Abstract
This paper aims to study the effect of externally applied longitudinal magnetic field on the transverse vibration of viscoelastic double-walled carbon nanotubes (visco-DWCNTs) embedded in a viscoelastic medium. The analyses are carried out based on the nonlocal viscoelastic model and Euler–Bernoulli beam theory. Governing equations are derived for the vibration of the embedded visco-DWCNT subjected to a magnetic field, where the Lorentz magnetic force, the surrounding viscoelastic medium, the intertube van der Waals forces and viscoelasticity of the DWCNT are taken into consideration. In this study, the transfer function method is employed to solve the governing equations, which enables one to obtain the natural frequencies and the corresponding mode shapes in closed form for the DWCNTs with arbitrary boundary conditions. Here the developed mechanics model is first compared with the existing techniques available in the literature in a few particular cases, where excellent agreement is achieved. The validation of the model is followed by a detailed parametric study of the effects of longitudinal magnetic field, nonlocal parameter, boundary conditions, structural damping coefficient and aspect ratio of the DWCNTs on their vibration. The study demonstrates the efficiency of the present technique designed for vibration analysis of a complicated multi-physics system comprising DWCNTs, the viscoelastic medium and a magnetic field in longitudinal direction.
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22 November 2018
In all the articles in Acta Mechanica Solida Sinica, Volume 31, Issues 1–4, the copyright is incorrectly displayed as “The Chinese Society of Theoretical and Applied Mechanics and Technology ” where it should be “The Chinese Society of Theoretical and Applied Mechanics”.
References
Iijima S. Helical microtubules of graphitic carbon. Nature. 1991;354:56–8.
Young KK, JianXin G, Se-Gyu J. Enhanced field emission of an electric field assisted singwalled carbon nanotube assembly in colloid interstices. Carbon. 2009;47:1555–60.
Tae-Won I, Young GJ. Enhanced electrical conductivity, mechanical modulus, and thermal stability of immiscible polylactide/polypropylene blends by the selective localization of multi-walled carbon nanotubes. Compos Sci Technol. 2014;103:78–84.
Kibalchenko M, Payne MC, Yates JR. Magnetic response of single-walled carbon nanotubes induced by an external magnetic field. ACS Nano. 2011;5(1):537–45.
Arani AG, Amir S, Dashti P, Yousefi M. Flow-induced vibration of double bonded visco-CNTs under magnetic fields considering surface effect. Compos Mater Sci. 2014;86:144–54.
Chang TP. Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Appl Math Model. 2012;36:1964–73.
Roche S, Saito R. Effects of magnetic field and disorder on the electronic properties of carbon nanotubes. Phys Rev B. 1999;59(7):5242.
Brian WS, Derrick RD. Magnetic field alignment and electrical properties of solution cast PET-carbon nanotube composite films. Polymer. 2009;50:898–904.
Camponeschi E, Vance R, Al-Haik M, Garmestani H, Tannenbaum R. Properties of carbon nanotube-polymer composites aligned in a magnetic field. Carbon. 2007;45:2037–46.
Andrey MP, Irina VL, Andrey AK, Yurii EL, Nikolai AP, Andrei IS, Sergey AV, Sergey VR. Force and magnetic field sensor based on measurement of tunneling conductance between ends of coaxial carbon nanotubes. Compos Mater Sci. 2014;92:84–91.
Kiani K. Magnetically affected single-walled carbon nanotubes as nanosensors. Mech Res Commun. 2014;60:33–9.
Murmu T, McCarthy MA, Adhikari S. Vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach. J Sound Vib. 2012;331:5069–86.
Arani AG, Roudbari MA, Amir S. Longitudinal magnetic field effect on wave propagation of fluid-conveyed SWCNT using Knudsen number and surface considerations. Appl Math Model. 2016;40(3):2025–38.
Wang H, Dong K, Men F, Yan YJ, Wang X. Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix. Appl Math Model. 2010;34:878–89.
Narendar S, Gupta SS, Gopalakrishnan S. Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory. Appl Math Model. 2012;36:4529–38.
Raju APA, Lewis A, Derby B, Young RJ. Wide-Area strain sensors based upon graphene-polymer composite coatings probed by Raman spectroscopy. Mater Views. 2014;24:2865–74.
Cooper CA, Young RJ, Halsall M. Investigation into the deformation of carbon nanotubes and their composites through the use of Raman spectroscopy. Compos Part A. 2001;32:401–11.
Ranjbartoreh AR, Wang G. Molecular dynamic investigation of mechanical properties of armchair and zigzag double-walled carbon nanotubes under various loading conditions. Phys Lett A. 2010;374(7):969–74.
Eringen AC. On differential equations of nonlocal elasticity and solution of screw dislocation and surface waves. J Appl Phys. 1983;54(9):4703–10.
Adhikari S, Gilchrist D, Murmu T, McCarthy MA. Nonlocal normal modes in nanoscale dynamical systems. Mech Syst Signal Process. 2015;60–61:583–603.
Mahmoud MA. Mass sensing of multiple particles adsorbed to microcantilever resonators. Microsyst Technol. 2015;23:1–10.
Kiani K. A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. Int J Mech Sci. 2010;52:1343–56.
Ke LL, Wang YS. Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory. Physica E. 2011;43:1031–9.
Eringen AC. Theory of nonlocal plasticity. Int J Eng Sci. 1983;21:741–51.
