Structural Engineering and Mechanics

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Frequency response of initially deflected nanotubes conveying fluid via a nonlinear NSGT model

  • Farajpour, Ali (School of Mechanical Engineering, University of Adelaide) ;
  • Ghayesh, Mergen H. (School of Mechanical Engineering, University of Adelaide) ;
  • Farokhi, Hamed (Department of Mechanical and Construction Engineering, Northumbria University)
  • Received : 2019.04.11
  • Accepted : 2019.05.13
  • Published : 2019.10.10
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Abstract

The objective of this paper is to develop a size-dependent nonlinear model of beams for fluid-conveying nanotubes with an initial deflection. The nonlinear frequency response of the nanotube is analysed via an Euler-Bernoulli model. Size influences on the behaviour of the nanosystem are described utilising the nonlocal strain gradient theory (NSGT). Relative motions at the inner wall of the nanotube is taken into consideration via Beskok-Karniadakis model. Formulating kinetic and elastic energies and then employing Hamilton's approach, the nonlinear motion equations are derived. Furthermore, Galerkin's approach is employed for discretisation, and then a continuation scheme is developed for obtaining numerical results. It is observed that an initial deflection significantly alters the frequency response of NSGT nanotubes conveying fluid. For small initial deflections, a hardening nonlinearity is found whereas a softening-hardening nonlinearity is observed for large initial deflections.

Keywords

References

  1. Ahouel, M., Houari, M.S.A., Bedia, E. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981. https://doi.org/10.12989/scs.2016.20.5.963.
  2. Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Comput. Theoretical Nanosci., 8(9), 1821-1827. https://doi.org/10.1166/jctn.2011.1888
  3. Akgoz, B. and Civalek, O. (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004.
  4. Akgoz, B. and Civalek, O. (2014), "Longitudinal vibration analysis for microbars based on strain gradient elasticity theory", J. Vib. Control, 20(4), 606-616. https://doi.org/10.1177%2F1077546312463752. https://doi.org/10.1177/1077546312463752
  5. Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.
  6. Al-Basyouni, K., Tounsi, A. and Mahmoud, S. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070.
  7. Amiri, A., Talebitooti, R. and Li, L. (2018), "Wave propagation in viscous-fluid-conveying piezoelectric nanotubes considering surface stress effects and Knudsen number based on nonlocal strain gradient theory", Europe Phys. J. Plus, 133(7), 252. https://doi.org/10.1140/epjp/i2018-12077-y.
  8. Arefi, M. (2019), "Static analysis of laminated piezo-magnetic size-dependent curved beam based on modified couple stress theory", Struct. Eng. Mech., 69(2), 145-153. https://doi.org/10.12989/sem.2019.69.2.145.
  9. Asemi, S.R. and Farajpour, A. (2014), "Vibration characteristics of double-piezoelectric-nanoplate-systems", IET Micro Nano Lett., 9(4), 280-285. https://doi.org/10.1049/mnl.2013.0741.
  10. Askari, H. and Esmailzadeh, E. (2017), "Forced vibration of fluid conveying carbon nanotubes considering thermal effect and nonlinear foundations", Compos. Part B Eng., 113, 31-43. https://doi.org/10.1016/j.compositesb.2016.12.046
  11. Bahaadini, R., Saidi, A.R. and Hosseini, M. (2018), "On dynamics of nanotubes conveying nanoflow", Int. J. Eng. Sci., 123, 181-196. https://doi.org/10.1016/j.ijengsci.2017.11.010.
  12. Bakhadda, B., Bouiadjra, M.B., Bourada, F., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2018), "Dynamic and bending analysis of carbon nanotube-reinforced composite plates with elastic foundation", Wind Struct., 27(5), 311-324. https://doi.org/10.12989/was.2018.27.5.311.
  13. Bedia, W.A., Houari, M.S.A., Bessaim, A., Bousahla, A.A., Tounsi, A., Saeed, T. and Alhodaly, M.S. (2019), "A New Hyperbolic Two-Unknown Beam Model for Bending and Buckling Analysis of a Nonlocal Strain Gradient Nanobeams", J. Nano Res., 57, 175-191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175.
