DOI QR코드

DOI QR Code

Bending of axially functionally graded carbon nanotubes reinforced composite nanobeams

  • Ahmed Drai (Department of Mechanical Engineering, Mustapha STAMBOULI University of Mascara) ;
  • Ahmed Amine Daikh (Department of Technology, University Centre of Naama) ;
  • Mohamed Oujedi Belarbi (Laboratoire de Recherche en Genie Civil, LRGC, Universite de Biskra) ;
  • Mohammed Sid Ahmed Houari (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Departement de Genie Civil, Faculte des Sciences et de la Technologie, Universite Mustapha Stambouli) ;
  • Benoumer Aour (LABAB Laboratory of ENPO) ;
  • Amin Hamdi (Department of Civil Engineering, Faculty of Engineering, King Abdulaziz University) ;
  • Mohamed A. Eltaher (Faculty of Engineering, Mechanical Design and Production Department, Zagazig University)
  • Received : 2021.09.08
  • Accepted : 2022.11.11
  • Published : 2023.03.25

Abstract

This work presents a modified analytical model for the bending behavior of axially functionally graded (AFG) carbon nanotubes reinforced composite (CNTRC) nanobeams. New higher order shear deformation beam theory is exploited to satisfy parabolic variation of shear through thickness direction and zero shears at the bottom and top surfaces.A Modified continuum nonlocal strain gradient theoryis employed to include the microstructure and the geometrical nano-size length scales. The extended rule of the mixture and the molecular dynamics simulations are exploited to evaluate the equivalent mechanical properties of FG-CNTRC beams. Carbon nanotubes reinforcements are distributed axially through the beam length direction with a new power graded function with two parameters. The equilibrium equations are derived with associated nonclassical boundary conditions, and Navier's procedure are used to solve the obtained differential equation and get the response of nanobeam under uniform, linear, or sinusoidal mechanical loadings. Numerical results are carried out to investigate the impact of inhomogeneity parameters, geometrical parameters, loadings type, nonlocal and length scale parameters on deflections and stresses of the AFG CNTRC nanobeams. The proposed model can be used in the design and analysis of MEMS and NEMS systems fabricated from carbon nanotubes reinforced composite nanobeam.

Keywords

Acknowledgement

This research was supported by the Algerian Directorate General of Scientific Research and Technological Development (DGRSDT) and Mustapha STAMBOULI university of Mascara (UMS Mascara) in Algeria. The authors gratefully acknowledge the scientific support of the laboratory of "Etude des Structures et de Mécanique des Matériaux" (UMS Mascara, Algeria), and the laboratory of applied biomechanics and biomaterials (LABAB, ENP Oran, Algeria).

References

  1. Abdelrahman, A.A., Esen, I., O zarpa, C. and Eltaher, M.A. (2021), "Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory", Appl. Math. Modell., 96, 215-235. https://doi.org/10.1016/j.apm.2021.03.008 
  2. Abdelrahman, A.A. and Eltaher, M.A. (2020), "On bending and buckling responses of perforated nanobeams including surface energy for different beams theories", Eng. Comput., 1-27. https://doi.org/10.1007/s00366-020-01211-8 
  3. Abo-Bakr, H.M., Abo-Bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020), "Weight optimization of axially functionally graded microbeams under buckling and vibration behaviors", Mech. Based Des. Struct., 1-22. https://doi.org/10.1080/15397734.2020.1838298 
  4. Abo-bakr, H.M., Abo-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2021), "Multi-objective shape optimization for axially functionally graded microbeams", Compos. Struct., 258, 113370. https://doi.org/10.1016/j.compstruct.2020.113370 
  5. Abo-Bakr, R.M., Eltaher, M.A. and Attia, M.A. (2020), "Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects", Eng. Comput., 1-22. https://doi.org/10.1007/s00366-020-01146-0 
  6. Akgoz, B. (2019), "Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions", Steel Compos. Struct., 33(1),133-142. http://dx.doi.org/10.12989/scs.2019.33.1.133 
  7. Alazwari, M.A., Daikh, A.A. and Eltaher, M.A. (2022), "Novel quasi 3D theory for mechanical responses of FG-CNTs reinforced composite nanoplates", Adv. Nano Res., 12(2). https://doi.org/https://doi.org/10.12989/anr.2022.12.2.000 
  8. Askes, H. and Aifantis, E.C. (2009), "Gradient elasticity and flexural wave dispersion in carbon nanotubes", Phys. Rev. B, 80(19), 195412. https://doi.org/10.1103/PhysRevB.80.195412 
  9. Bekhadda, A., Cheikh, A., Bensaid, I., Hadjoui, A. and Daikh, A. (2019), "A novel first order refined shear-deformation beam theory for vibration and buckling analysis of continuously graded beams", Advances in Aircraft and Spacecraft Science, 6(3), 189-206. https://doi.org/10.12989/aas.2019.6.3.189 
  10. Belarbi, M.O. and Tati, A. (2016), "Bending analysis of composite sandwich plates with laminated face sheets: New finite element formulation", J. Solid Mech., 8(2), 280-299. 
