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Nonlinear vibration analysis of carbon nanotube-reinforced composite beams resting on nonlinear viscoelastic foundation

  • M., Alimoradzadeh (Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University) ;
  • S.D., Akbas (Department of Civil Engineering, Bursa Technical University)
  • Received : 2022.06.30
  • Accepted : 2022.09.12
  • Published : 2023.01.25
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Abstract

Nonlinear vibration analysis of composite beam reinforced by carbon nanotubes resting on the nonlinear viscoelastic foundation is investigated in this study. The material properties of the composite beam is considered as a polymeric matrix by reinforced carbon nanotubes according to different distributions. With using Hamilton's principle, the governing nonlinear partial differential equations are derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The nonlinear natural frequency and the nonlinear free response of the system is obtained. In addition, the effects of different patterns of reinforcement, linear and nonlinear damping coefficients of the viscoelastic foundation on the nonlinear vibration responses and phase trajectory of the carbon nanotube reinforced composite beam are investigated.

Keywords

References

  1. Addou, F.Y., Meradjah, M., Bousahla, A.A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S.R. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347.
  2. Akbas, S.D. (2013), "Free vibration characteristics of edge cracked functionally graded beams by using finite element method", Int. J. Eng. Trends Technol., 4(10), 4590-4597.
  3. Akbas, S.D. (2014), "Free vibration of axially functionally graded beams in thermal environment", Int. J. Eng. Appl. Sci., 6(3), 37-51. https://doi.org/10.24107/ijeas.251224.
  4. Akbas, S.D. (2016), "Static analysis of a nano plate by using generalized differential quadrature method", Int. J. Eng. Appl. Sci., 8(2), 30-39. https://doi.org/10.24107/ijeas.252143.
  5. Akbas, S.D. (2018a), "Nonlinear thermal displacements of laminated composite beams", Coupled Syst. Mech., 7(6), 691-705. https://doi.org/10.12989/csm.2018.7.6.691.
  6. Akbas, S.D. (2018b), "Geometrically nonlinear analysis of a laminated composite beam", Struct. Eng. Mech., 66(1), 27-36. https://doi.org/10.12989/sem.2018.66.1.027.
  7. Akbas, S.D. (2018c), "Post-buckling responses of a laminated composite beam", Steel Compos. Struct., 26(6), 733-743. https://doi.org/10.12989/scs.2018.26.6.733.
  8. Akbas, S.D. (2018d), "Thermal post-buckling analysis of a laminated composite beam", Struct. Eng. Mech., 67(4), 337-346. https://doi.org/10.12989/sem.2018.67.4.337.
  9. Akbas, S.D. (2019a), "Hygrothermal post-buckling analysis of laminated composite beams", Int. J. Appl. Mech., 11(1), 1950009. https://doi.org/10.1142/S1758825119500091.
  10. Akbas, S.D. (2019b), "Nonlinear static analysis of laminated composite beams under hygro-thermal effect", Struct. Eng. Mech., 72(4), 433-441. https://doi.org/10.12989/sem.2019.72.4.433.
  11. Akbas, S.D. (2019c) "Longitudinal forced vibration analysis of porous A nanorod", Muhendislik Bilimleri ve Tasarim Dergisi, 7(4), 736-743. https://doi.org/10.21923/jesd.553328.
  12. Akbas, S.D. (2019d) "Axially forced vibration analysis of cracked a nanorod", J. Comput. Appl. Mech., 5(2), 477-485. https://doi.org/10.22059/JCAMECH.2019.281285.392.
  13. Akbas, S.D. (2020a), "Modal analysis of viscoelastic nanorods under an axially harmonic load", Adv. Nano Res., 8(4), 277-282. https://doi.org/10.12989/anr.2020.8.4.277.
  14. Akbas, S.D. (2020b), "Dynamic responses of laminated beams under a moving load in thermal environment", Steel Compos. Struct., 35(6), 729-737. https://doi.org/10.12989/scs.2020.35.6.729.
  15. Akbas, S.D. (2020c), "Dynamic analysis of a laminated composite beam under harmonic load", Coupled Syst. Mech., 9(6), 563-573. https://doi.org/10.12989/csm.2020.9.6.563.
  16. Akbas, S.D. (2021), "Dynamic analysis of axially functionally graded porous beams under a moving load", Steel Compos. Struct., 39(6), 811-821. https://doi.org/10.12989/scs.2021.39.6.811.
  17. Akbas, S.D. (2022), "Moving-load dynamic analysis of AFG beams under thermal effect", Steel Compos. Struct., 42(5), 649-655. https://doi.org/10.12989/scs.2022.42.5.649.
