Structural Engineering and Mechanics

  • 제73권1호
  • /
  • Pages.37-52
  • /
  • 2020
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Longitudinal vibration of double nanorod systems using doublet mechanics theory

  • Aydogdu, Metin (Department of Mechanical Engineering, Trakya University) ;
  • Gul, Ufuk (Department of Mechanical Engineering, Trakya University)
  • 투고 : 2018.06.28
  • 심사 : 2019.08.27
  • 발행 : 2020.01.10
⟨ 이전 논문 다음 논문 ⟩

초록

This paper investigates the free and forced longitudinal vibration of a double nanorod system using doublet mechanics theory. The doublet mechanics theory is a multiscale theory spanning between lattice dynamics and continuum mechanics. Equations of motion and boundary conditions for the double nanorod system are obtained using Hamilton's principle. Clamped-clamped and clamped-free boundary conditions are considered. Frequencies and dynamic displacements are determined to demonstrate the effects of length scale parameter of considered material and geometry of the nanorods. It is shown that frequencies obtained by the doublet mechanics theory are bounded from above (van Hove singularity) and unlike classical elasticity theory doublet mechanics theory predicts finite number of modes depending on the length of the nanotube. The present doublet mechanics results have been compared to molecular dynamics, experimental and nonlocal theory results and good agreement is observed between the present and other mentioned results. The difference between wave frequencies of graphite is less than 10% between doublet mechanics and experimental results near to the end of the first Brillouin zone.

