Advances in nano research

  • 제11권2호
  • /
  • Pages.157-171
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  • 2021
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  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Free vibration analysis of carbon nanotube RC nanobeams with variational approaches

  • Madenci, Emrah (Department of Civil Engineering, Necmettin Erbakan University)
  • 투고 : 2021.02.06
  • 심사 : 2021.05.22
  • 발행 : 2021.08.25
⟨ 이전 논문 다음 논문 ⟩

초록

There is not enough mixed finite element method (MFEM) model developed for dynamic analysis of carbon nanotube reinforced (CNTRC) composite beams in the literature. In the present study, free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) nanobeams is carried out in the framework of variational formulations. The rule of mixture is employed to estimate the effective material properties of single-walled CNT reinforced nanobeams. Four kinds of CNT distribution of un-axially aligned reinforcement material are investigated in the through-thickness direction of the nanobeams. There are the uniform distribution (UD) and functionally graded distributions FG-O, FG-X and FG-Ʌ of CNTs in the thickness direction of the nanobeams (z axis direction) are assumed here for the analysis. The Hamilton's principle is used to derive governing differential equations based on trigonometric shear deformation beam theory. The effective functional has been constituted for FG-CNTRC nanobeams through a scientific procedure based on the Gâteaux differential. A simple mixed finite element formulation is utilized for the formulation of free vibration problems of FG-CNTRC nanobeams with different boundary conditions. The results of the present method are compared with others from the literature where a good agreement has been found. An effective energy functional and the mixed finite element formulation for FG-CNTRC nanobeams are the original contributions of this study.

키워드

과제정보

The research described in this paper was no financially supported.

참고문헌

  1. Ahmed Houari, M.S., Benyoucef, S., Mechab, I., Tounsi, A. and Adda Bedia, E.A. (2011), "Two-variable refined plate theory for thermoelastic bending analysis of functionally graded sandwich plates", J. Therm. Stresses, 34(4), 315-334. https://doi.org/10.1080/01495739.2010.550806.
  2. Akbas, S.D. (2016), "Analytical solutions for static bending of edge cracked micro beams", Struct. Eng. Mech., 59(3), 579-599. http://doi.org/10.12989/sem.2016.59.3.579.
  3. Akbas, S.D. (2018), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano Res., 6(1), 39. http://doi.org/10.12989/anr.2018.6.1.039.
  4. Akoz, A. and Kadioglu, F. (1996), "The mixed finite element solution of circular beam on elastic foundation", Comput. Struct., 60(4), 643-651. https://doi.org/10.1016/0045-7949(95)00418-1.
  5. Akoz, Y. and Kadioglu, F. (1999), "The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic timoshenko beams", Int. J. Numer. Meth. Eng., 44(12), 1909-1932.https://doi.org/10.1002/(SICI)1097-0207(19990430)44:12<1909::AID-NME573>3.0.CO;2-P.
  6. Al-Furjan, M., Habibi, M., Chen, G., Safarpour, H., Safarpour, M. and Tounsi, A. (2020a), "Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM", Eng. Comput., 1-24. https://doi.org/10.1007/s00366-020-01144-2.
  7. Al-Furjan, M., Habibi, M., Ni, J., won Jung, D. and Tounsi, A. (2020b), "Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-020-01200-x.
  8. Al-Furjan, M., Safarpour, H., Habibi, M., Safarpour, M. and Tounsi, A. (2020c), "A comprehensive computational approach for nonlinear thermal instability of the electrically FG-GPLRC disk based on GDQ method", Eng. Comput., 1-18. https://doi.org/10.1007/s00366-020-01088-7.
  9. Al-Furjan, M., Habibi, M., Ghabussi, A., Safarpour, H., Safarpour, M. and Tounsi, A. (2021a), "Non-polynomial framework for stress and strain response of the FG-GPLRC disk using threedimensional refined higher-order theory", Eng. Struct., 228, 111496. https://doi.org/10.1016/j.engstruct.2020.111496.
  10. Al-Furjan, M., Habibi, M., Shan, L. and Tounsi, A. (2021b), "On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method", Compos. Struct., 257, 113150. https://doi.org/10.1016/j.compstruct.2020.113150.
  11. Alibeigloo, A. (2014), "Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity", Eur. J. Mech. A Solid, 44, 104-115. https://doi.org/10.1016/j.euromechsol.2013.10.002.
