DOI QR코드

DOI QR Code

Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method

  • Gao, Yang (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University) ;
  • Xiao, Wan-Shen (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University) ;
  • Zhu, Haiping (School of Computing, Engineering and Mathematics, Western Sydney University)
  • 투고 : 2018.08.24
  • 심사 : 2018.12.14
  • 발행 : 2019.01.25

초록

This paper analyzes nonlinear free vibration of the circular nano-tubes made of functionally graded materials in the framework of nonlocal strain gradient theory in conjunction with a refined higher order shear deformation beam model. The effective material properties of the tube related to the change of temperature are assumed to vary along the radius of tube based on the power law. The refined beam model is introduced which not only contains transverse shear deformation but also satisfies the stress boundary conditions where shear stress cancels each other out on the inner and outer surfaces. Moreover, it can degenerate the Euler beam model, the Timoshenko beam model and the Reddy beam model. By incorporating this model with Hamilton's principle, the nonlinear vibration equations are established. The equations, including a material length scale parameter as well as a nonlocal parameter, can describe the size-dependent in linear and nonlinear vibration of FGM nanotubes. Analytical solution is obtained by using a two-steps perturbation method. Several comparisons are performed to validate the present analysis. Eventually, the effects of various physical parameters on nonlinear and linear natural frequencies of FGM nanotubes are analyzed, such as inner radius, temperature, nonlocal parameter, strain gradient parameter, scale parameter ratio, slenderness ratio, volume indexes, different beam models.