키워드

참고문헌

  1. Aizawa, T., Souda, R., Otani, S., Ishizawa, Y. and Oshima, C. (1991), "Bond softening in monolayer graphite formed on transition-metal carbide surfaces", Phys. Rev. B, 43, https://doi.org/10.1103/PhysRevB.42.11469.
  2. Arani, A.G., Amir, S., Dashti, P. and Yousefi, M. (2014), "Flow-induced vibration of double bonded visco-CNTs under magnetic fields considering surface effect", Comput. Mater. Sci., 86, 144-154. https://doi.org/10.1016/j.commatsci.2014.01.047.
  3. Arani, A.G., Kolahchi, R. and Esmailpour, M. (2016), "Nonlinear vibration analysis of piezoelectric plates reinforced with carbon nanotubes using DQM", Smart Struct. Syst., 18(4), 787-800. http://dx.doi.org/10.12989/sss.2016.18.4.787.
  4. Aydogdu, M. (2009), "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E, 41(5), 861-864. https://doi.org/10.1016/j.physe.2009.01.007.
  5. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029.
  6. Cao, G., Chen, X. and Kysar, J.W. (2006), "Thermal vibration and apparent thermal contraction of single-walled carbon nanotubes", J. Mech. Phys. Solids, 54, 1206-1236. https://doi.org/10.1016/j.jmps.2005.12.003.
  7. Cao, Q., Xia, M., Kocabas, C., Shim, M., Rogers, J.A. and Rotkin, S.V. (2007), "Gate capacitance coupling of single-walled carbon nanotube thin-film transistors", Appl. Phys. Lett., 90, https://doi.org/10.1063/1.2431465.
  8. Cosserat, E. and Cosserat, F. (1909), Sur la theorie des corps deformables, Hermann et fils, Paris, France.
  9. Dirote, E.V. (2004) Trends in Nanotechnology Research, Nova Science Publishers, New York, USA.
  10. Ebbesen, T.W. (1997), Carbon Nanotubes Preparation and Properties, CRC Press, Florida, USA.
  11. Eringen, A.C. (1976), Nonlocal Polar Field Models, Academic Press, MA, USA.
  12. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface-waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803.
  13. Erol, H. and Gurgoze, M. (2004), "Longitudinal vibrations of a double-rod system coupled by springs and dampers", J. Sound Vib., 276, 431-438. https://doi.org/10.1016/j.jsv.2003.10.036
  14. Ferrari, M., Granik, V.T., Imam, A. and Nadeau, J. (1997), Advances in Doublet Mechanics, Springer, Berlin, Germany.
  15. Foroutan, S., Haghshenas, A., Hashemian, M., Eftekhari, S.A. and Toghraie, D. (2018), "Spatial buckling analysis of current-carrying nanowires in the presence of a longitudinal magnetic field accounting for both surface and nonlocal effects", Physica E, 97, 191-205. https://doi.org/10.1016/j.physe.2017.11.015.
  16. Granik, V.T. (1978), "Microstructural mechanics of granular media", Technique Report IM/MGU, Institute of Mechanics, Moscow State University; Russia. 78-241. https://doi.org/10.1016/0167-6636(93)90005-C.
  17. Granik, V.T. and Ferrari, M. (1993), "Microstructural mechanics of granular media", Mech. Mater., 15, 301-322. https://doi.org/10.1016/0167-6636(93)90005-C.
  18. Gul, U., Aydogdu, M. and Gaygusuzoglu, G. (2017), "Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics", Compos. Struct., 160, 1268-1278. https://doi.org/10.1016/j.compstruct.2016.11.023.
  19. Gul, U. and Aydogdu, M. (2018a), "Vibration and buckling analysis of nanotubes (nanofibers) embedded in an elastic medium using Doublet Mechanics", J. Eng. Math., 109(1), 85-111. https://doi.org/10.1007/s10665-017-9908-8.
  20. Gul, U. and Aydogdu, M. (2018b), "Structural modelling of nanorods and nanobeams using doublet mechanics theory", Int. J. Mech. Mater. Des., 14(2), 195-212. https://doi.org/10.1007/s10999-017-9371-8.
  21. Gul, U. and Aydogdu, M. (2017), "Wave propagation in double walled carbon nanotubes by using doublet mechanics theory", Physica E, 93, 345-357. https://doi.org/10.1016/j.physe.2017.07.003.
  22. Hove, V.L. (1953), "The occurrence of singularities in the elastic frequency distribution of a crystal", Phys. Rev., 89, 1189-1193. https://doi.org/10.1103/PhysRev.89.1189.
  23. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354, 56-58. https://doi.org/10.1038/354056a0.
  24. Jun, L., Hongxing, H. and Xiaobin, L. (2010), "Dynamic stiffness matrix of an axially loaded slender double-beam element", Struct. Eng. Mech., 35(6), 717-733. https://doi.org/10.12989/sem.2010.35.6.717.
  25. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems", Europ. J. Mech - A/Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005.
  26. Karlicic, D., Kozic, P. and Pavlovic, R. (2016), "Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium", Appl. Math. Model., 40(2), 1599-1614. https://doi.org/10.1016/j.apm.2015.06.036.
