Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Theoretical impact of Kelvin's theory for vibration of double walled carbon nanotubes

  • Hussain, Muzamal (Department of Mathematics, Government College University Faisalabad) ;
  • Naeem, Muhammad N. (Department of Mathematics, Government College University Faisalabad) ;
  • Asghar, Sehar (Department of Mathematics, Government College University Faisalabad) ;
  • Tounsi, Abdelouahed (Materials and Hydrology Laboratory, University of Sidi Bel Abbes, Algeria Faculty of Technology Civil Engineering Department)
  • Received : 2020.02.07
  • Accepted : 2020.04.30
  • Published : 2020.05.25
⟨ Previous Next ⟩

Abstract

In this article, free vibration of double-walled carbon nanotubes (DWNT) based on nonlocal Kelvin's model have been investigated. For this purpose, a nonlocal Kelvin's model is established to observe the small scale effect. The wave propagation is employed to frame the governing equations as eigenvalue system. The influence of nonlocal parameter subjected to different end supports has been overtly examined. The new set of inner and outer tubes radii investigated in detail against aspect ratio. The influence of boundary conditions via nonlocal parameter is shown graphically. Due to small scale effect fundamental frequency ratio decreases as length to diameter ratio increases. Small scale effect becomes negligible on all end supports for the higher values of aspect ratio. With the smaller inner tube radius double-walled CNT behaves more sensitive towards nonlocal parameter. The results generated furnish the evidence regarding applicability of nonlocal model and also verified by earlier published literature.

Keywords

Acknowledgement

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

  1. Adela, I. (2018), Computational Fluid Dynamics, Romania.
  2. Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Computat. Theor. Nanosci., 8, 1821-1827. https://doi.org/10.1166/jctn.2011.1888
  3. Akgoz, B. and Civalek, O. (2015), "A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory", Acta Mechanica, 226(7), 2277-2294. https://doi.org/10.1007/s00707-015-1308-4
  4. Amara, K., Tounsi, A., Mechab, I. and Adda-Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Mathe. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  5. Amnieh, H.B., Zamzam, M.S. and Kolahchi, R. (2018), "Dynamic analysis of non-homogeneous concrete blocks mixed by $SiO_2$ nanoparticles subjected to blast load experimentally and theoretically", Constr. Build. Mater., 174, 633-644. https://doi.org/10.1016/j.conbuildmat.2018.04.140
  6. Ansari, R. and Arash, B. (2013), "Nonlocal Flugge shell model for vibrations of double-walled carbon nanotubes with different boundary conditions", J. Appl. Mech., 80(2), 021006. https://doi.org/10.1115/1.4007432
  7. Ansari, R. and Rouhi, H. (2012), "Nonlocal analytical Flugge shell model for the axial buckling of double-walled carbon nanotubes with different end conditions", Int. J. Nano, 7, 1250081. https://doi.org/10.1142/S179329201250018X
  8. Ansari, R. and Rouhi, H. (2013), "Nonlocal analytical Flugge shell model forr the vibrations of double-walled carbon nanotubes with different end conditions", Int. J. Appl. Mech., 80, 021006-1. https://doi.org/10.1142/S179329201250018X
  9. Ansari, R., Sahmani, S. and Arash, B. (2010), "Nonlocal plate model for free vibrations of single-layered graphene sheets", Phy. Letters A., 375(1), 53-62. https://doi.org/10.1016/j.physleta.2010.10.028
  10. Ansari, R., Hemmatnezhad, M. and Rezapour, J. (2011), "The thermal effect on nonlinear oscillations of carbon nanotubes with arbitrary boundary conditions", Current Appl. Phys., 11(3), 692-697. https://doi.org/10.1016/j.cap.2010.11.034
  11. Arani, A.J. and Kolahchi, R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput. Concrete, Int. J., 17(5), 567-578. https://doi.org/10.12989/cac.2016.17.5.567
  12. Arefi, M., Mohammadi, M., Tabatabaeian, A., Dimitri, R. and Tornabene, F. (2018), "Two-dimensional thermo-elastic analysis of FG-CNTRC cylindrical pressure vessels", Steel Compos. Struct., Int. J., 27(4), 525-536. https://doi.org/10.12989/scs.2018.27.4.525
  13. Asghar, S., Hussain, M. and Naeem, M. (2019), "Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory", Physica E: Low-dimens. Syst. Nanostruct., 116, 113726. https://doi.org/10.1016/j.physe.2019.113726
  14. Avcar, M. (2015), "Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam", Struct. Eng. Mech., Int. J., 55(4), 871-884. https://doi.org/10.12989/sem.2015.55.4.871
  15. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
  16. Aydogdu, M.A. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E, 41, 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  17. Batou, B., Nebab, M., Bennai, R., Atmane, H.A., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., Int. J., 33(5), 699-716. https://doi.org/10.12989/scs.2019.33.5.699
  18. Behera, S. and Kumari, P. (2018), "Free vibration of Levy-type rectangular laminated plates using efficient zig-zag theory", Adv. Computat. Des., Int. J., 3(3), 213-232. https://doi.org/10.12989/acd.2018.3.3.213
  19. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical and buckling properties of zigzag double-walled carbon nanotubes", Composites Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  20. Bensattalah, T., Bouakkaz, K., Zidour, M. and Daouadji, T.H. (2018), "Critical buckling loads of carbon nanotube embedded in Kerr's medium", Adv. Nano Res., Int. J., 6(4), 339-356. https://doi.org/10.12989/anr.2018.6.4.339
  21. Bilouei, B.S., Kolahchi, R. and Bidgoli, M.R. (2016), "Buckling of concrete columns retrofitted with Nano-Fiber Reinforced Polymer (NFRP)", Comput. Concrete, Int. J., 18(5), 1053-1063. https://doi.org/10.12989/cac.2016.18.5.1053
  22. 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", Adv. Nano Res., Int. J., 6(2), 147-162. https://doi.org/10.12989/anr.2018.6.2.147
  23. Brischotto, S. (2015), "A continuum shell model including van der Waals interaction for free vibrations of double-walled carbon nanotubes", CMES, 104, 305-327.
