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Finite element bending and buckling analysis of functionally graded carbon nanotubes-reinforced composite beam under arbitrary boundary conditions

  • Belarbi, Mohamed-Ouejdi (Laboratoire de Recherche en Genie Civil, LRGC, Universite de Biskra) ;
  • Salami, Sattar Jedari (Department of Biomedical Engineering, Central Tehran branch, Islamic Azad University) ;
  • Garg, Aman (Department of Civil and Environmental Engineering, The NorthCap University) ;
  • Hirane, Hicham (Laboratoire de Recherche en Genie Civil, LRGC, Universite de Biskra) ;
  • Amine, Daikh Ahmed (Department of Technology, University Centre of Naama) ;
  • Houari, Mohammed Sid Ahmed (Laboratoire d'Etude des Structures et de Mécanique des Materiaux, Departement de Genie Civil, Faculté des Sciences et de la Technologie, Universite Mustapha Stambouli)
  • Received : 2020.10.28
  • Accepted : 2021.05.21
  • Published : 2022.08.25
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Abstract

In the present paper, the static bending and buckling responses of functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) beam under various boundary conditions are investigated within the framework of higher shear deformation theory. The significant feature of the proposed theory is that it provides an accurate parabolic distribution of transverse shear stress through the thickness satisfying the traction-free boundary conditions needless of any shear correction factor. Uniform (UD) and four graded distributions of CNTs which are FG-O, FG-X, FG- and FG-V are selected here for the analysis. The effective material properties of FG-CNTRC beams are estimated according to the rule of mixture. To model the FG-CNTRC beam realistically, an efficient Hermite-Lagrangian finite element formulation is successfully developed. The accuracy and efficiency of the present model are demonstrated by comparison with published benchmark results. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, and length-to-thickness ratio on the bending and buckling responses of FG-CNTRC beam. Several new referential results are also reported for the first time which will serve as a benchmark for future studies in a similar direction. It is concluded that the FG-X-CNTRC beam is the strongest beam that carries the lowest central deflection and is followed by the UD, V, Λ, and FG-O-CNTRC beam. Besides, the critical buckling load belonging to the FG-X-CNTRC beam is the highest, followed by UD and FG-O.

Keywords

References

  1. Alibeigloo, A. and Liew, K. (2015), "Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using differential quadrature method", Int. J. Appl. Mech., 7, 1550002. https://doi.org/10.1142/S1758825115400025.
  2. Belarbi, M.O., Garg, A., Houari, M.S.A., Hirane, H., Tounsi, A. and Chalak, H.D. (2021a), "A three-unknown refined shear beam element model for buckling analysis of functionally graded curved sandwich beams", Eng. Comput., https://doi.org/10.1007/s00366-021-01452-1.
  3. Belarbi, M.O., Houari, M.S.A., Daikh, A.A., Garg, A., Merzouki, T., Chalak, H.D. and Hirane, H. (2021b), "Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory", Compos. Struct., 264, 113712. https://doi.org/10.1016/j.compstruct.2021.113712.
  4. Belarbi, M.O., Li, L., Ahmed Houari, M.S., Garg, A., Chalak, H. D., Dimitri, R. and Tornabene, F. (2022), "Nonlocal vibration of functionally graded nanoplates using a layerwise theory", Mathematic. Mech. Solids. https://doi.org/10.1177/10812865221078571.
  5. Bouazza, M. and Zenkour, A.M. (2020), "Vibration of carbon nanotube-reinforced plates via refined nth-higher-order theory", Archive Appl. Mech., 90, 1755-1769. https://doi.org/10.1007/s00419-020-01694-3.
  6. Chareonsuk, J. and Vessakosol, P. (2011), "Numerical solutions for functionally graded solids under thermal and mechanical loads using a high-order control volume finite element method", Appl. Thermal Eng., 31, 213-227. https://doi.org/10.1016/j.applthermaleng.2010.09.001.
  7. Daikh, A.A., Drai, A., Houari, M.S.A. and Eltaher, M.A. (2020), "Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes", Steel Compos. Struct., 36, 643-656. http://dx.doi.org/10.12989/scs.2020.36.6.643.
  8. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Chakraverty, S. and Eltaher, M.A. (2021a), "Analysis of axially temperaturedependent functionally graded carbon nanotube reinforced composite plates", Eng. Comput., https://doi.org/10.1007/s00366-021-01413-8.