  14. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E. and Mahmoud, S. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct, 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063.
  15. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2017), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., 62(6), 695-702. https://doi.org/10.12989/sem.2017.62.6.695.
  16. Benahmed, A., Fahsi, B., Benzair, A., Zidour, M., Bourada, F. and Tounsi, A. (2019), "Critical buckling of functionally graded nanoscale beam with porosities using nonlocal higher-order shear deformation", Struct. Eng. Mech., 69(4), 457-466. https://doi.org/10.12989/sem.2019.69.4.457.
  17. Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., 19(6), 601-614. https://doi.org/10.12989/sss.2017.19.6.601.
  18. Bouadi, A., Bousahla, A.A., Houari, M.S.A., Heireche, H. and Tounsi, A. (2018), "A new nonlocal HSDT for analysis of stability of single layer graphene sheet", Advances in nano research, 6(2), 147-162. https://doi.org/10.12989/anr.2018.6.2.147.
  19. Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115.
  20. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227.
  21. Bourada, F., Amara, K., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2018), "A novel refined plate theory for stability analysis of hybrid and symmetric S-FGM plates", Struct. Eng. Mech., 68(6), 661-675. https://doi.org/10.12989/sem.2018.68.6.661.
  22. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425.
  23. Civalek, O. and Akgoz, B. (2013), "Vibration analysis of microscaled sector shaped graphene surrounded by an elastic matrix", Comput. Mater. Sci., 77, 295-303. https://doi.org/10.1016/j.commatsci.2013.04.055.
  24. Demir, C. and Civalek, O. (2017), "A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix", Compos. Struct., 168, 872-884. https://doi.org/10.1016/j.compstruct.2017.02.091.
  25. Demir, C. and Civalek, O. (2017), "On the analysis of microbeams", Int. J. Eng. Sci., 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  26. Demir, C., Mercan, K. and Civalek, O. (2016), "Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel", Compos. Part B Eng., 94, 1-10. https://doi.org/10.1016/j.compositesb.2016.03.031.
  27. Draoui, A., Zidour, M., Tounsi, A. and Adim, B. (2019), "Static and Dynamic Behavior of Nanotubes-Reinforced Sandwich Plates Using (FSDT)", J. Nano Res., 57, 117-135. https://doi.org/10.4028/www.scientific.net/JNanoR.57.117.
  28. Ebrahimi, F. and Barati, M.R. (2018), "A nonlocal strain gradient refined plate model for thermal vibration analysis of embedded graphene sheets via DQM", Struct. Eng. Mech., 66(6), 693-701. https://doi.org/10.12989/sem.2018.66.6.693.
  29. Ebrahimi, F. and Dabbagh, A. (2018), "NSGT-based acoustical wave dispersion characteristics of thermo-magnetically actuated double-nanobeam systems", Struct. Eng. Mech., 68(6), 701-711. https://doi.org/10.12989/sem.2018.68.6.701.
  30. Ebrahimi, F., Haghi, P. and Dabbagh, A. (2018), "Analytical wave dispersion modeling in advanced piezoelectric double-layered nanobeam systems", Struct. Eng. Mech., 67(2), 175-183. https://doi.org/10.12989/sem.2018.67.2.175.
  31. Eringen, A.C. and Edelen, D. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0.
  32. Farajpour, A., Ghayesh, M.H. and Farokhi, H. (2018), "A review on the mechanics of nanostructures", Int. J. Eng. Sci., 133, 231-263. https://doi.org/10.1016/j.ijengsci.2018.09.006.
  33. Farajpour, A., Ghayesh, M.H. and Farokhi, H. (2019), "Largeamplitude coupled scale-dependent behaviour of geometrically imperfect NSGT nanotubes", J. Mech. Sci., 150, 510-525. https://doi.org/10.1016/j.ijmecsci.2018.09.043.