키워드

참고문헌

  1. Abdelaziz, H.H., Meziane, M.A.A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of fgm sand wich plates with various boundary conditions", Steel Compos. Struct., 25(6), 693-704. https://doi.org/10.12989/SCS.2017.25.6.693
  2. Ahmed, H.M.S., Mokhtar, Y., Heireche, H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory", Smart Struct. Syst., 21(4) 397-405. https://doi.org/10.12989/SSS.2018.21.4.397
  3. Ahouel, M., Houari, M.S.A., Bedia, E.A.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981. https://doi.org/10.12989/scs.2016.20.5.963
  4. Aifantis, E.C. (1992), "On the role of gradients in the localization of deformation and fracture", Int. J. Eng. Sci., 30(10), 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3
  5. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  6. Ansari, R., Hasrati, E. and Gholami, R. (2015), "Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto-electro-thermo elastic nanobeams", Compos. Part B., 83, 226-241. https://doi.org/10.1016/j.compositesb.2015.08.038
  7. Attia, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2018), "A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations", Struct. Eng. Mech., 65(4), 453-464. https://doi.org/10.12989/SEM.2018.65.4.453
  8. Bekir, A. and Omer, C. (2011), "Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams", Int. J. Eng. Sci., 49(11), 1268-1280. https://doi.org/10.1016/j.ijengsci.2010.12.009
  9. Bekir, A. and Oumlmer, C. (2013), "Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM)", Compos. Part B: Eng., 55(55), 263-268. https://doi.org/10.1016/j.compositesb.2013.06.035
  10. Bekir, A. and Omer, C. (2014), "Longitudinal vibration analysis for microbars based on strain gradient elasticity theory", J. Vibr. Contr., 20(4), 606-616. https://doi.org/10.1177/1077546312463752
  11. Belabed, Z., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate", Earthq. Struct., 14(2), 103-115. https://doi.org/10.12989/EAS.2018.14.2.103
  12. Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position", J. Braz. Soc. Mech. Sci. Eng., 38(1), 265-275. https://doi.org/10.1007/s40430-015-0354-0
  13. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygro-thermo-mechanical bending of s-fgm plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755
  14. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (fgm) plates", Compos. Part B, 60(1), 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  15. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  16. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., 62(6), 695-702. https://doi.org/10.12989/SEM.2017.62.6.695
  17. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Compos. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  18. Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Free vibration analysis of embedded nanosize fg plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., 19(6), 601-614. https://doi.org/10.12989/SSS.2017.19.6.601
  19. Bouafia, K., Kaci, A. and Houari, M.S.A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115
  20. Boukhari, A., Atmane, H.A., Tounsi, A., Bedia, E.A.A. and Mahmoud, S.R. (2016), "An efficient shear deformation theory for wave propagation of functionally graded material plates", Struct. Eng. Mech., 57(5), 837-859. https://doi.org/10.12989/sem.2016.57.5.837
  21. Bouderba, B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., 58(3), 397-422. https://doi.org/10.12989/sem.2016.58.3.397
  22. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of fgm thick plates resting on winkler-pasternak elastic foundations", Steel Compos. Struct., 14(14), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  23. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  24. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  25. Bousahla, A.A., Mohammed, S.A.H., Abdelouahed, T. and Elabbas, A.B. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Meth., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  26. Bousahla, A.A., Benyoucef, S., Tounsi, A. and Mahmoud, S.R. (2016), "On thermal stability of plates with functionally graded coefficient of thermal expansion", Struct. Eng. Mech., 60(2), 313-335. https://doi.org/10.12989/sem.2016.60.2.313
  27. 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. https://doi.org/10.12989/scs.2015.18.2.425
  28. Dai, H., Yue, X. and Yuan, J. (2014), "A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity", J. Comput. Phys., 270(3), 214-237. https://doi.org/10.1016/j.jcp.2014.03.063
  29. Dai, H.L., Rao, Y.N. and Dai, T. (2016), "A review of recent researches on fgm cylindrical structures under coupled physical interactions, 2000-2015", Compos. Struct., 152, 199-225. https://doi.org/10.1016/j.compstruct.2016.05.042
  30. Dohmann, F. and Hartl, C. (1997)," Tube hydroforming-research and practical application", J. Mater. Proc. Tech., 71(1), 174-186. https://doi.org/10.1016/S0924-0136(97)00166-0
  31. Dresselhaus, M.S, Dresselhaus, G., Charlier, J.C. and Hernandez, E. (2004), "Electronic, thermal and mechanical properties of carbon nanotubes", Philosoph. Trans. Roy. Soc. A., 362(1823), 2065-2098. https://doi.org/10.1098/rsta.2004.1430
  32. Ebrahimi, F. and Barati, M.R. (2017), "Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory", Mech. Adv. Mater. Struct., 25(11), 953-963. https://doi.org/10.1080/15376494.2017.1329467
  33. Ebrahimi, F. and Javari, A. (2016), "Thermo-mechanical vibration analysis of temperature-dependent porous FG beams based on Timoshenko beam theory", Struct. Eng. Mech., 59(2), 343-371. https://doi.org/10.12989/sem.2016.59.2.343
  34. El-Haina, F., Bakora, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A simple analytical approach for thermal buckling of thick functionally graded sandwich plates", Struct. Eng. Mech., 63(5), 585-595. https://doi.org/10.12989/SEM.2017.63.5.585
  35. 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