  9. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Mohamed, S.A. and Eltaher, M.A. (2021b), "Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory", Defence Technol., https://doi.org/10.1016/j.dt.2021.09.011.
  10. Duc, N.D., Lee, J., Nguyen-Thoi, T. and Thang, P.T. (2017), "Static response and free vibration of functionally graded carbon nanotube-reinforced composite rectangular plates resting on Winkler-Pasternak elastic foundations", Aeros. Sci. Technol., 68, 391-402. https://doi.org/10.1016/j.ast.2017.05.032.
  11. Ebrahimi, F. and Farazamandnia, N. (2017), "Thermo-mechanical analysis of carbon nanotube-reinforced composite sandwich beams", Coupl. Syst. Mech., 6, 207-227. http://dx.doi.org/10.12989/csm.2017.6.2.207.
  12. Ebrahimi, F. and Farazmandnia, N. (2017), "Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory", Mech. Adv. Mater. Struct., 24, 820-829. https://doi.org/10.1080/15376494.2016.1196786.
  13. Ebrahimi, F. and Farazmandnia, N. (2018), "Vibration analysis of functionally graded carbon nanotubereinforced composite sandwich beams in thermal environment", Adv. Aircraft Spacecraft Sci., 5, 107. http://dx.doi.org/10.12989/aas.2018.5.1.107.
  14. Esawi, A.M. and Farag, M.M. (2007), "Carbon nanotube reinforced composites: potential and current challenges", Mater. Des., 28, 2394-2401. https://doi.org/10.1016/j.matdes.2006.09.022.
  15. Fan, Y. and Wang, H. (2015), "Nonlinear vibration of matrix cracked laminated beams containing carbon nanotube reinforced composite layers in thermal environments", Compos. Struct., 124, 35-43. https://doi.org/10.1016/j.compstruct.2014.12.050.
  16. Fan, Y. and Wang, H. (2016a), "The effects of matrix cracks on the nonlinear bending and thermal postbuckling of shear deformable laminated beams containing carbon nanotube reinforced composite layers and piezoelectric fiber reinforced composite layers", Compos. Part B: Eng., 106, 28-41. https://doi.org/10.1016/j.compositesb.2016.09.005.
  17. Fan, Y. and Wang, H. (2016b), "Nonlinear bending and postbuckling analysis of matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers in thermal environments", Compos. Part B: Eng., 86, 1-16. https://doi.org/10.1016/j.compositesb.2015.09.048.
  18. Garg, A., Belarbi, M.O., Chalak, H. and Chakrabarti, A. (2020), "A review of the analysis of sandwich FGM structures", Compos. Struct., 113427. https://doi.org/10.1016/j.compstruct.2020.113427.
  19. Garg, A., Chalak, H.D., Belarbi, M.O., Zenkour, A.M. and Sahoo, R. (2021a), "Estimation of carbon nanotubes and their applications as reinforcing composite materials-An engineering review", Compos. Struct., 272, 114234. https://doi.org/10.1016/j.compstruct.2021.114234.
  20. Garg, A., Chalak, H.D., Li, L., Belarbi, M.O., Sahoo, R. and Mukhopadhyay, T. (2022a), "Vibration and buckling analyses of sandwich plates containing functionally graded metal foam core", Acta Mech. Solida Sinica. https://doi.org/10.1007/s10338-021-00295-z.
  21. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Houari, M.S.A. (2021b), "A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures", Archiv. Comput. Meth. Eng., https://doi.org/10.1007/s11831-021-09652-0.
  22. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Sahoo, R. (2022b), "Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore", Thin-Wall. Struct., 170, 108626. https://doi.org/10.1016/j.tws.2021.108626.
  23. Ghannadpour, S., Mohammadi, B. and Fazilati, J. (2013), "Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method", Compos. Struct., 96, 584-589. https://doi.org/10.1016/j.compstruct.2012.08.024.
  24. Gibson, R.F., Ayorinde, E.O. and Wen, Y.F. (2007), "Vibrations of carbon nanotubes and their composites: a review", Compos. Sci. Technol., 67, 1-28. https://doi.org/10.1016/j.compscitech.2006.03.031.
  25. Han, Y. and Elliott, J. (2007), "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci., 39, 315-323. https://doi.org/10.1016/j.commatsci.2006.06.011.
  26. Hirane, H., Belarbi, M.O., Houari, M.S.A. and Tounsi, A. (2021), "On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates", Eng. Comput., https://doi.org/10.1007/s00366-020-01250-1.