  18. Al-Furjan, M.S.H., Safarpour, H., Habibi, M., Safarpour, M. and Tounsi, A. (2020a), "A comprehensive computational approach for nonlinear thermal instability of the electrically FG-GPLRC disk based on GDQ method", Eng. with Comput., 1-18. https://doi.org/10.1007/s00366-020-01088-7.
  19. Al-Furjan, M.S.H., Habibi, M., Sadeghi, S., Safarpour, H., Tounsi, A. and Chen, G. (2020b), "A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel", Eng. with Comput., 1-18. https://doi.org/10.1007/s00366-020-01130-8.
  20. Alimoradzadeh, M., Salehi, M. and Esfarjani, S.M. (2019), "Nonlinear dynamic response of an axially functionally graded (AFG) beam resting on nonlinear elastic foundation subjected to moving load", Nonlinear Eng., 8(1), 250-260. https://doi.org/10.1515/nleng-2018-0051.
  21. Alimoradzadeh, M., Salehi, M. and Esfarjani, S.M. (2020), "Nonlinear vibration analysis of axially functionally graded microbeams based on nonlinear elastic foundation using modified couple stress theory", Periodica Polytechnica Mech. Eng., 64(2), 97-108. https://doi.org/10.3311/PPme.11684.
  22. Alimoradzadeh M. and Akbas S.D. (2021a), "Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory", Eur. Phys. J. Plus, 136:536. https://doi.org/10.1140/epjp/s13360-021-01539-0.
  23. Alimoradzadeh, M., Akbas, S.D. and Esfrajani, S.M. (2021b), "Nonlinear dynamic and stability of a beam resting on the nonlinear elastic foundation under thermal effect based on the finite strain theory", Struct. Eng. Mech., 80(3), 275-284. https://doi.org/10.12989/sem.2021.80.3.275.
  24. Alimoradzadeh, M. and Akbas, S.D. (2022a), "Superharmonic and subharmonic resonances of a carbon nanotube-reinforced composite beam", Adv. Nano Res., 12(4), 353-363. https://doi.org/10.12989/anr.2022.12.4.353.
  25. Alimoradzadeh, M. and Akbas, S.D. (2022b), "Nonlinear dynamic behavior of functionally graded beams resting on nonlinear viscoelastic foundation under moving mass in thermal environment", Struct. Eng. Mech., 81(6), 705-714. https://doi.org/10.12989/sem.2022.81.6.705.
  26. Alimoradzadeh, M. and Akbas, S.D. (2022c), "Nonlinear dynamic responses of cracked atomic force microscopes", Struct. Eng. Mech., 82(6), 747-756. https://doi.org/10.12989/sem.2022.82.6.747.
  27. Ansari, M., Esmailzadeh, E. and Younesian, D. (2010), "Internalexternal resonance of beams on non-linear viscoelastic foundation traversed by moving load", Nonlinear Dynam., 61(1), 163-182. https://doi.org/10.1007/s11071-009-9639-0.
  28. Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S. and Tounsi, A. (2021), "Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory", Eng. with Comput., 1-22. https://doi.org/10.1007/s00366-021-01382-y.
  29. Babu Arumugam, A., Rajamohan, V., Bandaru, N., Sudhagar P,E. and Kumbhar, S.G. (2019), "Vibration analysis of a carbon nanotube reinforced uniform and tapered composite beams", Archiv. Acoust., 44(2), 309-320. https://doi.org/10.24425/aoa.2019.128494.
  30. Bellal, M., Hebali, H., Heireche, H., Bousahla, A.A., Tounsi, A., Bourada, F. and Tounsi, A. (2020), "Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model", Steel Compos. Struct., 34(5), 643-655. https://doi.org/10.12989/scs.2020.34.5.643.
  31. Bendenia, N., Zidour, M., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H. and Tounsi, A. (2020), "Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation", Comput. Concrete, 26(3), 213-226. https://doi.org/10.12989/cac.2020.26.3.213.
  32. Bourada, F., Bousahla, A.A., Tounsi, A., Bedia, E.A., Mahmoud, S.R., Benrahou, K.H. and Tounsi, A. (2020), "Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation", Comput. Concrete, 25(6), 485-495. https://doi.org/10.12989/cac.2020.25.6.485.
  33. Bousahla, A.A., Bourada, F., Mahmoud, S.R., Tounsi, A., Algarni, A., Bedia, E.A. and Tounsi, A. (2020), "Buckling and dynamic behavior of the simply supported CNT-RC beams using an integral-first shear deformation theory", Comput. Concrete, 25(2), 155-166. https://doi.org/10.12989/cac.2020.25.2.155.
