Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Analysis of boundary conditions effects on vibration of nanobeam in a polymeric matrix

  • Received : 2017.04.12
  • Accepted : 2018.06.07
  • Published : 2018.09.10
⟨ Previous Next ⟩

Abstract

In this study, we investigate the vibration of single-walled carbon nanotubes embedded in a polymeric matrix using nonlocal elasticity theories with account arbitrary boundary conditions effects. A Winkler type elastic foundation is employed to model the interaction of nanobeam and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, Winkler modulus parameter, vibration mode and aspect ratio of nanobeam on the vibration frequency are analyzed and discussed. The mechanical properties of carbon nanotubes and polymer matrix are treated and an analytical solution is derived using the governing equations of the nonlocal Euler-Bernoulli beam models. Solutions have been compared with those obtained in the literature and The results obtained show that the non-dimensional natural frequency is significantly affected by the small-scale coefficient, the vibrational mode number and the elastic medium.

Keywords

Acknowledgement

Supported by : Algerian National Thematic Agency of Research in Science and Technology (ATRST), university of Sidi Bel Abbes (UDL SBA)

References

  1. Abdel-Halim Hassan, I.H. (2002), "Different applications for the differential transformation in the differential equations", Appl. Math. Comput., 129(2-3), 183-201. https://doi.org/10.1016/S0096-3003(01)00037-6
  2. 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 sandwich plates with various boundary conditions", Steel Compos. Struct., 25(6), 693-704. https://doi.org/10.12989/SCS.2017.25.6.693
  3. Abualnour, M., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697. https://doi.org/10.1016/j.compstruct.2017.10.047
  4. Ahouel, M., Houari, M.S.A., Bedia, E.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
  5. Ait Amar Meziane, M., 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
  6. Ait Yahia, S., Ait Atmane, H., 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
  7. 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
  8. Behrouz, K., Davood, S. and Li, L. (2018), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Phys. E: Low-Dimens. Syst. Nanostruct., 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020
  9. 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
  10. 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: Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  11. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygrothermo-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
  12. Belkorissat, I., Houari, M.S.A., Tounsi, A., Adda Bedia, E.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
  13. Bellifa, H., Bakora, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017b), "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
  14. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017a), "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
  15. 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
  16. Benchohra, M., Driz, H., Bakora, A., Tounsi, A., Bedia E.A. and Mahmoud S.R. (2018), "A new quasi-3D sinusoidal shear deformation theory for functionally graded plates", Struct. Eng. Mech., 65(1), 19-31. https://doi.org/10.12989/SEM.2018.65.1.019
  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. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  18. Bensattalah, T., Daouadji, T.H., Zidour, M., Tounsi, A. and Adda Bedia, E.A. (2016), "Investigation of thermal and chirality effects on vibration of single-walled carbon nanotubes embedded in a polymeric matrix using nonlocal elasticity theories", Mech. Compos. Mater., 52(4), 555-568. https://doi.org/10.1007/s11029-016-9606-z
  19. 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
  20. Bouafia, K., Kaci, A., Houri, M.S.A., Benzair, A. and Tounsi, 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
  21. Bouazza, M., Amara, K., Zidour, M., Tounsi, A. and Adda Bedia, E.A. (2014), "Thermal effect on buckling of multi-walled carbon nanotubes using different gradient elasticity theories", Nanosci. Nanotechnol., 4(2), 27-33.
  22. Bouazza, M., Amara, K., Zidour, M., Tounsi, A. and Bedia, E.A. (2015), "Postbuckling analysis of nanobeams using trigonometric shear deformation theory", Appl. Sci. Rep., 10, 112-121.
  23. 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(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  24. Bouderba, B., Houari, M.S.A. and 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
  25. Bouhadra, A., Tounsi, A., Bousahla, A.A., Benyoucef, S. and Mahmoud, S.R. (2018), "Improved HSDT accounting for effect of thickness stretching in advanced composite plates", Struct. Eng. Mech., 66(1), 61-73. https://doi.org/10.12989/SEM.2018.66.1.061
  26. Boukhari, A., Ait Atmane, H., Tounsi, A., Adda Bedia, E.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
  27. Boumia, L., Zidour, M., Benzair, A. and Tounsi, A. (2014), "A Timoshenko beam model for vibration analysis of chiral singlewalled carbon nanotubes", Phys. E: Low-Dimens. Syst. Nanostruct., 59, 186-191. https://doi.org/10.1016/j.physe.2014.01.020
  28. 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
