Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory

Abstract

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 𝜎_𝑥 which can be tensile or compressive in which case 𝜎_𝑥 is less than the critical buckling load. We introduce a difference operator defined as

Δ𝑤_𝑖𝑗=𝑤_𝑖−𝑤_𝑗,(1) where 𝑤_𝑖 and 𝑤_𝑗 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]

𝐷_𝑎1𝑤₁,𝜑₁,𝑤₂=𝐿_𝑎1𝑤₁,𝜑₁−𝑐₁₂Δ𝑤₂₁+𝜂²𝑐₁₂𝜕²Δ𝑤₂₁𝜕𝑥²𝐷=0,(2)_𝑏1𝐷𝑤₁,𝜑₁=𝐿_𝑏1𝑤₁,𝜑₁=0,(3)_𝑎2𝑤₁,𝑤₂,𝜑₂,𝑤₃=𝐿_𝑎2𝑤₂,𝜑₂+𝑐₁₂Δ𝑤₂₁−𝑐₂₃Δ𝑤₃₂+𝜂²−𝑐¹²𝜕²Δ𝑤₂₁𝜕𝑥²+𝑐₂₃𝜕²Δ𝑤₃₂𝐷=0,𝜕𝑥²(4)_𝑏2𝐷𝑤₂,𝜑₂=⋮𝐿_𝑏2𝑤₂,𝜑₂=0,(5)_𝑎𝑖𝑤_𝑖−1,𝑤_𝑖,𝜑_𝑖,𝑤_𝑖+1=𝐿_𝑎𝑖𝑤_𝑖,𝜑_𝑖+𝑐_{(𝑖−1)𝑖}Δ𝑤_{𝑖(𝑖−1)}−𝑐_𝑖(𝑖+1)Δ𝑤_(𝑖+1)𝑖−𝜂²𝑐_{(𝑖−1)𝑖}𝜕²Δ𝑤_{𝑖(𝑖−1)}𝜕𝑥²+𝜂²𝑐_𝑖(𝑖+1)𝜕²Δ𝑤_(𝑖+1)𝑖𝜕𝑥²𝐷=0,(6)_𝑏𝑖𝐷⋮𝑤fo_𝑖,r𝜑𝑖_𝑖==3,𝐿4,_𝑏𝑖…𝑤,_𝑖𝑛,𝜑−_𝑖1,=0(7)_𝑎𝑛𝑤_𝑛−1,𝑤_𝑛,𝜑_𝑛=𝐿_𝑎𝑛𝑤_𝑛,𝜑_𝑛+𝑐_{(𝑛−1)𝑛}Δ𝑤_{𝑛(𝑛−1)}−𝜂²𝑐_{(𝑛−1)𝑛}𝜕²Δ𝑤_{𝑛(𝑛−1)}𝜕𝑥²𝐷=𝑓(𝑥,𝑡),(8)_𝑏𝑛𝑤_𝑛,𝜑_𝑛=𝐿_𝑏𝑛𝑤_𝑛,𝜑_𝑛=0,(9) where 𝜑_𝑖 is the angle of rotation and the operators 𝐿_𝑎𝑖(𝑤_𝑖,𝜑_𝑖)  and 𝐿_𝑏𝑖(𝑤_𝑖,𝜑_𝑖) are given by 𝐿_𝑎𝑖𝑤_𝑖,𝜑_𝑖=𝜌𝐴_𝑖𝜕²𝑤_𝑖𝜕𝑡²−𝜌𝐴_𝑖𝜂²𝜕⁴𝑤_𝑖𝜕𝑡²𝜕𝑥²+𝜅𝐺𝐴_𝑖𝜕𝜕𝑥𝜑^𝑖−𝜕𝑤^𝑖+𝐴_𝑖𝜕𝑥𝜎_𝑥𝜕²𝑤_𝑖𝜕𝑥²−𝐴_𝑖𝜎_𝑥𝜂²𝜕⁴𝑤_𝑖𝜕𝑥⁴+𝛿_𝑖𝑛𝑘𝑤^𝑛−𝑘𝜂²𝜕²𝑤_𝑛𝜕𝑥²,(10)𝐿_𝑏𝑖=𝑤_𝑖𝜌,𝐼𝜑_𝑖𝑖𝜕²𝜑_𝑖𝜕𝑡²−𝜌𝐼_𝑖𝜂²𝜕⁴𝜑_𝑖𝜕𝑡²𝜕𝑥²+𝜅𝐺𝐴_𝑖𝜑^𝑖−𝜕𝑤^𝑖𝜕𝑥−𝐸𝐼^𝑖𝜕²𝜑_𝑖𝜕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 0≤𝑥≤𝐿. In (8) 𝑓(𝑥,𝑡) is a forcing function, and in (10) 𝛿_𝑖𝑛 is the Kronecker’s delta with 𝛿_𝑖𝑛=0 for 𝑖≠𝑛 and 𝛿_𝑛𝑛=1. In (10) and (11), E is the Young’s modulus, G is the shear modulus, 𝜅 is the shear correction factor, 𝐼_𝑖 is the moment of inertia, 𝐴_𝑖 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(𝑤_𝑖,𝜑_𝑖) can be constructed as follows with the motion taking place between the initial time 𝑡₁ and the final time 𝑡₂