  36. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. App. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  37. Faghidian, S.A. (2018), "Integro-differential nonlocal theory of elasticity", Int. J. Eng. Sci., 129, 96-110. https://doi.org/10.1016/j.ijengsci.2018.04.007
  38. Fernandez-Saez, J., Zaera, R. and Loya, J.A. (2016), "Bending of Euler-Bernoulli beams using Eringen's integral formulation: A paradox resolved", Int. J. Eng. Sci., 99, 107-116. https://doi.org/10.1016/j.ijengsci.2015.10.013
  39. Fleck, N.A. and Hutchinson, J.W. (2001), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Sol., 41(12), 1825-1857. https://doi.org/10.1016/0022-5096(93)90072-N
  40. Gan, B.S. (2016), "Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation", Struct. Eng. Mech., 58(3), 515-532. https://doi.org/10.12989/sem.2016.58.3.515
  41. Ghadiri, M., Rajabpour, A. and Akbarshahi, A. (2017), "Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects", Appl. Math. Model., 50, 676-694. https://doi.org/10.1016/j.apm.2017.06.019
  42. Ghiasian, S.E., Kiani, Y. and Sadighi, M. (2014), "Thermal buckling of shear deformable temperature dependent circular/annular FGM plates", Int. J. Mech. Sci., 81(4), 137-148. https://doi.org/10.1016/j.ijmecsci.2014.02.007
  43. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  44. Hamzacherif, R., Meradjah, M., Zidour, M., Tounsi, A., Belmahi, S. and Bensattalah, T. (2018), "Vibration analysis of nanobeam using differential transform method including thermal effect", J. Nano Res., 54, 1-14. https://doi.org/10.4028/www.scientific.net/JNanoR.54.1
  45. Hall, K.C., Thomas, J.P. and Clark, W.S. (2002), "Computation of unsteady nonlinear flows in cascades using a harmonic balance technique", AIAA. J., 40(5), 879-886. https://doi.org/10.2514/2.1754
  46. Hichem, B., Bakora, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S. R. (2017), "An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates", Steel Compos. Struct., 25(3), 257-270. https://doi.org/10.12989/SCS.2017.25.3.257
  47. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257
  48. Hu, W., Song, M. and Deng, Z. (2017), "Axial dynamic buckling analysis of embedded single-walled carbon nanotube by complex structure-preserving method", Appl. Math. Model., 52, 15-27. https://doi.org/10.1016/j.apm.2017.06.040
  49. Huang, Y. and Li, X.F. (2010), "Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity", Int. J. Mech. Sci., 52(4), 595-601. https://doi.org/10.1016/j.ijmecsci.2009.12.008
  50. Jha, D.K., Kant, T. and Singh, R.K. (2013), "A critical review of recent research on functionally graded plates", Compos. Struct., 96(4), 833-849. https://doi.org/10.1016/j.compstruct.2012.09.001
  51. Kaci, A., Houari, M., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory", Struct. Eng. Mech., 65(5), 621-631. https://doi.org/10.12989/SEM.2018.65.5.621
  52. Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025
  53. Karami, B., Janghorban, M. and Tounsi, A. (2018b), "Nonlocal strain gradient 3d elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216. https://doi.org/10.12989/SCS.2018.27.2.201
  54. Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. https://doi.org/10.12989/SCS.2017.25.3.361
  55. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize fg plates", Struct. Eng. Mech., 64(4), 391-402. https://doi.org/10.12989/SEM.2017.64.4.391
  56. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B: Eng., 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9
  57. Khodabakhshi, P. and Reddy, J.N. (2015), "A unified integro-differential nonlocal model", Int. J. Eng. Sci., 95, 60-75. https://doi.org/10.1016/j.ijengsci.2015.06.006
  58. Lam, D.C.C., Yang, F. and Chong, A.C.M. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Sol., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  59. Li, X., Li, L. and Hu, Y. (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
  60. Li, L., Hu, Y. and Li, X. (2016), "Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory", Int. J. Mech. Sci., 115, 135-144. https://doi.org/10.1016/j.ijmecsci.2016.06.011
  61. Li, Y.S. and Pan, E. (2015), "Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory", Int. J. Eng. Sci., 97, 40-59. https://doi.org/10.1016/j.ijengsci.2015.08.009
  62. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Sol., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  63. Liu, L., Dowell, E.H. and Hall, K.C. (2007), "A novel harmonic balance analysis for the van der pol oscillator", Int. J. Nonlin. Mech., 42(1), 2-12. https://doi.org/10.1016/j.ijnonlinmec.2006.09.004
  64. Liu, L. and Dowell, E.H. (2004), "The secondary bifurcation of an aeroelastic airfoil motion: Effect of high harmonics", Nonlin. Dyn., 37(1), 31-49. https://doi.org/10.1023/B:NODY.0000040033.85421.4d
  65. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Sol., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  66. Lu, L., Guo, X. and Zhao, J. (2017), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", Int. J. Eng. Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006
  67. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Sol., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  68. Mahi, A., Bedia, E.A.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  69. Menasria, A., Bouhadra, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017), "A new and simple HSDT for thermal stability analysis of FG sandwich plates", Steel Compos. Struct., 25(2), 157-175. https://doi.org/10.12989/SCS.2017.25.2.157
  70. Meziane, M.A.A., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  71. Merdaci, S., Tounsi, A. and Bakora, A. (2016), "A novel four variable refined plate theory for laminated composite plates", Steel Compos. Struct., 22(4), 713-732. https://doi.org/10.12989/scs.2016.22.4.713