  12. Ambartsumian, S. (1958), "On the theory of bending plates", Izv Otd Tech Nauk AN SSSR, 5(5), 69-77.
  13. Arani, A.G., Pourjamshidian, M. and Arefi, M. (2018), "Nonlinear free and forced vibration analysis of sandwich nano-beam with FG-CNTRC face-sheets based on nonlocal strain gradient theory", Smart Struct. Sys., 22(1), 105-120. http://doi.org/10.12989/sss.2018.22.1.105.
  14. Arani, A.G., Pourjamshidian, M., Arefi, M. and Arani, M. (2019), "Thermal, electrical and mechanical buckling loads of sandwich nano-beams made of FG-CNTRC resting on Pasternak's foundation based on higher order shear deformation theory", Struct. Eng. Mech., 69(4), 439-455. http://doi.org/10.12989/sem.2019.69.4.439.
  15. 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., 27(4), 525-536. http://doi.org/10.12989/scs.2018.27.4.525.
  16. Arefi, M., Pourjamshidian, M. and Arani, A.G. (2019), "Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory", Steel Compos. Struct., 32(2), 157-171. http://doi.org/10.12989/scs.2019.32.2.151.
  17. Aribas, U.N., Ermis, M., Eratli, N. and Omurtag, M.H. (2019), "The static and dynamic analyses of warping included composite exact conical helix by mixed FEM", Compos. Part B Eng., 160, 285-297. https://doi.org/10.1016/j.compositesb.2018.10.018.
  18. 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. Comput., 1-22. https://doi.org/10.1007/s00366-021-01382-y.
  19. Asghar, S., Naeem, M.N., Hussain, M., Taj, M. and Tounsi, A. (2020a), "Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis", Comput. Concrete, 25(2), 133-144. http://doi.org/10.12989/cac.2020.25.2.133.
  20. Asghar, S., Naeem, M.N., Khadimallah, M.A., Hussain, M., Iqbal, Z. and Tounsi, A. (2020b), "Effect of chiral structure for free vibration of DWCNTs: Modal analysis", Adv. Concrete Construct., 9(6), 577-588. https://doi.org/10.12989/acc.2020.9.6.577.
  21. Aydogdu, M. (2009), "A new shear deformation theory for laminated composite plates", Compos. Struct., 89(1), 94-101. https://doi.org/10.1016/j.compstruct.2008.07.008.
  22. Balubaid, M., Tounsi, A., Dakhel, B. and Mahmoud, S. (2019), "Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory" Comput. Concrete, 24(6), 579-586. https://doi.org/10.12989/cac.2019.24.6.579.
  23. Bellal, M., Hebali, H., Heireche, H., Bousahla, A.A., Tounsi, A., Bourada, F., Mahmoud, S., Bedia, E. 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.
  24. Bendenia, N., Zidour, M., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H., Bedia, E., Mahmoud, S. 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.
  25. Berghouti, H., Adda Bedia, E., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., 7(5), 351-364. http://dx.doi.org/10.12989/anr.2019.7.5.351.
  26. Bouazza, M. and Zenkour, A.M. (2020), "Vibration of carbon nanotube-reinforced plates via refined nth-higher-order theory", Arch. Appl. Mech., 90, 1755-1769. https://doi.org/10.1007/s00419-020-01694-3.
  27. Bouderba, B., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., 58(3), 397-422. http://doi.org/10.12989/sem.2016.58.3.397.
  28. Boukhari, A., Atmane, H.A., Tounsi, A., Adda Bedia, E. and Mahmoud, S. (2016), "An efficient shear deformation theory for wave propagation of functionally graded material plates", Struct. Eng. Mech., 57(5), 837-859. http://doi.org/10.12989/sem.2016.57.5.837.
  29. Bourada, F., Bousahla, A.A., Tounsi, A., Bedia, E., Mahmoud, S., 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.
  30. Bousahla, A.A., Bourada, F., Mahmoud, S., Tounsi, A., Algarni, A., Bedia, E. 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.
  31. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 191. http://doi.org/10.12989/anr.2019.7.3.191.
  32. Carrera, E. (1998), "Mixed layer-wise models for multilayered plates analysis", Compos. Struct., 43(1), 57-70. https://doi.org/10.1016/S0263-8223(98)00097-X.