  24. Chemi, A., Zidour, M., Heireche, H., Rakrak, K. and Bousahla, A.A. (2018), "Critical Buckling Load of Chiral Double-Walled Carbon Nanotubes Embedded in an Elastic Medium", Mech. Compos. Mater., 53(6), 827-836. https://doi.org/10.1007/s11029-018-9708-x
  25. Das, B., Mandal, M., Upadhyay, A., Chattopadhyay, P. and Karak, N. (2013), "Bio-based hyperbranched polyurethane/Fe3O4 nanocomposites: smart antibacterial biomaterials for biomedical devices and implants", Biomed. Mater., 8(3), 035003. https://doi.org/10.1088/1748-6041/8/3/035003
  26. Do, Q.C., Pham, D.N., Vu, D.Q., Vu, T.T.A. and Nguyen, D.D. (2019), "Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load", Steel Compos. Struct., Int. J., 31(3), 243-259. https://doi.org/10.12989/scs.2019.31.3.243
  27. Ebrahimi, F. and Mahmoodi, F. (2018), "Vibration analysis of carbon nanotubes with multiple cracks in thermal environment", Adv. Nano Res., Int. J., 6(1), 57-80. https://doi.org/10.12989/anr.2018.6.1.057
  28. Ehyaei, J. and Daman, M. (2017), "Free vibration analysis of double walled carbon nanotubes embedded in an elastic medium with initial imperfection", Adv. Nano Res., Int. J., 5(2), 179-192. https://doi.org/10.12989/anr.2017.5.2.179
  29. Eltaher, M., Emam, S.A. and Mahmoud, F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030
  30. Eringen, A.C. (1972), "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Eng. Sci., 10(5), 425-435. https://doi.org/10.1016/0020-7225(72)90050-X
  31. 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/10.1063/1.33280
  32. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science & Business Media.
  33. Fakhrabadi, M.M.S., Rastgoo, A. and Ahmadian, M.T. (2015), "Application of electrostatically actuated carbon nanotubes in nanofluidic and bio-nanofluidic sensors and actuators", Measurement, 73, 127-136. https://doi.org/10.1016/j.measurement.2015.05.009
  34. Fatahi-Vajari. A., Azimzadeh, Z. and Hussain. M. (2019), "Nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes using Galerkin and Homotopy perturbation method", Micro Nano Lett., 14(14), 1366-1371. https://doi.org/10.1049/mnl.2019.0203