  72. Mouffoki, A., Bedia, E.A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory", Smart Struct. Syst., 20(3), 369-383. https://doi.org/10.12989/SSS.2017.20.3.369
  73. Mouffoki, A., Bedia, E.A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory", Smart Struct. Syst., 20(3), 369-383. https://doi.org/10.12989/SSS.2017.20.3.369
  74. Mook, D. and Nayfeh, A. (1979), Nonlinear Oscillations, John Wiley & Sons, New York, U.S.A.
  75. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., 16(1), 51-78. https://doi.org/10.1007/BF00248490
  76. Mindlin, R.D. (1965), "Second gradient of strain and surface-tension in linear elasticity", Int. J. Sol. Struct., 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5
  77. Nazemnezhad, R. and Hosseini-Hashemi, S. (2014), "Nonlocal nonlinear free vibration of functionally graded nanobeams", Compos. Struct., 110(110), 192-199. https://doi.org/10.1016/j.compstruct.2013.12.006
  78. Rahmani, O. and Pedram, O. (2014), "Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory", Int. J. Eng. Sci., 77(7), 55-70. https://doi.org/10.1016/j.ijengsci.2013.12.003
  79. Rahaeifard, M. (2015), "Size-dependent torsion of functionally graded bars", Compos. Part B, 82, 205-211. https://doi.org/10.1016/j.compositesb.2015.08.011
  80. Reddy, J.N., Romanoff, J. and Loya, J.A. (2015), "Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory", Eur. J. Mech. A/Sol., 56, 92-104. https://doi.org/10.1016/j.euromechsol.2015.11.001
  81. Reddy, J.N. and Chin, C.D. (1998), "Thermomechanical analysis of functionally graded cylinders and plates", J. Therm. Stress., 21(6), 593-626. https://doi.org/10.1080/01495739808956165
  82. Salehipour, H., Shahidi, A.R. and Nahvi, H. (2015), "Modified nonlocal elasticity theory for functionally graded materials", Int. J. Eng. Sci., 90, 44-57. https://doi.org/10.1016/j.ijengsci.2015.01.005
  83. Sahmani, S., Aghdam, M.M. and Rabczuk, T. (2018), "Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs", Compos. Struct., 198, 51-62. https://doi.org/10.1016/j.compstruct.2018.05.031
  84. She, G.L., Ren, Y.R. and Yuan, F.G. (2018), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  85. She, G.L., Yuan, F.G. and Ren, Y.R. (2017), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  86. She, G.L, Yuan, F.G. and Ren, Y.R. (2018), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  87. Shen, H.S. and Wang, Z.X. (2014), "Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments", Int. J. Mech. Sci., 81(4), 195-206. https://doi.org/10.1016/j.ijmecsci.2014.02.020
  88. Simsek, M., Aydin, M. and Yurtcu, H. (2015), "Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory", Acta. Mech., 226(11), 3807-3822. https://doi.org/10.1007/s00707-015-1437-9
  89. Tounsi, A., Houari, M.S.A. and Benyoucef, S. (2013), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  90. Tuna, M. and Kirca, M. (2016), "Exact solution of Eringen's nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams", Int. J. Eng. Sci., 105, 80-92. https://doi.org/10.1016/j.ijengsci.2016.05.001
  91. Tsiatas, G.C. (2009), "A new Kirchhoff plate model based on a modified couple stress theory", Int. J. Sol. Struct., 46(13), 2757-2764. https://doi.org/10.1016/j.ijsolstr.2009.03.004
  92. Yang, F., Chong, A.C.M. and Lam, D.C.C. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Sol. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  93. Yang, F., Chong, A.C.M. and Lam, D.C.C. (2009), "Couple stress based strain gradient theory for elasticity", Int. J. Sol. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  94. Yazid, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Houari, M.S.A. (2018), "A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium", Smart Struct. Syst., 21(1), 15-25. https://doi.org/10.12989/SSS.2018.21.1.015
  95. Yahia, S.A., Atmane, H.A., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  96. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams including surface stress effects", Smart Struct. Syst., 21(1), 65-74. https://doi.org/10.12989/SSS.2018.21.1.065
  97. Younsi, A., Tounsi, A., Zaoui, F.Z., Bousahla, A.A. and Mahmoud, S.R. (2018), "Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates", Geomech. Eng., 14(6), 519-532. https://doi.org/10.12989/GAE.2018.14.6.519
  98. Zaoui, F.Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. Part B, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051
  99. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: An assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693
  100. Zhang, P. and Fu, Y. (2013), "A higher-order beam model for tubes", Eur. J. Mech. A/Sol., 38, 12-19. https://doi.org/10.1016/j.euromechsol.2012.09.009
  101. Zhang, X.M., Liu, G.R. and Lam, K.Y. (2001), "Vibration analysis of thin cylindrical shells using wave propagation approach", J. Sound Vibr., 239(3), 397-403. https://doi.org/10.1006/jsvi.2000.3139
  102. Zhong, J., Fu, Y. and Wan, D. (2016), "Nonlinear bending and vibration of functionally graded tubes resting on elastic foundations in thermal environment based on a refined beam model", Appl. Math. Mod., 40(17-18), 7601-7614. https://doi.org/10.1016/j.apm.2016.03.031
  103. Zidi, M., Tounsi, A., Houari, M.S.A., Bedia, E.A.A. and Beg, O.A. (2014), "Bending analysis of fgm plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34(4), 24-34. https://doi.org/10.1016/j.ast.2014.02.001
  104. Zidi, M., Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2017), "A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams", Struct. Eng. Mech., 64(2), 145-153 https://doi.org/10.12989/sem.2017.64.2.145
  105. Zouatnia, N., Hadji, L. and Kassoul, A. (2017), "A refined hyperbolic shear deformation theory for bending of functionally graded beams based on neutral surface position", Struct. Eng. Mech., 63(5), 683-689. https://doi.org/10.12989/SEM.2017.63.5.683

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