  33. Civalek, O., Uzun, B. and Yayli, M.O. (2020), "Frequency, bending and buckling loads of nanobeams with different cross sections", Adv. Nano Res., 9(2), 91-104. http://doi.org/10.12989/anr.2020.9.2.091.
  34. Desai, Y., Ramtekkar, G. and Shah, A. (2003), "Dynamic analysis of laminated composite plates using a layer-wise mixed finite element model", Compos. Struct., 59(2), 237-249. https://doi.org/10.1016/S0263-8223(02)00121-6.
  35. Di Sciuva, M. and Sorrenti, M. (2019), "Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory", Compos. Struct., 227, 111324. https://doi.org/10.1016/j.compstruct.2019.111324.
  36. Ebrahimi, F. and Rostami, P. (2018), "Propagation of elastic waves in thermally affected embedded carbon-nanotube-reinforced composite beams via various shear deformation plate theories", Struct. Eng. Mech., 66(4), 495-504. http://doi.org/10.12989/sem.2018.66.4.495.
  37. Ebrahimi, F., Nouraei, M., Dabbagh, A. and Civalek, O. (2019), "Buckling analysis of graphene oxide powder-reinforced nanocomposite beams subjected to non-uniform magnetic field", Struct. Eng. Mech., 71(4), 351-361. http://doi.org/10.12989/sem.2019.71.4.351.
  38. Emdadi, M., Mohammadimehr, M. and Navi, B.R. (2019), "Free vibration of an annular sandwich plate with CNTRC facesheets and FG porous cores using Ritz method", Adv. Nano Res., 7(2), 109. http://doi.org/10.12989/anr.2019.7.2.109.
  39. Farazin, A. and Mohammadimehr, M. (2020), "Nano research for investigating the effect of SWCNTs dimensions on the properties of the simulated nanocomposites: a molecular dynamics simulation", Adv. Nano Res., 9(2), 83-90. http://doi.org/10.12989/anr.2020.9.2.083.
  40. Feng, H., Shen, D. and Tahouneh, V. (2020), "Vibration analysis of sandwich sector plate with porous core and functionally graded wavy carbon nanotube-reinforced layers", Steel Compos. Struct., 37(6), 711. https://doi.org/10.12989/scs.2020.37.6.711.
  41. Gemi, L., Yazman, S., Uludag, M., Dispinar, D. and Tiryakioglu, M. (2017), "The effect of 0.5 wt% additions of carbon nanotubes & ceramic nanoparticles on tensile properties of epoxy-matrix composites: a comparative study", Mater. Sci. Nanotechnol., 1(2), 15-22. http://doi.org/10.35841/nanotechnology.1.2.15-22.
  42. Grover, N., Maiti, D. and Singh, B. (2013), "A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates", Compos. Struct., 95, 667-675. https://doi.org/10.1016/j.compstruct.2012.08.012.
  43. Grover, N., Maiti, D. and Singh, B. (2014), "An efficient C 0 finite element modeling of an inverse hyperbolic shear deformation theory for the flexural and stability analysis of laminated composite and sandwich plates", Finite Elem. Anal. Des., 80, 11-22. https://doi.org/10.1016/j.finel.2013.11.003.
  44. Hadji, L., Atmane, H.A., Tounsi, A., Mechab, I. and Bedia, E.A. (2011), "Free vibration of functionally graded sandwich plates using four-variable refined plate theory", Appl. Math. Mech., 32(7), 925-942. https://doi.org/10.1007/s10483-011-1470-9.
  45. Heidari, F., Afsari, A. and Janghorban, M. (2020), "Several models for bending and buckling behaviors of FG-CNTRCs with piezoelectric layers including size effects", Adv. Nano Res., 9(3), 193-210. http://doi.org/10.12989/anr.2020.9.3.193.
  46. 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.
  47. Hu, H., Onyebueke, L. and Abatan, A. (2010), "Characterizing and modeling mechanical properties of nanocomposites-review and evaluation", J. Miner. Mater. Charact. Eng., 9(4), 275. https://doi.org/10.4236/jmmce.2010.94022.
  48. Huang, X., Hao, H., Oslub, K., Habibi, M. and Tounsi, A. (2021), "Dynamic stability/instability simulation of the rotary sizedependent functionally graded microsystem", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-021-01399-3.
  49. Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., 7(6), 431-442. http://doi.org/10.12989/anr.2019.7.6.431.