𝑉𝑤_𝑖,𝜑_𝑖=+𝑉₁⋯𝑤+₁,𝑉𝜑₁,_𝑛−1𝑤₂+𝑉₂𝑤₁,𝑤₂,𝜑₂,𝑤₃+𝑉_𝑛𝑤_𝑛−2,𝑤_𝑛−1,𝜑_𝑛−1,𝑤_𝑛𝑤_𝑛−1,𝑤_𝑛,𝜑_𝑛,(12) where

𝑉₁𝑤₁,𝜑₁,𝑤₂=𝑈₁𝑤₁,𝜑₁+^𝑡2_𝑡1^𝐿0𝑉𝐹₁𝑤₁,𝑤₂𝑑𝑥𝑑𝑡,₂𝑉𝑤₁,𝑤₂,𝜑₂,𝑤₃=𝑈₂𝑤₂,𝜑₂+^𝑡2_𝑡1^𝐿0𝐹₂𝑤₁,𝑤₂,𝑤₃𝑑𝑥𝑑𝑡,_𝑖𝑤𝑉_𝑖−1,𝑤_𝑖,𝜑_𝑖,𝑤_𝑖+1=𝑈_𝑖𝑤_𝑖,𝜑_𝑖+for^𝑡2_𝑡1𝑖=^𝐿03𝐹,_𝑖4,𝑤…,_𝑖−1𝑛,𝑤−_𝑖1,𝑤,_𝑖+1𝑛𝑑𝑥𝑑𝑡𝑤_𝑛−1,𝑤_𝑛,𝜑_𝑛=+𝑈_𝑛1𝑤_𝑛,𝜑_𝑛2^𝑡2_𝑡1^𝐿0𝑘𝑤^2𝑛+𝑘𝜂²+𝜕𝜕𝑤𝑥^𝑛²𝑑𝑥𝑑𝑡_𝑡2𝑡1^𝐿0−𝑓𝑤_𝑛+𝐹_𝑛𝑤_𝑛−1,𝑤_𝑛𝑑𝑥𝑑𝑡(13) with 𝑈_𝑖(𝑤_𝑖,𝜑_𝑖) given by

𝑈=_𝑖1𝑤_𝑖,𝜑_𝑖2^𝑡2_𝑡1^𝐿0𝜅𝐺𝐴^𝑖𝜑^𝑖−𝜕𝑤^𝑖𝜕𝑥²+𝐸𝐼_𝑖𝜕𝜕𝜑𝑥^𝑖²−𝐴_𝑖𝜎_𝑥𝜕𝜕𝑤𝑥^𝑖²−𝐴_𝑖𝜎_𝑥𝜂²𝜕²𝑤_𝑖𝜕𝑥²²+1⎞⎟⎟⎠𝑑𝑥𝑑𝑡2^𝑡2_𝑡1^𝐿0⎛⎜⎜⎝−𝜌𝐴_𝑖𝜕𝜕𝑤𝑡^𝑖²−𝜌𝐴_𝑖𝜂²𝜕²𝑤_𝑖𝜕𝑡𝜕𝑥²−𝜌𝐼_𝑖𝜕𝜕𝜑𝑡^𝑖²−𝜌𝐼_𝑖𝜂²𝜕²𝜑_𝑖𝜕𝑡𝜕𝑥²⎞⎟⎟⎠𝑑𝑥𝑑𝑡,(14) where 𝑖=1,2,…,𝑛 and 𝐹_𝑖(𝑤_𝑖−1,𝑤_𝑖,𝑤_𝑖+1) denotes the unknown functions of 𝑤_𝑖 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