Eringen AC. A unified continuum theory of electrodynamics of liquid crystals. Int J Eng Sci. 1997;35:1137–57.
Peddieson J, Buchanan GR, McNitt RP. Application of nonlocal continuum models to nanotechnology. Int J Eng Sci. 2003;41:305–12.
Sudak LJ. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys. 2003;94:72–81.
Lei Y, Murmu T, Adhikari S, Friswell MI. Dynamic characteristics of damped viscoelastic nonlocal Euler–Bernoulli beams. Eur J Mech A/Solids. 2013;42:125–36.
Lei Y, Adhikari S, Friswell MI. Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int J Eng Sci. 2013;66–67:1–13.
Wang B, Deng Z, Ouyang H, Zhang K. Wave characteristics of single-walled fluid-conveying carbon nanotubes subjected to multi-physical fields. Physica E. 2013;52:97–105.
Ghasemi A, Dardel M, Ghasemi MH, Barzegari MM. Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes. Appl Math Model. 2013;37:4972–92.
Güven U. Transverse vibrations of single-walled carbon nanotubes with initial stress under magnetic field. Compos Struct. 2014;114:92–8.
Hoseinzadeh MS, Khadem SE. A nonlocal shell theory model for evaluation of thermoelastic damping in the vibration of a double-walled carbon nanotube. Physica E. 2014;57:6–11.
Ansari R, Rouhi H, Sahmani S. Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int J Mech Sci. 2011;53:786–92.
Kiani K. Vibration and instability of a single-walled carbon nanotube in a three-dimensional magnetic field. J Phys Chem Solids. 2014;75:15–22.
Kiani K. Stability and vibrations of doubly parallel current-carrying nanowires immersed in a longitudinal magnetic field. Phys Lett A. 2015;379:348–60.
Kiani K. Column buckling of doubly parallel slender nanowires carrying electric current acted upon by a magnetic field. J Phys Chem Solids. 2016;95:89–97.
Thamviratnam D, Zhuge Y. Free vibration analysis of beams on elastic foundation. Comput Struct. 1996;60:971–80.
Lei Y. Finite element analysis of beams with nonlocal foundations. In: 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference. Newport, Rhode Island 2006, pp 1–11.
Friswell MI, Adhikari S, Lei Y. Vibration analysis of beams with non-local foundations using the finite element method. Int J Numer Methods Eng. 2007;71(11):1365–86.
Kiani K. Transverse wave propagation in elastically confined single-walled carbon nanotubes subjected to longitudinal magnetic fields using nonlocal elasticity models. Physica E. 2012;45:86–96.
Kiani K. Longitudinally varying magnetic field influenced transverse vibration of embedded double-walled carbon nanotubes. Int J Mech Sci. 2014;87:179–99.
Kiani K. Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories. Int J Mech Sci. 2013;68:16–34.
Arani AG, Zarei MS. Nonlocal vibration of Y-shaped CNT conveying nano-magnetic viscous fluid under magnetic field. Ain Shams Eng J. 2015;6:565–75.
Wang X, Shen JX, Liu Y, Shen GG, Lu G. Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field. Appl Math Model. 2012;36:648–56.
Kazemi-Lari MA, Fazelzadeh SA, Ghavanloo E. Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation. Physica E. 2012;44:1623–30.
Soltani P, Taherian MM, Farshidianfar A. Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium. J Phys D Appl Phys. 2010;43:425401.
Arash B, Wang Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci. 2012;51:303–13.
Ghavanloo E, Fazelzadeh SA. Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid. Physica E. 2011;44:17–24.
Liang F, Su Y. Stability analysis of a single-walled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect. Appl Math Model. 2013;37:6821–8.
Xu KY, Alnefaie KA, Abu-Hamdeh NH, Almitani KH, Aifantis EC. Free transverse vibrations of a double-walled carbon nanotube: gradient and internal inertia effects. Acta Mech Solida Sin. 2014;27(4):345–52.
Cigeroglu E, Samandari H. Nonlinear free vibration of double walled carbon nanotubes by using describing function method with multiple trial functions. Physica E. 2012;46:160–73.
Yang B, Tan CA. Transfer functions of one-dimensional distributed parameter system. Transl ASME J Appl Mech. 1992;59(4):1009–14.
Shen ZB, Li XF, Sheng LP, Tang GJ. Transverse vibration of nanotube-based micro-mass sensor via nonlocal Timoshenko beam theory. Comput Mater Sci. 2012;53:340–6.
Shen ZB, Tang GJ, Zhang L, Li XF. Vibration of double-walled carbon nanotube based nanomechanical sensor with initial axial stress. Comput Mater Sci. 2012;58:51–8.
Yoon J, Ru CQ, Mioduchowski A. Vibration of an embedded multiwall carbon nanotube. Compos Sci Technol. 2003;63:1533–42.
Ke LL, Xiang Y, Yang J, Kitipornchai S. Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Compos Sci Technol. 2009;47:409–17.
Murmu T, McCarthy MA, Adhikari S. In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach. Compos Struct. 2013;96:57–63.
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant Nos. 11272348 and 11302254).
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Zhang, D., Lei, Y. & Shen, Z. Effect of Longitudinal Magnetic Field on Vibration Response of Double-Walled Carbon Nanotubes Embedded in Viscoelastic Medium. Acta Mech. Solida Sin. 31, 187–206 (2018). https://doi.org/10.1007/s10338-018-0006-x
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DOI: https://doi.org/10.1007/s10338-018-0006-x