  50. Hussain, M., Naeem, M.N. and Tounsi, A. (2020), "Numerical Study for nonlocal vibration of orthotropic SWCNTs based on Kelvin's model", Adv. Concrete Construct., 9(3), 301-312. https://doi.org/10.12989/acc.2020.9.3.301.
  51. Iijima, S. and Ichihashi, T. (1993), "Single-shell carbon nanotubes of 1-nm diameter", Nature, 363(6430), 603-605. https://doi.org/10.1038/363603a0.
  52. Jamali, M., Shojaee, T., Mohammadi, B. and Kolahchi, R. (2019), "Cut out effect on nonlinear post-buckling behavior of FGCNTRC micro plate subjected to magnetic field via FSDT", Adv. Nano Res., 7(6), 405-417. http://dx.doi.org/10.12989/anr.2019.7.6.405.
  53. Karama, M., Afaq, K. and Mistou, S. (2003), "Mechanical behaviour of laminated composite beam by the new multilayered laminated composite structures model with transverse shear stress continuity", Int. J. Solid. Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9.
  54. Khater, H. and Abd el Gawaad, H. (2016), "Characterization of alkali activated geopolymer mortar doped with MWCNT", Construct. Building Mater., 102, 329-337. https://doi.org/10.1016/j.conbuildmat.2015.10.121.
  55. Kim, Y., Kim, D., Choi, H., Yu, S. and Park, K. (2017), "Fatigue performance of deepwater steel catenary riser considering nonlinear soil", Struct. Eng. Mech., 61(6), 737-746. http://doi.org/10.12989/sem.2017.61.6.737.
  56. Kolahdouzan, F., Arani, A.G. and Abdollahian, M. (2018), "Buckling and free vibration analysis of FG-CNTRC-micro sandwich plate", Steel Compos. Struct., 26(3), 273-287. https://doi.org/10.12989/scs.2018.26.3.273.
  57. Kumar, P. and Srinivas, J. (2017), "Free vibration, bending and buckling of a FG-CNT reinforced composite beam: Comparative analysis with hybrid laminated composite beam", Multidiscipline Model. Mater. Struct., 13(4), 590-611. https://doi.org/10.1108/MMMS-05-2017-0032.
  58. Lage, R.G., Soares, C.M., Soares, C.M. and Reddy, J. (2004), "Analysis of adaptive plate structures by mixed layerwise finite elements", Compos. Struct., 66(1), 269-276. https://doi.org/10.1016/j.compstruct.2004.04.048.
  59. 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.
  60. Lin, F. and Xiang, Y. (2014), "Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories", Appl. Math. Model., 38(15), 3741-3754. https://doi.org/10.1016/j.apm.2014.02.008.
  61. Madenci, E. (2019), "A refined functional and mixed formulation to static analyses of fgm beams", Struct. Eng. Mech., 69(4), 427-437. https://doi.org/10.12989/sem.2019.69.4.427.
  62. Madenci, E. and Gulcu, S. (2020), "Optimization of flexure stiffness of FGM beams via artificial neural networks by mixed FEM", Struct. Eng. Mech., 75(5), 633-642. https://doi.org/10.12989/sem.2020.75.5.633.
  63. Madenci, E. and Ozutok, A. (2020), "Variational approximate for high order bending analysis of laminated composite plates", Struct. Eng. Mech., 73(1), 97-108. https://doi.org/10.12989/sem.2020.73.1.097.
  64. Madenci, E., Ozkilic, Y.O. and Gemi, L. (2020), "Experimental and theoretical investigation on flexure performance of pultruded GFRP composite beams with damage analyses", Compos. Struct., 242, 112162. https://doi.org/10.1016/j.compstruct.2020.112162.
  65. 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.
  66. Mahesh, V. and Harursampath, D. (2021), "Large deflection analysis of functionally graded magneto-electro-elastic porous flat panels", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-020-01270-x.
  67. Matouk, H., Bousahla, A.A., Heireche, H., Bourada, F., Bedia, E., Tounsi, A., Mahmoud, S., Tounsi, A. and Benrahou, K. (2020), "Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory", Adv. Nano Res., 8(4), 293-305. https://doi.org/10.12989/anr.2020.8.4.293.
  68. 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(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595.
  69. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017), "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.
  70. Mehrabadi, S.J., Aragh, B.S., Khoshkhahesh, V. and Taherpour, A. (2012), "Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes", Compos. Part B. Eng., 43(4), 2031-2040. https://doi.org/10.1016/j.compositesb.2012.01.067.
