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 [48] 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 [1115] 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 [1827]. 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 [2833] 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, 2325].

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 [3842] 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=𝑤𝑖𝑤𝑗,(1) 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]

𝐷𝑎1𝑤1,𝜑1,𝑤2=𝐿𝑎1𝑤1,𝜑1𝑐12Δ𝑤21+𝜂2𝑐12𝜕2Δ𝑤21𝜕𝑥2𝐷=0,(2)𝑏1𝑤1,𝜑1=𝐿𝑏1𝑤1,𝜑1𝐷=0,(3)𝑎2𝑤1,𝑤2,𝜑2,𝑤3=𝐿𝑎2𝑤2,𝜑2+𝑐12Δ𝑤21𝑐23Δ𝑤32+𝜂2𝑐12𝜕2Δ𝑤21𝜕𝑥2+𝑐23𝜕2Δ𝑤32𝜕𝑥2𝐷=0,(4)𝑏2𝑤2,𝜑2=𝐿𝑏2𝑤2,𝜑2𝐷=0,(5)𝑎𝑖𝑤𝑖1,𝑤𝑖,𝜑𝑖,𝑤𝑖+1=𝐿𝑎𝑖𝑤𝑖,𝜑𝑖+𝑐(𝑖1)𝑖Δ𝑤𝑖(𝑖1)𝑐𝑖(𝑖+1)Δ𝑤(𝑖+1)𝑖𝜂2𝑐(𝑖1)𝑖𝜕2Δ𝑤𝑖(𝑖1)𝜕𝑥2+𝜂2𝑐𝑖(𝑖+1)𝜕2Δ𝑤(𝑖+1)𝑖𝜕𝑥2𝐷=0,(6)𝑏𝑖𝑤𝑖,𝜑𝑖=𝐿𝑏𝑖𝑤𝑖,𝜑𝑖𝐷=0for𝑖=3,4,,𝑛1,(7)𝑎𝑛𝑤𝑛1,𝑤𝑛,𝜑𝑛=𝐿𝑎𝑛𝑤𝑛,𝜑𝑛+𝑐(𝑛1)𝑛Δ𝑤𝑛(𝑛1)𝜂2𝑐(𝑛1)𝑛𝜕2Δ𝑤𝑛(𝑛1)𝜕𝑥2𝐷=𝑓(𝑥,𝑡),(8)𝑏𝑛𝑤𝑛,𝜑𝑛=𝐿𝑏𝑛𝑤𝑛,𝜑𝑛=0,(9) where 𝜑𝑖 is the angle of rotation and the operators 𝐿𝑎𝑖(𝑤𝑖,𝜑𝑖)  and 𝐿𝑏𝑖(𝑤𝑖,𝜑𝑖) are given by 𝐿𝑎𝑖𝑤𝑖,𝜑𝑖=𝜌𝐴𝑖𝜕2𝑤𝑖𝜕𝑡2𝜌𝐴𝑖𝜂2𝜕4𝑤𝑖𝜕𝑡2𝜕𝑥2+𝜅𝐺𝐴𝑖𝜕𝜑𝜕𝑥𝑖𝜕𝑤𝑖𝜕𝑥+𝐴𝑖𝜎𝑥𝜕2𝑤𝑖𝜕𝑥2𝐴𝑖𝜎𝑥𝜂2𝜕4𝑤𝑖𝜕𝑥4+𝛿𝑖𝑛𝑘𝑤𝑛𝑘𝜂2𝜕2𝑤𝑛𝜕𝑥2,(10)𝐿𝑏𝑖𝑤𝑖,𝜑𝑖=𝜌𝐼𝑖𝜕2𝜑𝑖𝜕𝑡2𝜌𝐼𝑖𝜂2𝜕4𝜑𝑖𝜕𝑡2𝜕𝑥2+𝜅𝐺𝐴𝑖𝜑𝑖𝜕𝑤𝑖𝜕𝑥𝐸𝐼𝑖𝜕2𝜑𝑖𝜕𝑥2,(11) 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 0𝑥𝐿. In (8) 𝑓(𝑥,𝑡) is a forcing function, and in (10) 𝛿𝑖𝑛 is the Kronecker’s delta with 𝛿𝑖𝑛=0 for 𝑖𝑛 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(𝑖1)𝑖 is the interaction coefficient of van der Waals forces between the (𝑖1)th and ith nanotubes with 𝑖=2,,𝑛 [710, 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 [1726].