  34. Farajpour, A., Rastgoo, A. and Farajpour, M. (2017), "Nonlinear buckling analysis of magneto-electro-elastic CNT-MT hybrid nanoshells based on the nonlocal continuum mechanics", Compos. Struct., 180, 179-191. https://doi.org/10.1016/j.compstruct.2017.07.100.
  35. Farajpour, M., Shahidi, A. and Farajpour, A. (2018), "A nonlocal continuum model for the biaxial buckling analysis of composite nanoplates with shape memory alloy nanowires", Mater. Res. Exp., 5(3), 035026. https://doi.org/10.1088/2053-1591/aab3a9.
  36. Farajpour, M., Shahidi, A., Tabataba'i-Nasab, F. and Farajpour, A. (2018), "Vibration of initially stressed carbon nanotubes under magneto-thermal environment for nanoparticle delivery via higher-order nonlocal strain gradient theory", Europe Phys. J. Plus, 133(6), 219. https://doi.org/10.1140/epjp/i2018-12039-5.
  37. Farajpour, M.R., Shahidi, A.R., Tabataba'i-Nasab, F. and Farajpour, A. (2018), "Vibration of initially stressed carbon nanotubes under magneto-thermal environment for nanoparticle delivery via higher-order nonlocal strain gradient theory", European Phys. J. Plus, 133(6), 219. https://doi.org/10.1140/epjp/i2018-12039-5.
  38. Farajpour, M.R., Shahidi, A.R., Hadi, A. and Farajpour, A. (2019), "Influence of initial edge displacement on the nonlinear vibration, electrical and magnetic instabilities of magnetoelectro-elastic nanofilms", Mech. Adv. Mater. Struct., 26(17), 1469-1481. https://doi.org/10.1080/15376494.2018.1432820.
  39. Farajpour, M.R., Shahidi, A.R. and Farajpour, A. (2019), "Influence of shear preload on wave propagation in small-scale plates with nanofibers", Struct. Eng. Mech., 70(4), 407-420. https://doi.org/10.12989/sem.2019.70.4.407.
  40. Farajpour, A., Farokhi, H., Ghayesh, M.H. and Hussain, S. (2018), "Nonlinear mechanics of nanotubes conveying fluid", J. Eng. Sci., 133, 132-143. https://doi.org/10.1016/j.ijengsci.2018.08.009.
  41. Farajpour, A., Ghayesh, M.H. and Farokhi, H. (2019), "Largeamplitude coupled scale-dependent behaviour of geometrically imperfect NSGT nanotubes", J. Mech. Sci., 150, 510-525. https://doi.org/10.1016/j.ijmecsci.2018.09.043.
  42. Farokhi, H. and Ghayesh, M.H. (2015), "Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory", J. Mech. Sci., 90, 133-144. https://doi.org/10.1016/j.ijmecsci.2014.11.002.
  43. Farokhi, H. and Ghayesh, M.H. (2018), "Nonlinear mechanical behaviour of microshells", Int. J. Eng. Sci., 127, 127-144. https://doi.org/10.1016/j.ijengsci.2018.02.009.
  44. Farokhi, H. and Ghayesh, M.H. (2018), "Nonlinear mechanics of electrically actuated microplates", Int. J. Eng. Sci., 123, 197-213. https://doi.org/10.1016/j.ijengsci.2017.08.017.
  45. Farokhi, H. and Ghayesh, M.H. (2018), "Supercritical nonlinear parametric dynamics of Timoshenko microbeams", Communication Nonlinear Sci. Numeric. Simul., 59, 592-605. https://doi.org/10.1016/j.cnsns.2017.11.033.
  46. Farokhi, H., Ghayesh, M.H. and Hussain, S. (2016), "Largeamplitude dynamical behaviour of microcantilevers", Int. J. Eng. Sci., 106, 29-41. https://doi.org/10.1016/j.ijengsci.2016.03.002.
  47. Farokhi, H. and Ghayesh, M.H. (2017), "Nonlinear resonant response of imperfect extensible Timoshenko microbeams", J. Mech. Mater. Design, 13(1), 43-55. https://doi.org/10.1007/s10999-015-9316-z.