  27. Jedari Salami, S. (2018), "Free vibration analysis of sandwich beams with carbon nanotube reinforced face sheets based on extended high-order sandwich panel theory", J. Sandw. Struct. Mater., 20, 219-248. https://doi.org/10.1177/1099636216649788.
  28. Jedari Salami, S. (2016), "Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets", Physica E: Low-Dimens. Syst. Nanostruct., 76, 187-197. https://doi.org/10.1016/j.physe.2015.10.015.
  29. Karami, B. and Janghorban, M. (2020), "On the mechanics of functionally graded nanoshells", Int. J. Eng. Sci., 153, 103309. https://doi.org/10.1016/j.ijengsci.2020.103309.
  30. Karami, B., Janghorban, M., Shahsavari, D., Dimitri, R. and Tornabene, F. (2019a), "Nonlocal buckling analysis of composite curved beams reinforced with functionally graded carbon nanotubes", Molecules, 24, 2750. https://doi.org/10.3390/molecules24152750.
  31. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A.J. (2018a), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28, 99-110. https://doi.org/10.12989/scs.2018.28.1.099.
  32. Karami, B., Shahsavari, D. and Janghorban, M. (2018b), "A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates", Aeros. Sci. Technol., 82-83, 499-512. https://doi.org/10.1016/j.ast.2018.10.001.
  33. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019b), "Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates", Mater. Res. Express, 6, 0950a9. https://doi.org/10.1088/2053-1591/ab3474.
  34. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019c), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Int. J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036.
  35. Ke, L.L., Yang, J. and Kitipornchai, S. (2013), "Dynamic stability of functionally graded carbon nanotube-reinforced composite beams", Mech. Adv. Mater. Struct., 20, 28-37. https://doi.org/10.1080/15376494.2011.581412.
  36. Keshtegar, B., Kolahchi, R., Eyvazian, A. and Trung, N.T. (2021), "Dynamic stability analysis in hybrid nanocomposite polymer beams reinforced by carbon fibers and carbon nanotubes", Polymers, 13, 106. https://doi.org/10.1080/15376494.2011.581412.
  37. Khadimallah, M.A. and Hussain, M. (2020), "Effect of power law index for vibration of armchair and zigzag single walled carbon nanotubes", Steel Compos. Struct, 37, 621-632. http://dx.doi.org/10.12989/scs.2020.37.5.621.
  38. Khaniki, H.B. and Ghayesh, M.H. (2020), "A review on the mechanics of carbon nanotube strengthened deformable structures", Eng. Struct., 220, 110711. https://doi.org/10.1016/j.engstruct.2020.110711.
  39. Khelifa, Z., Hadji, L., Daouadji, T.H. and Bourada, M. (2018), "Buckling response with stretching effect of carbon nanotubereinforced composite beams resting on elastic foundation", Struct. Eng. Mech., 67, 125-130. http://dx.doi.org/10.12989/sem.2018.67.2.125.
  40. Kiani, Y. and Mirzaei, M. (2019), "Nonlinear stability of sandwich beams with carbon nanotube reinforced faces on elastic foundation under thermal loading", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233, 1701-1712. https://doi.org/10.1177/0954406218772613.
  41. Kumar, A., Sharma, K. and Dixit, A.R. (2020), "Carbon nanotubeand graphene-reinforced multiphasepolymeric composites: review on their properties and applications", J. Mater. Sci., 1-43. https://doi.org/10.1007/s10853-019-04196-y.
  42. Liew, K., Lei, Z. and Zhang, L. (2015), "Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review", Compos. Struct., 120, 90-97. https://doi.org/10.1016/j.compstruct.2014.09.041.
  43. Mayandi, K. and Jeyaraj, P. (2015), "Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load", Proceedings of the Institution of Mechanical Engineers, Part L: J. Mater. Des. Appl., 229, 13-28. https://doi.org/10.1177/1464420713493720.
  44. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle", Steel Compos. Struct, 32, 595-610. http://dx.doi.org/10.12989/scs.2019.32.5.595.
  45. Mirzaei, M. and Kiani, Y. (2015), "Snap-through phenomenon in a thermally postbuckled temperature dependent sandwich beam with FG-CNTRC face sheets", Compos. Struct., 134, 1004-1013. https://doi.org/10.1016/j.compstruct.2015.09.003.
