Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamilton's principle applicable to this case is also given.

1. Introduction

In the present study, the variational principles and the natural boundary conditions are derived for multiwalled carbon nanotubes undergoing the transverse vibrations. The governing equations are based on the nonlocal theory of elasticity which gives more accurate results than local elastic theory by taking the small scale effects into account in the formulation. Variational principles applicable to the multiwalled nanotubes undergoing vibrations and the related boundary conditions were derived in [1] using a continuum model based on the nonlocal theory of Euler-Bernoulli beams. In the present study these results are extended to the case of multiwalled nanotubes undergoing transverse vibrations and the Hamilton’s principle is derived.

The laws of continuum mechanics are known to be robust enough to treat intrinsically discrete objects only a few atoms in diameter [2]. Subsequent studies established the accuracy of continuum-based approaches to the mechanics of nanotubes. A study of the range of applicability of elastic beam theory to model nanotubes and nanorods was given in [3]. Beam models used to study the buckling and vibration behavior of carbon nanotubes (CNTs) mostly employed the Euler-Bernoulli or Timoshenko beam theories. The equation for an Euler-Bernoulli beam is expressed in terms of only one unknown, namely, the deflection of the beam, and neglects the effect of transverse shear deformation. However, for nanotubes with low length to diameter ratio, shear deformation can be substantial and can be taken into account using a Timoshenko beam model. In this case the governing equations have two dependent variables, namely, the slope and deflection of the beam and are able to predict the mechanical behavior of CNTs more accurately. Several studies on the buckling of nanotubes used these two beam models with the Euler-Bernoulli beam model used in [4–8] and the Timoshenko model in [9]. Vibration of multiwalled nanotubes was studied in [10] using a Timoshenko beam model.

However, small scale effects were not taken into account in these papers. The importance of size effects for nanosized structures has been emphasized in [11–15] where properties of nano materials have been obtained. Beam theories capable of taking the small scale effects into account are based on the nonlocal theory of elasticity which was developed in early seventies [16, 17]. The nonlocal theory was applied to the study of nanoscale Euler-Bernoulli and Timoshenko beams in a number of papers [18–27]. Variational formulations for various nonlocal beam models were given in [23]. The nonlocal Euler-Bernoulli and Timoshenko beam models were employed to investigate the buckling and vibration characteristics of CNTs in [28–33] and comparisons between the two models were given in these papers. These studies considered single and double-walled nanotubes involving mostly simply supported boundary conditions and analytical solutions of the differential equations. Variational formulations allow the implementation of approximate and numerical methods of solutions and facilitate the consideration of complicated boundary conditions, especially in the case of multiwalled nanotubes. Recently variational principles and the natural boundary conditions were derived for multiwalled CNTs modeled as nonlocal Euler-Bernoulli beams in a number of studies with CNTs subject to vibrations [1] and a buckling load [34] where the linear elastic theory was employed. Variational principles were derived for CNTs undergoing nonlinear vibrations in [35] using a local Euler-Bernoulli beam CNT model.

Present study differs from the studies [1, 34, 35] where CNTs were modeled as Euler-Bernoulli beams with the nonlocal elastic theory employed in the case of CTNs undergoing linear vibrations [1] and buckling [34]. In the case of CTNs undergoing nonlinear vibrations again Euler-Bernoulli beam was used as a model which was based on the local elastic theory [35]. Euler-Bernoulli models are mostly applicable to nanotubes with a large length to diameter ratio and become inaccurate as the nanotubes become shorter. In the present study multiwalled CNTs are modeled as nonlocal Timoshenko beams which are applicable to nanotubes with a small length to diameter ratio and as such give accurate solutions for short CNTs [9, 10, 23–25].

The approach used in the present study to derive the variational principles is the semi-inverse method developed by He [36, 37]. Several examples of variational principles for systems of differential equations obtained by this method can be found in [38–42] and in the references therein. In the present study first the coupled differential equations governing the vibrations of multiwalled nanotubes based on nonlocal Timoshenko beam theory are given. Next a trial variational functional is formulated and a set of integrability conditions is derived which ensure that a classical variational principle can be obtained for the problem. Finally the variational principle and the Hamilton’s principle are obtained by the semi-inverse method and natural and geometric boundary conditions are derived.