  48. Farokhi, H., Ghayesh, M.H. and Gholipour, A. (2017), "Dynamics of functionally graded micro-cantilevers", J. Eng. Sci., 115, 117-130. https://doi.org/10.1016/j.ijengsci.2017.01.004.
  49. Filiz, S. and Aydogdu, M. (2015), "Wave propagation analysis of embedded (coupled) functionally graded nanotubes conveying fluid", Compos. Struct., 132, 1260-1273. https://doi.org/10.1016/j.compstruct.2015.07.043.
  50. Gao, Y., Xiao, W.S. and Zhu, H. (2019), "Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method", Struct. Eng. Mech., 69(2), 205-219. https://doi.org/10.12989/sem.2019.69.2.205.
  51. Ghayesh, M.H. (2018), "Dynamics of functionally graded viscoelastic microbeams", Int. J. Eng. Sci., 124, 115-131. https://doi.org/10.1016/j.ijengsci.2017.11.004.
  52. Ghayesh, M.H. (2018), "Functionally graded microbeams: Simultaneous presence of imperfection and viscoelasticity", J. Mech. Sci., 140, 339-350. https://doi.org/10.1016/j.ijmecsci.2018.02.037.
  53. Ghayesh, M.H. (2018), "Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams", Applied Mathematical Modelling, 59, 583-596. https://doi.org/10.1016/j.apm.2018.02.017.
  54. Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013), "Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory", Int. J. Eng. Sci., 63, 52-60. https://doi.org/10.1016/j.ijengsci.2012.12.001.
  55. Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013), "Threedimensional nonlinear size-dependent behaviour of Timoshenko microbeams", Int. J. Eng. Sci., 71, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.003.
  56. Ghayesh, M.H. and Farajpour, A. (2018), "Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory", Int. J. Eng. Sci., 129, 84-95. https://doi.org/10.1016/j.ijengsci.2018.04.003.
  57. Ghayesh, M.H. and Farokhi, H. (2015), "Chaotic motion of a parametrically excited microbeam", Int. J. Eng. Sci., 96, 34-45. https://doi.org/10.1016/j.ijengsci.2015.07.004.
  58. Ghayesh, M.H., Farokhi, H. and Alici, G. (2016), "Size-dependent performance of microgyroscopes", Int. J. Eng. Sci., 100, 99-111. https://doi.org/10.1016/j.ijengsci.2015.11.003.
  59. Gholipour, A., Farokhi, H. and Ghayesh, M.H. (2015), "In-plane and out-of-plane nonlinear size-dependent dynamics of microplates", Nonlinear Dynam., 79(3), 1771-1785. https://doi.org/10.1007/s11071-014-1773-7.
  60. Ghayesh, M.H., Farokhi, H. and Alici, G. (2015), "Subcritical parametric dynamics of microbeams", J. Eng. Sci., 95, 36-48. https://doi.org/10.1016/j.ijengsci.2015.06.001.
  61. Ghayesh, M.H. (2012), "Subharmonic dynamics of an axially accelerating beam", Archive Appl. Mech., 82(9), 1169-1181. https://doi.org/10.1007/s00419-012-0609-5.
  62. Ghayesh, M.H. and Farajpour, A. (2019), "A review on the mechanics of functionally graded nanoscale and microscale structures", J. Eng. Sci., 137, 8-36. https://doi.org/10.1016/j.ijengsci.2018.12.001.
  63. Ghayesh, M.H., Farokhi, H. and Farajpour, A. (2019), "Global dynamics of fluid conveying nanotubes", J. Eng. Sci., 135, 37-57. https://doi.org/10.1016/j.ijengsci.2018.11.003.
  64. Hamza-Cherif, R., Meradjah, M., Zidour, M., Tounsi, A., Belmahi, S. and Bensattalah, T. (2018), "Vibration analysis of nano beam using differential transform method including thermal effect", J. Nano Res., 54, 1-14. https://doi.org/10.4028/www.scientific.net/JNanoR.54.1.