  29. 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
  30. 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
  31. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (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
  32. Chemi, A., Heireche, H., Zidour, M., Rakrak, K. and Bousahla, A.A. (2015), "Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity", Adv. Nano Res., 3(4), 193-206. https://doi.org/10.12989/anr.2015.3.4.193
  33. Chemi, A., Zidour, M., Heireche, H., Rakrak, K. and Bousahla, A.A. (2018), "Critical buckling load of chiral double-walled carbon nanotubes embedded in an elastic medium", Mech. Compos. Mater., 53(6), 827-836. https://doi.org/10.1007/s11029-018-9708-x
  34. Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), "Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., 19(3), 289-297. https://doi.org/10.12989/sss.2017.19.3.289
  35. Dai, H., Hafner, J.H., Rinzler, A.G., Colbert, D.T. and Smalley, R.E. (1996), "Nanotubes as nanoprobes in scanning probe microscopy", Nat., 384(6605), 147-150. https://doi.org/10.1038/384147a0
  36. Dihaj, A., Zidour, M., Meradjah, M., Rakrak, K., Heireche, H. and Chemi, A. (2018), "Free vibration analysis of chiral doublewalled carbon nanotube embedded in an elastic medium using non-local elasticity theory and Euler Bernoulli beam model", Struct. Eng. Mech., 65(3), 335-342. https://doi.org/10.12989/SEM.2018.65.3.335
  37. Draiche, K., Tounsi, A. and Mahmoud, S.R. (2016), "A refined theory with stretching effect for the flexure analysis of laminated composite plates", Geomech. Eng., 11(5), 671-690. https://doi.org/10.12989/gae.2016.11.5.671
  38. Dresselhaus, M.S. and Avouris, P. (2001), Carbon Nanotubes: Synthesis, Structure, Properties and Application, Springer-Verlag, Berlin Heidelberg, New York, U.S.A.
  39. 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
  40. Eringen, A.C. (1967), "Theory of micropolar plates", Z Angew. Math. Phys., 18(1), 12-30. https://doi.org/10.1007/BF01593891
  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. Feldman, E. and Aboudi, J. (1997), "Buckling analysis of functionally graded plates subjected to uniaxial loading", Compos. Struct., 38(1-4), 29-36. https://doi.org/10.1016/S0263-8223(97)00038-X
  44. Fourn, H., Atmane, H.A., Bourada, M., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel four variable refined plate theory for wave propagation in functionally graded material plates", Steel Compos. Struct., 27(1), 109-122. https://doi.org/10.12989/SCS.2018.27.1.109
  45. Ganji, D.D., Sadighi, A. and Khatami, I. (2008), "Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences", Phys. Lett. A, 372(24), 4399-4406. https://doi.org/10.1016/j.physleta.2008.04.004
  46. Ghorbanpour Arani, .A, Rahmani, R. and Arefmanesh, A. (2008), "Elastic buckling analysis of single-walled carbon nanotube under combined loading by using the ANSYS software", Phys. E, 40(7), 2390-2395. https://doi.org/10.1016/j.physe.2007.11.011
  47. Ghorbanpour Arani, A., Zarei, M.S., Mohammadimehr, M., Arefmanesh, A. and Mozdianfard, M.R. (2011), "The thermal effect on buckling analysis of a DWCNT embedded on the Pasternak foundation", Phys. E, 43(9), 1642-1648. https://doi.org/10.1016/j.physe.2011.05.014
  48. Hajnayeb, A. and Khadem, S.E. (2015), "An analytical study on the nonlinear vibration of a double walled carbon nanotube", Struct. Eng. Mech., 54(5), 987-998. https://doi.org/10.12989/sem.2015.54.5.987
  49. 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
  50. Hamidi, A., Zidour, M., Bouakkaz, K. and Bensattalah, T. (2018), "Thermal and small-scale effects on vibration of embedded armchair single-walled carbon nanotubes", J. Nano Res., 51, 24-38. https://doi.org/10.4028/www.scientific.net/JNanoR.51.24
  51. He, J.H. (2005), "Homotopy perturbation method for bifurcation of nonlinear problems", Int. J. Nonlin. Sci. Numer. Simulat., 6(2), 207-208.
  52. Hebali, H., Tounsi, A., Houari, M.S.A. Bessaim, A. and Adda Bedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  53. 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
  54. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nat., 354, 56-58. https://doi.org/10.1038/354056a0
  55. Javed, S., Viswanathan, K.K. and Aziz, Z.A. (2016), "Free vibration analysis of composite cylindrical shells with nonuniform thickness walls", Steel Compos. Struct., 20(5), 1087-1102. https://doi.org/10.12989/scs.2016.20.5.1087
  56. Kaci, A., Houari, M.S.A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "Post-buckling analysis of sheardeformable composite beams using a novel simple twounknown beam theory", Struct. Eng. Mech., 65(5), 621-631. https://doi.org/10.12989/SEM.2018.65.5.621
  57. Karami, B. and Janghorban, M. (2016), "Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory", Mod. Phys. Lett. B, 30(36), 1650421. https://doi.org/10.1142/S0217984916504212
  58. Karami, B., Janghorban, M. and Li, L. (2018a), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronaut., 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011
  59. Karami, B., Janghorban, M. and Tounsi, A. (2017a), "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
  60. Karami, B., Janghorban, M. and Tounsi, A. (2018e), "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
