1 Introduction

Structures with characteristic dimension shrinking down to microscale or nanoscale have wide applications in modern science and technology^[1]. Experiments and numerical simulations, however,have demonstrated that the mechanical behaviors of the microstructures and nanostructures are size-dependent when the characteristic size is comparable to the internal length scale^[2,3,4]. To model the phenomena,a great deal of advancements and developments have been revealed in the applications of size-dependent continuum mechanics for the modeling and analysis of nanomaterials and nanostructures such as multi-walled carbon nanotubes (MWCNTs) and graphene nanoribbons (GNs)^{[5,6,7,8,9,10,11]}. The above-mentioned size-dependent continuum theories such as the strain gradient theory,the nonlocal elasticity theory,the surface elasticity theory,the couple stress theory,and the micropolar elasticity theory are found to be valid by comparing the molecular simulation results with the experimental data.

The nonlocal elasticity,initiated by Eringen^[12] by assuming that the stress at an arbitrary point is not only a function of the strain at that point but also a function of the strain of the occupied domain,has been extensively used in the past decade to analyze the carbonbased nanostructures^[13]. The buckling and vibration problems modeled by the nonlocal Euler- Bernoulli beam^[7] and the nonlocal Timoshenko beam^[8,14] are studied in detail with different boundary conditions. The analytical results for such problems seem so complex that great demand for the establishment of variational formulations that allow one to construct approximate numerical methods for the solutions is required. Based on the above idea,variational principles and corresponding boundary conditions have been derived for the MWCNTs modeled by nonlocal beams^{[15,16,17,18]} and the GNs modeled by nonlocal plates^[19] in numerous works recently. The derivation of the adopted variational principles is the semi-inverse method proposed by He^[20]. It will be used in the present paper.

The strain gradient theory is also used to investigate the buckling and wave propagation of carbon nanotubes (CNs)^[6,21,22,23]. Kumar^[21] adopted a variational approach to obtain the variationally consistent boundary conditions,and studied the buckling of the CNs modeled by the strain gradient Euler-Bernoulli beam. By comparing with the molecular dynamics (MD) simulation results,Wang et al.^[22] concluded that the fact that the MD simulation cannot give a proper dispersion relation when the wave number approaches a certain value might be explained by the proposed strain gradient model. However,the vibration of the CNTs modeled by the strain gradient theory is lacking in the open literature.

The present paper focuses on the derivation of the variational principles for the buckling and vibration of MWCNTs modeled by the strain gradient theory. A variational function is established, and the variationally consistent equations with the boundary conditions coupled with small scale parameters are determined. In the numerical simulation,the Rayleigh-Ritz method is used to analyze the small scale effect on the buckling and vibration of single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs) with three typical boundary conditions. 2 Problem formulation 2.1 Governing equations

Consider a multi-walled concentric system of n nanotubes with the uniform length L,where the adjacent tubes are coupled with van der Waals (vdW) interaction. The concentric system of the n nanotubes lies on a Winkler foundation with the modulus k,and is subject to a uniform axial stress σ (positive for compression). Then,the governing equations for the multi-walled concentric system are readily derived based on Wang et al.^[22] as follows:

where i = 2,3,· · · ,n−1,x is the longitudinal coordinate,and t is the time. wj is the transverse displacement of the jth tube,where j = 1,2,· · · ,n,j = 1 indicates the innermost nanotube, and j = n represents the outermost nanotube. f(x,t) is the external force (per unit axial length) acting on the outermost tube. ci−1,i is the vdW force constant between the (i − 1)th and the ith nanotubes. The differential operator yet undefined is expressed as where E is the Young’s modulus of the beam system,I_j and A_j are the second moment inertia and the cross-section area of the jth tube,respectively. r is a graded parameter related to the strain gradient theory^[6,21,23,24]. ρ is the density of the system. The prime and over-dot denote partial differential with respect to the coordinate x and the time t,respectively. δjn is Kronecker’s delta function. 2.2 Variational formulation

In this subsection,the variational principle of the multi-walled system is derived based on the semi-inverse method developed by He^[20]. To begin,a trial variational functional V (wj,t) is assumed as follows:

where in which From Eqs. (1)−(3) and (5),we can obtain F_j (w_j−1,w_j ,w_j+1) as follows: Substitute Eqs. (10)-(12) into (5). Then,we can obtain the total potential energy of the concentric system V. From δV = 0,the governing equations (1)-(3) can be obtained. 2.3 Hamilton’s principle

According to Eqs. (5) and (10)-(12),the Hamilton’s principle can be written as follows:

where In the above equations,T is the kinetic energy,WE is the external work,UE1 is the potential energy,and UE2 is the vdW interaction energy. 2.4 Buckling

Buckling is a kind of static problems,which means that the terms related to time in Eq. (13) must vanish. Therefore,we have

Then,the Rayleigh quotient for the buckling stress can be obtained by inserting Eqs. (15)-(17) into Eq. (18): 2.5 Vibration