  46. Mohammadimehr, M. and Alimirzaei, S. (2017), "Buckling and free vibration analysis of tapered FG-CNTRC micro Reddy beam under longitudinal magnetic field using FEM", Smart Struct. Syst., 19, 309-322. http://dx.doi.org/10.12989/sss.2017.19.3.309.
  47. Mohammadimehr, M., Mohammadi-Dehabadi, A., Akhavan Alavi, S., Alambeigi, K., Bamdad, M., Yazdani, R. and Hanifehlou, S. (2018), "Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocomposite", Steel Compos. Struct., 29, 405-422. http://dx.doi.org/10.12989/scs.2018.29.3.405.
  48. Mohseni, A. and Shakouri, M. (2019), "Vibration and stability analysis of functionally graded CNT-reinforced composite beams with variable thickness on elastic foundation", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 233, 2478-2489. https://doi.org/10.1177/1464420719866222.
  49. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018), "Bending analysis of bi-directional functionally graded EulerBernoulli nano-beams using integral form of Eringen's nonlocal elasticity theory", Struct. Eng. Mech., 67, 417-425. http://dx.doi.org/10.12989/sem.2018.67.4.417.
  50. Nejati, M., Eslampanah, A. and Najafizadeh, M. (2016), "Buckling and vibration analysis of functionally graded carbon nanotubereinforced beam under axial load", Int. J. Appl. Mech., 8, 1650008. https://doi.org/10.1142/S1758825116500083.
  51. Nguyen, T.K. and Nguyen, B.D. (2015), "A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams", J. Sandw. Struct. Mater., 17, 613-631. https://doi.org/10.1177/1099636215589237.
  52. Phung-Van, P., Lieu, Q.X., Nguyen-Xuan, H. and Wahab, M.A. (2017), "Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates", Compos. Struct., 166, 120-135. https://doi.org/10.1016/j.compstruct.2017.01.049.
  53. Pouresmaeeli, S. and Fazelzadeh, S. (2017), "Uncertain buckling and sensitivity analysis of functionally graded carbon nanotubereinforced composite beam", Int. J. Appl. Mech., 9, 1750071. https://doi.org/10.1142/S1758825117500715.
  54. Safari, M., Mohammadimehr, M. and Ashrafi, H. (2021), "Free vibration of electro-magneto-thermo sandwich Timoshenko beam made of porous core and GPLRC", Adv. Nano Res., 10, 115-128. http://dx.doi.org/10.12989/anr.2021.10.2.11.
  55. Salvetat-Delmotte, J.P. and Rubio, A. (2002), "Mechanical properties of carbon nanotubes: a fiber digest for beginners", Carbon, 40, 1729-1734. https://doi.org/10.1016/S0008-6223(02)00012-X.
  56. Sharma, R., Jadon, V. and Singh, B. (2015), "A Review on the Finite Element Methods for Heat Conduction in Functionally Graded Materials", J. Instit. Eng. (India): Series C, 96, 73-81. https://doi.org/10.1007/s40032-014-0125-1.
  57. She, G.L., Liu, H.B. and Karami, B. (2020), "On resonance behavior of porous FG curved nanobeams", Steel Compos. Struct, 36, 179-186. https://doi.org/10.12989/scs.2020.36.2.179.
  58. She, G.L., Liu, H.B. and Karami, B. (2021), "Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets", Thin-Wall. Struct., 160, 107407. https://doi.org/10.1016/j.tws.2020.107407.
  59. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91, 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026.
  60. Shen, H.S. (2015), "Nonlinear analysis of functionally graded fiber reinforced composite laminated beams in hygrothermal environments, Part I: Theory and solutions", Compos. Struct., 125, 698-705. https://doi.org/10.1016/j.compstruct.2014.12.024
  61. 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. Non-Linear Mech., 91, 69-75. https://doi.org/10.1016/j.ijnonlinmec.2017.02.010.
  62. 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.
  63. 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", Sci. Reports, 7, 1-18. https://doi.org/10.1038/s41598-017-12596-w.
  64. Singh, S.D. and Sahoo, R. (2020), "Static and free vibration analysis of functionally graded CNT reinforced composite plates using trigonometric shear deformation theory", Struct., 28, 685-696. https://doi.org/10.1016/j.istruc.2020.09.008.
  65. Sobhy, M. and Zenkour, A.M. (2018), "Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets on a viscoelastic substrate", Compos. Part B: Eng., 154, 492-506. https://doi.org/10.1016/j.compositesb.2018.09.011.