  11. Belarbi, M.O., Garg, A., Houari, M.S.A., Hirane, H., Tounsi, A. and Chalak, H.D. (2021), "A three-unknown refined shear beam element model for buckling analysis of functionally graded curved sandwich beams" Eng. Comput., 1-28. https://doi.org/10.1007/s00366-021-01452-1 
  12. 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", Math. Mech. Solids, 10812865221078571. https://doi.org/10.1177/10812865221078571 
  13. Belarbi, M.O., Houari, M.S.A., Hirane, H., Daikh, A.A. and Bordas, S.P.A. (2022a), "On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory", Compos. Struct., 279, 114715. https://doi.org/10.1016/j.compstruct.2021.114715 
  14. Belarbi, M.O., Houari, M.S.A. Daikh, A.A. Garg, A. Merzouki, T. Chalak, H.D. and Hirane, H. (2021), "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 
  15. Belarbi, M.O., Zenkour, A.M. Tati, A. Salami, S.J. Khechai, A. and Houari, M.S.A. (2020), "An efficient eight-node quadrilateral element for free vibration analysis of multilayer sandwich plates", Int. J. Numer. Method Eng., 122(9), 2360-2387. https://doi.org/10.1002/nme.6624. 
  16. Bensaid, I., Daikh, A.A. and Drai, A. (2019), "Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects", J. Mech. Eng. Sci., 234(18). https://doi.org/10.1177/0954406220916481 
  17. Cao, D., Gao, Y. Yao, M.and Zhang, W. (2018), "Free vibration of axially functionally graded beams using the asymptotic development method", Eng. Struct., 173, 442-448. https://doi.org/10.1016/j.engstruct.2018.06.111 
  18. Chaht, F.L., Kaci, A. Houari, M.S.A., Tounsi, A. Beg, O.A. and Mahmoud, S.R. (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. http://doi.org/10.12989/scs.2015.18.2.425 
  19. Civalek, O ., Dastjerdi, S., Akbas S. and Akgoz B. (2021), "Vibration analysis of carbon nanotube-reinforced composite microbeams", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.7069 
  20. Daikh, A.A. (2019), "Temperature dependent vibration analysis of functionally graded sandwich plates resting on Winkler/ Pasternak/Kerr foundation", Mater. Res. Express, 6, 065702. https://doi.org/10.1088/2053-1591/ab097b 
  21. Daikh, A.A. and Zenkour, A.M. (2019a), "Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory". Mater. Res. Express ,6, 115707. https://doi.org/10.1088/2053-1591/ab48a9 
  22. Daikh, A.A. and Zenkour, A.M. (2019b), "Effect of porosity on the bending analysis of various functionally graded sandwich plates", Mater Res Express, 6, 065703. https://doi.org/10.1088/2053-1591/ab0971 
  23. Daikh, A.A., Guerroudj, M., Elajrami, M., Megueni, A., (2019a), "Thermal buckling of functionally graded sandwich beams", Adv. Mater. Res., 1156, 43-59. https://doi.org/10.4028/www.scientific.net/AMR.1156.43 
  24. Daikh, A.A., Houari, M.S.A. and Tounsi, A. (2019b), "Buckling analysis of porous FGM sandwich nanoplates due to heat conduction via nonlocal strain gradient theory", Eng. Res. Express, 1, 015022. https://doi.org/10.1088/2631-8695/ab38f9 
  25. Daikh, A.A. and Zenkour, A.M. (2020), "Bending of functionally graded sandwich nanoplates resting on Pasternak foundation under different boundary conditions", J. Appl. Comput. Mech., 6(Special Issue), 1245-1259. https://doi.org/10.22055/JACM.2020.33136.2166 
  26. Daikh, A.A., Drai, A. Bensaid, I. Houari, M.S.A. and Tounsi, A. (2020a), "On vibration of functionally graded sandwich nanoplates in the thermal environment", J. Sandw. Struct. Mater., https://doi.org/10.1177/1099636220909790 
  27. Daikh, A.A., Bachiri, A. Houari, M.S.A. and Tounsi, A. (2020b), "Size dependent free vibration and buckling of multilayered carbon nanotubes reinforced composite nanoplates in thermal environment", Mech. Based Des. Struct., 50(4), 1371-1399. https://doi.org/10.1080/15397734.2020.1752232 
  28. Daikh, A.A., Houari, M.S.A. and Eltaher, M.A. (2020c), "A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates", Compos. Struct., 262, 113347. https://doi.org/10.1016/j.compstruct.2020.113347 