𝐿_𝑎1𝑤₁,𝜑₁+^2𝑗=1𝛿𝐹_𝑗𝛿𝑤₁=+𝐿_𝑎1𝑤_12𝑗=1,𝜑₁𝜕𝜕𝑤𝐹^𝑗₁−𝜕𝐿𝜕𝑥𝜕𝜕𝑤𝐹^𝑗_1𝑥−𝜕𝜕𝑡𝜕𝜕𝑤𝐹^𝑗_1𝑡+⋯=0,_𝑎2𝑤₂,𝜑₂+^3𝑗=1𝛿𝐹_𝑗𝛿𝑤₂=+𝐿_𝑎2𝑤_23𝑗=1,𝜑₂𝜕𝜕𝑤𝐹^𝑗₂−𝜕𝐿𝜕𝑥𝜕𝜕𝑤𝐹^𝑗_2𝑥−𝜕𝜕𝑡𝜕𝜕𝑤𝐹^𝑗_1𝑡+⋯=0,_𝑎𝑖𝑤_𝑖,𝜑_𝑖+^{𝑖+1𝑗=𝑖−1}𝛿𝐹_𝑗𝛿𝑤_𝑖=+𝐿_𝑎𝑖𝑤_𝑖,𝜑_{𝑖𝑖+1𝑗=𝑖−1}𝜕𝜕𝑤𝐹^𝑗_𝑖−𝜕𝐿𝜕𝑥fo𝜕𝜕r𝑤𝐹𝑖^𝑗_𝑖𝑥=3−,4,𝜕𝜕…𝑡,𝜕𝑛𝜕𝑤𝐹−^𝑗_𝑖𝑡1,_𝑎𝑛+⋯=0,𝑤_𝑛,𝜑_𝑛+^{𝑛𝑗=𝑛−1}𝛿𝐹_𝑗𝛿𝑤_𝑛=+𝐿_𝑎𝑛𝑤_𝑛,_{𝑛𝑗=𝑛−1}𝜑_𝑛𝜕𝜕𝑤𝐹^𝑗_𝑛−𝜕𝐿𝜕𝑥𝜕𝜕𝑤𝐹^𝑗_𝑛𝑥−𝜕𝜕𝑡𝜕𝜕𝑤𝐹^𝑗_𝑛𝑡+⋯=0,_𝑏𝑖𝑤_𝑖,𝜑_𝑖=0for𝑖=1,2,…,𝑛,(15) where the subscripts 𝑥 and 𝑡 denote differentiation with respect to 𝑥 and 𝑡, and the variational derivative 𝛿𝐹_𝑖/𝛿𝑤_𝑖 is defined as [36, 37]

𝛿𝐹_𝑖𝛿𝑤_𝑖=𝜕𝐹^𝑖𝜕𝑤_𝑖−𝜕+𝜕₂𝜕𝑥𝜕𝜕𝑤𝐹^𝑖_𝑖𝑥−𝜕𝜕𝑥𝜕𝜕𝑤𝐹^𝑖_𝑖𝑡+𝜕𝜕𝑥²²𝜕𝜕𝑤𝐹^𝑖_𝑖𝑥𝑥𝜕𝑥𝜕𝑡𝜕𝜕𝑤𝐹^𝑖_𝑖𝑥𝑡⋯−𝜕𝜕𝑡𝜕𝜕𝑤𝐹^𝑖_𝑖𝑡+𝜕𝜕𝑡²²𝜕𝜕𝑤𝐹^𝑖_𝑖𝑡𝑡+⋯.(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=1𝛿𝐹_𝑗𝛿𝑤₁=−𝑐₁₂Δ𝑤₂₁+𝜂²𝑐₁₂𝜕²Δ𝑤₂₁𝜕𝑥²_3𝑗=1,𝛿𝐹_𝑗𝛿𝑤₂=𝑐₁₂Δ𝑤₂₁−𝑐₂₃Δ𝑤₃₂−𝜂²𝑐₁₂𝜕²Δ𝑤₂₁𝜕𝑥²+𝜂²𝑐₂₃𝜕²Δ𝑤₃₂𝜕𝑥²_{𝑖+1𝑗=𝑖−1},𝛿𝐹_𝑗𝛿𝑤_𝑖=𝑐_{(𝑖−1)𝑖}Δ𝑤_{𝑖(𝑖−1)}−𝑐_𝑖(𝑖+1)Δ𝑤_(𝑖+1)𝑖−𝜂²𝑐_{(𝑖−1)𝑖}𝜕²Δ𝑤_{𝑖(𝑖−1)}𝜕𝑥²+𝜂²𝑐_𝑖(𝑖+1)𝜕²Δ𝑤_(𝑖+1)𝑖𝜕𝑥²_{𝑛𝑗=𝑛−1},𝛿𝐹_𝑗𝛿𝑤_𝑛=𝑐_{(𝑛−1)𝑛}Δ𝑤_{𝑛(𝑛−1)}−𝜂²𝑐_{(𝑛−1)𝑛}𝜕²Δ𝑤_{𝑛(𝑛−1)}𝜕𝑥².(17) Integrability relations between these equations can be obtained by noting that