  65. Kadari, B., Bessaim, A., Tounsi, A., Heireche, H., Bousahla, A.A. and Houari, M.S.A. (2018), "Buckling analysis of orthotropic nanoscale plates resting on elastic foundations", J. Nano Res., 55, 42-56. https://doi.org/10.4028/www.scientific.net/JNanoR.55.42.
  66. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099.
  67. Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. https://doi.org/10.12989/scs.2017.25.3.361.
  68. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-018-0664-9.
  69. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201.
  70. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Walled Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025.
  71. Karami, B., Janghorban, M. and Tounsi, A. (2019), "On exact wave propagation analysis of triclinic material using threedimensional bi-Helmholtz gradient plate model", Struct. Eng. Mech., 69(5), 487-497. https://doi.org/10.12989/sem.2019.69.5.487.
  72. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036
  73. Kazemirad, S., Ghayesh, M.H. and Amabili, M. (2013), "Thermomechanical nonlinear dynamics of a buckled axially moving beam", Archive Appl. Mech., 83(1), 25-42. https://doi.org/10.1007/s00419-012-0630-8.
  74. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Struct. Eng. Mech., 64(4), 391-402. https://doi.org/10.12989/sem.2017.64.4.391.
  75. Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos. Struct., 165, 250-265. https://doi.org/10.1016/j.compstruct.2017.01.032.
  76. Liang, F. and Su, Y. (2013), "Stability analysis of a single-walled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect", Appl. Math. Modell., 37(10-11), 6821-6828. https://doi.org/10.1016/j.apm.2013.01.053.
  77. Lim, C., Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solid., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  78. Malekzadeh, P., Haghighi, M.G. and Shojaee, M. (2014), "Nonlinear free vibration of skew nanoplates with surface and small scale effects", Thin Wall Struct., 78, 48-56. https://doi.org/10.1016/j.tws.2013.10.027.
  79. Malekzadeh, P. and Shojaee, M. (2015), "A two-variable firstorder shear deformation theory coupled with surface and nonlocal effects for free vibration of nanoplates", J. Vib. Control, 21(14), 2755-2772. https://doi.org/10.1177/1077546313516667.
  80. Malekzadeh, P. and Farajpour, A. (2012), "Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium", Acta Mechanica, 223(11), 2311-2330. https://doi.org/10.1007/s00707-012-0706-0.
  81. Malikan, M. (2017), "Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory", Appl. Math. Modell., 48, 196-207. https://doi.org/10.1016/j.apm.2017.03.065.
  82. Maraghi, Z.K., Arani, A.G., Kolahchi, R., Amir, S. and Bagheri, M. (2013), "Nonlocal vibration and instability of embedded DWBNNT conveying viscose fluid", Compos. Part B Eng., 45(1), 423-432. https://doi.org/10.1016/j.compositesb.2012.04.066.
  83. Mercan, K. and Civalek, O. (2016), "DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix", Compos. Struct., 143, 300-309. https://doi.org/10.1016/j.compstruct.2016.02.040.
  84. Mercan, K. and Civalek, O. (2017), "Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ", Compos. Part B Eng., 114, 34-45. https://doi.org/10.1016/j.compositesb.2017.01.067.
  85. Mohammadi, M. and Rastgoo, A. (2019), "Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core", Struct. Eng. Mech., 69(2), 131-143. https://doi.org/10.12989/sem.2019.69.2.131.
  86. Mohammadi, M., Farajpour, A., Goodarzi, M. and Heydarshenas, R. (2013), "Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium", J. Solid Mech., 5(2), 116-132.
  87. Mohammadi, M., Farajpour, A., Goodarzi, M., Dinari, F. (2014), "Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium", Latin American J. Solids Struct., 11(4), 659-682. http://dx.doi.org/10.1590/S1679-78252014000400007.
  88. Mokhtar, Y., Heireche, H., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2018), "A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory", Smart Struct. Syst., 21(4), 397-405. https://doi.org/10.12989/sss.2018.21.4.397.
  89. Mouffoki, A., Bedia, E., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new twounknown trigonometric shear deformation beam theory", Smart Struct. Syst., 20(3), 369-383. https://doi.org/10.12989/sss.2017.20.3.369.