  71. Merdaci, S., Tounsi, A., Houari, M.S.A., Mechab, I., Hebali, H. and Benyoucef, S. (2011), "Two new refined shear displacement models for functionally graded sandwich plates", Arch. Appl. Mech., 81(11), 1507-1522. https://doi.org/10.1007/s00419-010-0497-5.
  72. 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(3), 309-322. https://doi.org/10.12989/sss.2017.19.3.309.
  73. Moradi-Dastjerdi, R. (2016), "Wave propagation in functionally graded composite cylinders reinforced by aggregated carbon nanotube", Struct. Eng. Mech., 57(3), 441-456. http://doi.org/10.12989/sem.2016.57.3.441.
  74. Oden, J.T. and Reddy, J.N. (1976), "On mixed finite element approximations", SIAM J. Numer. Anal., 13(3), 393-404. https://doi.org/10.1137/0713035.
  75. Ouakad, H.M., Sedighi, H.M. and Al-Qahtani, H.M. (2020), "Forward and backward whirling of a spinning nanotube nanorotor assuming gyroscopic effects", Adv. Nano Res., 8(3), 245-254. http://doi.org/10.12989/anr.2020.8.3.245.
  76. O zutok, A. and Madenci, E. (2013), "Free vibration analysis of cross-ply laminated composite beams by mixed finite element formulation", Int. J. Struct. Stabil. Dynam., 13(2), 1250056. https://doi.org/10.1142/S0219455412500563.
  77. Ozutok, A., Madenci, E. and Kadioglu, F. (2014), "Free vibration analysis of angle-ply laminate composite beams by mixed finite element formulation using the Gateaux differential", Sci. Eng. Compos. Mater., 21(2), 257-266. https://doi.org/10.1515/secm-2013-0043.
  78. O zutok, A. and Madenci, E. (2017), "Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method", Int. J. Mech. Sci., 130, 234-243. https://doi.org/10.1016/j.ijmecsci.2017.06.013.
  79. Phung-Van, P., Abdel-Wahab, M., Liew, K., Bordas, S. and Nguyen-Xuan, H. (2015), "Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory", Compos. Struct., 123, 137-149. https://doi.org/10.1016/j.compstruct.2014.12.021.
  80. Rajabi, J. and Mohammadimehr, M. (2019), "Bending analysis of a micro sandwich skew plate using extended Kantorovich method based on Eshelby-Mori-Tanaka approach", Comput. Concrete, 23(5), 361-376. https://doi.org/10.12989/cac.2019.23.5.361.
  81. Reddy, J.N. (1984), "A Simple Higher-Order Theory for Laminated Composite Plates", J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719.
  82. Reissner, E. (1975), "On transverse bending of plates, including the effect of transverse shear deformation", Int. J. Solids Struct., 11(5), 569-573. https://doi.org/10.1016/0020-7683(75)90030-X.
  83. Rouabhia, A., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A., Kouider Halim, B., Tounsi, A. and Al-Zahrani, M.M. (2020), "Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory", Steel Compos. Struct., 37(6), 695-709. https://doi.org/10.12989/scs.2020.37.6.695.
  84. 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. http://doi.org/10.12989/scs.2017.24.1.065.
  85. Soldatos, K. and Elishakoff, I. (1992), "A transverse shear and normal deformable orthotropic beam theory", J. Sound Vib., 155(3), 528-533. https://doi.org/10.1016/0022-460X(92)90717-C.
  86. Tahouneh, V. (2017), "Using modified Halpin-Tsai approach for vibrational analysis of thick functionally graded multi-walled carbon nanotube plates", Steel Compos. Struct., 23(6), 657-668. https://doi.org/10.12989/scs.2017.23.6.657.
  87. Taibi, F.Z., Benyoucef, S., Tounsi, A., Bachir Bouiadjra, R., Adda Bedia, E.A. and Mahmoud, S. (2015), "A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 17(2), 99-129. https://doi.org/10.1177/1099636214554904.
  88. Torabi, J. and Ansari, R. (2018), "Thermally induced mechanical analysis of temperature-dependent FG-CNTRC conical shells", Struct. Eng. Mech., 68(3), 313-323. http://doi.org/10.12989/sem.2018.68.3.313.
  89. Tounsi, A., Benguediab, S., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. http://doi.org/10.12989/anr.2013.1.1.001.