In this subsection,the resonant frequency is obtained by the Hamilton’s principle. The displacement of the jth nanotube can be written as

where ω is the angular frequency. Substituting Eq. (20) into Eqs. (4) and (9) without considering the external force f(x,t),after some mathematical manipulations,we have Other functions are simply transformed by replacing the displacement w_j with W_j . For example, Fj(w_j-1,w_j ,w_j+1) is replaced by F_j(W_j-1,W_j,W_j+1). Then,the Rayleigh quotient for the resonant frequency can be obtained by substituting Eqs. (14)-(17) into Eq. (13): 2.6 Boundary conditions

To derive the geometric and natural boundary conditions of the governing equations (1)- (3),the variations of the functional V (w_j,t) in Eq. (5) with respect to the displacement wj can be derived. Without loss of generality,the variations of w_j at the initial and final times are assumed to be zero,i.e.,

δwj(x,t₀) = δwj (x,t₁) = 0. After integration by parts,the variation of V (wj,t) with respect to wj can be obtained. Denote the variation by δV,j . Then,we have

where in which Q_j ,M_j,and H_j represent the gradient shear force,moment,and higher-order bending moment,respectively,expressed by It is seen from Eqs. (27) and (28) that the boundary conditions involve totally six independent linear algebraic equations for r ≠ 0,which are independent of time and are different from those obtained within the framework of the nonlocal elasticity theory^[15,16]. The physical interpretation of the boundary conditions has been given in Ref. ^[25]. However,explicitly solving the buckling and vibration problems in the strain gradient framework faces mathematical challenge, even for the single beam. Therefore,the approximation method,which is our present interest, must be developed for the sake of science and engineering. 3 Static and dynamic analysis

In this section,the multi-walled concentric system of n nanotubes with three typical boundary conditions,i.e.,clamped-free (CF),simple-supported (SS),and doubly-clamped (CC),are investigated. The buckling and resonant frequencies are obtained by the Rayleigh-Ritz method using the trial function ψ(x) with deflections of the nanotubes given by W_j(x) = a_jψ(x). The mode shape functions selected in the numerical simulation are listed in Table 1 for different boundary conditions.

3.1 Buckling analysis

We first consider a double-walled concentric system with deflections of the inner and outer nanotubes given by

W₁(x) = a₁ψ(x),W₂(x) = a₂ψ(x).

Substituting W₁(x) and W₂(x) into Eq. (13) and neglecting the terms containing the time derivative w˙n,external force f(x,t),and elastic foundation k of the outermost nanotube,we arrive at the following variation functional:

The equilibrium profiles require the potential of the system to be a minimum,i.e.,

resulting in a system of two linear homogeneous equations of a1 and a2. The existence of a nonzero solution of (a1,a2) requires

where the non-dimensional parameters are defined by

Alternatively,Eq. (30) can be rewritten as

It must be pointed out that Eq. (31) can be reduced to the single beam modeled by the strain gradient theory by assuming that the vdW interaction between the adjacent layers is infinity (c₁₂ →∞),giving where I and A are the total inertia moment and the total cross-sectional area of the concentric beam system,respectively. 3.2 Vibration analysis

Following a similar procedure as in the buckling case,substituting W1(x) and W2(x) into Eq. (13) and neglecting the terms containing the external force f(x,t) and the elastic foundation k of the outermost nanotube,we can derive the following functional:

The equilibrium profiles require that the potential of the system must be a minimum,i.e.,

resulting in a system of two linear homogeneous equations for a1 and a2. The precondition for the existence of a nonzero solution of (a1,a2) is given by

Then,the resonant frequency of the mode m is determined by where and ωm0 and ω_m1 are the lower frequency (natural frequency) and the higher frequency (intertube frequency),respectively. The associated ratio of the amplitude of the inner tube to that of the outer tube with the initial stress is given by where It is readily verified from Eq. (37) that the amplitude ratio a1/a2 is positive (in-phase) for the natural frequency and negative (out-of-phase) for the intertube frequency,regardless of the selected mode shapes and boundary conditions. In other words,the vibration of the inner tube and that of the outer tube are coaxial for the natural frequency and noncoaxial for the intertube frequency.

In addition,it is pointed out that Eq. (34) can be reduced to the single beam result with the strain gradient theory by assuming that the vdW interaction between the layers is infinity (c₁₂ →∞),giving

4 Numerical results and discussion 4.1 Validation

To validate the present result against the previous works,the buckling load of a single beam modeled by the strain gradient theory with different boundary conditions is compared with the published results^[21]. Figure 1 shows the variations of the nondimensional buckling load σ₀/σ_0cr with the nondimensional strain gradient parameter η,in which σ0cr is the conventional fundamental buckling stress of a single beam with the SS boundary conditions. As shown in Fig. 1,the present results obtained by Eq. (32) agree excellently with those of the published analytical solutions,indicating that the selected buckling mode shapes are of adequate accuracy.

