vibrational characteristics. One of the well-known models for vibrations of CNTRs considering vdW force was developed by Ke et al. ⁷⁴ In their model, they have used nonlocal Timoshenko beam theory to model vibrations of CNTRs. Their obtained governing equation for CNTRs is as follows

$$E{A}_i(\frac{{\partial }^2{U}_i}{\partial x^2}+\frac{\partial w_i}{\partial x}\frac{{\partial }^2{w}_i}{\partial x^2})=\rho A_i\frac{{\partial }^2}{\partial t^2}[U_i-{(e_{0}a)}^2\frac{{\partial }^2{U}_i}{\partial x^2}]$$

(79)

$$\begin{array}{c}\multicolumn{1}{c}{K_{\mathrm{si}}(\frac{{\partial }^2{w}_i}{\partial x^2}+\frac{\partial {\Phi }_i}{\partial x})+Z_i-(e_{0}a{)}^2{R}_i+n_0{c}_{1}D}\\ \multicolumn{1}{c}{+\frac{\partial \Psi }{\partial x}-m_{0}k[w_2-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial x^2}]}\\ \multicolumn{1}{c}{=\rho A_i\frac{{\partial }^2}{\partial t^2}[w_i-{(e_{0}a)}^2\frac{{\partial }^{2}w}{\partial t^2}]-\rho A_i(e_{0}a{)}^2\frac{{\partial }^2}{\partial t^2}}\\ \multicolumn{1}{c}{[\frac{{\partial }^2{U}_i}{{\partial }^{2}x}-{(e_{0}a)}^2\frac{{\partial }^4{U}_i}{\partial x^4}]}\end{array}$$

(80)

and

$$E{I}_i\frac{\partial {\Phi }_i}{\partial x^2}-K_{\mathrm{si}}\mathrm{GA}(\frac{\partial w_i}{\partial x}+{\Phi }_i)=\rho I_i\frac{{\partial }^2}{\partial t^2}[{\Phi }_i-{(e_{0}a)}^2\frac{{\partial }^2{\Phi }_i}{\partial x^2}]$$

(81)

In the above equation, $c_1$ and D are related to the vdW force and formulated as below

$$c_i=\frac{320(2r_i)\mathrm{erg}/\mathrm{c}{\mathrm{m}}^2}{0.16{\Delta }^2},\Delta =0.142\mathrm{nm}$$

(82)

and

$$D=[(W_2-W_1)-{(e_{0}a)}^2(\frac{{\partial }^2{w}_2}{\partial x^2}-\frac{{\partial }^2{w}_1}{\partial x^2})]$$

(83)

As we illustrated before, vdW interaction can be a potential reason for emerging the nonlinearity in the vibrations of CNTRs. It may cause geometrical nonlinearity in CNTRs or interacting force as follows ¹¹²

$$c_3[w_2-w_1{]}^3-c_1[w_2-w_1]$$

(84)

where

$$c_1=\frac{{\partial }^{2}U}{\partial {\delta }^2}2R,\delta ={\delta }_0$$

(85)

$$c_3=\frac{{\partial }^{4}U}{\partial {\delta }^4}2R,\delta ={\delta }_0$$

(86)

where U is the interlayer potential per unit area, which can be expressed in terms of the interlayer spacing $\delta$ as

$$U(\delta )=K[{\left(\frac{{\delta }_0}{\delta }\right)}^2-0.4{\left(\frac{{\delta }_0}{\delta }\right)}^{1}0]$$

(87)

where $K=61.665\mathrm{meV}/\mathrm{atom}$ , and $0=0.34\mathrm{nm}$ is the equilibrium interlayer spacing. As an interesting example of considering nonlinear vdW interaction, we can refer to a paper by Ansari et al. ¹¹³ and Ansari and Hemmatnezhad ¹¹⁴ in which they analyzed vibrations of CNTs assuming the nonlinear interaction force.