  29. Daikh, A.A., Bensaid, I. and Zenkour, A.M. (2020d), "Temperature dependent thermomechanical bending response offunctionally graded sandwich plates", Eng. Res. Express, 2, 015006. https://doi.org/10.1088/2631-8695/ab638c 
  30. Daikh, A.A., Bensaid, I., Bachiri, A., Houari, M.S.A. Tounsi, A., Merzouki, T. (2020e), "On static bending of multilayered carbon nanotube-reinforced composite plates", Comput. Concr., 26(2), 137-150. https://doi.org/10.12989/cac.2020.26.2.137 
  31. Daikh, A.A., Drai, A., Houari, M.S.A. and Eltaher, M.A. (2020f), "Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes", Steel Compos. Struct., 36(6), 643-656. http://doi.org/10.12989/scs.2020.36.6.643 
  32. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Chakraverty, S. and Eltaher, M.A. (2021a), "Analysis of axially temperature- dependent functionally graded carbon nanotube reinforced composite plates", Eng. Comput., 38(Suppl 3), 2533-2554. https://doi.org/10.1007/s00366-021-01413-8 
  33. Daikh, A.A., Houari, M.S.A., Karami, B., Eltaher, M.A., Dimitri, R. and Tornabene, F. (2021b), "Buckling analysis of CNTRC curved sandwich nanobeams in thermal Environment", Appl. Sci., 11, 3250. https://doi.org/10.3390/app11073250 
  34. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Mohamed, S.A. and Eltaher, M.A. (2021c), "Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory", Defence Technology, 18(10), 1778-1809. https://doi.org/10.1016/j.dt.2021.09.011 
  35. 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", Aerosp. Sci. Technol., 68, 391-402. https://doi.org/10.1016/j.ast.2017.05.032 
  36. Ebrahimi, F. and Barati, M.R. (2018), "Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium", J. Mech. Eng. Sci., 232(11), 2067-2078. https://doi.org/10.1177/0954406217713518 
  37. El-Ashmawy, A.M. and Xu, Y. (2021), "Combined effect of carbon nanotubes distribution and orientation on functionally graded nanocomposite beams using finite element analysis", Mater. Res. Express, 8(1), 015012. https://doi.org/10.1088/2053-1591/abc773 
  38. Eltaher, M.A. and Mohamed, N. (2020a), "Nonlinear stability and vibration of imperfect CNTs by doublet mechanics" Appl. Math. Comput., 382, 125311. https://doi.org/10.1016/j.amc.2020.125311 
  39. Eltaher, M.A., El-Borgi, S. and Reddy, J.N. (2016), "Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs", Compos. Struct., 153, 902-913. https://doi.org/10.1016/j.compstruct.2016.07.013 
  40. Eltaher, M.A., Mohamed, N. and Mohamed, S.A. (2020b), "Nonlinear buckling and free vibration of curved CNTs by doublet mechanics", Smart Struct. Syst., 26(2), 213-226. http://doi.org/10.12989/sss.2020.26.2.213 
  41. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5 
  42. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803 
  43. Esen, I., Daikh, A.A. and Eltaher, M.A. (2021), "Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubesunder moving point load", Eur. Phys. J. Plus, 136, 458. https://doi.org/10.1140/epjp/s13360-021-01419-7 
  44. Esen, I., O zarpa, C. and Eltaher, M.A. (2021), "Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment", Compos. Struct., 261, 113552. https://doi.org/10.1016/j.compstruct.2021.113552 
  45. 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 
  46. Fan, Y. and Wang, H. (2016), "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 
  47. Fan, Y. and Wang, H. (2016), "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 
  48. Ferreira, A., Castro, L.M. and Bertoluzza, S. (2009), "A high order collocation method for the static and vibration analysis of composite plates using a first-order theory", Compos. Struct., 89(3), 424-432. https://doi.org/10.1016/j.compstruct.2008.09.006 
  49. Fiedler, B., Gojny, F.H. Wichmann, M.H. Nolte, M.C. and Schulte, K. (2006), "Fundamental aspects of nano-reinforced composites", Compos. Sci. Technol., 66(16), 3115-3125. https://doi.org/10.1016/j.compscitech.2005.01.014 
