Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory
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.
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  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 . 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 . 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 . Vibration of multiwalled nanotubes was studied in  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 . 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  and a buckling load  where the linear elastic theory was employed. Variational principles were derived for CNTs undergoing nonlinear vibrations in  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  and buckling . 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 . 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 consisting of n nanotubes of cylindrical shape is considered. It lies on a Winkler foundation of modulus and is subject to an axial stress which can be tensile or compressive in which case is less than the critical buckling load. We introduce a difference operator defined as
where and 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]
where is the angle of rotation and the operators and are given by where the index refers to the order of the nanotubes with the innermost nanotube indicated by and the outermost nanotube by with . In (8) is a forcing function, and in (10) is the Kronecker’s delta with for and . In (10) and (11), E is the Young’s modulus, G is the shear modulus, is the shear correction factor, is the moment of inertia, is the cross-sectional area of the ith nanotube and is the density. The coefficient is the interaction coefficient of van der Waals forces between the and ith nanotubes with [7–10, 28]. The parameter appears in the nonlocal theory of beams and helps define the small scale effects accurately where is a constant for adjusting the model by experimental results and a is an internal characteristic length [17–26].
3. Variational Formulation
with given by
where and denotes the unknown functions of 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
Integrability relations between these equations can be obtained by noting that
4. Hamilton’s Principle
The Hamilton’ principle can be expressed as
5. Boundary Conditions
Next the variations of the functional in (12) are evaluated with respect to and in order to derive the natural and geometric boundary conditions. Let and denote the variations of and such that . The first variations of with respect to and , denoted by and , respectively, can be obtained by integration by parts and expressed as
where and are the boundary terms defined as
where is the derivative of with respect to . Thus the boundary conditions at are given by
Note that for , the boundary condition (30) is not needed. It is observed that for the small scale parameter (nonlocal theory) the natural boundary conditions are coupled and time derivative appears in the boundary conditions. These boundary conditions uncouple for (local theory) and time derivatives drop out.
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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