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

𝑉𝑤𝑖,𝜑𝑖=𝑉1𝑤1,𝜑1,𝑤2+𝑉2𝑤1,𝑤2,𝜑2,𝑤3++𝑉𝑛1𝑤𝑛2,𝑤𝑛1,𝜑𝑛1,𝑤𝑛+𝑉𝑛𝑤𝑛1,𝑤𝑛,𝜑𝑛,(12) where

𝑉1𝑤1,𝜑1,𝑤2=𝑈1𝑤1,𝜑1+𝑡2𝑡1𝐿0𝐹1𝑤1,𝑤2𝑉𝑑𝑥𝑑𝑡,2𝑤1,𝑤2,𝜑2,𝑤3=𝑈2𝑤2,𝜑2+𝑡2𝑡1𝐿0𝐹2𝑤1,𝑤2,𝑤3𝑉𝑑𝑥𝑑𝑡,𝑖𝑤𝑖1,𝑤𝑖,𝜑𝑖,𝑤𝑖+1=𝑈𝑖𝑤𝑖,𝜑𝑖+𝑡2𝑡1𝐿0𝐹𝑖𝑤𝑖1,𝑤𝑖,𝑤𝑖+1𝑉𝑑𝑥𝑑𝑡for𝑖=3,4,,𝑛1,𝑛𝑤𝑛1,𝑤𝑛,𝜑𝑛=𝑈𝑛𝑤𝑛,𝜑𝑛+12𝑡2𝑡1𝐿0𝑘𝑤2𝑛+𝑘𝜂2𝜕𝑤𝑛𝜕𝑥2+𝑑𝑥𝑑𝑡𝑡2𝑡1𝐿0𝑓𝑤𝑛+𝐹𝑛𝑤𝑛1,𝑤𝑛𝑑𝑥𝑑𝑡(13) with 𝑈𝑖(𝑤𝑖,𝜑𝑖) given by

𝑈𝑖𝑤𝑖,𝜑𝑖=12𝑡2𝑡1𝐿0𝜅𝐺𝐴𝑖𝜑𝑖𝜕𝑤𝑖𝜕𝑥2+𝐸𝐼𝑖𝜕𝜑𝑖𝜕𝑥2𝐴𝑖𝜎𝑥𝜕𝑤𝑖𝜕𝑥2𝐴𝑖𝜎𝑥𝜂2𝜕2𝑤𝑖𝜕𝑥22+1𝑑𝑥𝑑𝑡2𝑡2𝑡1𝐿0𝜌𝐴𝑖𝜕𝑤𝑖𝜕𝑡2𝜌𝐴𝑖𝜂2𝜕2𝑤𝑖𝜕𝑡𝜕𝑥2𝜌𝐼𝑖𝜕𝜑𝑖𝜕𝑡2𝜌𝐼𝑖𝜂2𝜕2𝜑𝑖𝜕𝑡𝜕𝑥2𝑑𝑥𝑑𝑡,(14) where 𝑖=1,2,,𝑛 and 𝐹𝑖(𝑤𝑖1,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

