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Research Papers

Torsional Vibration Analysis of Carbon Nanotubes Based on the Strain Gradient Theory and Molecular Dynamic Simulations

[+] Author and Article Information
R. Ansari

e-mail: r_ansari@guilan.ac.ir

S. Ajori

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received December 5, 2011; final manuscript received February 20, 2013; published online June 18, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(5), 051016 (Jun 18, 2013) (6 pages) Paper No: VIB-11-1293; doi: 10.1115/1.4024208 History: Received December 05, 2011; Revised February 20, 2013

In the current study, the torsional vibration of carbon nanotubes is examined using the strain gradient theory and molecular dynamic simulations. The model developed based on this gradient theory enables us to interpret size effect through introducing material length scale parameters. The model accommodates the modified couple stress and classical models when two or all material length scale parameters are set to zero, respectively. Using Hamilton's principle, the governing equation and higher-order boundary conditions of carbon nanotubes are obtained. The generalized differential quadrature method is utilized to discretize the governing differential equation of the present model along with two boundary conditions. Then, molecular dynamic simulations are performed for a series of carbon nanotubes with different aspect ratios and boundary conditions, the results of which are matched with those of the present strain gradient model to extract the appropriate value of the length scale parameter. It is found that the present model with properly calibrated value of length scale parameter has a good capability to predict the torsional vibration behavior of carbon nanotubes.