It is generally received that the small scale effect is significant for microstructures and nanostructures,e.g.,CNs and graphene sheets. However,in the published literature,the value of the small scale parameter (e₀) has been discussed extensively,which is 0.39 given by Eringen^[12], 0.82 extracted by Zhang et al.^[26],10.56 proposed by Wang^[27],and 1.48 extracted by Wang et al.^[28]. However,for the nanotubes modeled by the strain gradient theory,the small scale parameter (e₀) is 0.289^[5]. Jafari et al.^[24] studied the dispersion of waves in the CNs with the nonlocal shell theory and the strain gradient shell theory in detail^[24]. However,the similar work relating to the nonlocal beam theory and the strain gradient beam theory has not,to our best knowledge,been compared. The buckling stress and resonant frequency given in the present work are compared with the previous results based on the nonlocal Euler-Bernoulli beam theory with the SS boundary conditions. As a comparison with the subsequent numerical simulation,the material parameters are as follows^[29]:

where r_i is the inner radii,r_o is the outer radius,L/r_o is the length to radius ratio,h is the effective thickness,and ρ is the mass density. In comparison,a dimensional axial strain γ = σ/E is defined.

Figure 2 depicts the nondimensional buckling strain γ as a function of the mode m in comparison with the nonlocal Euler-Bernoulli beam model for the SS boundary conditions. It is noted that γ (= σ/E) is the lower root of Eq. (31) (see Fig. 2(a)). As shown in Fig. 2,γ, calculated based on the two continuum beam models,is insensitive to the buckling mode m for small scale parameters,but has an essential effect for larger scale parameters,especially at higher mode numbers. γ is 0.140 4 when m = 5 and r/l₀ = 10,which is 1.8% larger than the present work; while γ increases to 0.253 5 when m = 10 and r/l₀ = 10 based on the nonlocal beam model^[29],which is 60.13% larger than the present work. In addition,the use of the nonlocal beam model overestimates the nondimensional buckling stress,regardless of the buckling mode and nonzero small scale parameters.

Figure 3 shows the natural frequencies for various small scale parameters. Similar to Fig. 2, the frequencies predicted by the two beam models are found to have significant differences for higher mode numbers and larger scale parameters. The detailed information of the observation in Figs. 2 and 3 is provided by Jafari et al.^[24].

4.2 Buckling with different boundary conditions

The fundamental buckling strain is presented in Table 2 for different aspect ratios,small scale parameters,and boundary conditions. It is seen that the buckling strain γ decreases as r/l₀ increases,and it is close to that of the single beam model,in particular for large aspect ratios. It is also seen that the buckling strain predicted by the single beam model is a little higher than that predicted by the double beam system coupled by the vdW interaction,and the difference is weakened for large aspect ratio and compliant boundary conditions (e.g.,the CF boundary conditions). In other words,the vdW interaction plays an important role for the beam system with small aspect ratio and strengthened boundary conditions.

4.3 Vibration with different boundary conditions

The fundamental angular frequency is listed in Table 3 for different aspect ratios,small scale parameters,and boundary conditions. The main conclusions are similar to those of Table 2. In addition,the intertube frequency is found to be insensitive to the aspect ratio and the scale parameter for compliant boundary conditions. While for strengthened boundary conditions (e.g.,the CC boundary conditions),it varies distinctly,especially for small aspect ratio.

In order to know the effect of the vdW interaction on the resonant frequency of the nanotubes, the amplitude ratio is plotted in Fig. 4. It is observed that,for the in-phase mode, the amplitude ratio is close to unity for small mode number,and deviates from unity for large mode number. The amplitude ratio of the in-phase mode may be larger or smaller than unity, depending on the used small scale parameter. When m = 5,the amplitude ratio is 1.721 as r/l₀ = 0 and 0.411 as r/l₀ = 50. For the out-of-phase mode,however,the amplitude ratio is in the vicinity of −2 for small mode number,and varies for large mode number. In addition,it is also seen that the amplitude ratio,which decreases as the small scale parameter increases,is strongly related to the small scale parameter,mode number,and boundary conditions. Comparing the results for the three different boundaries shown in Fig. 4,we can conclude that the amplitude ratio is enlarged for strengthened boundary conditions at higher mode,indicating,as anticipated,the significant influence of the vdW interaction between the adjacent nanotubes. In contrast to Table 3 that the natural frequency does not converge to that of the corresponding single beam,it is concluded that the actual displacement under strengthened boundary conditions is sharply reduced compared with that under compliant boundary conditions.

5 Conclusions

A variational formulation for the buckling and vibration of the strain gradient MWCNTs is given by the semi-inverse method. Using this formulation,one can derive not only the governing equations of the MWCNTs but also the variationally consistent sets of the coupled boundary conditions. Based on the Rayleigh-Ritz method,the buckling stress,strain,and vibration frequency of the SWCNTs and DWCNTs are analytically obtained. These results are compared with those in published works. Two elastic beam models are also compared to show that the strain gradient beam model is more flexible than the nonlocal beam model when the small scale parameter is the same. In addition,the effects of the small scale parameter,aspect ratio,and boundary conditions on the buckling and vibration of the SWCNTs and DWCNTs are studied in detail. The results also indicate that the coupled vdW interaction between the adjacent nanotubes has prominent influence on the vibration of nanotubes.

Acknowledgements We would like to thank the anonymous reviewers for their carefully reading of the manuscript and valuable suggestions.