𝐿𝑎1𝑤1,𝜑1+2𝑗=1𝛿𝐹𝑗𝛿𝑤1=𝐿𝑎1𝑤1,𝜑1+2𝑗=1𝜕𝐹𝑗𝜕𝑤1𝜕𝜕𝑥𝜕𝐹𝑗𝜕𝑤1𝑥𝜕𝜕𝑡𝜕𝐹𝑗𝜕𝑤1𝑡𝐿+=0,𝑎2𝑤2,𝜑2+3𝑗=1𝛿𝐹𝑗𝛿𝑤2=𝐿𝑎2𝑤2,𝜑2+3𝑗=1𝜕𝐹𝑗𝜕𝑤2𝜕𝜕𝑥𝜕𝐹𝑗𝜕𝑤2𝑥𝜕𝜕𝑡𝜕𝐹𝑗𝜕𝑤1𝑡𝐿+=0,𝑎𝑖𝑤𝑖,𝜑𝑖+𝑖+1𝑗=𝑖1𝛿𝐹𝑗𝛿𝑤𝑖=𝐿𝑎𝑖𝑤𝑖,𝜑𝑖+𝑖+1𝑗=𝑖1𝜕𝐹𝑗𝜕𝑤𝑖𝜕𝜕𝑥𝜕𝐹𝑗𝜕𝑤𝑖𝑥𝜕𝜕𝑡𝜕𝐹𝑗𝜕𝑤𝑖𝑡𝐿+=0,for𝑖=3,4,,𝑛1,𝑎𝑛𝑤𝑛,𝜑𝑛+𝑛𝑗=𝑛1𝛿𝐹𝑗𝛿𝑤𝑛=𝐿𝑎𝑛𝑤𝑛,𝜑𝑛+𝑛𝑗=𝑛1𝜕𝐹𝑗𝜕𝑤𝑛𝜕𝜕𝑥𝜕𝐹𝑗𝜕𝑤𝑛𝑥𝜕𝜕𝑡𝜕𝐹𝑗𝜕𝑤𝑛𝑡𝐿+=0,𝑏𝑖𝑤𝑖,𝜑𝑖=0for𝑖=1,2,,𝑛,(15) 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+.(16) 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)

2𝑗=1𝛿𝐹𝑗𝛿𝑤1=𝑐12Δ𝑤21+𝜂2𝑐12𝜕2Δ𝑤21𝜕𝑥2,3𝑗=1𝛿𝐹𝑗𝛿𝑤2=𝑐12Δ𝑤21𝑐23Δ𝑤32𝜂2𝑐12𝜕2Δ𝑤21𝜕𝑥2+𝜂2𝑐23𝜕2Δ𝑤32𝜕𝑥2,𝑖+1𝑗=𝑖1𝛿𝐹𝑗𝛿𝑤𝑖=𝑐(𝑖1)𝑖Δ𝑤𝑖(𝑖1)𝑐𝑖(𝑖+1)Δ𝑤(𝑖+1)𝑖𝜂2𝑐(𝑖1)𝑖𝜕2Δ𝑤𝑖(𝑖1)𝜕𝑥2+𝜂2𝑐𝑖(𝑖+1)𝜕2Δ𝑤(𝑖+1)𝑖𝜕𝑥2,𝑛𝑗=𝑛1𝛿𝐹𝑗𝛿𝑤𝑛=𝑐(𝑛1)𝑛Δ𝑤𝑛(𝑛1)𝜂2𝑐(𝑛1)𝑛𝜕2Δ𝑤𝑛(𝑛1)𝜕𝑥2.(17) Integrability relations between these equations can be obtained by noting that

𝜕𝜕𝑤2+𝜕𝜕𝑤2𝑥𝑥2𝑗=1𝛿𝐹𝑗𝛿𝑤1=𝑐12+𝜂2𝑐12,𝜕(18)𝜕𝑤1+𝜕𝜕𝑤1𝑥𝑥3𝑗=1𝛿𝐹𝑗𝛿𝑤2=𝑐12+𝜂2𝑐12𝜕,(19)𝜕𝑤𝑖+1+𝜕𝜕𝑤(𝑖+1)𝑥𝑥𝑖+1𝑗=𝑖1𝛿𝐹𝑗𝛿𝑤𝑖=𝑐𝑖(𝑖+1)+𝜂2𝑐𝑖(𝑖+1)𝜕,(20)𝜕𝑤𝑖+𝜕𝜕𝑤𝑖𝑥𝑥𝑖+2𝑗=𝑖𝛿𝐹𝑗𝛿𝑤𝑖+1=𝑐𝑖(𝑖+1)+𝜂2𝑐𝑖(𝑖+1)𝜕,(21)𝜕𝑤𝑛+𝜕𝜕𝑤𝑛𝑥𝑥𝑛𝑗=𝑛2𝛿𝐹𝑗𝛿𝑤𝑛1=𝑐(𝑛1)𝑛+𝜂2𝑐(𝑛1)𝑛𝜕,(22)𝜕𝑤𝑛1+𝜕𝜕𝑤(𝑛1)𝑥𝑥𝑛𝑗=𝑛1𝛿𝐹𝑗𝛿𝑤𝑛=𝑐(𝑛1)𝑛+𝜂2𝑐(𝑛1)𝑛.(23) 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