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References

Zhang, W. D., Wen, Y., Liu, S. M., Tjiu, W. C., Xu, G. Q., and Gan, L. M., 2002, “Synthesis of Vertically Aligned Carbon Nanotubes on Metal Deposited Quartz Plates,” Carbon, 40, pp. 1981–1989. [CrossRef]
Zhao, C., Song, Y., Ren, J., and Qu, X., 2009, “A DNA Nanomachine Induced by Single-Walled Carbon Nanotubes on Gold Surface,” Biomaterials, 30, pp. 1739–1745. [CrossRef] [PubMed]
Qin, C., Shen, J., Hu, Y., and Ye, M., 2009, “Facile Attachment of Magnetic Nanoparticles to Carbon Nanotubes Via Robust Linkages and Its Fabrication of Magnetic Nanocomposites,” Compos. Sci. Technol., 69, pp. 427–431. [CrossRef]
Yan, X. B., Chen, X. J., Tay, B. K., and Khor, K. A., 2007, “Transparent and Flexible Glucose Via Layer-by-Layer Assembly of Multi-Wall Carbon Nanotubes and Glucose Oxidase,” Electrochem. Commun., 9, pp. 1269–1275. [CrossRef]
Liu, L., and Zhang, Y., 2004, “Multi-Wall Carbon Nanotube as a New Infrared Detected Material,” Sensors Actuators A, 116, pp. 394–397. [CrossRef]
Chen, Y. L., Liu, B., Wu, J., Huang, Y., Jiang, H., and Hwang, K. C., 2008, “Mechanics of Hydrogen Storage in Carbon Nanotubes,” J. Mech. Phys. Solids, 56, pp. 3224–3241. [CrossRef]
Ansari, R., Sahmani, S., and Arash, B., 2010, “Nonlocal Plate Model for Free Vibrations of Single-Layered Graphene Sheets,” Phys. Lett. A, 375, pp. 53–62. [CrossRef]
Arash, B., and Ansari, R., 2010, “Evaluation of Nonlocal Parameter in the Vibrations of Single-Walled Carbon Nanotubes With Initial Strain,” Physica E, 42, pp. 2058–2064. [CrossRef]
Yang, J., Ke, L. L., and Kitipornchai, S., 2010, “Nonlinear Free Vibration of Single-Walled Carbon Nanotubes Using Nonlocal Timoshenko Beam Theory,” Physica E, 42, pp. 1727–1735. [CrossRef]
Narendar, S., and Gopalakrishnan, S., 2010, “Terahertz Wave Characteristics of a Single-Walled Carbon Nanotube Containing a Fluid Flow Using the Nonlocal Timoshenko Beam Model,” Physica E, 42(5), pp. 1706–1712. [CrossRef]
Ansari, R., Sahmani, S., and Rouhi, H., 2011, “Rayleigh-Ritz Axial Buckling Analysis of Single-Walled Carbon Nanotubes With Different Boundary Conditions,” Phys. Lett. A, 375, pp. 1255–1263. [CrossRef]
Ansari, R., and Sahmani, S., 2011, “Bending Behavior and Buckling of Nanobeams Including Surface Stress Effects Corresponding to Different Beam Theories,” Int. J. Eng. Sci., 49(11), pp. 1244–1255. [CrossRef]
Mindlin, R. D., and Tiersten, H. F., 1962, “Effects of Couple-Stresses in Linear Elasticity,” Arch. Rat. Mech. Anal., 11, pp. 415–448. [CrossRef]
Koiter, W. T., 1964, “Couple Stresses in the Theory of Elasticity I and II,” Proc. Koninklijke Nederlandse Akad. van Wetenschappen (B), 67, pp. 17–44.
Eringen, A. C., and Suhubi, E. S., 1964, “Nonlinear Theory of Simple Microelastic Solid—I,” Int. J. Eng. Sci., 2, pp. 189–203. [CrossRef]
Eringen, A. C., and Suhubi, E. S., 1964, “Nonlinear Theory of Simple Microelastic Solid—II,” Int. J. Eng. Sci., 2, pp. 389–404. [CrossRef]
Mindlin, R. D., 1964, “Micro-Structure in Linear Elasticity,” Arch. Rat. Mech. Anal., 16, pp. 51–78. [CrossRef]
Toupin, R. A., 1964, “Theory of Elasticity With Couple Stresses,” Arch. Rat. Mech. Anal., 17, pp. 85–112. [CrossRef]
Mindlin, R. D., 1965, “Second Gradient of Strain and Surface Tension in Linear Elasticity,” Int. J. Solids Struct., 1, pp. 417–438. [CrossRef]
Mindlin, R. D., and Eshel, N. N., 1968, “On First Strain-Gradient Theories in Linear Elasticity,” Int. J. Solids Struct., 4, pp. 109–124. [CrossRef]
Eringen, A. C., 1983, “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” J. Appl. Phys., 54, pp. 4703–4710. [CrossRef]
Vardoulaksi, I., Exadaktylos, G., and Kourkoulis, S. K., 1998, “Bending of Marble With Intrinsic Length Scales: A Gradient Theory With Surface Energy and Size Effects,” J. Phys. IV, 8, pp. 399–406. [CrossRef]
Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., 2002, “Couple Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., 39, pp. 2731–2743. [CrossRef]
Akgöz, B., and Civalek, Ö., 2011, “Stability Analysis of Carbon Nanotubes (CNTS) Based on Modified Couple Stress Theory,” Int. Adv. Technol. Symp., 6, pp. 71–74. [CrossRef]
Ke, L. L., and Wang, Y. S., 2011, “Flow-Induced Vibration and Instability of Embedded Double-Walled Carbon Nanotubes Based on a Modified Couple Stress Theory,” Physica E, 43, pp. 1031–1039. [CrossRef]
Gheshlaghi, B., Hasheminejad, S. M., and Abbasian, S., 2010, “Size Dependent Torsional Vibration of Nanotubes,” Physica E, 43, pp. 45–48. [CrossRef]
Fleck, N. A., and Hutchinson, J. W., 1993, “Phenomenological Theory for Strain Gradient Effects in Plasticity,” J. Mech. Phys. Solids, 41, pp. 1825–1857. [CrossRef]
Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., and Tong, P., 2003, “Experiments and Theory in Strain Gradient Elasticity,” J. Mech. Phys. Solids, 51, pp. 1477–1508. [CrossRef]
Kong, S., Zhou, S., Nie, Z., and Wang, K., 2009, “Static and Dynamic Analysis of Micro Beams Based on Strain Gradient Elasticity Theory,” Int. J. Eng. Sci., 47, pp. 487–498. [CrossRef]
Wang, B., Zhao, J., and Zhou, S., 2010, “A Micro Scale Timoshenko Beam Model Based on Strain Gradient Elasticity Theory,” Eur. J. Mech. A, 29, pp. 591–599. [CrossRef]
Ansari, R., Gholami, R., and Sahmani, S., 2011, “Free Vibration of Size-Dependent Functionally Graded Microbeams Based on a Strain Gradient Theory,” Compos. Struct., 94, pp. 221–228. [CrossRef]
Rao, S. S., 2007, Vibration of Continuous Systems, Wiley, Hoboken, NJ.
Shu, C., Chen, W., and Du, H., 2000, “Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method,” J. Comput. Phys., 163(2), pp. 452–466. [CrossRef]
Haftchenari, H., Darvizeh, M., Darvizeh, A., Ansari, R., and Sharma, C. B., 2007, “Dynamic Analysis of Composite Cylindrical Shells Using Differential Quadrature Method (DQM),” Compos. Struct., 78(2), pp. 292–298. [CrossRef]
De Rosa, M. A., Auciello, N. M., and Lippiello, M., 2008, “Dynamic Stability Analysis and DQM for Beams With Variable Cross-Section,” Mech. Res. Commun., 35(3), pp. 187–192. [CrossRef]
Hu, Y. J., Zhu, Y. Y., and Cheng, C. J., 2009, “QM for Dynamic Response of Fluid-Saturated Visco-Elastic Porous Media,” Int. J. Solids Struct., 46(7–8), pp. 1667–1675. [CrossRef]
Sepahi, O., Forouzan, M. R., and Malekzadeh, P., 2010, “Large Deflection Analysis of Thermo-Mechanical Loaded Annular FGM Plates on Nonlinear Elastic Foundation Via DQM,” Compos. Struct., 92(10), pp. 2369–2378. [CrossRef]
Pradhan, S. C., and Murmu, T., 2010, “Application of Nonlocal Elasticity and DQM in the Flapwise Bending Vibration of a Rotating Nanocantilever,” Physica E, 42(7), pp. 1944–1949. [CrossRef]
Tersoff, J., 1988, “New Empirical Approach for the Structure and Energy of Covalent Systems,” Phys. Rev. B, 37, pp. 6991–7000. [CrossRef]
Brenner, D. W., 1990, “Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films,” Phys. Rev. B, 42, pp. 9458–9471. [CrossRef]
Allen, M. P., and Tildesley, D. J., 1986, Computer Simulation of Liquids, Oxford Science Publication, New York.
Hoover, W. G., 1985, “Canonical Dynamics: Equilibrium Phase-Space Distributions,” Phys. Rev. A, 31, pp. 1695–1697. [CrossRef] [PubMed]
Shen, L., and Li, J., 2004, “Transversely Isotropic Elastic Properties of Single-Walled Carbon Nanotubes,” Phys. Rev. B, 69, p. 045414. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic view of an armchair SWCNT with C-F boundary conditions (a) before torsional deformation and (b) after torsional deformation

Grahic Jump Location
Fig. 3

Comparison of torsional vibration frequencies from the strain gradient model for C-C and C-F (6,6) SWCNTs with those of MD simulations (l/h = 0.43)

Grahic Jump Location
Fig. 2

Effect of aspect ratio on the torsional vibrations of a C-C (6, 6) SWCNT for different dimensionless length scale parameters l/h

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