  27. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2009), "Static and dynamic analysis of micro beams based on strain gradient theory", Int. J. Eng. Sci., 47(4), 487-498. https://doi.org/10.1016/j.ijengsci.2008.08.008.
  28. Mao, Q. and Wattanasakulpong, N. (2015), "Vibration and stability of a double-beam system interconnected by an elastic foundation under conservative and nonconservative axial forces", Int. J. Mech. Sci., 93, 1-7. https://doi.org/10.1016/j.ijmecsci.2014.12.019.
  29. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., 16, 51-78. https://doi.org/10.1007/BF00248490
  30. Mirkalantari, S.A., Hashemian, M., Eftekhari, S.A. and Toghraie, D. (2017), "Pull-in instability analysis of rectangular nanoplate based on strain gradient theory considering surface stress effects", Physica B, 519, 1-14. https://doi.org/10.1016/j.physb.2017.05.028.
  31. Murmu, T. and Adhikari, S. (2010), "Nonlocal effects in the longitudinal vibration of double-nanorod systems", Physica E, 43, 415-422. https://doi.org/10.1016/j.physe.2010.08.023.
  32. Murmu, T. and Adhikari, S. (2012), "Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems", Europ. J. Mech. A/Solids, 34, 52-62. https://doi.org/10.1016/j.euromechsol.2011.11.010.
  33. Narendar, S. and Gopalakrishnan, S. (2011), "Axial wave propagation in coupled nanorod system with nonlocal small scale effects", Compos. Part B, 42, 2013-2023. https://doi.org/10.1016/j.compositesb.2011.05.021.
  34. Oniszczuk, Z. (2000), "Free transverse vibrations of elastically connected simply supported double-beams complex system", J. Sound Vib., 232, 419-430. https://doi.org/10.1006/jsvi.1999.2744.
  35. Oveissi, S., Eftekhari, S.A. and Toghraie, D. (2016a), "Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure", Physica E, 83, 164-173. https://doi.org/10.1016/j.physe.2016.05.010.
  36. Oveissi, S., Toghraie, D. and Eftekhari, S.A. (2016b), "Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid", Physica E, 83, 275-283. https://doi.org/10.1016/j.physe.2016.05.004.
  37. Oveissi, S., Toghraie, D. and Eftekhari, S.A. (2017), "Analysis of transverse vibrational response and instabilities of axially moving CNT conveying fluid", Int. J. Fluid Mech. Res., 44(2), 115-129. https://doi.org/10.1615/InterJFluidMechRes.2017016740.
  38. Oveissi, S., Toghraie, D.S. and Eftekhari, S.A. (2018), "Investigation on the effect of axially moving carbon nanotube, nanoflow, and Knudsen number on the vibrational behavior of the system", Int. J. Fluid Mech. Res., 45(2), 171-186. https://doi.org/10.1615/InterJFluidMechRes.2018021036.
  39. Oveissi, S., Nahvi, H. and Toghraie, D. (2015), "Longitudinal wave propagation analysis of stationary and axially moving carbon nanotubes conveying fluid", J. Solid Mech. Eng., 8(2), 107-115.
  40. Pajand, M.R., Sani, A.A. and Hozhabrossadati, S.M. (2018), "Vibration suppression of a double-beam system by a two-degree-of-freedom mass-spring system", Smart Struct. Syst., 21, 349-358. https://doi.org/10.12989/sss.2018.21.3.349.
  41. Pavlovic, I.R., Karlicic, D., Pavlovic, R., Janevski, G. and Ciric, I. (2016), "Stochastic stability of multi-nanobeam systems", Int. J. Eng. Sci., 109, 88-105. https://doi.org/10.1016/j.ijengsci.2016.09.006.
  42. Pradhan, S.C. and Phadikar, J.K. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., 33(2), 193-213. https://doi.org/10.12989/sem.2009.33.2.193.
  43. Rosa, M.A.D. and Lippiello, M. (2007), "Non-classical boundary conditions and DQM for double-beams", Mech. Res. Commun., 34, 538-544. https://doi.org/10.1016/j.mechrescom.2007.08.003.
  44. Saffari, S., Hashemian, M. and Toghraie, D. (2017), "Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects", Physica B, 520, 97-105. https://doi.org/10.1016/j.physb.2017.06.029.
  45. Vajari, A.F. and Imam, A. (2016), "Axial vibration of single-walled carbon nanotubes using doublet mechanics", Indian J. Phys., 90(4), 447-455. https://doi.org/10.1007/s12648-015-0775-8.
  46. Vajari, A.F. and Imam, A. (2016), "Torsional vibration of single-walled carbon nanotubes using doublet mechanics", Z. Angew. Math. Phy., 67, 81. https://doi.org/10.1007/s00033-016-0675-6.
  47. Wang, D.H. and Wang, G.F. (2011), "Surface effects on the vibration and buckling of double-nanobeam-systems", J. Nanomater., 2011, https://doi.org/10.1155/2011/518706.
  48. Wu, J. and Layman, C. (2004), "Wave equations, dispersion relations, and van Hove singularities for applications of doublet mechanics to ultrasound propagation in bio-and nanomaterials", J. Acoust. Soc. Am., 115(2), 893-900. https://doi.org/10.1121/1.1642620.