Research Article | Open Access

Ismail Kucuk, Ibrahim S. Sadek, Sarp Adali, "Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory", *Journal of Nanomaterials*, vol. 2010, Article ID 461252, 7 pages, 2010. https://doi.org/10.1155/2010/461252

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

**Academic Editor:**Bohua Sun

#### 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 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 *i*th and *j*th 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 *i*th nanotube and is the density. The coefficient is the interaction coefficient of van der Waals forces between the and *i*th 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

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

where

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

where the subscripts and denote differentiation with respect to and , and the variational derivative is defined as [36, 37]

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)

Integrability relations between these equations can be obtained by noting that

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

With , given by (24), we observe that (15) are equivalent to (2)–(9).

#### 4. Hamilton’s Principle

The Hamilton’ principle can be expressed as

where In (25)–(26), is the kinetic energy, is the work done by external forces, is the potential energy of deformation and is the potential energy due to van der Waals forces between the nanotubes.

#### 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.

#### 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 2*n* 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.

#### References

- S. Adali, “Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal euler-bernoulli beam model,”
*Nano Letters*, vol. 9, no. 5, pp. 1737–1741, 2009. View at: Publisher Site | Google Scholar - B. I. Yakobson and R. E. Smalley, “Fullerene nanotubes: C1,000,000 and beyond,”
*American Scientist*, vol. 85, no. 4, pp. 324–337, 1997. View at: Google Scholar - V. M. Harik, “Ranges of applicability for the continuum beam model in the mechanics of carbon nanotubes and nanorods,”
*Solid State Communications*, vol. 120, no. 7-8, pp. 331–335, 2001. View at: Publisher Site | Google Scholar - C. Q. Ru, “Column buckling of multiwalled carbon nanotubes with interlayer radial displacements,”
*Physical Review B*, vol. 62, no. 24, pp. 16962–16967, 2000. View at: Publisher Site | Google Scholar - Q. Wang and V. K. Varadan, “Stability analysis of carbon nanotubes via continuum models,”
*Smart Materials and Structures*, vol. 14, no. 1, pp. 281–286, 2005. View at: Publisher Site | Google Scholar - Q. Wang, T. Hu, G. Chen, and Q. Jiang, “Bending instability characteristics of double-walled carbon nanotubes,”
*Physical Review B*, vol. 71, no. 4, Article ID :045403, 8 pages, 2005. View at: Publisher Site | Google Scholar - A. Sears and R. C. Batra, “Buckling of multiwalled carbon nanotubes under axial compression,”
*Physical Review B*, vol. 73, no. 8, Article ID 085410, 11 pages, 2006. View at: Publisher Site | Google Scholar - Y. Q. Zhang, X. Liu, and J. H. Zhao, “Influence of temperature change on column buckling of multiwalled carbon nanotubes,”
*Physics Letters A*, vol. 372, no. 10, pp. 1676–1681, 2008. View at: Publisher Site | Google Scholar - Y. Y. Zhang, C. M. Wang, and V. B. C. Tan, “Buckling of multiwalled carbon nanotubes using Timoshenko beam theory,”
*Journal of Engineering Mechanics*, vol. 132, no. 9, pp. 952–958, 2006. View at: Google Scholar - C. M. Wang, V. B. C. Tan, and Y. Y. Zhang, “Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes,”
*Journal of Sound and Vibration*, vol. 294, no. 4, pp. 1060–1072, 2006. View at: Publisher Site | Google Scholar - R. E. Miller and V. B. Shenoy, “Size-dependent elastic properties of nanosized structural elements,”
*Nanotechnology*, vol. 11, no. 3, pp. 139–147, 2000. View at: Publisher Site | Google Scholar - T. Chang and H. Gao, “Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model,”
*Journal of the Mechanics and Physics of Solids*, vol. 51, no. 6, pp. 1059–1074, 2003. View at: Publisher Site | Google Scholar - C. T. Sun and H. Zhang, “Size-dependent elastic moduli of platelike nanomaterials,”
*Journal of Applied Physics*, vol. 93, no. 2, pp. 1212–1218, 2003. View at: Publisher Site | Google Scholar - C. W. Lim and L. H. He, “Size-dependent nonlinear response of thin elastic films with nano-scale thickness,”
*International Journal of Mechanical Sciences*, vol. 46, no. 11, pp. 1715–1726, 2004. View at: Publisher Site | Google Scholar - D. W. Huang, “Size-dependent response of ultra-thin films with surface effects,”
*International Journal of Solids and Structures*, vol. 45, no. 2, pp. 568–579, 2008. View at: Publisher Site | Google Scholar - D. G. B. Edelen and N. Laws, “On the thermodynamics of systems with nonlocality,”
*Archive for Rational Mechanics and Analysis*, vol. 43, no. 1, pp. 24–35, 1971. View at: Publisher Site | Google Scholar - A. C. Eringen, “Linear theory of nonlocal elasticity and dispersion of plane waves,”