2. Multiwalled Carbon Nanotubes

A multiwalled carbon nanotube of length L consisting of n nanotubes of cylindrical shape is considered. It lies on a Winkler foundation of modulus k and is subject to an axial stress σx which can be tensile or compressive in which case σx is less than the critical buckling load. We introduce a difference operator defined as

Δwij=wi-wj,
where wi and wj are the deflections of the ith and jth nanotubes. The differential equations governing the vibrations of multiwalled nanotubes based on the nonlocal Timoshenko beam theory can be expressed as [10, 24]

Da1(w1,φ1,w2)=La1(w1,φ1)-c12Δw21+η2c12∂2Δw21∂x2=0,Db1(w1,φ1)=Lb1(w1,φ1)=0,Da2(w1,w2,φ2,w3)=La2(w2,φ2)+c12Δw21-c23Δw32+η2(-c12∂2Δw21∂x2+c23∂2Δw32∂x2)=0,Db2(w2,φ2)=Lb2(w2,φ2)=0,⋮Dai(wi-1,wi,φi,wi+1)=Lai(wi,φi)+c(i-1)iΔwi(i-1)-ci(i+1)Δw(i+1)i-η2c(i-1)i∂2Δwi(i-1)∂x2+η2ci(i+1)∂2Δw(i+1)i∂x2=0,Dbi(wi,φi)=Lbi(wi,φi)=0fori=3,4,…,n-1,⋮Dan(wn-1,wn,φn)=Lan(wn,φn)+c(n-1)nΔwn(n-1)-η2c(n-1)n∂2Δwn(n-1)∂x2=f(x,t),Dbn(wn,φn)=Lbn(wn,φn)=0,
where φi is the angle of rotation and the operators Lai(wi,φi) and Lbi(wi,φi) are given by
Lai(wi,φi)=ρAi∂2wi∂t2-ρAiη2∂4wi∂t2∂x2+κGAi∂∂x(φi-∂wi∂x)+Aiσx∂2wi∂x2-Aiσxη2∂4wi∂x4+δin(kwn-kη2∂2wn∂x2),Lbi(wi,φi)=ρIi∂2φi∂t2-ρIiη2∂4φi∂t2∂x2+κGAi(φi-∂wi∂x)-EIi∂2φi∂x2,
where the index i=1,2,…,n refers to the order of the nanotubes with the innermost nanotube indicated by i=1 and the outermost nanotube by i=n with 0≤x≤L. In (8) f(x,t) is a forcing function, and in (10) δin is the Kronecker’s delta with δin=0 for i≠n and δnn=1. In (10) and (11), E is the Young’s modulus, G is the shear modulus, κ is the shear correction factor, Ii is the moment of inertia, Ai is the cross-sectional area of the ith nanotube and ρ is the density. The coefficient c(i-1)i is the interaction coefficient of van der Waals forces between the (i-1)th and ith nanotubes with i=2,…,n [7–10, 28]. The parameter η=e0a appears in the nonlocal theory of beams and helps define the small scale effects accurately where e0 is a constant for adjusting the model by experimental results and a is an internal characteristic length [17–26].

3. Variational Formulation

According to the semi-inverse method [36, 37], a variational trial-functional V(wi,φi) can be constructed as follows with the motion taking place between the initial time t1 and the final time t2

V(wi,φi)=V1(w1,φ1,w2)+V2(w1,w2,φ2,w3)+⋯+Vn-1(wn-2,wn-1,φn-1,wn)+Vn(wn-1,wn,φn),
where