𝐹1𝑤1,𝑤2=𝑐124Δ𝑤221+𝑐124𝜂2𝜕Δ𝑤21𝜕𝑥2,𝐹𝑖𝑤𝑖1,𝑤𝑖,𝑤𝑖+1=𝑐(𝑖1)𝑖4Δ𝑤2𝑖(𝑖1)+𝑐𝑖(𝑖+1)4Δ𝑤2(𝑖+1)𝑖+𝜂2𝑐(𝑖1)𝑖4𝜕Δ𝑤𝑖(𝑖1)𝜕𝑥2+𝜂2𝑐𝑖(𝑖+1)4𝜕Δ𝑤(𝑖+1)𝑖𝜕𝑥2𝐹for𝑖=2,3,,𝑛1,𝑛𝑤𝑛1,𝑤𝑛=𝑐(𝑛1)𝑛4Δ𝑤2𝑛(𝑛1)+𝜂2𝑐(𝑛1)𝑛4𝜕Δ𝑤𝑛(𝑛1)𝜕𝑥2.(24) 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

𝑡2𝑡1𝛿KE(𝑡)𝛿𝑊𝐸(𝑡)+𝛿PE1(𝑡)+𝛿PE2(𝑡)𝑑𝑡=0,(25) where 1KE(𝑡)=2𝑖=𝑛𝑖=1𝐿0𝜌𝐴𝑖𝜕𝑤𝑖𝜕𝑡2+𝜂2𝜌𝐴𝑖𝜕2𝑤𝑖𝜕𝑥𝜕𝑡2𝜌𝐼𝑖𝜕𝜑𝑖𝜕𝑡2+𝜂2𝜌𝐼𝑖𝜕2𝜑𝑖𝜕𝑡𝜕𝑥2𝑊𝑑𝑥,𝐸(1𝑡)=2𝑖=𝑛𝑖=1𝐿0𝐴𝑖𝜎𝑥𝜕𝑤𝑖𝜕𝑥2𝜂2𝐴𝑖𝜎𝑥𝜕2𝑤𝑖𝜕𝑥22𝑓(𝑥,𝑡)𝑤𝑛(𝑥,𝑡)𝑑𝑥,PE11(𝑡)=2𝑖=𝑛𝑖=1𝐿0𝜅𝐺𝐴𝑖𝜑𝑖𝜕𝑤𝑖𝜕𝑥2+𝐸𝐼𝑖𝜕𝜑𝑖𝜕𝑥2+𝑘𝑤2𝑛+𝑘𝜂2𝜕𝑤𝑛𝜕𝑥2𝑑𝑥,PE21(𝑡)=2𝑖=𝑛𝑖=1𝐿0𝑐(𝑖1)𝑖𝑤𝑖𝑤𝑖12+𝜂2𝑐(𝑖1)𝑖𝜕𝑤𝑖𝜕𝑥𝜕𝑤𝑖1𝜕𝑥2𝑑𝑥.(26) 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=𝑡2𝑡1𝐿0𝐷𝑎1𝑤1,𝜑1,𝑤2𝛿𝑤1𝑑𝑥𝑑𝑡+𝜕Ω𝑎1𝛿(0,𝐿,𝑡),𝜑1𝑉=𝛿𝜑1𝑉1=𝑡2𝑡1𝐿0𝐷𝑏1𝑤1,𝜑1𝛿𝜑1𝑑𝑥𝑑𝑡+𝜕Ω𝑏1(𝛿0,𝐿,𝑡),𝑤𝑖𝑉=𝑖+1𝑗=𝑖1𝛿𝑤𝑖𝑉𝑗=𝑡2𝑡1𝐿0𝐷𝑎𝑖𝑤𝑖1,𝑤𝑖,𝜑𝑖,𝑤𝑖+1𝛿𝑤𝑖𝑑𝑥𝑑𝑡+𝜕Ω𝑎𝑖𝛿(0,𝐿,𝑡)for𝑖=2,,𝑛1,𝜑𝑖𝑉=𝛿𝜑𝑖𝑉𝑖=𝑡2𝑡1𝐿0𝐷𝑏𝑖𝑤𝑖,𝜑𝑖𝛿𝜑𝑖𝑑𝑥𝑑𝑡+𝜕Ω𝑏𝑖𝛿(0,𝐿,𝑡)for𝑖=2,,𝑛1,𝑤𝑛𝑉=𝛿𝑤𝑛𝑉𝑛1+𝛿𝑤𝑛𝑉𝑛=𝑡2𝑡1𝐿0𝐷𝑎𝑛𝑤𝑛1,𝑤𝑛,𝜑𝑛𝛿𝑤𝑛𝑑𝑥𝑑𝑡+𝜕Ω𝑎𝑛𝛿(0,𝐿,𝑡),𝜑𝑛𝑉=𝛿𝜑𝑛𝑉𝑛=𝑡2𝑡1𝐿0𝐷𝑏𝑛𝑤𝑛,𝜑𝑛𝛿𝜑𝑛𝑑𝑥𝑑𝑡+𝜕Ω𝑏𝑛(0,𝐿,𝑡),(27) where 𝜕Ω𝑖𝑎(0,𝐿,𝑡) and 𝜕Ω𝑖𝑏(0,𝐿,𝑡) are the boundary terms defined as

