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Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. I: Theoretical formulations. (English) Zbl 1398.74145
Summary: The current work suggests mathematical models for the vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal classical and shear deformable beam theories. The van der Waals interaction forces between atoms of the innermost and outermost tubes are modeled by an elastic layer. The equations of motion are derived for the nonlocal double body Euler-Bernoulli, Timoshenko and higher-order beams connected by a flexible layer under excitation of a moving nanoparticle. Analytical solutions of the problem are provided for the aforementioned nonlocal beam models with simply supported boundary conditions. The dynamical deflections and nonlocal bending moments of the innermost and outermost tubes are then obtained during the courses of excitation and free vibration. Finally, the critical velocities of the moving nanoparticle associated with the nonlocal beam theories are expressed in terms of small-scale effect parameter, geometry, and material properties of DWCNTs. This article is lovingly dedicated to my father and mother, Amrullah Kiani and Kobra Ahmadi, whose love and encouragements I feel every day of my life.

MSC:
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M25 Micromechanics of solids
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[1] Coleman J.N., Khan U., Blau W.J., Gunko Y.K.: Small but strong: a review of the mechanical properties of carbon nanotubepolymer composites. Carbon 44, 1624–1652 (2006)
[2] Gibson R.F., Ayorinde E.O., Wen Y.F.: Vibrations of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 67, 1–28 (2007)
[3] Hummer G., Rasaiah J.C., Noworyta J.P.: Water conduction through the hydrophobic channel of a carbon nanotube. Nature 414, 188–190 (2001)
[4] Supple S., Quirke N.: Rapid imbibition of fluids in carbon nanotubes. Phys. Rev. Lett. 90, 214501 (2003)
[5] Majumder M., Chopra N., Andrews R., Hinds B.J.: Nanoscale hydrodynamics enhanced flow in carbon nanotubes. Nature 438, 444 (2005)
[6] Regan, B.C., Aloni, S., Huard, B., Fennimore, A., Ritchie, R.O., Zettl, A.: Nanowicks: nanotubes as tracks for mass transfer. In: Kuzmany, H., Fink, J., Mehring, M., Roth, S. (eds.) Molecular Nanostructures. AIP Conference Proceedings, vol. 685, pp. 612–615 (2003)
[7] Regan B.C., Aloni S., Ritchie R.O., Dahmen U., Zettl A.: Carbon nanotubes as nanoscale mass conveyors. Nature 428, 924–927 (2004)
[8] Babu S., Ndungu P., Bradley J.C., Rossi M.P., Gogotsi Y.: Guiding water into carbon nanopipes with the aid of bipolar electrochemistry. Microfluid. Nanofluid. 1, 284–288 (2005)
[9] Gogotsi Y.: In situ multiphase fluid experiments in hydrothermal CNTs. Appl. Phys. Lett. 79, 1021–1023 (2001)
[10] Waghe A., Rasaiah J.C.: Filling and emptying kinetics of carbon nanotubes in water. J. Chem. Phys. 117, 10790–10795 (2002)
[11] Tuzun R.E., Noid D.W., Sumpter B.G., Merkle R.C.: Dynamics of fluid flow inside CNTs. Nanotechnology 7, 241–246 (1996)
[12] Mao Z., Sinnott S.B.: A computational study of molecular diffusion and dynamics flow through CNTs. J. Phys. Chem. B 104, 4618–4624 (2000)
[13] Sokhan V.P., Nicholson D., Quirke N.: Fluid flow in nanopores: an examination of hydrodynamic boundary conditions. J. Chem. Phys. 115, 3878–3887 (2001)
[14] Yoon J., Ru C.Q., Mioduchowski A.: Vibration and instability of CNTs conveying fluid. Compos. Sci. Technol. 65, 1326–1336 (2005)
[15] Yoon J., Ru C.Q., Mioduchowski A.: Flow-induced flutter instability of cantilever CNTs. Int. J. Solids. Struct. 43, 3337–3349 (2006) · Zbl 1121.74385
[16] Yana Y., He X.Q., Zhang L.X., Wang C.M.: Dynamic behavior of triple-walled carbon nanotubes conveying fluid. J. Sound. Vib. 319, 1003–1018 (2009)
[17] Ru C.Q.: Effect of van der Waals forces on axial buckling of a double-wall carbon nanotube. J. Appl. Phys. 87, 7227–7231 (2000)
[18] Ru C.Q.: Axially compressed buckling of a double walled carbon nanotube embedded in an elastic medium. J. Mech. Phys. Solids 49, 1265–1279 (2001) · Zbl 1015.74014
[19] Li C., Chou T.W.: Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces. Compos. Sci. Technol. 63, 1517–1524 (2003)
[20] Yoon J., Ru C.Q., Mioduchowski A.: Timoshenko-beam effects on transverse wave propagation in carbon nanotubes. Composites B 35, 87–93 (2004)
[21] He X.Q., Kitipornchai S., Liew K.M.: Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J. Mech. Phys. Solids 53, 303–326 (2005) · Zbl 1162.74354
[22] Xu C.L., Wang X.: Matrix effects on the breathing modes of multiwall carbon nanotubes. Compos. Struct. 80, 73–81 (2007)
[23] Gupta S.S., Batra R.C.: Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes. Comput. Mater. Sci. 43, 715–723 (2008)
[24] Peddieson J., Buchanan G.R., McNitt R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
[25] Sudak L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)
[26] Wang Q., Liew K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)
[27] Ece M.C., Aydogdu M.: Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta. Mech. 190, 185–195 (2007) · Zbl 1117.74022
[28] Schoen P.A.E., Walther J.H., Arcidiacono S., Poulikakos D., Koumoutsakos P.: Nanoparticle traffic on helical tracks: thermophoretic mass transport through carbon nanotubes. Nano. Lett. 6, 1910–1917 (2006)
[29] Schoen P.A.E., Walther J.H., Poulikakos D., Koumoutsakos P.: Phonon assisted thermophoretic motion of gold nanoparticles inside carbon nanotubes. Appl. Phys. Lett. 90, 253116 (2007)
[30] Kiani K., Mehri B.: Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J. Sound. Vib. 329, 2241–2264 (2010)
[31] Lennard-Jones J.E.: The determination of molecular fields: from the variation of the viscosity of a gas with temperature. Proc. Roy. Soc. Lond. Ser. A 106, 441–462 (1924)
[32] Girifalco L.A., Lad R.A.: Energy of cohesion, compressibility and the potential energy function of graphite system. J. Chem. Phys. 25, 693–697 (1956)
[33] Saito R., Dresselhaus G., Dresselhaus M.S.: Physical Properties of Carbon Nanotubes. Imperial College, London (1998) · Zbl 0752.68069
[34] Frýba L.: Vibration of Solids and Structures Under Moving Loads. Thomas Telford, London (1999)
[35] Eringen A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972) · Zbl 0229.73006
[36] Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
[37] Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002) · Zbl 1023.74003
[38] Timoshenko S.: Vibration Problems in Engineering. Van Nostrand, New Jersey (1955) · JFM 63.1305.03
[39] Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables. Dover, New York (1972) · Zbl 0543.33001
[40] Reddy J.N.: Mechanics of Laminated Composite Plates. CRC press, Florida (1997) · Zbl 0899.73002
[41] Kiani K., Nikkhoo A., Mehri B.: Prediction capabilities of classical and shear deformable beam models excited by a moving mass. J. Sound. Vib. 320, 632–648 (2009)
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