  35. Flugge, W. (1962), Statik und Dynamik der Scahlen, Springer, Berlin, Germany.
  36. Flugge, S. (1973), Stresses in Shells, Springer, 2nd Edition, Berlin, Germany.
  37. Fu, Y.M., Hong, J.W. and Wang, X.Q. (2006), "Analysis of nonlinear vibration for embedded carbon nanotubes", J. Sound Vib., 296(4-5), 746-756. https://doi.org/10.1016/j.jsv.2006.02.024
  38. Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M. and Bensattalah, T. (2020), "Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle", Adv. Nano Res., Int. J., 8(1), 37-47. https://doi.org/10.12989/anr.2020.8.1.037
  39. Gao, Y. and An, L. (2010), "A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm", Physica E: Low-dimens. Syst. Nanostruct., 42(9), 2406-2415. https://doi.org/10.1016/j.physe.2010.05r.022
  40. Ghodrati, B., Yaghootian, A., Ghanbar Zadeh, A. and Mohammad-Sedighi, H. (2018), "Lamb wave extraction of dispersion curves in micro/nano-plates using couple stress theories", Waves Random Complex Media, 28(1), 15-34. https://doi.org/10.1080/17455030.2017.1308582
  41. Hadji, L., Zouatnia, N. and Bernard, F. (2019), "An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models", Struct. Eng. Mech., Int. J., 69(2), 231-241. https://doi.org/10.12989/sem.2019.69.2.231
  42. Hajmohammad, M.H., Farrokhian, A. and Kolahchi, R. (2018a), "Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory", Aerosp. Sci. Technol., 78, 260-270. https://doi.org/10.1016/j.ast.2018.04.030
  43. Hajmohammad, M.H., Maleki, M. and Kolahchi, R. (2018b), "Seismic response of underwater concrete pipes conveying fluid covered with nano-fiber reinforced polymer layer", Soil Dyn. Earthq. Eng., 110, 18-27. https://doi.org/10.1016/j.soildyn.2018.04.002
  44. Hajmohammad, M.H., Kolahchi, R., Zarei, M.S. and Maleki, M. (2018c), "Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects", Compos. Struct., 187, 498-508. https://doi.org/10.1016/j.compstruct.2017.12.004
  45. Hajmohammad, M.H., Kolahchi, R., Zarei, M.S. and Nouri, A.H. (2019), "Dynamic response of auxetic honeycomb plates integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal theory", Int. J. Mech. Sci., 153, 391-401. https://doi.org/10.1016/j.ijmecsci.2019.02.008
  46. Hajnayeb, A. and Khadem, S.E. (2015), "An analytical study on the nonlinear vibration of a double walled carbon nanotube", Struct. Eng. Mech., Int. J., 54(5), 987-998. https://doi.org/10.12989/sem.2015.54.5.987
  47. Hao, M.J., Guo, X.M. and Wang, Q. (2010), "Small-scale effect on torsional buckling of multi-walled carbon nanotubes", Eur. J. Mech. A/Solids, 29(1), 49-55. https://doi.org/10.1016/j.euromechsol.2009.05.008
  48. Hernandez, E., Goze, C., Bemier, P. and Rubio, A. (1998), "Elastic properties of C and BxCyNz composite nanotubes", Phys. Rev. Lett., 80, 4502-4505. https://doi.org/10.1103/PhysRevLett.80.4502
  49. Heydarpour, Y., Aghdam, M.M. and Malekzadeh, P. (2014), "Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells", Compos. Struct., 117, 187-200. https://doi.org/10.1016/j.compstruct.2014.06.023
  50. Hosseini, H. and Kolahchi, R. (2018), "Seismic response of functionally graded-carbon nanotubes-reinforced submerged viscoelastic cylindrical shell in hygrothermal environment", Physica E: Low-dimens. Syst. Nanostruct., 102, 101-109. https://doi.org/10.1016/j.physe.2018.04.037
  51. Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q. and Yakobson, B.I. (2008), "Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes", J. Mech. Phy. Solids, 56, 3475-3485. https://doi.org/10.1016/j.jmps.2008.08.010
  52. Hu, Y.G., Liew, K.M. and Wang, Q. (2012), "Modeling of vibrations of carbon nanotubes", Procedia Eng., 31, 343-347. https://doi.org/10.1016/j.proeng.2012.01.1034
  53. Hussain, M. and Naeem, M.N. (2017), "Vibration analysis of single-walled carbon nanotubes using wave propagation approach", Mech. Sci., 8(1), 155-164. https://doi.org/10.5194/ms-8-155-2017
  54. Hussain, M. and Naeem, M. (2018a), "Vibration of single-walled carbon nanotubes based on Donnell shell theory using wave propagation approach", Chapter, Intechopen, In: Novel Nanomaterials - Synthesis and Applications. ISBN 978-953-51-5896-7 https://doi.org/10.5772 /intechopen.73503
  55. Hussain, M. and Naeem, M.N. (2018b), "Effect of various edge conditions on free vibration characteristics of rectangular plates", Chapter, Intechopen, In: Advance Testing and Engineering. ISBN 978-953-51-6706-8
  56. Hussain, M. and Naeem, M.N. (2019a), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes". Appl. Math. Modeling, 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039
  57. Hussain, M. and Naeem, M.N. (2019b), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos. Part B: Eng., 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144