  90. Murmu, T. and Adhikari, S. (2013), "Nonlocal mass nanosensors based on vibrating monolayer graphene sheets", Sensor Actuator B Chem., 188, 1319-1327. https://doi.org/10.1016/j.snb.2013.07.051.
  91. Nebab, M., Atmane, H.A., Bennai, R., Tounsi, A. and Adda Bedia, E.A. (2019), "Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT", Struct. Eng. Mech., 69(5), 511-525. https://doi.org/10.12989/sem.2019.69.5.511.
  92. Nejad, M.Z., Hadi, A. and Farajpour, A. (2017), "Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials", Struct. Eng. Mech., 63(2), 161-169. https://doi.org/10.12989/sem.2017.63.2.161.
  93. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018), "Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory", Struct. Eng. Mech., 67(4), 417-425. https://doi.org/10.12989/SEM.2018.67.4.417
  94. Numanoglu, H.M., Akgoz, B. and Civalek, O. (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  95. Paidoussis, M.P. (1998), Fluid-structure interactions: slender structures and axial flow, Academic press, MA, USA.
  96. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0.
  97. Rahmani, O., Refaeinejad, V. and Hosseini, S. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct, 23(3), 339-350. https://doi.org/10.12989/scs.2017.23.3.339.
  98. Reddy, J. (2010), "Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates", Int. J. Eng. Sci., 48(11), 1507-1518. https://doi.org/10.1016/j.ijengsci.2010.09.020.
  99. Romano, G., Barretta, R. and Diaco, M. (2017), "On nonlocal integral models for elastic nano-beams", J. Mech. Sci., 131, 490-499. https://doi.org/10.1016/j.ijmecsci.2017.07.013.
  100. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., 7(2), 89-98. https://doi.org/10.12989/anr.2019.7.2.089.
  101. Shen, H.-S. (2011), "Nonlinear analysis of lipid tubules by nonlocal beam model", J. Theoretical Bio., 276(1), 50-56. https://doi.org/10.1016/j.jtbi.2011.02.001.
  102. Şimsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013.
  103. Şimsek, M. and Reddy, J. (2013), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002.
  104. Soltani, P., Taherian, M. and Farshidianfar, A. (2010), "Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium", J. Phys. D Appl. Phys., 43(42), 425401. https://doi.org/10.1088/0022-3727/43/42/425401.
  105. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A., Bousahla, A.A. and Mahmoud, S. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., 69(6), 637-649. https://doi.org/10.12989/sem.2019.69.6.637.
  106. Tounsi, A., Heireche, H., Berrabah, H., Benzair, A. and Boumia, L. (2008), "Effect of small size on wave propagation in doublewalled carbon nanotubes under temperature field", J. Appl. Phys., 104(10), 104301. https://doi.org/10.1063/1.3018330.
  107. Wang, Y.-Z., Li, F.-M. and Kishimoto, K. (2010), "Wave propagation characteristics in fluid-conveying double-walled nanotubes with scale effects", Comput. Mater. Sci., 48(2), 413-418. https://doi.org/10.1016/j.commatsci.2010.01.034.
  108. Yazid, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Ahmed Houari, M.S. (2018), "A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium", Smart Struct. Syst., 21(1), 15-25. https://doi.org/10.12989/sss.2018.21.1.015.
  109. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams including surface stress effects", Smart Struct. Syst., 21(1), 65-74. https://doi.org/10.12989/sss.2018.21.1.065.
  110. Zenkour, A. and Sobhy, M. (2013), "Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler-Pasternak elastic substrate medium", Physica E Low-dimensional Syst. Nanostruct., 53, 251-259. https://doi.org/10.1016/j.physe.2013.04.022.
  111. Zhang, B., He, Y., Liu, D., Gan, Z. and Shen, L. (2014), "Nonclassical Timoshenko beam element based on the strain gradient elasticity theory", Finite Elements Anal. Design, 79, 22-39. https://doi.org/10.1016/j.finel.2013.10.004.