V1(w1,φ1,w2)=U1(w1,φ1)+∫t1t2∫0LF1(w1,w2)dxdt,V2(w1,w2,φ2,w3)=U2(w2,φ2)+∫t1t2∫0LF2(w1,w2,w3)dxdt,Vi(wi-1,wi,φi,wi+1)=Ui(wi,φi)+∫t1t2∫0LFi(wi-1,wi,wi+1)dxdtfori=3,4,…,n-1,Vn(wn-1,wn,φn)=Un(wn,φn)+12∫t1t2∫0L(kwn2+kη2(∂wn∂x)2)dxdt+∫t1t2∫0L(-fwn+Fn(wn-1,wn))dxdt
with Ui(wi,φi) given by

Ui(wi,φi)=12∫t1t2∫0L(κGAi(φi-∂wi∂x)2+EIi(∂φi∂x)2-Aiσx(∂wi∂x)2-Aiσxη2(∂2wi∂x2)2)dxdt+12∫t1t2∫0L(-ρAi(∂wi∂t)2-ρAiη2(∂2wi∂t∂x)2-ρIi(∂φi∂t)2-ρIiη2(∂2φi∂t∂x)2)dxdt,
where i=1,2,…,n and Fi(wi-1,wi,wi+1) denotes the unknown functions of wi and its derivatives to be determined such that the differential equations (2)–(9) correspond to the Euler-Lagrange equations of the variational functional (12). These equations are given by

La1(w1,φ1)+∑j=12δFjδw1=La1(w1,φ1)+∑j=12[∂Fj∂w1-∂∂x(∂Fj∂w1x)-∂∂t(∂Fj∂w1t)+⋯]=0,La2(w2,φ2)+∑j=13δFjδw2=La2(w2,φ2)+∑j=13[∂Fj∂w2-∂∂x(∂Fj∂w2x)-∂∂t(∂Fj∂w1t)+⋯]=0,Lai(wi,φi)+∑j=i-1i+1δFjδwi=Lai(wi,φi)+∑j=i-1i+1[∂Fj∂wi-∂∂x(∂Fj∂wix)-∂∂t(∂Fj∂wit)+⋯]=0,fori=3,4,…,n-1,Lan(wn,φn)+∑j=n-1nδFjδwn=Lan(wn,φn)+∑j=n-1n[∂Fj∂wn-∂∂x(∂Fj∂wnx)-∂∂t(∂Fj∂wnt)+⋯]=0,Lbi(wi,φi)=0fori=1,2,…,n,
where the subscripts x and t denote differentiation with respect to x and t, and the variational derivative δFi/δwi is defined as [36, 37]

δFiδwi=∂Fi∂wi-∂∂x(∂Fi∂wix)-∂∂x(∂Fi∂wit)+∂2∂x2(∂Fi∂wixx)+∂2∂x∂t(∂Fi∂wixt)⋯-∂∂t(∂Fi∂wit)+∂2∂t2(∂Fi∂witt)+⋯.
Comparison of (15) with (2)–(9) indicates that the following equations have to be satisfied for Euler-Lagrange equations to represent the governing (2)–(9)

∑j=12δFjδw1=-c12Δw21+η2c12∂2Δw21∂x2,∑j=13δFjδw2=c12Δw21-c23Δw32-η2c12∂2Δw21∂x2+η2c23∂2Δw32∂x2,∑j=i-1i+1δFjδwi=c(i-1)iΔwi(i-1)-ci(i+1)Δw(i+1)i-η2c(i-1)i∂2Δwi(i-1)∂x2+η2ci(i+1)∂2Δw(i+1)i∂x2,∑j=n-1nδFjδwn=c(n-1)nΔwn(n-1)-η2c(n-1)n∂2Δwn(n-1)∂x2.
Integrability relations between these equations can be obtained by noting that