  58. Hussain, M. and Naeem, M.N. (2019c), "Vibration characteristics of zigzag and chiral functionally graded material rotating carbon nanotubes sandwich with ring supports", J. Mech. Eng. Sci., Part C, 233(16), 5763-5780. https://doi.org/10.1177/0954406219855095
  59. Hussain, M. and Naeem, M. (2019d), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Mathe. Model., 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039
  60. Hussain, M. and Naeem, M. (2019e), "Vibration characteristics of single-walled carbon nanotubes based on non-local elasticity theory using wave propagation approach (WPA) including chirality", Chapter, Intechopen, In: Perspective of Carbon Nanotubes. https://doi.org/10.5772/intechopen.85948
  61. Hussain, M. and Naeem, M.N. (2020a), "Mass density effect on vibration of zigzag and chiral SWCNTs", J. Sandw. Struct. Mater. https://doi.org/10.1177/1099636220906257
  62. Hussain, M. and Naeem, M.N. (2020b), "Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners", Indian J. Phys. [In Press]
  63. Hussain, M., Naeem, M.N., Shahzad, A. and He, M. (2017), "Vibrational behavior of single-walled carbon nanotubes based on cylindrical shell model using wave propagation approach", AIP Advances, 7(4), 045114. https://doi.org/10.1063/1.4979112
  64. Hussain, M., Naeem, M., Shahzad, A. and He, M. (2018a), "Vibration characteristics of fluid-filled functionally graded cylindrical material with ring supports", Chapter, Intechopen, Computational Fluid Dynamics. ISBN 978-953-51-5706-9 https://doi.org/10.5772 /intechopen.72172
  65. Hussain, M., Naeem, M.N., Shahzad, A., He, M. and Habib, S. (2018b), "Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach", IMechE Part C: J. Mech. Eng. Sci., 232(23), 4342-4356. https://doi.org/10.1177/0954406218802320
  66. Hussain, M., Naeem, M.N. and Isvandzibaei, M. (2018c), "Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical sheel", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232(24), 4564-4577. https://doi.org/10.1177/0954406217753459
  67. Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019a), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., Int. J., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431
  68. Hussain, M., Naeem, M.N. and Taj, M. (2019b), "Effect of length and thickness variations on the vibration of SWCNTs based on Flugge's shell model", Micro Nano Lett., 15(1), 1-6. https://doi.org/10.1049/mnl.2019.0309
  69. Hussain, M., Naeem, M.N. and Tounsi, A. (2020a), "Simulating vibration of single-walled carbon nanotube using Rayleigh-Ritz's method", 8(3), 215-228. https://doi.org/10.12989/anr.2020.8.3.215
  70. Hussain, M., Naeem, M.N. and Tounsi, A. (2020b), "On mixing the Rayleigh-Ritz formulation with Hankel's function for vibration of fluid-filled Fluid-filled cylindrical shell", Adv. Computat. Des., Int. J. [In Press]
  71. Hussain, M., Naeem, M.N. and Tounsi, A. (2020c), "Numerical Study for nonlocal vibration of orthotropic SWCNTs based on Kelvin's model", Adv. Concrete Constr., Int. J., 9(3), 301-312. https://doi.org/10.12989/acc.2020.9.3.301
  72. Hussain, M., Naeem, M.N. and Tounsi, A. (2020d), "Response of orthotropic Kelvin modeling for single-walled carbon nanotubes: Frequency analysis", Adv. Nano Res., Int. J., 8(3), 229-244. https://doi.org/10.12989/anr.2020.8.3.229
  73. Hussain, M., Naeem, M.N., Sehar, A. and Tounsi, A. (2020e), "Eringen's nonlocal model sandwich with Kelvin's theory for vibration of DWCNT", Comput. Concrete, Int. J., 25(4), 343-354. https://doi.org/10.12989/cac.2020.25.4.343
  74. Hussain, M., Naeem, M.N., Khan, M.S. and Tounsi, A. (year), "Computer-aided approach for modelling of FG cylindrical shell sandwich with ring supports", Comput. Concrete, Int. J., 25(5), 411-425. https://doi.org/10.12989/cac.2020.25.5.411
  75. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(7), 56-58. https://doi.org/10.1038/354056a0
  76. Iijima, S., Brabec, C., Maiti, A. and Bernholc, J. (1996), "Structural flexibility of carbon nanotubes", J. Chem. Phys., 104(5), 2089. https://doi.org/10.1063/1.470966
  77. Jassas, M.R., Bidgoli, M.R. and Kolahchi, R. (2019), "Forced vibration analysis of concrete slabs reinforced by agglomerated SiO2 nanoparticles based on numerical methods", Constr. Build. Mater., 211, 796-806. https://doi.org/10.1016/j.conbuildmat.2019.03.263
  78. 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-Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025
  79. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019), "Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates", Mater. Res. Express, 6(9), 0950a9. https://doi.org/10.1088/2053-1591/ab3474
  80. Ke, L.L., Xiang, Y., Yang, J. and Kitipornchai, S. (2009), "Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory", Computat. Mater. Sci., 47(2), 409-417. https://doi.org/10.1016/j.commatsci.2009.09.002