𝜕𝑤𝜕₂+𝜕𝜕𝑤_2𝑥𝑥^2𝑗=1𝛿𝐹_𝑗𝛿𝑤₁𝜕=−𝑐₁₂+𝜂²𝑐₁₂,(18)𝜕𝑤₁+𝜕𝜕𝑤_1𝑥𝑥^3𝑗=1𝛿𝐹_𝑗𝛿𝑤₂𝜕=−𝑐₁₂+𝜂²𝑐₁₂,(19)𝜕𝑤_𝑖+1+𝜕𝜕𝑤_{(𝑖+1)𝑥𝑥}^{𝑖+1𝑗=𝑖−1}𝛿𝐹_𝑗𝛿𝑤_𝑖𝜕=−𝑐_𝑖(𝑖+1)+𝜂²𝑐_𝑖(𝑖+1),(20)𝜕𝑤_𝑖+𝜕𝜕𝑤_𝑖𝑥𝑥^{𝑖+2𝑗=𝑖}𝛿𝐹_𝑗𝛿𝑤_𝑖+1𝜕=−𝑐_𝑖(𝑖+1)+𝜂²𝑐_𝑖(𝑖+1),(21)𝜕𝑤_𝑛+𝜕𝜕𝑤_𝑛𝑥𝑥^{𝑛𝑗=𝑛−2}𝛿𝐹_𝑗𝛿𝑤_𝑛−1𝜕=−𝑐_{(𝑛−1)𝑛}+𝜂²𝑐_{(𝑛−1)𝑛},(22)𝜕𝑤_𝑛−1+𝜕𝜕𝑤_{(𝑛−1)𝑥𝑥}^{𝑛𝑗=𝑛−1}𝛿𝐹_𝑗𝛿𝑤_𝑛=−𝑐_{(𝑛−1)𝑛}+𝜂²𝑐_{(𝑛−1)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

𝐹₁𝑤₁,𝑤₂=𝑐¹²4Δ𝑤²²¹+𝑐¹²4𝜂²𝜕Δ𝜕𝑤𝑥²¹²𝐹_𝑖,𝑤_𝑖−1,𝑤_𝑖,𝑤_𝑖+1=𝑐4^{(𝑖−1)𝑖}Δ𝑤^{2𝑖(𝑖−1)}+𝑐4^𝑖(𝑖+1)Δ𝑤^{2(𝑖+1)𝑖}+𝜂²𝑐_{(𝑖−1)𝑖}4𝜕Δ𝑤𝜕𝑥^{𝑖(𝑖−1)}²+𝜂²𝑐_𝑖(𝑖+1)4𝜕Δ𝑤𝜕𝑥^(𝑖+1)𝑖²𝐹for𝑖=2,3,…,𝑛−1,_𝑛𝑤_𝑛−1,𝑤_𝑛=𝑐4^{(𝑛−1)𝑛}Δ𝑤^{2𝑛(𝑛−1)}+𝜂²𝑐_{(𝑛−1)𝑛}4𝜕Δ𝑤𝜕𝑥^{𝑛(𝑛−1)}².(24) With 𝐹_𝑖, 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