  50. Garg, A. and Chalak, H. (2020), "Novel higher-order zigzag theory for analysis of laminated sandwich beams", J. Mater. Des. Appl., 235(1), 176-194. https://doi.org/10.1177/1464420720957045 
  51. Garg, A., Chalak, H.D., Belarbi, M.O., Zenkour, A.M. and Sahoo, R. (2021), "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 
  52. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Houari, M.S.A. (2021a), "A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures", Arch. Comput. Methods Eng., 1-34. https://doi.org/10.1007/s11831-021-09652-0 
  53. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Sahoo, R. (2022), "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 
  54. 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 Mechanica Solida Sinica, 1-16. https://doi.org/10.1007/s10338-021-00295-z 
  55. Garg, A., Belarbi, M.O. Chalak, H. and Chakrabarti, A. (2020a), "A review of the analysis of sandwich FGM structures", Compos. Struct., 113427. https://doi.org/10.1016/j.compstruct.2020.113427 
  56. Garg, A., Chalak, H.D. and Chakrabarti, A. (2020b), "Bending analysis of functionally graded sandwich plates using HOZT including transverse displacement effects", Mech. Based Des. Struct., 1-15. https://doi.org/10.1080/15397734.2020.1814157 
  57. 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 
  58. Ghayesh, M.H. and Farajpour, A. (2019), "A review on the mechanics of functionally graded nanoscale and microscale structures", Int. J. Eng. Sci., 137, 8-36. https://doi.org/10.1016/j.ijengsci.2018.12.001 
  59. Ghandourah, E.E., Daikh, A.A., Alhawsawi, A.M., Fallatah, O.A., Eltaher, M.A. (2022), "Bending and buckling of FG-GRNC laminated plates via Quasi-3D nonlocal strain gradient theory", Mathematics, 10(8), 1321. https://doi.org/10.3390/math10081321 
  60. Ghayesh, M.H. and Farajpour, A. (2019), "Vibrations of shear deformable FG viscoelastic microbeams", Microsyst. Technol., 25(4), 1387-1400. https://doi.org/10.1007/s00542-018-4184-8 
  61. Hamed, M.A., Abo-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020), "Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core", Eng. Comput., 36(4), 1929-1946. https://doi.org/10.1007/s00366-020-01023-w 
  62. Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (2019), "Effects of porosity models on static behavior of size dependent functionally graded beam", Struct. Eng. Mech., 71(1), 89-98. http://doi.org/10.12989/sem.2019.71.1.089 
  63. 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., 1-29. https://doi.org/10.1007/s00366-020-01250-1 
  64. Houari, M.S.A., Bessaim, A. Bernard, F. Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., 28(1), 13-24. http://doi.org/10.12989/scs.2018.28.1.013 
  65. Karamanli, A. and Vo, T.P. (2021), "Finite element model for carbon nanotube-reinforced and graphene nanoplateletreinforced composite beams", Compos. Struct., 264, 113739. https://doi.org/10.1016/j.compstruct.2021.113739 
  66. Jankowski, P., Zur, K.K. and Farajpour, A. (2022), "Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect", Eng. Anal. Bound. Elem., 136, 266-289. https://doi.org/10.1016/j.enganabound.2022.01.007 
  67. Karami, B., Janghorban, M. Shahsavari, D., Dimitri, R. and Tornabene, F. (2019), "Nonlocal buckling analysis of composite curved beams reinforced with functionally graded carbon nanotubes", Molecules, 24(15), 2750. https://doi.org/10.3390/molecules24152750 
  68. 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(1), 106. https://doi.org/10.3390/polym13010106 
  69. 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 
  70. Khaniki, H.B. and Ghayesh, M.H. (2020), "On the dynamics of axially functionally graded CNT strengthened deformable beams", Eur. Phys. J. Plus, 135(5), 415. https://doi.org/10.1140/epjp/s13360-020-00433-5 
  71. Khdair, A., Daikh, A.A. and Eltaher, M.A. (2021), "novel fourunknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates", Adv. Nano Res., 11(6), 621-640. https://doi.org/10.12989/anr.2021.11.6.621 