  81. Khosrazadeh, A. and Hajabasi, M.A. (2012), "Free vibrations of embedded doube-walled carbon nanotubes considering nonlinear interlayer van der Waals forces", Appl. Mathe. Model., 36(3), 997-1007 https://doi.org/10.1016/j.apm.2011.07.063
  82. Kolahchi, R. (2017), "A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods", Aerosp. Sci. Technol., 66, 235-248. https://doi.org/10.1016/j.ast.2017.03.016
  83. Kolahchi, R. and Bidgoli, A.M. (2016), "Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes", Appl. Mathe. Mech., 37(2), 265-274. https://doi.org/10.1007/s10483-016-2030-8
  84. Kolahchi, R. and Cheraghbak, A. (2017), "Agglomeration effects on the dynamic buckling of viscoelastic microplates reinforced with SWCNTs using Bolotin method", Nonlinear Dyn., 90(1), 479-492. https://doi.org/10.1007/s11071-017-3676-x
  85. Kolahchi, R., Hosseini, H. and Esmailpour, M. (2016a), "Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories", Compos. Struct., 157, 174-186. https://doi.org/10.1016/j.compstruct.2016.08.032
  86. Kolahchi, R., Safari, M. and Esmailpour, M. (2016b), "Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium", Compos. Struct., 150, 255-265. https://doi.org/10.1016/j.compstruct.2016.05.023
  87. Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Nouri, A. (2017a), "Wave propagation of embedded viscoelastic FG-CNTreinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory", Int. J. Mech. Sci., 130, 534-545. https://doi.org/10.1016/j.ijmecsci.2017.06.039
  88. Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Oskouei, A.N. (2017b), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin-Wall. Struct., 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016
  89. Kolahchi, R., Hosseini, H., Fakhar, M.H., Taherifar, R. and Mahmoudi, M. (2019), "A numerical method for magneto-hygrothermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions", Comput. Mathe. Applicat., 78(6), 2018-2034. https://doi.org/10.1016/j.camwa.2019.03.042
  90. Kolahchi, R., Keshtegar, B. and Fakhar, M.H. (2020), "Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-viscopiezoelasticity theories using Grey Wolf algorithm", J. Sandw. Struct. Mater., 22(1), 3-27. https://doi.org/10.1177/1099636217731071
  91. Kroner, E. (1967), "Elasticity theory of materials with long range cohesive forces", Int. J. Solids Struct., 3(5),731-742. https://doi.org/10.1016/0020-7683(67)90049-2
  92. Kumar, B.R. (2018), "Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces", Adv. Nano Res., Int. J., 6(2), 135. https://doi.org/10.12989/anr.2018.6.2.135
  93. Lal, A., Jagtap, K.R. and Singh, B.N. (2017), "Thermomechanically induced finite element based nonlinear static response of elastically supported functionally graded plate with random system properties", Adv. Computat. Des., Int. J., 2(3), 165-194. https://doi.org/10.12989/acd.2017.2.3.165
  94. Lei, Z. and Zhang, Y. (2018), "Characterizing buckling behavior of matrix-cracked hybrid plates containing CNTR-FG layers", Steel Compos. Struct., Int. J., 28(4), 495-508. https://doi.org/10.12989/scs.2018.28.4.495
  95. Li, C. and Chou, T.W. (2003), "A structural mechanics approach for the analysis of carbon nanotubes", Int. J. Solids Struct., 40(10), 2487-2499. https://doi.org/10.1016/S0020-7683(03)00056-8
  96. Madani, H., Hosseini, H. and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNTreinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., Int. J., 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889
  97. Mehar, K. and Panda, S.K. (2016a), "Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field", Compos. Struct., 143, 336-346. https://doi.org/10.1016/j.compstruct.2016.02.038
  98. Mehar, K. and Panda, S.K. (2016b), "Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory", Proceedings of IOP Conference Series: Materials Science and Engineering, 115(1), 012014. https://doi.org/10.1088/1757-899X/115/1/012014
  99. Mehar, K. and Panda, S.K. (2018a), "Dynamic response of functionally graded carbon nanotube reinforced sandwich plate", Proceedings of IOP Conference Series: Materials Science and Engineering, 338(1), p. 012017. https://doi.org/10.1088/1757-899X/338/1/012017
  100. Mehar, K. and Panda, S.K. (2018b), "Thermal free vibration behavior of FG-CNT reinforced sandwich curved panel using finite element method", Polym. Compos., 39(8), 2751-2764. https://doi.org/10.1002/pc.24266
  101. Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., Int. J., 7(3), 181-190. https://doi.org/10.12989/anr.2019.7.3.181
  102. Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324
  103. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Eur. J. Mech.- A/Solids, 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005
  104. Mehar, K., Panda, S.K., Bui, T.Q. and Mahapatra, T.R. (2017b), "Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM", J. Thermal Stress., 40(7), 899-916. https://doi.org/10.1080/01495739.2017.1318689