^𝑡2_𝑡1𝛿KE(𝑡)−𝛿𝑊_𝐸(𝑡)+𝛿PE₁(𝑡)+𝛿PE₂(𝑡)𝑑𝑡=0,(25) where KE(𝑡)=12^{𝑖=𝑛𝑖=1}^𝐿0⎛⎜⎜⎝𝜌𝐴_𝑖𝜕𝜕𝑤𝑡^𝑖²+𝜂²𝜌𝐴_𝑖𝜕²𝑤_𝑖𝜕𝑥𝜕𝑡²−𝜌𝐼_𝑖𝜕𝜕𝜑𝑡^𝑖²+𝜂²𝜌𝐼_𝑖𝜕²𝜑_𝑖𝜕𝑡𝜕𝑥²𝑊⎞⎟⎟⎠𝑑𝑥,_𝐸(𝑡)=12^{𝑖=𝑛𝑖=1}^𝐿0⎛⎜⎜⎝−𝐴_𝑖𝜎_𝑥𝜕𝜕𝑤𝑥^𝑖²−𝜂²𝐴_𝑖𝜎_𝑥𝜕²𝑤_𝑖𝜕𝑥²²PE₁−𝑓(𝑥,𝑡)𝑤_𝑛(𝑥,𝑡)⎞⎟⎟⎠𝑑𝑥,(𝑡)=12^{𝑖=𝑛𝑖=1}^𝐿0𝜅𝐺𝐴^𝑖𝜑^𝑖−𝜕𝑤^𝑖𝜕𝑥²+𝐸𝐼_𝑖𝜕𝜕𝜑𝑥^𝑖²+𝑘𝑤^2𝑛+𝑘𝜂²PE₂𝜕𝜕𝑤𝑥^𝑛²𝑑𝑥,(𝑡)=12^{𝑖=𝑛𝑖=1}^𝐿0⎛⎜⎜⎝𝑐_{(𝑖−1)𝑖}𝑤_𝑖−𝑤_𝑖−1²+𝜂²𝑐_{(𝑖−1)𝑖}𝜕𝜕𝑤𝑥^𝑖−𝜕𝑤𝜕𝑥^𝑖−1²𝑑𝑥.(26) In (25)–(26), KE is the kinetic energy, 𝑊_𝐸 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(𝑤_𝑖,𝜑_𝑖) in (12) are evaluated with respect to 𝑤_𝑖 and 𝜑_𝑖 in order to derive the natural and geometric boundary conditions. Let 𝛿𝑤_𝑖 and 𝛿𝜑_𝑖  denote the variations of 𝑤_𝑖 and 𝜑_𝑖 such that 𝛿𝑤_𝑖(𝑥,𝑡₁)=𝛿𝑤_𝑖(𝑥,𝑡₂)=𝛿𝜑_𝑖(𝑥,𝑡₁)=𝛿𝜑_𝑖(𝑥,𝑡₂)=0. The first variations of 𝑉(𝑤_𝑖,𝜑_𝑖) with respect to 𝑤_𝑖 and 𝜑_𝑖, denoted by 𝛿_𝑤𝑖𝑉 and 𝛿_𝜑𝑖𝑉, respectively, can be obtained by integration by parts and expressed as