  66. Tagrara, S., 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, 1259-1277. http://dx.doi.org/10.12989/scs.2015.19.5.1259.
  67. Talebizadehsardari, P., Eyvazian, A., Asmael, M., Karami, B., Shahsavari, D. and Mahani, R. B. (2020), "Static bending analysis of functionally graded polymer composite curved beams reinforced with carbon nanotubes", Thin-Wall. Struct., 157, 107139. https://doi.org/10.1016/j.tws.2020.107139.
  68. Thai, C.H., Zenkour, A., Wahab, M.A. and Nguyen-Xuan, H. (2016), "A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis", Compos. Struct., 139, 77-95. https://doi.org/10.1016/j.compstruct.2015.11.066.
  69. Thanh, C.L., Phung-Van, P., Thai, C.H., Nguyen-Xuan, H. and Abdel Wahab, M. (2018), "Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory", Compos. Struct.,184, 633-649. https://doi.org/10.1016/j.compstruct.2017.10.025.
  70. Thostenson, E.T., Ren, Z. and Chou, T.W. (2001), "Advances in the science and technology of carbon nanotubes and their composites: a review", Compos. Sci. Technol., 61, 1899-1912. https://doi.org/10.1016/S0266-3538(01)00094-X.
  71. Tong, G., Liu, Y., Cheng, Q. and Dai, J. (2020), "Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect", Europ. J. Mech. A/Solids, 80, 103929. https://doi.org/10.1016/j.euromechsol.2019.103929.
  72. Vinh, P.V., Belarbi, M.O. and Tounsi, A. (2022), "Wave propagation analysis of functionally graded nanoplates using nonlocal higher-order shear deformation theory with spatial variation of the nonlocal parameters", Waves Random Complex Media, 1-21. https://doi.org/10.1080/17455030.2022.2036387.
  73. Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), "A quasi-3D theory for vibration and buckling of functionally graded sandwich beams", Compos. Struct., 119, 1-12. https://doi.org/10.1016/j.compstruct.2014.08.006.
  74. 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.
  75. Wu, H., Kitipornchai, S. and Yang, J. (2015), "Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets", Int. J. Struct. Stab., 15, 1540011. https://doi.org/10.1142/S0219455415400118.
  76. Xu, Y. (2020), "Combined effect of carbon nanotubes distribution and orientation on functionally graded nanocomposite beams using finite element analysis", Mater. Res. Express. https://doi.org/10.1088/2053-1591/abc773.
  77. Yang, J., Ke, L.L. and Feng, C. (2015), "Dynamic buckling of thermo-electro-mechanically loaded FG-CNTRC beams", Int. J. Struct. Stab., 15, 1540017. https://doi.org/10.1142/S0219455415400179.
  78. Yas, M. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Pressure Vessels Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012.
  79. Yengejeh, S.I., Kazemi, S.A. and Ochsner, A. (2017), "Carbon nanotubes as reinforcement in composites: a review of the analytical, numerical and experimental approaches", Comput. Mater. Sci., 136, 85-101. https://doi.org/10.1016/j.commatsci.2017.04.023.
  80. Yu, Y. and Shen, H.S. (2020), "A Comparison of nonlinear bending and vibration of hybrid metal/CNTRC laminated beams with positive and negative poisson's ratios", Int. J. Struct. Stab. Dyn., 2043007. https://doi.org/10.1142/S0219455420430075.
  81. Zenkour, A.M. (2018), "Nonlocal elasticity and shear deformation effects on thermal buckling of a CNT embedded in a viscoelastic medium", Europ. Phys. J. Plus, 133, 196. https://doi.org/10.1140/epjp/i2018-12014-2.
  82. Zghal, S., Ataoui, D. and Dammak, F. (2020a), "Static bending analysis of beams made of functionally graded porous materials", Mech. Based Des. Struct. Mach., 1-18. https://doi.org/10.1080/15397734.2020.1748053.
  83. Zghal, S., Frikha, A. and Dammak, F. (2017), "Static analysis of functionally graded carbon nanotube-reinforced plate and shell structures", Compos. Struct., 176, 1107-1123. https://doi.org/10.1016/j.compstruct.2017.06.015.
  84. Zhang, H., Gao, C., Li, H., Pang, F., Zou, T., Wang, H. and Wang, N. (2020), "Analysis of functionally graded carbon nanotubereinforced composite structures: A review", Nanotechnol. Rev., 9, 1408-1426. https://doi:10.1515/ntrev-2020-0110.