  34. Chikr, S.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Bedia, E.A. and Tounsi, A. (2020), "A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin's approach", Geomech. Eng., 21(5), 471-487. https://doi.org/10.12989/gae.2020.21.5.471.
  35. Chu, H., Li, Y., Wang, C., Zhang, H. and Li, D. (2020), "Recent investigations on nonlinear absorption properties of carbon nanotubes", Nanophotonics, 9(4), 761-781. https://doi.org/10.1515/nanoph-2020-0085.
  36. Civalek, O., Akbas, S.D., Akgoz, B. and Dastjerdi, S. (2021), "Forced vibration analysis of composite beams reinforced by carbon nanotubes", Nanomaterials, 11(3), 571. https://doi.org/10.3390/nano11030571.
  37. Ebrahimi, F., Shaghaghi, G.R. and Boreiry, M. (2016), "An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes", Struct. Eng. Mech., 57(1), 179-200. https://doi.org/10.12989/sem.2016.57.1.179.
  38. Eshraghi, I., Jalali, S.K. and Pugno, N.M. (2016), "Imperfection sensitivity of nonlinear vibration of curved single-walled carbon nanotubes based on nonlocal timoshenko beam theory", Materials, 9(9), 786. https://doi.org/10.20944/preprints201609.0060.v1.
  39. Fan, Y., Xiang, Y., Shen, H.S. and Wang, H. (2018), "Lowvelocity impact response of FG-GRC laminated beams resting on visco-elastic foundations", Int. J. Mech. Sci., 141, 117-126. https://doi.org/10.1016/j.ijmecsci.2018.04.007.
  40. Fan, Y., Xiang, Y. and Shen, H.S. (2020), "Nonlinear dynamics of temperature-dependent FG-GRC laminated beams resting on visco-Pasternak foundations", Int. J. Struct. Stab. Dynam., 20(1), 2050012. https://doi.org/10.1142/S0219455420500121.
  41. Fernandes, R., Mousavi, S.M. and El-Borgi, S. (2016), "Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory", Acta Mechanica, 227(9), 2657-2670. https://doi.org/10.1007/s00707-016-1646-x.
  42. Fu, Y., Zhong, J., Shao, X. and Tao, C. (2016), "Analysis of nonlinear dynamic stability for carbon nanotube-reinforced composite plates resting on elastic foundations", Mech. Adv. Mater. Struct., 23(11), 1284-1289. https://doi.org/10.1080/15376494.2015.1068404.
  43. Ghayesh, M.H. (2009), "Stability characteristics of an axially accelerating string supported by an elastic foundation", Mechanism Machine Theory, 44(10), 1964-1979. https://doi.org/10.1016/j.mechmachtheory.2009.05.004.
  44. Ghayesh, M.H. (2012), "Nonlinear dynamic response of a simplysupported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring", Nonlinear Analysis: Real World Applications, 13(3), 1319-1333. https://doi.org/10.1016/j.nonrwa.2011.10.009.
  45. Ghayesh, M.H., Amabili, M. and Paidoussis, M.P. (2012a), "Thermo-mechanical phase-shift determination in Coriolis mass-flowmeters with added masses", J. Fluid. Struct., 34, 1-13. https://doi.org/10.1016/j.jfluidstructs.2012.05.003.
  46. Ghayesh, M.H., Kazemirad, S. and Reid, T. (2012b), "Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure", Appl. Math. Model., 36(7), 3299-3311. https://doi.org/10.1016/j.apm.2011.09.084.
  47. Ghayesh, M.H. (2018a), "Dynamics of functionally graded viscoelastic microbeams", Int. J. Eng. Sci., 124, 115-131. https://doi.org/10.1016/j.ijengsci.2017.11.004.
  48. Ghayesh, M.H. (2018b), "Nonlinear vibrations of axially functionally graded Timoshenko tapered beams", J. Comput. Nonlinear Dynam., 13(4), 041002. https://doi.org/10.1115/1.4039191.
  49. Guellil, M., Saidi, H., Bourada, F., Bousahla, A.A., Tounsi, A., Al-Zahrani, M.M. and Mahmoud, S.R. (2021), "Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation", Steel Compos. Struct., 38(1), 1-15. https://doi.org/10.12989/scs.2021.38.1.001.
  50. Guo, X.Y. and Zhang, W. (2016), "Nonlinear vibrations of a reinforced composite plate with carbon nanotubes", Compos. Struct., 135, 96-108. https://doi.org/10.1016/j.compstruct.2015.08.063.