(∂∂w2+∂∂w2xx)∑j=12δFjδw1=-c12+η2c12,(∂∂w1+∂∂w1xx)∑j=13δFjδw2=-c12+η2c12,(∂∂wi+1+∂∂w(i+1)xx)∑j=i-1i+1δFjδwi=-ci(i+1)+η2ci(i+1),(∂∂wi+∂∂wixx)∑j=ii+2δFjδwi+1=-ci(i+1)+η2ci(i+1),(∂∂wn+∂∂wnxx)∑j=n-2nδFjδwn-1=-c(n-1)n+η2c(n-1)n,(∂∂wn-1+∂∂w(n-1)xx)∑j=n-1nδFjδwn=-c(n-1)n+η2c(n-1)n.
Having (18)-(19), (20)-(21), and (22)-(23) with the same right-hand sides ensures that the variational principle can be derived for the present problem. From (17), it follows that

F1(w1,w2)=c124Δw212+c124η2(∂Δw21∂x)2,Fi(wi-1,wi,wi+1)=c(i-1)i4Δwi(i-1)2+ci(i+1)4Δw(i+1)i2+η2c(i-1)i4(∂Δwi(i-1)∂x)2+η2ci(i+1)4(∂Δw(i+1)i∂x)2fori=2,3,…,n-1,Fn(wn-1,wn)=c(n-1)n4Δwn(n-1)2+η2c(n-1)n4(∂Δwn(n-1)∂x)2.
With Fi, i=1,2,…,n given by (24), we observe that (15) are equivalent to (2)–(9).

4. Hamilton’s Principle

The Hamilton’ principle can be expressed as

∫t1t2(δKE(t)-(δWE(t)+δPE1(t)+δPE2(t)))dt=0,
where
KE(t)=12∑i=1i=n∫0L(ρAi(∂wi∂t)2+η2ρAi(∂2wi∂x∂t)2-ρIi(∂φi∂t)2+η2ρIi(∂2φi∂t∂x)2)dx,WE(t)=12∑i=1i=n∫0L(-Aiσx(∂wi∂x)2-η2Aiσx(∂2wi∂x2)2-f(x,t)wn(x,t)(∂2wi∂x2)2)dx,PE1(t)=12∑i=1i=n∫0L(κGAi(φi-∂wi∂x)2+EIi(∂φi∂x)2+kwn2+kη2(∂wn∂x)2)dx,PE2(t)=12∑i=1i=n∫0L((∂2wi∂x2)2c(i-1)i(wi-wi-1)2+η2c(i-1)i(∂wi∂x-∂wi-1∂x)2)dx.
In (25)–(26), KE is the kinetic energy, WE is the work done by external forces, PE1 is the potential energy of deformation and PE2 is the potential energy due to van der Waals forces between the nanotubes.

5. Boundary Conditions

Next the variations of the functional V(wi,φi) in (12) are evaluated with respect to wi and φi in order to derive the natural and geometric boundary conditions. Let δwi and δφi denote the variations of wi and φi such that δwi(x,t1)=δwi(x,t2)=δφi(x,t1)=δφi(x,t2)=0. The first variations of V(wi,φi) with respect to wi and φi, denoted by δwiV and δφiV, respectively, can be obtained by integration by parts and expressed as

δw1V=δw1V1+δw1V2=∫t1t2∫0LDa1(w1,φ1,w2)δw1dxdt+∂Ωa1(0,L,t),δφ1V=δφ1V1=∫t1t2∫0LDb1(w1,φ1)δφ1dxdt+∂Ωb1(0,L,t),δwiV=∑j=i-1i+1δwiVj=∫t1t2∫0LDai(wi-1,wi,φi,wi+1)δwidxdt+∂Ωai(0,L,t)fori=2,…,n-1,δφiV=δφiVi=∫t1t2∫0LDbi(wi,φi)δφidxdt+∂Ωbi(0,L,t)fori=2,…,n-1,δwnV=δwnVn-1+δwnVn=∫t1t2∫0LDan(wn-1,wn,φn)δwndxdt+∂Ωan(0,L,t),δφnV=δφnVn=∫t1t2∫0LDbn(wn,φn)δφndxdt+∂Ωbn(0,L,t),
where ∂Ωia(0,L,t) and ∂Ωib(0,L,t) are the boundary terms defined as