  105. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017c), "Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure", Int. J. Mech. Sci., 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057
  106. Mehar, K., Panda, S.K. and Patle, B.K. (2017d), "Thermoelastic vibration and flexural behavior of FG-CNT reinforced composite curved panel", Int. J. Appl. Mech., 9(4), 1750046. https://doi.org/10.1142/S1758825117500466
  107. Mehar, K., Panda, S.K. and Patle, B.K. (2018a), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792-3809. https://doi.org/10.1002/pc.24409
  108. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018b), "Nonlinear frequency responses of functionally graded carbon nanotubereinforced sandwich curved panel under uniform temperature field", Int. J. Appl. Mech., 10(3), 1850028. https://doi.org/10.1142/S175882511850028X
  109. Mehar, K., Mahapatra, T.R., Panda, S.K., Katariya, P.V. and Tompe, U.K. (2018c), "Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure", J. Eng. Mech., 144(9), 04018094. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001519
  110. Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216, 406-414. https://doi.org/10.1016/j.compstruct.2019.03.002
  111. 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
  112. Mohsen, M. and Eyvazian, A. (2020), "Post-buckling analysis of Mindlin Cut out-plate reinforced by FG-CNTs", Steel Compos. Struct., Int. J., 34(2), 289-297. https://doi.org/10.12989/scs.2020.34.2.289
  113. Moradi-Dastjerdi, R. and Payganeh, G. (2017), "Transient heat transfer analysis of functionally graded CNT reinforced cylinders with various boundary conditions", Steel Compos. Struct., Int. J., 24(3), 359-367. https://doi.org/10.12989/scs.2017.24.3.359
  114. Motezaker, M. and Eyvazian, A. (2020), "Buckling load optimization of beam reinforced by nanoparticles", Struct. Eng. Mech., Int. J., 73(5), 481-486. https://doi.org/10.12989/sem.2020.73.5.481
  115. Motezaker, M. and Kolahchi, R. (2017a), "Seismic response of concrete columns with nanofiber reinforced polymer layer", Comput. Concrete, Int. J., 20(3), 361-368. https://doi.org/10.12989/cac.2017.20.3.361
  116. Motezaker, M. and Kolahchi, R. (2017b), "Seismic response of $SiO_2$ nanoparticles-reinforced concrete pipes based on DQ and newmark methods", Comput. Concrete, Int. J., 19(6), 745-753. https://doi.org/10.12989/cac.2017.19.6.745
  117. Motezaker, M., Jamali, M. and Kolahchi, R. (2020), "Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocalpiezoelasticity theory", J. Computat. Appl. Mathe., 369, 112625. https://doi.org/10.1016/j.cam.2019.112625
  118. Narwariya, M., Choudhury, A. and Sharma, A.K. (2018), "Harmonic analysis of moderately thick symmetric cross-ply laminated composite plate using FEM", Adv. Computat. Des., Int. J., 3(2), 113-132. https://doi.org/10.12989/acd.2018.3.2.113
  119. Natsuki, T., Endo, M. and Tsuda, H. (2006), "Vibration analysis of embedded carbon nanotubes using wave propagation approach", J. Appl. Phys., 99(3), 034311. https://doi.org/10.1063/1.2170418
  120. Natsuki, T., Ni, Q.Q. and Endo, M. (2007), "Wave propagation in single-walled and double-walled carbon nanotubes filled with fluids", J. Appl Phys., 101(3), 034319-034319-5. https://doi.org/10.1063/1.2432025
  121. Zou, R.D. and Foster, C.G. (1995), "Simple solution for buckling of orthotropic circular cylindrical shells", Thin-Wall. Struct., 22(3), 143-158. https://doi.org/10.1016/0263-8231(94)00026-V
  122. Paliwal, D.N., Kanagasabapathy, H. and Gupta, K.M. (1995), "The large deflection of an orthotropic cylindrical shell on a Pasternak foundation", Compos. Struct., 31(1), 31-37. https://doi.org/10.1016/0263-8223(94)00068-9
  123. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of Nonlocal Continuum Models to Nanotechnology", Int. J. Eng. Sei., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  124. Pradhan, S.C. and Phadikar, J.K. (2009), "Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models", Phy. Lett. A, 373(11), 1062-1069. https://doi.org/10.1016/j.physleta.2009.01.030
  125. Qian, D., Wagner, G.J., Liu, W.K., Yu, M.F. and Ruoff, R.S. (2002), "Mechanics of carbon nanotubes", Appl. Mech. Rev., 55(6), 495-533. https://doi.org/10.1115/1.1490129
  126. Rakrak, K., Zidour, M., Heireche, H., Bousahla, A.A. and Chemi, A. (2016), "Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory", Adv. Nano Res., Int. J., 4(1), 31-44. https://doi.org/10.12989/anr.2016.4.1.031
  127. 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
  128. Rouhi, H., Ansari, R. and Arash, B. (2013), "Vibrational analysis of double-walled carbon nanotubes based on the nonlocal Donnell shell theory via a new numerical approach", Int J. Mech. Sei., 37, 91-105.