  51. Heidari, M. and Arvin, H. (2019), "Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes", J. Vib. Control, 25(14), 2063-2078. https://doi.org/10.1016/j.compstruct.2016.12.009.
  52. Heidari, F., Taheri, K., Sheybani, M., Janghorban, M. and Tounsi, A. (2021), "On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes", Steel Compos. Struct., 38(5), 533-545. https://doi.org/10.12989/scs.2021.38.5.533.
  53. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(56-58), 56-58. https://doi.org/10.1038/354056a0.
  54. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Bedia, E.A. and Al-Osta, M.A. (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: Bending and free vibration analysis", Comput. Concrete, 25(1), 37-57. https://doi.org/10.12989/cac.2020.25.1.037.
  55. Ke, L.L., Yang, J. and Kitipornchai, S. (2010), "Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams", Compos. Struct., 92(3), 676-683. https://doi.org/10.1016/j.compstruct.2009.09.024.
  56. Kirlangic, O. and Akbas, S.D. (2020), "Comparison study between layered and functionally graded composite beams for static deflection and stress analyses", J. Comput. Appl. Mech., 51(2), 294-301. https://doi.org/10.22059/JCAMECH.2020.296319.473.
  57. Kirlangic, O. and Akbas, S.D. (2021), "Dynamic responses of functionally graded and layered composite beams", Smart Struct. Syst., 27(1), 115-122. https://doi.org/10.12989/sss.2021.27.1.115
  58. Lee, S.Y. and Hwang, J.G. (2019), "Finite element nonlinear transient modelling of carbon nanotubes reinforced fiber/polymer composite spherical shells with a cutout", Nanotechnol. Rev., 8(1), 444-451. https://doi.org/10.1515/ntrev2019-0039.
  59. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle", Steel Compos. Struct., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595.
  60. Mehdipour, I., Soltani, P., Ganji, D.D. and Farshidianfar, A. (2011), "Nonlinear vibration and bending instability of a singlewalled carbon nanotube using nonlocal elastic beam theory", Int. J. Nanosci., 10(3), 447-453. https://doi.org/10.1142/S0219581X11008216.
  61. Merazka, B., Bouhadra, A., Menasria, A., Selim, M.M., Bousahla, A.A., Bourada, F. and Al-Zahrani, M.M. (2021), "Hygrothermo-mechanical bending response of FG plates resting on elastic foundations", Steel Compos. Struct., 39(5), 631-643. https://doi.org/10.12989/scs.2021.39.5.631.
  62. Mudhaffar, I.M., Tounsi, A., Chikh, A., Al-Osta, M.A., AlZahrani, M.M. and Al-Dulaijan, S.U. (2021), "Hygro-thermomechanical bending behavior of advanced functionally graded ceramic metal plate resting on a viscoelastic foundation", Structures, 33, 2177-2189. https://doi.org/10.1016/j.istruc.2021.05.090.
  63. Nayfeh, A.H., Mook, D.T. and Holmes, P. (1980), "Nonlinear oscillations", ASME. J. Appl. Mech., 47(3), 692. https://doi.org/10.1115/1.3153771.
  64. Ponnusami, S.A., Gupta, M. and Harursampath, D. (2019), "Asymptotic modeling of nonlinear bending and buckling behavior of carbon nanotubes", AIAA J., 57(10), 4132-4140. https://doi.org/10.2514/1.J057564.
  65. Rabhi, M., Benrahou, K.H., Kaci, A., Houari, M.S.A., Bourada, F., Bousahla, A.A. and Tounsi, A. (2020), "A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions", Geomech. Eng., 22(2), 119-132. https://doi.org/10.12989/gae.2020.22.2.119.
  66. Rafiee, M., He, X.Q. and Liew, K.M. (2014), "Non-linear dynamic stability of piezoelectric functionally graded carbon nanotubereinforced composite plates with initial geometric imperfection", Int. J. Nonlinear Mech., 59, 37-51. https://doi.org/10.1016/j.ijnonlinmec.2013.10.011.
  67. Refrafi, S., Bousahla, A.A., Bouhadra, A., Menasria, A., Bourada, F., Tounsi, A. and Tounsi, A. (2020), "Effects of hygro-thermomechanical conditions on the buckling of FG sandwich plates resting on elastic foundations", Comput. Concrete, 25(4), 311-325. https://doi.org/10.12989/cac.2020.25.4.311.
  68. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065.
  69. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026.
  70. Shen, H.S. and Xiang, Y. (2012), "Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments", Comput. Method. Appl. M., 213(2012), 196-205. https://doi.org/10.1016/j.cma.2011.11.025.