𝜕Ω𝑎1(0,𝐿,𝑡)=𝐴1𝜎𝑥𝜂2𝜕2𝑤1𝜕𝑥2𝛿𝑤1||||𝑥=𝐿𝑥=0+𝐴1𝜎𝑥𝜂2𝜕3𝑤1𝜕𝑥3𝛿𝑤1||||𝑥=𝐿𝑥=0+𝜌𝐴1𝜂2𝜕3𝑤1𝜕𝑥𝜕𝑡2𝛿𝑤1||||𝑥=𝐿𝑥=0+𝜅𝐺𝐴1𝜑1𝜕𝑤1+𝜕𝑥𝐴1𝜎𝑥+𝜂2𝑐12𝜕𝑤1𝜕𝑥𝜂2𝑐12𝜕𝑤2𝜕𝑥𝛿𝑤1||||𝑥=𝐿𝑥=0,𝜕Ω𝑎𝑖(0,𝐿,𝑡)=𝐴𝑖𝜎𝑥𝜂2𝜕2𝑤𝑖𝜕𝑥2𝛿𝑤1||||𝑥=𝐿𝑥=0+𝐴𝑖𝜎𝑥𝜂2𝜕3𝑤𝑖𝜕𝑥3𝛿𝑤𝑖||||𝑥=𝐿𝑥=0+𝜌𝐴𝑖𝜂2𝜕3𝑤𝑖𝜕𝑥𝜕𝑡2𝛿𝑤𝑖||||𝑥=𝐿𝑥=0+𝜅𝐺𝐴𝑖𝜑𝑖𝜕𝑤𝑖+𝜕𝑥𝐴𝑖𝜎𝑥+𝜂2𝑐(𝑖1)𝑖+𝑐𝑖(𝑖+1)×𝜕𝑤𝑖𝜕𝑥𝜂2𝑐(𝑖1)𝑖𝜕𝑤𝑖1𝜕𝑥+𝑐𝑖(𝑖+1)𝜕𝑤𝑖+1𝜕𝑥𝛿𝑤𝑖||||𝑥=𝐿𝑥=0for𝑖=2,3,,𝑛1𝜕Ω𝑎𝑛(0,𝐿,𝑡)=𝐴𝑛𝜎𝑥𝜂2𝜕2𝑤𝑛𝜕𝑥2𝛿𝑤𝑛||||𝑥=𝐿𝑥=0+𝐴𝑛𝜎𝑥𝜂2𝜕3𝑤𝑛𝜕𝑥3𝛿𝑤𝑛||||𝑥=𝐿𝑥=0+𝜌𝐴𝑛𝜂2𝜕3𝑤𝑛𝜕𝑥𝜕𝑡2𝛿𝑤𝑛||||𝑥=𝐿𝑥=0+𝜅𝐺𝐴𝑛𝜑𝑛𝜕𝑤𝑛+𝜕𝑥𝐴𝑛𝜎𝑥+𝜂2𝑐(𝑛1)𝑛+𝑘×𝜕𝑤𝑛𝜂𝜕𝑥2𝑐(𝑛1)𝑛𝜕𝑤𝑛1𝜕𝑥𝛿𝑤𝑛||||𝑥=𝐿𝑥=0,𝜕Ω𝑏𝑖(=0,𝐿,𝑡)𝐸𝐼𝑖𝜕𝜑𝑖𝜕𝑥+𝜌𝐼𝑖𝜂2𝜕3𝜑𝑖𝜕𝑥𝜕𝑡2𝛿𝜑𝑖|||||𝑥=𝐿𝑥=0for𝑖=1,2,,𝑛,(28) 