  129. Rouhi, H., BazdidVahdati, M. and Ansari, R. (2015), "Rayleigh-Rits vibrational analysis of multi-walled carbon nanotubes based on the non-local Flugge shell theory", J. Compos., 750392. https://doi.org/10.1155/2015/750392
  130. Safa, A., Hadji, L., Bourada, M. and Zouatnia, N. (2019), "Thermal vibration analysis of FGM beams using an efficient shear deformation beam theory", Earthq. Struct., Int. J., 17(3), 329-336. https://doi.org/10.12989/eas.2019.17.3.329
  131. Sahouane, A., Hadji, L. and Bourada, M. (2019), "Numerical analysis for free vibration of functionally graded beams using an original HSDBT", Earthq. Struct., Int. J., 17(1), 31-37. https://doi.org/10.12989/eas.2019.17.1.031
  132. Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A. and Tounsi, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., Int. J., 33(6), 805-822. https://doi.org/10.12989/scs.2019.33.6.805
  133. Sanchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A. and Ordejon, P. (1999), "Ab-initio structural, elastic, and Vibrational Properties of Carbon Nanotubes", Phys. Rev. B, 59, 12678-2688. http://dx.doi.org/10.1103/PhysRevB.59.12678
  134. Sedighi, H.M. and Sheikhanzadeh, A. (2017), "Static and dynamic pull-in instability of nano-beams resting on elastic foundation based on the nonlocal elasticity theory", Chin. J. Mech. Eng., 30, 385-397. https://doi.org/10.1007/s10033-017-0079-3
  135. Sehar, A., Hussain, M., Naeem, M.N. and Tounsi, A. (2020), "Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis", Comput. Concrete, Int. J., 25(2), 133-144. https://doi.org/10.12989/cac.2020.25.2.133
  136. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., Int. J., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065
  137. Sharma, P., Singh, R. and Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. https://doi.org/10.1177/0954406219888234
  138. She, G.L., Ren, Y.R. and Yuan, F.G. (2019), "Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems", Steel Compos. Struct., Int. J., 31(6), 641-653. https://doi.org/10.12989/scs.2019.31.6.641
  139. Shen, H.S. and Zhang, C.L. (2010), "Torsional buckling and post buckling of double-walled carbon nanotubes by nonlocal shear deformable shell model", Compos. Struct., 92(5), 1073-1084. https://doi.org/10.1016/j.compstruct.2009.10.002
  140. Soldano, C. (2015), "Hybrid metal-based carbon nanotubes", "Novel platform for multifunctional applications", Progress in Mater. Sci., 69, 183-212. https://doi.org/10.1016/j.pmatsci.2014.11.001
  141. Sosa, E.D., Darlington, T.K., Hanos, B.A. and O'Rourke, M.J.E. (2014), "Multifunctional thermally remendable nanocomposites", J. Compos., 12 p. http://dx.doi.org/10.1155/2014/705687
  142. Sudak, L.J. (2003), "Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94, 7281-7287. https://doi.org/10.1063/1.1625437
  143. Sun, C.T. and Zhang, H. (2002), "Size-dependent elastic moduli of plate like nanomaterials", J. Appl. Phys., 93, 212-1218. https://doi.org/10.1063/1.1530365
  144. Tahouneh, V. (2017), "Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate", Steel Compos. Struct., Int. J., 25(6), 649-661. https://doi.org/10.12989/scs.2017.25.6.649
  145. Taj, M., Safeer, M., Hussain, M., Naeem, M.N., Majeed, A., Ahmad, M., Khan, H.U. and Tounsi, A. (2020a), "Non-local orthotropic elastic shell model for vibration analysis of protein microtubules", Comput. Concrete, Int. J., 25(3), 245-253. https://doi.org/10.12989/cac.2020.25.3.245
  146. Taj, M., Safeer, M., Hussain, M., Naeem, M.N., Ahmad, M., Abbas, K., Khan, A.Q. and Tounsi, A. (2020b), "Effect of external force on buckling of cytoskeleton intermediate filaments within viscoelastic media", Comput. Concrete, Int. J., 25(3), 205-214. https://doi.org/10.12989/cac.2020.25.3.205
  147. Taj, M., Hussain, M., Naeem, M.N. and Tounsi, A. (2020c), "Effects of elastic medium on buckling of microtubules due to bending and torsion", Adv. Concrete Constr., Int. J., 9(5), 491-501. https://doi.org/10.12989/acc.2020.9.5.491
  148. Tounsi, A., Benguediab, S., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of doublewalled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  149. Usuki, T. and Yogo, K. (2009), "Beam equations for multi-walled carbon nanotubes derived from Flugge shell theory", Proceedings of Royal Society A, 465(2104). https://doi.org/10.1098/rspa.2008.0394
  150. Vodenitcharova, T. and Zhang, L.C. (2003), "Effective wall thickness of single walled carbon nanotubes", Phy. Rev. B, 68, 165401. https://doi.org/10.1103/PhysRevB.68.165401
  151. Wang, J. and Gao, Y. (2016), "Nonlocal orthotropic shell model applied on wave propagation in microtubules", Appl. Mathe. Model., 40(11-12), 5731-5744. https://doi.org/10.1016/j.apm.2016.01.013
  152. Wang, C.Y. and Zhang, L.C. (2007), "Modeling the free vibration of single-walled carbon nanotubes", Proceedings of the 5th Australasian Congress on Applied Mechanics, ACAM, Brisbane, Australia, pp. 252-257.
