Coupled Axial-Radial Vibration of Single-Walled Carbon Nanotubes Via Doublet Mechanics

Document Type : Research Paper


Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran



This paper investigates the coupled axial-radial (CAR) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a scale parameter. Two coupled forth order partial differential equations that govern the CAR vibration of SWCNTs are derived. It is the first time that DM is used to model the CAR vibration of SWCNTs. To obtain the natural frequency and dynamic response of the CAR vibration, the equations of motion are solved and the relation between natural frequencies and scale parameter is derived. It is found that there are two frequencies in the frequency spectrum and these CAR vibrational frequencies are complicated due to coupling between two vibration modes. The advantage of these analytical formulas is that they are explicitly dependent to scale parameter and chirality effect. The influence of changing some geometrical and mechanical parameters of SWCNT on its CAR frequencies has been investigated, too. It is shown that the chirality and scale parameter play significant role in the CAR vibration response of SWCNTs. The scale parameter decreases the higher band CAR frequency compared to the predictions of the classical continuum models. However, with increase in tube radius and length, the effect of the scale parameter on the natural frequencies decreases. The lower band CAR frequency is nearly independent to scale effect and tube diameter. The CAR frequencies of SWCNTs decrease as the length of the tube increases. This decreasing is higher for higher band CAR frequency. To show the accuracy and ability of this method, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed.