  71. Shen, H.S. and Xiang, Y. (2013), "Nonlinear analysis of nanotubereinforced composite beams resting on elastic foundations in thermal environments", Eng. Struct., 56, 698-708. https://doi.org/10.1016/j.engstruct.2013.06.002.
  72. Shen, H.S., He, X.Q. and Yang, D.Q. (2017), "Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations", Int. J. Nonlinear Mech., 91, 69-75. https://doi.org/10.1016/j.ijnonlinmec.2017.02.010.
  73. Shi, Z., Yao, X., Pang, F. and Wang, Q. (2017), "An exact solution for the free-vibration analysis of functionally graded carbonnanotube-reinforced composite beams with arbitrary boundary conditions", Scientific Reports, 7(1), 1-18. https://doi.org/10.1038/s41598-017-12596-w.
  74. Simsek, M. (2014), "Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He's variational method", Compos. Struct., 112, 264-272. https://doi.org/10.1016/j.compstruct.2014.02.010.
  75. Taati, E., Borjalilou, V., Fallah, and, F. and Ahmadian, M.T. (2020), "On size-dependent nonlinear free vibration of carbon nanotube-reinforced beams based on the nonlocal elasticity theory: Perturbation technique", Mech. Based Des. Struct., 1-23. https://doi.org/10.1080/15397734.2020.1772087.
  76. Tagrara, S.H., Benachour, A., Bouiadjra, M.B. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259.
  77. Tahir, S.I., Tounsi, A., Chikh, A., Al-Osta, M.A., Al-Dulaijan, S. U. and Al-Zahrani, M.M. (2021), "An integral four-variable hyperbolic HSDT for the wave propagation investigation of a ceramic-metal FGM plate with various porosity distributions resting on a viscoelastic foundation", Waves in Random and Complex Media, 1-24. https://doi.org/10.1080/17455030.2021.1942310.
  78. Thang, P.T., Nguyen, T.T. and Lee, J. (2017), "A new approach for nonlinear buckling analysis of imperfect functionally graded carbon nanotube-reinforced composite plates", Compos. Part B: Eng., 127, 166-174. https://doi.org/10.1016/j.compositesb.2016.12.002.
  79. Ton-That, H.L. (2020), "The linear and nonlinear bending analyses of functionally graded carbon nanotube-reinforced composite plates based on the novel four-node quadrilateral element", Eur. J. Comput. Mech., 139-172. https://doi.org/10.13052/ejcm2642-2085.2915.
  80. Tornabene, F., Bacciocchi, M., Fantuzzi, N. and Reddy, J.N. (2019), "Multiscale approach for three-phase CNT/polymer/fiber laminated nanocomposite structures", Polymer composites, 40(1), 102-126. https://doi.org/10.1002/pc.24520.
  81. Tounsi, A., Al-Dulaijan, S.U., Al-Osta, M.A., Chikh, A., AlZahrani, M.M., Sharif, A. andTounsi, A. (2020), "A four variable trigonometric integral plate theory for hygro-thermomechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation", Steel Compos. Struct., 34(4), 511-524. https://doi.org/10.12989/scs.2020.34.4.511.
  82. Van Do, V.N., Jeon, J.T. and Lee, C.H. (2020), "Dynamic analysis of carbon nanotube reinforced composite plates by using Bezier extraction based isogeometric finite element combined with higher-order shear deformation theory", Mech. Mater., 142, 103307. https://doi.org/10.1016/j.mechmat.2019.103307.
  83. Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71, 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028
  84. Wu, C.P., Chen, Y.H., Hong, Z.L. and Lin, C.H. (2018), "Nonlinear vibration analysis of an embedded multi-walled carbon nanotube", Adv. Nano Res., 6(2), 163-183. https://doi.org/10.12989/anr.2018.6.2.163.
  85. Yang, J., Huang, X.H. and Shen, H.S. (2020), "Nonlinear vibration of temperature-dependent FG-CNTRC laminated beams with negative Poisson's ratio", Int. J. Struct. Stab. Dynam., 20(4), 2050043. https://doi.org/10.1142/S0219455420500431.
  86. Yas, M.H. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Press. Vessel. Pip., 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012.
  87. Yayli, M.O. (2018), "On the torsional vibrations of restrained nanotubes embedded in an elastic medium", J. Braz. Soc. Mech. Sci. Eng., 40(9), 1-12. https://doi.org/10.1007/s40430-018-1346-7.
  88. Zerrouki, R., Karas, A., Zidour, M., Bousahla, A.A., Tounsi, A., Bourada, F. and Mahmoud, S.R. (2021), "Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam", Struct. Eng. Mech., 78(2), 117-124. https://doi.org/10.12989/sem.2021.78.2.117.