  153. Wang, Q., Varadan, V.K. and Quek, S.T. (2006a), "Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models", Phys. Lett. A, 357(2), 130-135. https://doi.org/10.1016/j.physleta.2006.04.026
  154. Wang, Q., Zhou, G.Y. and Lin, K.C. (2006b), "Scale effect on wave propagation of double-walled carbon nanotubes", Int. J. Solids Struct., 43, 6071-6084. https://doi.org/10.1016/j.ijsolstr.2005.11.005
  155. Xiaobin, L., Shuangxi, X., Weiguo, W. and Jun, L. (2014), "An exact dynamic stiffness matrix for axially loaded double-beam systems", Sadhana, 39(3), 607-623. https://doi.org/10.1007/s12046-013-0214-5
  156. Xu, K.U., Aifantis, E.C. and Yan, Y.H. (2008), "Vibrations of double-walled carbon nanotubes with different boundary conditions between inner and outer tubes", J. Appl. Mech., 75(2), 021013-1. https://doi.org/10.1115/1.2793133
  157. Yakobson, B.I., Brabec, C.J. and Bernholc, J. (1996), "Nanomechanics of carbon tubes: instabilities beyond linear response", Phy. Rev. Lett, 76, 2511-2514. https://doi.org/10.1103/PhysRevLett.76.2511
  158. Yakobson, B.I., Campbell, M.P., Brabec, C.J. and Bemholc, J. (1997), "High strain rate fracture and C-chain unravelling in carbon nanotubes", Comput. Mater. Sei., 8(4), 341-348. https://doi.org/10.1016/S0927-0256(97)00047-5
  159. Yoon, J., Ru, C.Q. and Mioduchowski, A. (2003), "Vibration of an embedded multiwall carbon nanotube", Compos. Sei. Tech., 63(11), 1533-1542. https://doi.org/10.1016/S0266-3538(03)00058-7
  160. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams surface stress effects", Smart Struct. Syst., Int. J., 21(1), 65-74. https://doi.org/10.12989/sss.2018.21.1.065
  161. Zamanian, M., Kolahchi, R. and Bidgoli, M.R. (2017), "Agglomeration effects on the buckling behaviour of embedded concrete columns reinforced with $SiO_2$ nano-particles", Wind Struct., Int. J., 24(1), 43-57. https://doi.org/10.12989/was.2017.24.1.043
  162. Zarei, M.S., Kolahchi, R., Hajmohammad, M.H. and Maleki, M. (2017), "Seismic response of underwater fluid-conveying concrete pipes reinforced with $SiO_2$ nanoparticles and fiber reinforced polymer (FRP) layer", Soil Dyn. Earthq. Eng., 103, 76-85. https://doi.org/10.1016/j.soildyn.2017.09.009
  163. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., Int. J., 54(4), 693-710. http://dx.doi.org/10.12989/sem.2015.54.4.693
  164. Zidour, M., Daouadji, T.H., Benrahou, K.H., Tounsi, A., Bedia, E.A.A. and Hadji, L. (2014), "Buckling analysis of chiral singlewalled carbon nanotubes by using the nonlocal Timoshenko beam theory", Mech. Compos. Mater., 50(1), 95-104. https://doi.org/10.1007/s11029-014-9396-0
  165. Zouatnia, N. and Hadji, L. (2019), "Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear deformation theory", Earthq. Struct., Int. J., 16(2), 177-183. https://doi.org/10.12989/eas.2019.16.2.177