∂Ωa1(0,L,t)=-A1σxη2∂2w1∂x2δw1′|x=0x=L+A1σxη2∂3w1∂x3δw1|x=0x=L+ρA1η2∂3w1∂x∂t2δw1|x=0x=L+(-κGA1(φ1-∂w1∂x)+(-A1σx+η2c12)∂w1∂x-η2c12∂w2∂x)δw1|x=0x=L,∂Ωai(0,L,t)=-Aiσxη2∂2wi∂x2δw1′|x=0x=L+Aiσxη2∂3wi∂x3δwi|x=0x=L+ρAiη2∂3wi∂x∂t2δwi|x=0x=L+[-κGAi(φi-∂wi∂x)+(-Aiσx+η2(c(i-1)i+ci(i+1)))×∂wi∂x-η2(c(i-1)i∂wi-1∂x+ci(i+1)∂wi+1∂x)]δwi|x=0x=Lfori=2,3,…,n-1∂Ωan(0,L,t)=-Anσxη2∂2wn∂x2δwn′|x=0x=L+Anσxη2∂3wn∂x3δwn|x=0x=L+ρAnη2∂3wn∂x∂t2δwn|x=0x=L+[-κGAn(φn-∂wn∂x)+(-Anσx+η2(c(n-1)n+k))×-∂wn∂xη2c(n-1)n∂wn-1∂x]δwn|x=0x=L,∂Ωbi(0,L,t)=(EIi∂φi∂x+ρIiη2∂3φi∂x∂t2)δφi|x=0x=Lfori=1,2,…,n,
where δwi′ is the derivative of δwi with respect to x. Thus the boundary conditions at x=0,L are given by

EIi∂φi∂x+ρIiη2∂3φi∂x∂t2=0orφi=0fori=1,2,…,n,(-Aiσxη2)∂2wi∂x2=0or∂wi∂x=0forσx≠0,i=1,2,…,n,A1σxη2∂3w1∂x3+ρA1η2∂3w1∂x∂t2-κGA1(φ1-∂w1∂x)+(-A1σx+η2c12)∂w1∂x-η2c12∂w2∂x=0orw1=0,Aiσxη2∂3wi∂x3+ρAiη2∂3wi∂x∂t2-κGAi(φi-∂wi∂x)+(-Aiσx+η2(c(i-1)i+ci(i+1)))∂wi∂x-η2(c(i-1)i∂wi-1∂x+ci(+1)∂wi+1∂x)=0orwi=0fori=2,…,n-1Anσxη2∂3wn∂x3+ρAnη2∂3wn∂x∂t2-κGAn(φn-∂wn∂x)+(-Anσx+η2(c(n-1)n+k))∂wn∂x-η2c(n-1)n∂wn-1∂x=0orwn=0.
Note that for σx=0, the boundary condition (30) is not needed. It is observed that for the small scale parameter η>0 (nonlocal theory) the natural boundary conditions are coupled and time derivative appears in the boundary conditions. These boundary conditions uncouple for η=0 (local theory) and time derivatives drop out.

6. Conclusions

Variational principles are derived using a semi-inverse variational method for multiwalled CNTs undergoing vibrations and modeled as nonlocal Timoshenko beams. Variational formulation of the problem facilitates the implementation of a number of computational approaches which, in most cases, simplify the method of solution as compared to the solution of a system of 2n differential equations. The nonlocal elasticity theory accounts for small scale effects applicable to nanosized objects and Timoshenko beam model takes shear deformation into account which is not negligible in the case of nanotubes with small length-to-diameter ratio. As such they provide a more accurate model as compared to the Euler-Bernoulli model in the case of short nanotubes as pointed out in the papers [9, 10, 23–25]. The corresponding Hamilton’s principle as well as the natural and geometric boundary conditions are derived. It is observed that the natural boundary conditions are coupled at the free end due to small scale effects being taken into account. The integrability conditions are also obtained which indicate whether a variational principle in the classical sense exists for the system of differential equations governing the vibrations of multiwalled nanotubes.

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