  72. Kumar, A., Sharma, K. and Dixit, A.R. (2020), "Carbon nanotubeand graphene-reinforced multiphase polymeric composites: Review on their properties and applications", J. Mater. Sci., 1-43. https://doi.org/10.1007/s10853-019-04196-y 
  73. Li, C., Zheng, S. and Chen, D. (2020), "Size-dependent isogeometric analysis of bi-directional functionally graded microbeams reinforced by graphene nanoplatelets", J. Mech. Based Des. Struct., 1-19. https://doi.org/10.1080/15397734.2020.1848591 
  74. 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 
  75. 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 
  76. Lim, C.W., Zhang, G. and Redd, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001 
  77. Mindlin, R.D. (1963), "Microstructure in linear elasticity", Technical Report No. AD0424156, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, U.S.A. https://doi.org/10.1007/BF00248490 
  78. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model", Eng. Comput., 37, 2823-2836. https://doi.org/10.1007/s00366-020-00976-2 
  79. Mahesh, V. (2021), "Nonlinear damped transient vibrations of carbon nanotube-reinforced magneto-electro-elastic shells with different electromagnetic circuits", J. Vib. Eng. Technol., 1-24. https://doi.org/10.1007/s42417-021-00380-0 
  80. Mahesh, V. (2021), "Nonlinear pyrocoupled deflection of viscoelastic sandwich shell with CNT reinforced magnetoelectro-elastic facing subjected to electromagnetic loads in thermal environment", Eur. Phys. J. Plus, 136(8), 1-30. https://doi.org/10.1140/epjp/s13360-021-01751-y 
  81. Mahesh, V. (2022), "Effect of carbon nanotube-reinforced magneto-electro-elastic facings on the pyrocoupled nonlinear deflection of viscoelastic sandwich skew plates in thermal environment", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 236(1), 200-221. https://doi.org/10.1177/14644207211044093 
  82. Mahesh, V. (2020), "Nonlinear deflection of carbon nanotube reinforced multiphase magnetoelectro-elastic plates in thermal environment considering pyrocoupling effects", Math. Method Appl. Sci., https://doi.org/10.1177/14644207211044093 
  83. Mahesh, V. and Harursampath, D. (2020), "Nonlinear deflection analysis of CNT/magneto-electro-elastic smart shells under multi-physics loading", Mech. Adv. Mater. Struct., 1-25. https://doi.org/10.1080/15376494.2020.1805059 
  84. Mahesh, V. and Harursampath, D. (2020), "Nonlinear vibration of functionally graded magneto-electro-elastic higher order plates reinforced by CNTs using FEM", Eng. Comput., 1-23. https://doi.org/10.1007/s00366-020-01098-5 
  85. Melaibari, A., Daikh, A.A., Basha, M., Abdalla, A.W., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A. and Eltaher, M.A. (2022a), "A dynamic analysis of randomly oriented functionally graded carbon nanotubes/fiber-reinforced composite laminated shells with different geometries", Mathematics, 10, 408. https://doi.org/10.3390/math10030408 
  86. Melaibari, A., Daikh, A.A., Basha, M., Abdalla, A.W., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A. and Eltaher, M.A. (2022b), "Free vibration of FG-CNTRCs nano-plates/ shells with temperature-dependent properties", Mathematics, 10, 583. https://doi.org/10.3390/math10040583 
  87. 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(4), 417-425. https://doi.org/10.12989/sem.2018.67.4.417 
  88. Nejad, M.Z., Hadi, A. and Farajpour, A. (2017), "Consistent couple-stress theory for free vibration analysis of EulerBernoulli 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 
  89. 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(6), 613-631. https://doi.org/10.1177/1099636215589237 
  90. Nguyen, V.H., Nguyen, T.K. Thai, H.T. and Vo, T.P. (2014), "A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates", Compos. Part B Eng., 66, 233-246. https://doi.org/10.1016/j.compositesb.2014.05.012 
  91. Papargyri-Beskou, S., Tsepoura, K. Polyzos, D. and Beskos, D. (2003), "Bending and stability analysis of gradient elastic beams", Int. J. Solids Struct., 40(2), 385-400. https://doi.org/10.1016/S0020-7683(02)00522-X 
