Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear vibration analysis of a nonlocal sinusoidal shear deformation carbon nanotube using differential quadrature method

  • Pour, Hasan Rahimi (Young Researchers and Elite Club, Khomeyni Shahr Branch, Islamic Azad University) ;
  • Vossough, Hossein (Department of Civil Engineering, Jasb Branch, Islamic Azad University) ;
  • Heydari, Mohammad Mehdi (Department of Civil Engineering, Jasb Branch, Islamic Azad University) ;
  • Beygipoor, Gholamhossein (Young Researchers and Elite Club, Bandarabbas Branch, Islamic Azad University) ;
  • Azimzadeh, Alireza (Department of Mechanical Engineering, Kashan Branch, Islamic Azad University)
  • Received : 2014.10.14
  • Accepted : 2015.02.21
  • Published : 2015.06.25
⟨ Previous Next ⟩

Abstract

This paper presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single walled carbon nanotubes (CNTs). The present model is capable of capturing both small scale effect and transverse shear deformation effects of CNTs, and does not require shear correction factors. The surrounding elastic medium is simulated based on Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the CNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the natural frequency is presented for different boundary conditions, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, boundary condition, aspect ratio on the frequency of CNTs are considered. The comparison firmly establishes that the present beam theory can accurately predict the vibration responses of CNTs.

Keywords

References

  1. Arash, B. and Ansari, R. (2010), "Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain", Physica E: Low Dimen. Syst. Nanostruct., 42, 2058-2064. https://doi.org/10.1016/j.physe.2010.03.028
  2. Arash, B. and Wang, Q. (2012), "A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes", Comput. Mater. Sci., 51, 303-313. https://doi.org/10.1016/j.commatsci.2011.07.040
  3. Thai, H.T. and Vo, T.P. (2012), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 54, 58-66. https://doi.org/10.1016/j.ijengsci.2012.01.009
  4. Aydogdu, M. (2009). "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E: Low Dimen. Syst. Nanostruct., 41, 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  5. Civalek, O. and Demir, C. (2011), "Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory", Appl. Math. Model., 35, 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004
  6. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  7. 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
  8. Li, X., Bhushan, B., Takashima, K., Baek, C.W. and Kim, Y.K. (2003), "Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques", Ultramicroscopy, 97, 481-494. https://doi.org/10.1016/S0304-3991(03)00077-9
  9. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56, 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  10. Murmu, T. and Pradhan, S.C. (2009), "Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Physica E: Low Dimen. Syst. Nanostruct., 41, 1232-1239. https://doi.org/10.1016/j.physe.2009.02.004
  11. Nix, W.D. and Gao, H. (1998), "Indentation size effects in crystalline materials: A law for strain gradient plasticity", J. Mech. Phys. Solid., 46, 411-425. https://doi.org/10.1016/S0022-5096(97)00086-0
  12. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  13. Reddy, J.N. (2002), Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons Inc.
  14. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45, 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  15. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103, 023511.
  16. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  17. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29, 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y
  18. Wang, Q. (2005), Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98, 124301.
  19. Wang, C.M., Kitipornchai, S., Lim, C.W. and Eisenberger, M. (2008), "Beam bending solutions based on nonlocal Timoshenko beam theory", J. Eng. Mech., 134, 475-481. https://doi.org/10.1061/(ASCE)0733-9399(2008)134:6(475)
  20. Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures", Phys. Lett. A, 363, 236-242. https://doi.org/10.1016/j.physleta.2006.10.093
  21. Wang, Q. and Varadan, V.K. (2006), "Vibration of carbon nanotubes studied using nonlocal continuum mechanics", Smart Mater. Struct., 15, 659-666. https://doi.org/10.1088/0964-1726/15/2/050
  22. Wang, Q., Varadan, V.K. and Quek, S.T. (2006), "Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models", Phys. Lett. A, 357, 130-135. https://doi.org/10.1016/j.physleta.2006.04.026
  23. Wang, Q. and Wang, C. (2007), "The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes", Nanotechnology, 18, 075702. https://doi.org/10.1088/0957-4484/18/7/075702
  24. Wang, C.M., Zhang, Y.Y. and He, X.Q. (2007), "Vibration of nonlocal Timoshenko beams", Nanotechnology, 18, 105401. https://doi.org/10.1088/0957-4484/18/10/105401
  25. Wang, C.M., Zhang, Y.Y., Ramesh, S.S. and Kitipornchai, S. (2006), "Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 39, 3904-3909. https://doi.org/10.1088/0022-3727/39/17/029
  26. Yakobson, B.I., Brabec, C.J. and Bernholc, J. (1996), "Nanomechanics of carbon tubes: Instabilities beyond linear response", Phys. Rev. Lett., 76, 2511-2514. https://doi.org/10.1103/PhysRevLett.76.2511
  27. Zenkour, A.M. (2004), "Analytical solution for bending of cross-ply laminated plates under thermomechanical loading", Compos. Struct., 65, 367-379. https://doi.org/10.1016/j.compstruct.2003.11.012
  28. Zenkour, A.M. (2005a), "A comprehensive analysis of functionally graded sandwich plates: Part 1-Deflection and stresses", Int. J. Solid. Struct., 42, 5224-5242. https://doi.org/10.1016/j.ijsolstr.2005.02.015
  29. Zenkour, A.M. (2005b), "A comprehensive analysis of functionally graded sandwich plates: Part 2-Buckling and free vibration", Int. J. Solid. Struct., 42, 5243-5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016
  30. Zenkour, A.M. (2006), "Generalized shear deformation theory for bending analysis of functionally graded plates", Appl. Math. Model., 30, 67-84. https://doi.org/10.1016/j.apm.2005.03.009
  31. Zhang, Y.Q., Liu, G.R. and Xie, X.Y. (2005), "Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity", Phys. Rev. B, 71, 195404 https://doi.org/10.1103/PhysRevB.71.195404