  90. Tounsi, A., Houari, M.S.A. and Bessaim, A. (2016), "A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate", Struct. Eng. Mech., 60(4), 547-565. https://doi.org/10.12989/sem.2016.60.4.547.
  91. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y.
  92. Vinyas, M. (2019a), "A higher-order free vibration analysis of carbon nanotube-reinforced magneto-electro-elastic plates using finite element methods", Compos. Part B Eng., 158, 286-301. https://doi.org/10.1016/j.compositesb.2018.09.086.
  93. Vinyas, M. (2019b), "Vibration control of skew magneto-electro-elastic plates using active constrained layer damping", Compos. Struct., 208, 600-617. https://doi.org/10.1016/j.compstruct.2018.10.046.
  94. Vinyas, M. (2020a), "Nonlinear deflection of carbon nanotube reinforced multiphase magneto-electro-elastic plates in thermal environment considering pyrocoupling effects", Math. Method Appl. Sci. https://doi.org/10.1002/mma.6858.
  95. Vinyas, M. (2020b), "On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT", Compos. Struct., 240, 112044. https://doi.org/10.1016/j.compstruct.2020.112044.
  96. Vinyas, M. (2021), "Porosity effect on the nonlinear deflection of functionally graded magneto-electro-elastic smart shells under combined loading", Mech. Adv. Mater. Struct., 1-27. https://doi.org/10.1080/15376494.2021.1875086.
  97. Vinyas, M. and Dineshkumar, H. (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.
  98. Vinyas, M. and Harursampath, D. (2020), "Nonlinear vibrations of magneto-electro-elastic doubly curved shells reinforced with carbon nanotubes", Compos. Struct., 253, 112749. https://doi.org/10.1016/j.compstruct.2020.112749.
  99. Vinyas, M., Nischith, G., Loja, M., Ebrahimi, F. and Duc, N. (2019a), "Numerical analysis of the vibration response of skew magneto-electro-elastic plates based on the higher-order shear deformation theory", Compos. Struct., 214, 132-142. https://doi.org/10.1016/j.compstruct.2019.02.010.
  100. Vinyas, M., Sunny, K., Harursampath, D., Nguyen-Thoi, T. and Loja, M. (2019b), "Influence of interphase on the multi-physics coupled frequency of three-phase smart magneto-electro-elastic composite plates", Compos. Struct., 226, 111254. https://doi.org/10.1016/j.compstruct.2019.111254.
  101. Vinyas, M., Harursampath, D. and Thoi, T.N. (2021), "A higher order coupled frequency characteristics study of smart magneto-electro-elastic composite plates with cut-outs using finite element methods", Defence Technology, 17(1), 100-118. https://doi.org/10.1016/j.dt.2020.02.009.
  102. Vodenitcharova, T. and Zhang, L.C. (2006), "Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube", Int. J. Solids Struct., 43(10), 3006-3024. https://doi.org/10.1016/j.ijsolstr.2005.05.014.
  103. Wu, C.P., Chen, Y.H., Hong, Z.L. and Lin, C.H. (2018a), "Nonlinear vibration analysis of an embedded multi-walled carbon nanotube", Adv. Nano Res., 6(2), 163-182. http://dx.doi.org/10.12989/anr.2018.6.2.163.
  104. Wu, H., Kitipornchai, S. and Yang, J. (2018b), "Free vibration of thermo-electro-mechanically postbuckled FG-CNTRC beams with geometric imperfections", Steel Compos. Struct., 29(3), 319-332. https://doi.org/10.12989/scs.2018.29.3.319.
  105. Wuite, J. and Adali, S. (2005), "Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis", , Compos. Struct., 71(3-4), 388-396. https://doi.org/10.1016/j.compstruct.2005.09.011.
  106. Zerrouki, R., Karas, A., Zidour, M., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Benrahou, K.H. and Mahmoud, S. (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.
  107. Zhou, C., Zhang, Z., Zhang, J., Fang, Y. and Tahouneh, V. (2020), "Vibration analysis of FG porous rectangular plates reinforced by graphene platelets", Steel Compos. Struct., 34(2), 215-226. https://doi.org/10.12989/scs.2020.34.2.215.
  108. Zhu, P., Lei, Z. and Liew, K.M. (2012), "Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory", Compos. Struct., 94(4), 1450-1460. https://doi.org/10.1016/j.compstruct.2011.11.010.