  92. Rajasekaran, S. and Bakhshi Khaniki, H. (2019), "Finite element static and dynamic analysis of axially functionally graded nonuniform small-scale beams based on nonlocal strain gradient theory", Mech. Adv. Mater. Struct., 26(14), 1245-1259. https://doi.org/10.1080/15376494.2018.1432797 
  93. Sarkar, K. and Ganguli, R. (2014), "Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed-fixed boundary condition", Compos. Part B Eng., 58, 361-370. https://doi.org/10.1016/j.compositesb.2013.10.077 
  94. Sayyad, A.S. and Ghugal, Y.M. (2019), "Modeling and analysis of functionally graded sandwich beams: A review", Mech. Adv. Mater. Struct., 26(21), 1776-1795. https://doi.org/10.1080/15376494.2018.1447178 
  95. Shafiei, N., Kazemi, M. Safi, M. and Ghadiri, M. (2016), "Nonlinear vibration of axially functionally graded non-uniform nanobeams", Int. J. Eng. Sci., 106, 77-94. https://doi.org/10.1016/j.ijengsci.2016.05.009 
  96. Shariati, A., Mohammad-Sedighi, H. Zur, K.K. Habibi, M. and Safa, M. (2020), "On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams", Materials, 13(7), 1707. https://doi.org/10.3390/ma13071707 
  97. 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 
  98. 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 
  99. 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 
  100. 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 
  101. Simsek, M. (2015), "Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He's variational method", Compos. Struct., 131, 207-214. https://doi.org/10.1016/j.compstruct.2015.05.004 
  102. Simsek, M. (2019), "Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory", Compos. Struct., 224, 111041. https://doi.org/10.1016/j.compstruct.2019.111041 
  103. 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 
  104. 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 
  105. 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(13), 1899-1912. https://doi.org/10.1016/S0266-3538(01)00094-X 
  106. 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 
  107. Wang, Y. and Wu, D. (2016), "Thermal effect on the dynamic response of axially functionally graded beam subjected to a moving harmonic load", Acta Astronautica, 127, 171-181. https://doi.org/10.1016/j.actaastro.2016.05.030 
  108. Wang, Y., Ren, H. Fu, T.and Shi, C. (2020), "Hygrothermal mechanical behaviors of axially functionally graded microbeams using a refined first order shear deformation theory", Acta Astronautica, 166, 306-316. https://doi.org/10.1016/j.actaastro.2019.10.036 
  109. 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. http://doi.org/10.1016/j.commatsci.2013.01.028 
  110. Yang, F., Chong, A. Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solid Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X 
  111. Yengejeh, S.I., Kazemi, S.A. and O chsner, 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 
  112. 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. Stabil. Dyn., 2043007. https://doi.org/10.1142/S0219455420430075 
  113. Zenkour, A. and Radwan, A. (2020), "Bending and buckling analysis of FGM plates resting on elastic foundations in hygrothermal environment", Arch. Civil Mech. Eng., 20(4), 1-23. https://doi.org/10.1007/s43452-020-00116-z 
  114. Zhen, Y.X., Wen, S.L. and Tang, Y. (2019), "Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model", Physica E, 105, 116-124. https://doi.org/10.1016/j.physe.2018.09.005 
  115. Zheng, S., Chen, D. and Wang, H. (2019), "Size dependent nonlinear free vibration of axially functionally graded tapered microbeams using finite element method", Thin Wall. Struct., 139, 46-52. https://doi.org/10.1016/j.tws.2019.02.033 
  116. Zur, K.K., Farajpour, A., Lim, C.W. and Jankowski, P. (2021), "On the nonlinear dynamics of porous composite nanobeams connected with fullerenes", Compos. Struct., 274, 114356. https://doi.org/10.1016/j.compstruct.2021.114356