𝛿_𝑤1𝑉=𝛿_𝑤1𝑉₁+𝛿_𝑤1𝑉₂=^𝑡2_𝑡1^𝐿0𝛿𝐷_𝑎1𝑤₁,𝜑₁,𝑤₂𝛿𝑤₁𝑑𝑥𝑑𝑡+𝜕Ω_𝑎1(0,𝐿,𝑡),_𝜑1𝑉=𝛿_𝜑1𝑉₁=^𝑡2_𝑡1^𝐿0𝛿𝐷_𝑏1𝑤₁,𝜑₁𝛿𝜑₁𝑑𝑥𝑑𝑡+𝜕Ω_𝑏1(0,𝐿,𝑡),_𝑤𝑖𝑉=^{𝑖+1𝑗=𝑖−1}𝛿_𝑤𝑖𝑉_𝑗=^𝑡2_𝑡1^𝐿0+𝜕Ω_𝑎𝑖𝐷_𝑎𝑖𝑤_𝑖−1,𝑤_𝑖,𝜑_𝑖,𝑤_𝑖+1𝛿𝑤_𝑖𝑑𝑥𝑑𝑡𝛿(0,𝐿,𝑡)for𝑖=2,…,𝑛−1,_𝜑𝑖𝑉=𝛿_𝜑𝑖𝑉_𝑖=^𝑡2_𝑡1^𝐿0+𝜕Ω_𝑏𝑖𝐷_𝑏𝑖𝑤_𝑖,𝜑_𝑖𝛿𝜑_𝑖𝑑𝑥𝑑𝑡𝛿(0,𝐿,𝑡)for𝑖=2,…,𝑛−1,_𝑤𝑛𝑉=𝛿_𝑤𝑛𝑉_𝑛−1+𝛿_𝑤𝑛𝑉_𝑛=^𝑡2_𝑡1^𝐿0+𝜕Ω_𝑎𝑛𝐷_𝑎𝑛𝑤_𝑛−1,𝑤_𝑛,𝜑_𝑛𝛿𝑤_𝑛𝑑𝑥𝑑𝑡𝛿(0,𝐿,𝑡),_𝜑𝑛𝑉=𝛿_𝜑𝑛𝑉_𝑛=^𝑡2_𝑡1^𝐿0𝐷_𝑏𝑛𝑤_𝑛,𝜑_𝑛𝛿𝜑_𝑛𝑑𝑥𝑑𝑡+𝜕Ω_𝑏𝑛(0,𝐿,𝑡),(27) where 𝜕Ω_𝑖𝑎(0,𝐿,𝑡) and 𝜕Ω_𝑖𝑏(0,𝐿,𝑡) are the boundary terms defined as

𝜕=Ω−_𝑎1𝐴(0₁,𝐿,𝑡)𝜎_𝑥𝜂²𝜕²𝑤₁𝜕𝑥²𝛿𝑤^1||||^{𝑥=𝐿𝑥=0}+𝐴₁𝜎_𝑥𝜂²𝜕³𝑤₁𝜕𝑥³𝛿𝑤₁||||^{𝑥=𝐿𝑥=0}+𝜌𝐴₁𝜂²𝜕³𝑤₁𝜕𝑥𝜕𝑡²𝛿𝑤₁||||^{𝑥=𝐿𝑥=0}+−𝜅𝐺𝐴₁𝜑¹−𝜕𝑤¹𝜕𝑥+−𝐴¹𝜎_𝑥+𝜂²𝑐₁₂𝜕𝑤¹−𝜂₂𝜕𝑥𝑐₁₂𝜕𝑤₂𝜕𝑥𝛿𝑤¹||||^{𝑥=𝐿𝑥=0}𝜕Ω_𝑎𝑖,=−𝐴(0_𝑖,𝐿,𝑡)𝜎_𝑥𝜂²𝜕²𝑤_𝑖𝜕𝑥²𝛿𝑤^1||||^{𝑥=𝐿𝑥=0}+𝐴_𝑖𝜎_𝑥𝜂²𝜕³𝑤_𝑖𝜕𝑥³𝛿𝑤_𝑖||||^{𝑥=𝐿𝑥=0}+𝜌𝐴_𝑖𝜂²𝜕³𝑤_𝑖𝜕𝑥𝜕𝑡²𝛿𝑤_𝑖||||^{𝑥=𝐿𝑥=0}+−𝜅𝐺𝐴_𝑖𝜑^𝑖−𝜕𝑤^𝑖𝜕𝑥+−𝐴^𝑖×𝜕𝑤_𝑖𝜎_𝑥+𝜂²𝑐_{(𝑖−1)𝑖}+𝑐_𝑖(𝑖+1)𝜕𝑥−𝜂²𝑐^{(𝑖−1)𝑖}𝜕𝑤_𝑖−1𝜕𝑥+𝑐^𝑖(𝑖+1)𝜕𝑤_𝑖+1𝜕𝑥𝛿𝑤^𝑖||||^{𝑥=𝐿𝑥=0}𝜕Ω_𝑎𝑛for𝑖=2,3,…,𝑛−1=−𝐴(0_𝑛,𝐿,𝑡)𝜎_𝑥𝜂²𝜕²𝑤_𝑛𝜕𝑥²𝛿𝑤^𝑛||||^{𝑥=𝐿𝑥=0}+𝐴_𝑛𝜎_𝑥𝜂²𝜕³𝑤_𝑛𝜕𝑥³𝛿𝑤_𝑛||||^{𝑥=𝐿𝑥=0}+𝜌𝐴_𝑛𝜂²𝜕³𝑤_𝑛𝜕𝑥𝜕𝑡²𝛿𝑤_𝑛||||^{𝑥=𝐿𝑥=0}+−𝜅𝐺𝐴_𝑛𝜑^𝑛−𝜕𝑤^𝑛𝜕𝑥+−𝐴^𝑛×−𝜕𝑤_𝑛𝜎_𝑥+𝜂²𝑐_{(𝑛−1)𝑛}+𝑘𝜕𝑥𝜂²𝑐_{(𝑛−1)𝑛}𝜕𝑤_𝑛−1𝜕𝑥𝛿𝑤^𝑛||||^{𝑥=𝐿𝑥=0}𝜕Ω_𝑏𝑖,=(𝐸0,𝐼𝐿_𝑖,𝑡)𝜕𝜑_𝑖𝜕𝑥+𝜌𝐼^𝑖𝜂²𝜕³𝜑_𝑖𝜕𝑥𝜕𝑡²𝛿𝜑^𝑖|||||^{𝑥=𝐿𝑥=0}for𝑖=1,2,…,𝑛,(28) where 𝛿𝑤^𝑖 is the derivative of 𝛿𝑤_𝑖 with respect to 𝑥. Thus the boundary conditions at 𝑥=0,𝐿 are given by