*International Journal of Engineering Science*, vol. 10, no. 5, pp. 425–435, 1972. View at: Google Scholar - J. Peddieson, G. R. Buchanan, and R. P. McNitt, “Application of nonlocal continuum models to nanotechnology,”
*International Journal of Engineering Science*, vol. 41, no. 3–5, pp. 305–312, 2003. View at: Publisher Site | Google Scholar - M. Xu, “Free transverse vibrations of nano-to-micron scale beams,”
*Proceedings of the Royal Society A*, vol. 462, no. 2074, pp. 2977–2995, 2006. View at: Publisher Site | Google Scholar | MathSciNet - C. M. Wang, Y. Y. Zhang, S. S. Ramesh, and S. Kitipornchai, “Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory,”
*Journal of Physics D*, vol. 39, no. 17, pp. 3904–3909, 2006. View at: Publisher Site | Google Scholar - Q. Wang and Y. Shindo, “Nonlocal continuum models for carbon nanotubes subjected to static loading,”
*Journal of Mechanics of Materials and Structures*, vol. 1, no. 4, pp. 663–680, 2006. View at: Google Scholar - Q. Wang and K. M. Liew, “Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures,”
*Physics Letters A*, vol. 363, no. 3, pp. 236–242, 2007. View at: Publisher Site | Google Scholar - J. N. Reddy, “Nonlocal theories for bending, buckling and vibration of beams,”
*International Journal of Engineering Science*, vol. 45, no. 2–8, pp. 288–307, 2007. View at: Publisher Site | Google Scholar - C. M. Wang, Y. Y. Zhang, and X. Q. He, “Vibration of nonlocal Timoshenko beams,”
*Nanotechnology*, vol. 18, no. 10, Article ID 105401, 9 pages, 2007. View at: Publisher Site | Google Scholar - C. M. Wang, S. Kitipornchai, C. W. Lim, and M. Eisenberger, “Beam bending solutions based on nonlocal Timoshenko beam theory,”
*Journal of Engineering Mechanics*, vol. 134, no. 6, pp. 475–481, 2008. View at: Publisher Site | Google Scholar - R. Artan and A. Tepe, “The initial values method for buckling of nonlocal bars with application in nanotechnology,”
*European Journal of Mechanics A*, vol. 27, no. 3, pp. 469–477, 2008. View at: Publisher Site | Google Scholar - J.-C. Hsu, R.-P. Chang, and W.-J. Chang, “Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory,”
*Physics Letters A*, vol. 372, no. 16, pp. 2757–2759, 2008. View at: Publisher Site | Google Scholar - L. J. Sudak, “Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics,”
*Journal of Applied Physics*, vol. 94, no. 11, pp. 7281–7287, 2003. View at: Publisher Site | Google Scholar - Q. Wang, “Wave propagation in carbon nanotubes via nonlocal continuum mechanics,”
*Journal of Applied Physics*, vol. 98, no. 12, Article ID 124301, 6 pages, 2005. View at: Publisher Site | Google Scholar - L. Wang and H. Hu, “Flexural wave propagation in single-walled carbon nanotubes,”
*Physical Review B*, vol. 71, no. 19, Article ID 195412, 7 pages, 2005. View at: Publisher Site | Google Scholar - Q. Wang, G. Y. Zhou, and K. C. Lin, “Scale effect on wave propagation of double-walled carbon nanotubes,”
*International Journal of Solids and Structures*, vol. 43, no. 20, pp. 6071–6084, 2006. View at: Publisher Site | Google Scholar - P. Lu, H. P. Lee, C. Lu, and P. Q. Zhang, “Application of nonlocal beam models for carbon nanotubes,”
*International Journal of Solids and Structures*, vol. 44, no. 16, pp. 5289–5300, 2007. View at: Publisher Site | Google Scholar - H. Heireche, A. Tounsi, A. Benzair, M. Maachou, and E. A. Adda Bedia, “Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity,”
*Physica E*, vol. 40, no. 8, pp. 2791–2799, 2008. View at: Publisher Site | Google Scholar - S. Adali, “Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory,”
*Physics Letters A*, vol. 372, no. 35, pp. 5701–5705, 2008. View at: Publisher Site | Google Scholar - S. Adali, “Variational principles for multi-walled carbon nanotubes undergoing nonlinear vibrations by semi-inverse method,”
*Micro and Nano Letters*, vol. 4, no. 4, pp. 198–203, 2009. View at: Google Scholar - J.-H. He, “Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics,”
*International Journal of Turbo and Jet Engines*, vol. 14, no. 1, pp. 23–28, 1997. View at: Google Scholar - J.-H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,”
*Chaos, Solitons and Fractals*, vol. 19, no. 4, pp. 847–851, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Variational approach to$(2+1)$-dimensional dispersive long water equations,”
*Physics Letters A*, vol. 335, no. 2-3, pp. 182–184, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-M. Liu, “Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method,”
*Chaos, Solitons and Fractals*, vol. 23, no. 2, pp. 573–576, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J.-H. He, “Variational theory for one-dimensional longitudinal beam dynamics,”
*Physics Letters A*, vol. 352, no. 4-5, pp. 276–277, 2006. View at: Publisher Site | Google Scholar - X.-W. Zhou, “Variational approach to the Broer-Kaup-Kupershmidt equation,”
*Physics Letters A*, vol. 363, no. 1-2, pp. 108–109, 2007. View at: Publisher Site | Google Scholar | MathSciNet - J.-H. He, “Variational principle for two-dimensional incompressible inviscid flow,”
*Physics Letters A*, vol. 371, no. 1-2, pp. 39–40, 2007. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2010 Ismail Kucuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.