Main Subjects

[1] Fatahi-Vajari A., Imam A., 2016, Torsional vibration of single-walled carbon nanotubes using doublet mechanics, Zeitschrift für angewandte Mathematik und Physik 67:81.
[2] Banks H. T., Hu S., Kenz Z. R., 2011, A brief review of elasticity and viscoelasticity for solids, Advances in Applied Mathematics and Mechanics 3(1): 1-51.
[3] Granik V.T., Ferrari M., 1993, Microstructural mechanics of granular media, Mechanics of Materials 15: 301-322.
[4] Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: l-16.
[5] Yang Y., Lim C.W., 2012, Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure, International Journal of Mechanical Sciences 54: 57-68.
[6] Mindlin R.D., Eshel N.N., 1968, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures 4: 109-124.
[7] Dell’Isola F., Della Corte A., Giorgio I., 2017, Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives, Mathematics and Mechanics of Solids 22(4): 852-872.
[8] Polizzotto C., 2014, Stress gradient versus strain gradient constitutive models within elasticity, International Journal of Solids and Structures 51: 1809-1818.
[9] Ramasubramaniam A., Carter E.A., 2007, Coupled quantum–atomistic and quantum–continuum mechanics methods, Materials Research 32: 913-918.
[10] Dove M.T., 2007, An introduction to atomistic simulation methods, Seminarios de la SEM 4: 7-37.
[11] Carcaterra A., 2015, Quantum euler beam—QUEB: modeling nanobeams vibration, Continuum Mechanics and Thermodynamics 27: 145-156.
[12] Friak M., Hickel T. Grabowski B., Lymperakis L., Udyansky A., Dick A., Ma D., Roters F., Zhu L.F., Schlieter A., Kuhn U., Ebrahimi Z., Lebensohn R.A., Holec D., Eckert J., Emmerich H., Raabe D., Neugebauer J., 2011, Methodological challenges in combining quantum-mechanical and continuum approaches for materials science applications, The European Physical Journal Plus 126(101): 1-22.
[13] Khodabakhshi P., Reddy J. N., 2016, A unified beam theory with strain gradient effect and the von Karman nonlinearity, ZAMM 97: 70-91.
[14] Beheshti A., 2017, Generalization of strain-gradient theory to finite elastic deformation for isotropic materials, Continuum Mechanics and Thermodynamics 29: 493-507.
[15] Kiani K., 2014, Axial buckling analysis of vertically aligned ensembles of single-walled carbon nanotubes using nonlocal discrete and continuous models, Acta Meccanica 225: 3569-3589.
[16] Fatahi-Vajari A. , Azimzadeh Z., 2018, Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method, Indian Journal of Physics 92: 1425-1438.
[17] Kiani K., 2014, In- and out-of-plane dynamic flexural behaviors of two-dimensional ensembles of vertically aligned single-walled carbon nanotubes, Physica B, Condensed Matter 449: 164-180.
[18] Aydogdu M., 2012, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanics Research Communications 43: 34-40.
[19] Kiani K., 2018, Application of nonlocal higher-order beam theory to transverse wave analysis of magnetically affected forests of single-walled carbon nanotubes, International Journal of Mechanical Sciences 138-139: 1-16.
[20] Ferrari M., Granik V.T., Imam A., Nadeau J., 1997, Advances in Doublet Mechanics, Springer-Verlag, Berlin.
[21] Fatahi-Vajari A., Imam A., 2016, Axial vibration of single-walled carbon nanotubes using doublet mechanics, Indian Journal of Physics 90(4): 447-455.
[22] Granik V.T., 1978, Microstructural Mechanics of Granular Media, Institute of Mechanics of Moscow State University, Russian.
[23] Kojic M., Vlastelica I., Decuzzi P., Granik V.T., Ferrari M., 2011, A finite element formulation for the doublet mechanics modeling of microstructural materials, Computer Methods in Applied Mechanical Engineering 200: 1446-1454.
[24] Xin J., Zhou L.X., Ru W.J., 2009, Ultrasound attenuation in biological tissue predicted by the modified doublet mechanics model, Chinese Physics Letters 26(7): 074301.1-074301.4.
[25] Gentile F., Sakamoto J., Righetti R., Decuzzi P., Ferrari M., 2011, A doublet mechanics model for the ultrasound characterization of malignant tissues, Journal of Biomedical Science and Engineering 4: 362-374.
[26] Fang J.Y., Jue Z., Jing F., Ferrari M., 2004, Dispersion analysis of wave propagation in cubic-tetrahedral assembly by doublet mechanics, Chinese Physics Letters 21(8): 1562-1565.
[27] Sadd M.H., Dai Q., 2005, A comparison of micro-mechanical modeling of asphalt materials using finite elements and doublet mechanics, Mechanics of Materials 37: 641-662.
[28] Fatahi-Vajari A., Imam A., 2016, Lateral vibrations of single-layered graphene sheets using doublet mechanics, Journal of Solid Mechanics 8(4): 875-894.
[29] Lin S.S., Shen Y.C., 2005, Stress fields of a half-plane caused by moving loads-resolved using doublet mechanics, Soil Dynamics and Earthquake Engineering 25: 893-904.
[30] Sadd M.H., 2005, Elasticity Theory, Applications, and Numeric, Elsevier Butterworth-Heinemann, Burlington.
[31] Lee A.P., Lee J., Ferrari M., 2006, BioMEMS and Biomedical Nanotechnology, Biological and Biomedical Nanotechnology, Springer, New York.
[32] Fatahi-Vajari A., 2018, A new method for evaluating the natural frequency in radial breathing like mode vibration of double-walled carbon nanotubes, ZAMM 98(2): 255-269.
[33] Maultzsch J., Telg H., Reich S., Thomsen C., 2005, Radial breathing mode of single-walled carbon nanotubes: Optical transition energies and chiral-index assignment, Physical Review B 72: 205438.1-205438.16.
[34] Basirjafari S., EsmaielzadehKhadem S., Malekfar R., 2013,Validation of shell theory for modeling the radial breathing mode of a single-walled carbon nanotube, International Journal of Engineering: Transactions A 26(4): 447-454.
[35] Szabó A., Perri C., Csató A., Giordano G., Vuono D., Nagy J.B., 2010, Synthesis methods of carbon nanotubes and related materials, Materials 3: 3092-3140.
[36] Prasek J., Drbohlavova J., Chomoucka J., Hubalek J., Jasek O., Adamc V., Kizek R., 2011, Methods for carbon nanotubes synthesis, Journal of Materials Chemistry 21: 15872-15884.
[37] Hongjie D., 2002, Carbon nanotubes: synthesis, integration, and properties, American Chemical Society 35: 1035-1044.
[38] Zhang Y. Y., Wang C. M., Tan V. B. C., 2009, Assessment of timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics, Advances in Applied Mathematics and Mechanics 1(1): 89-106.
[39] Kiani K., 2018, Nonlocal free dynamic analysis of periodic arrays of single-walled carbon nanotubes in the presence of longitudinal thermal and magnetic fields, Computers and Mathematics with Applications 75: 3849-3872.
[40] Ghorbanpour Arani A., Mosallaie Barzoki A. A., Kolahchi R., Loghman A., 2011, Pasternak foundation effect on the axial and torsional waves propagation in embedded DWCNTs using nonlocal elasticity cylindrical shell theory, Journal of Mechanical Science and Technology 25(9): 2385-2391.
[41] Basirjafari S., EsmaeilzadehKhadem S., Malekfar R., 2013, Radial breathing mode frequencies of carbon nanotubes for determination of their diameters, Current Applied Physics 13: 599-609.
[42] Kiani K., 2014, Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes, Current Applied Physics 14(8): 1116-1139.
[43] Das S.L., Mandal T., Gupta S.S., 2013, Inextensional vibration of zig-zag single-walled carbon nanotubes using nonlocal elasticity theories, International Journal of Solids and Structures 50: 2792-2797.
[44] Fatahi-Vajari A., Imam A., 2016, Analysis of radial breathing mode vibration of single-walled carbon nanotubes via doublet mechanics, ZAMM 96(9): 1020-1032.
[45] Li C.F., Zhou S.H., Liu J., Wen B.C., 2014, Coupled lateral-torsional-axial vibrations of a helical gear-rotor-bearing system, Acta Mechanica Sinica 30(5): 746-761.
[46] Kiani K., 2014, Nanoparticle delivery via stocky single-walled carbon nanotubes: A nonlinear-nonlocal continuum-based scrutiny, Composite Structures 116: 254-272.
[47] Gupta S.S., Batra R.C., 2008, Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes, Computational Materials Science 43: 715-723.
[48] Lin S.Y., 1995, Coupled vibration and natural frequency analysis of isotropic cylinders or disks of finite dimensions, Journal of Sound and Vibration 185(2): 193-199.
[49] Ren F., Wang B., Chen S., Yao Z., Bai B., 2016, Nonlinear model and qualitative analysis for coupled axial/torsional vibrations of drill string, Shock and Vibration 2016: 1646814.
[50] Subramaniyan A.K., Sun C.T., 2008, Continuum interpretation of virial stress in molecular simulations, International Journal of Solids and Structures 45: 4340-4346.