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,(29)𝐴𝑖𝜎𝑥𝜂2𝜕2𝑤𝑖𝜕𝑥2=0or𝜕𝑤𝑖𝜕𝑥=0for𝜎𝑥𝐴0,𝑖=1,2,,𝑛,(30)1𝜎𝑥𝜂2𝜕3𝑤1𝜕𝑥3+𝜌𝐴1𝜂2𝜕3𝑤1𝜕𝑥𝜕𝑡2𝜅𝐺𝐴1𝜑1𝜕𝑤1+𝜕𝑥𝐴1𝜎𝑥+𝜂2𝑐12𝜕𝑤1𝜕𝑥𝜂2𝑐12𝜕𝑤2𝜕𝑥=0or𝑤1𝐴=0,(31)𝑖𝜎𝑥𝜂2𝜕3𝑤𝑖𝜕𝑥3+𝜌𝐴𝑖𝜂2𝜕3𝑤𝑖𝜕𝑥𝜕𝑡2𝜅𝐺𝐴𝑖𝜑𝑖𝜕𝑤𝑖+𝜕𝑥𝐴𝑖𝜎𝑥+𝜂2𝑐(𝑖1)𝑖+𝑐𝑖(𝑖+1)𝜕𝑤𝑖𝜕𝑥𝜂2𝑐(𝑖1)𝑖𝜕𝑤𝑖1𝜕𝑥+𝑐𝑖(+1)𝜕𝑤𝑖+1𝜕𝑥=0or𝑤𝑖𝐴=0for𝑖=2,,𝑛1(32)𝑛𝜎𝑥𝜂2𝜕3𝑤𝑛𝜕𝑥3+𝜌𝐴𝑛𝜂2𝜕3𝑤𝑛𝜕𝑥𝜕𝑡2𝜅𝐺𝐴𝑛𝜑𝑛𝜕𝑤𝑛+𝜕𝑥𝐴𝑛𝜎𝑥+𝜂2𝑐(𝑛1)𝑛+𝑘𝜕𝑤𝑛𝜕𝑥𝜂2𝑐(𝑛1)𝑛𝜕𝑤𝑛1𝜕𝑥=0or𝑤𝑛=0.(33) 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, 2325]. 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.