𝐸𝐼_𝑖𝜕𝜑_𝑖𝜕𝑥+𝜌𝐼^𝑖𝜂²𝜕³𝜑_𝑖𝜕𝑥𝜕𝑡²−𝐴_𝑖for𝑖==10,2,o…r,𝜑𝑛_𝑖,=0(29)𝜎_𝑥𝜂²𝜕²𝑤_𝑖𝜕𝑥²=0or𝜕𝑤^𝑖for𝜎_𝑥𝜕𝑥=0𝐴≠0,𝑖=1,2,…,𝑛,(30)₁𝜎_𝑥𝜂²𝜕³𝑤₁𝜕𝑥³+𝜌𝐴₁𝜂²𝜕³𝑤₁𝜕𝑥𝜕𝑡²−𝜅𝐺𝐴₁𝜑¹−𝜕𝑤¹+−𝐴₁𝜕𝑥𝜎_𝑥+𝜂²𝑐₁₂𝜕𝑤¹−𝜂₂𝜕𝑥𝑐₁₂𝜕𝑤₂𝜕𝑥=0or𝑤¹𝐴=0,(31)_𝑖𝜎_𝑥𝜂²𝜕³𝑤_𝑖𝜕𝑥³+𝜌𝐴_𝑖𝜂²𝜕³𝑤_𝑖𝜕𝑥𝜕𝑡²−𝜅𝐺𝐴_𝑖𝜑^𝑖−𝜕𝑤^𝑖+−𝐴_𝑖𝜕𝑥𝜎_𝑥+𝜂²𝑐_{(𝑖−1)𝑖}+𝑐_𝑖(𝑖+1)𝜕𝑤^𝑖−𝜂₂𝜕𝑥𝑐^{(𝑖−1)𝑖}𝜕𝑤_𝑖−1𝜕𝑥+𝑐^𝑖(+1)𝜕𝑤_𝑖+1or𝑤_𝑖𝜕𝑥=0𝐴=0for𝑖=2,…,𝑛−1(32)_𝑛𝜎_𝑥𝜂²𝜕³𝑤_𝑛𝜕𝑥³+𝜌𝐴_𝑛𝜂²𝜕³𝑤_𝑛𝜕𝑥𝜕𝑡²−𝜅𝐺𝐴_𝑛𝜑^𝑛−𝜕𝑤^𝑛+−𝐴_𝑛𝜕𝑥𝜎_𝑥+𝜂²𝑐_{(𝑛−1)𝑛}+𝑘𝜕𝑤^𝑛−𝜂₂𝜕𝑥𝑐_{(𝑛−1)𝑛}𝜕𝑤_𝑛−1𝜕𝑥=0or𝑤^𝑛=0.(33) Note that for 𝜎_𝑥=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.