Cited by

  1. Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept vol.20, pp.5, 2016, https://doi.org/10.12989/scs.2016.20.5.963
  2. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory vol.58, pp.3, 2016, https://doi.org/10.12989/sem.2016.58.3.397
  3. A review of continuum mechanics models for size-dependent analysis of beams and plates vol.177, 2017, https://doi.org/10.1016/j.compstruct.2017.06.040
  4. Surface effects on vibration and buckling behavior of embedded nanoarches vol.64, pp.1, 2017, https://doi.org/10.12989/sem.2017.64.1.001
  5. Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment vol.64, pp.1, 2015, https://doi.org/10.12989/sem.2017.64.1.121
  6. Wave dispersion characteristics of nonlocal strain gradient double-layered graphene sheets in hygro-thermal environments vol.65, pp.6, 2018, https://doi.org/10.12989/sem.2018.65.6.645
  7. A unified formulation for modeling of inhomogeneous nonlocal beams vol.66, pp.3, 2015, https://doi.org/10.12989/sem.2018.66.3.369
  8. Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory vol.67, pp.4, 2015, https://doi.org/10.12989/sem.2018.67.4.417
  9. Analysis of boundary conditions effects on vibration of nanobeam in a polymeric matrix vol.67, pp.5, 2015, https://doi.org/10.12989/sem.2018.67.5.517
  10. Finite strain nonlinear longitudinal vibration of nanorods vol.6, pp.4, 2018, https://doi.org/10.12989/anr.2018.6.4.323
  11. Nonlinear behavior of fiber reinforced cracked composite beams vol.30, pp.4, 2019, https://doi.org/10.12989/scs.2019.30.4.327
  12. Static and Dynamic Behavior of Nanotubes-Reinforced Sandwich Plates Using (FSDT) vol.57, pp.None, 2015, https://doi.org/10.4028/www.scientific.net/jnanor.57.117
  13. Flexoelectric effects on dynamic response characteristics of nonlocal piezoelectric material beam vol.8, pp.4, 2015, https://doi.org/10.12989/amr.2019.8.4.259
  14. Mechanical buckling of FG-CNTs reinforced composite plate with parabolic distribution using Hamilton's energy principle vol.8, pp.2, 2015, https://doi.org/10.12989/anr.2020.8.2.135
  15. A comprehensive review on the modeling of smart piezoelectric nanostructures vol.74, pp.5, 2015, https://doi.org/10.12989/sem.2020.74.5.611
  16. On scale-dependent stability analysis of functionally graded magneto-electro-thermo-elastic cylindrical nanoshells vol.75, pp.6, 2015, https://doi.org/10.12989/sem.2020.75.6.659
  17. Large amplitude free torsional vibration analysis of size-dependent circular nanobars using elliptic functions vol.77, pp.4, 2021, https://doi.org/10.12989/sem.2021.77.4.535