Conclusion and future works

A comprehensive review was provided about vibrations of the CNTRs considering different effects based on the nonlocal continuum theory. Three different nonlocal beam theories including Timoshenko, Euler–Bernoulli, and Rayleigh were illustrated and their potential for modeling the vibrations of CNTRs was shown. Several mathematical models were tabulated considering different parameters such as nonlinearity, thermal effect, fluid–solid interaction, and also vdW forces. It was shown how the above parameters can be influential on the dynamical behavior of CNTRs. Despite many published articles based on the nonlocal theory for the vibrations analysis of the nanoresonators, there is lack of knowledge about a few specific and critical subjects in the field of vibration analysis of nanoresonators, and researchers can focus on these topics. In the last lines of these review articles, we provide a list of topics which can be useful for future research:

1. It is necessary to have an accurate mathematical model from the curvature of CNTRs, and there is lack of models which can predict the waviness of CNTRs accurately. Recently, Askari et al. ¹¹⁵ proposed a non-uniform rational basis spline (NURBS)-based mathematical model which can be considered by the researchers to use it for mathematical modeling of CNTRs waviness.

2. Chaotic vibrations analysis of CNTRs is another subject which needs more attention. ¹¹⁶ Chaos may emerge in the vibrations of CNTRs due to the axial load, thermal effect, and also FSI.

3. The use of CNTRs for small mass sensing is a hot topic and rapidly growing. Many researchers have focused on the application of CNTRs in mass sensing such as Cho et al. ¹¹⁷ and Murmu and Adhikari. ¹¹⁸ But still, there is a high demand for having a precise mathematical model of nanosensing instrument with ultra-high resolutions based on CNTRs.

4. In spite of many published articles about vibrations of CNTRs, there is lack of research about control and vibration suppression of these highly sensitive resonators. There is a high demand for designing passive and adaptive controller to suppress CNTRs vibrations due to their high sensitivity to very small perturbation.

5. Vibration analysis of other types of nanoresonators such as nanowires is another interesting topic for the researchers. It is critical to have an accurate mathematical model for the vibrations of nanowires. The area of researches about vibrations of nanowires is rapidly growing. The use of nanowires for energy harvesting still needs more research.

6. Also, researchers can focus on the vibration analysis of piezoelectric CNTRs. Nonlocal theories are currently being used for the vibration analysis of CNTRs and the field is steadily expanding.

7. Mathematical modeling of vibrations of CNTRs for drug delivery, gas sensing, and also bio-sensing might be considered as an interesting topic for the researchers in the field of vibrations of the nanoresonators.

8. Nonlocal shell theory has been used so far by many researchers to investigate vibrations of CNTs. A few crucial examples include works done by Ansari and colleagues,^38,39,119 Wang and Varadan, ¹²⁰ Ansari and Arjangpay, ¹²¹ Soltani et al., ¹²² Arash and Ansari, ¹²³ Fazelzadeh and Ghavanloo, ¹²⁴ Ansari et al., ¹²⁵ and Ansari and Arash. ¹²⁶ Shell theory is a highly efficient method for vibration analyzing of CNTs, and comparison with beam theories can be an interesting topic for future research.

9. As we reported in the paper, the differential form of the Eringen nonlocal theory of the elasticity has been widely exploited in the vibration analysis of nanostructures. There are some inconsistencies in the models based on the above-mentioned differential. Recently, Fernndez-Saez et al. ¹²⁷ developed a new static bending model with combination of Euler–Bernoulli and Eringen integral constitutive equations. The novel model can be implemented by researchers to overcome the above-mentioned issue pertinent to the application of the differential form of the Eringen nonlocal theory in the dynamic analysis of nanostructures. Recently published papers by Khodabakhshi and Reddy ¹²⁸ and Tuna and Kirca ¹²⁹ include detailed analysis about the Eringen integral constitutive equation.

10. Higher-order gradient elasticity theories have been exploited by many researchers such as Ansari et al.^113,130,131 for vibration modeling of CNTs. These theories have the potential to be implemented by researchers for modeling of nanoresonators with sensing applications. In addition, a comprehensive review article about the applications of the higher-order gradient elasticity theories in nanoresonators is highly beneficial for future works.

The field of vibration analysis of nanoresonators is beyond the above-mentioned topics, and every day, a new concept emerges in this field.