  61. Karami, B., Janghorban, M. and Tounsi, A. (2018f), "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
  62. Karami, B., Shahsavari, D. and Janghorban, M. (2017b), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 25(12), 1047-1057.
  63. Karami, B., Shahsavari, D. and Li, L. (2018d), "Temperaturedependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field", J. Therm. Stress., 41(4), 483-499. https://doi.org/10.1080/01495739.2017.1393781
  64. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2018c), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111. https://doi.org/10.1016/j.tws.2017.10.004
  65. Karami, B., Shahsavari, D., Li, L., Karami, M. and Janghorban, M. (2018b), "Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory", Proceedings of the Institution of Mechanical Engineers.
  66. 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
  67. Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. 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
  68. Liao, S.J. (2004), "On the homotopy analysis method for nonlinear problems", Appl. Math. Comput., 147(2), 499-513. https://doi.org/10.1016/S0096-3003(02)00790-7
  69. Mahi, A., Adda Bedia, E.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
  70. 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
  71. Mokhtar, Y., Heireche, H., Bousahla, A.A., Houari, M.S.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
  72. Moradi-Dastjerdi, R. (2016), "Wave propagation in functionally graded composite cylinders reinforced by aggregated carbon nanotube", Struct. Eng. Mech., 57(3), 441-456. https://doi.org/10.12989/sem.2016.57.3.441
  73. Mouffoki, A., Bedia, E.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. Mustapha, K.B. and Zhong, Z.W. (2012), "Stability of singlewalled carbon nanotubes and single-walled carbon nanocones under self-weight and an axial tip force", Int. J. Eng. Sci., 50(1), 268-278. https://doi.org/10.1016/j.ijengsci.2010.12.006
  75. Naceri, M., Zidour, M., Semmah, A., Houari, M. S. A., Benzair, A. and Tounsi, A. (2011), "Sound wave propagation in armchair single walled carbon nanotubes under thermal environment", J. Appl. Phys., 110(12), 124322. https://doi.org/10.1063/1.3671636
  76. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  77. Pour, H.R., Vossough, H., Heydari, M.M., Beygipoor, G. and Azimzadeh, A. (2015), "Nonlinear vibration analysis of a nonlocal sinusoidal shear deformation carbon nanotube using differential quadrature method", Struct. Eng. Mech., 54(6), 1061-1073. https://doi.org/10.12989/sem.2015.54.6.1061
  78. Pradhan, S.C. and Phadikar J.K. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., 33(2), 193-213. https://doi.org/10.12989/sem.2009.33.2.193
  79. Rakrak, K., Zidour, M., Heireche, H., Bousahla, A.A. and Chemi, A. (2016), "Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory", Adv. Nano Res., 4(1), 31-44. https://doi.org/10.12989/anr.2016.4.1.031
  80. Ranjbartoreh, A.R., Ghorbanpour, A. and Soltani, B. (2007), "Double-walled carbon nanotube with surrounding elastic medium under axial pressure", Phys. E, 39(2), 230-239. https://doi.org/10.1016/j.physe.2007.04.010
  81. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103(2), 023511. https://doi.org/10.1063/1.2833431
  82. Sakhaee-Pour, A., Ahmadian, M.T. and Vafai, A. (2009), "Vibrational analysis of single-walled carbon nanotubes using beam element", Thin-Wall. Struct., 47(6-7), 646-652. https://doi.org/10.1016/j.tws.2008.11.002
  83. Shahsavari, D. and Janghorban, M. (2017), "Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load", J. Braz. Soc. Mech. Sci. Eng., 39(10), 3849-3861. https://doi.org/10.1007/s40430-017-0863-0
  84. Shahsavari, D., Karami, B. and Mansouri, S. (2018a), "Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories", Eur. J. Mech.-A/Sol., 67, 200-214. https://doi.org/10.1016/j.euromechsol.2017.09.004
  85. Shahsavari, D., Karami, B., Janghorban, M. and Li, L. (2017), "Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment", Mater. Res. Expr., 4(8), 085013. https://doi.org/10.1088/2053-1591/aa7d89
  86. Shahsavari, D., Shahsavari, M., Li, L. and Karami, B. (2018b), "A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation", Aerosp. Sci. Technol., 72, 134-149. https://doi.org/10.1016/j.ast.2017.11.004
  87. Thostenson, E.T., Ren, Z. and Chou, T.W. (2001), "Advances in the science and technology of carbon nanotubes and their composites ", Compos. Sci. Technol., 61(13), 1899-1912. https://doi.org/10.1016/S0266-3538(01)00094-X
  88. 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
  89. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (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. Wang, C.M., Zhang Y.Y. and He, X.Q. (2007), "Vibration of nonlocal Timoshenko beams", Nanotechnol., 18(10), 105-401.
  91. Wang, C.M., Zhang, Y.Y., Ramesh, S.S. and Kitipornchai, S. (2006), "Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys. 39(17), 3904-3909. https://doi.org/10.1088/0022-3727/39/17/029
  92. Wang, X. and Yang, H.K. (2005), "Axially critical load of multiwall carbon nanotubes under thermal environment", J. Therm. Stress., 28(2), 185-196. https://doi.org/10.1080/014957390900067
  93. Wazwaz, A.M. (2002), Partial Differential Equations: Methods and Applications, the Netherlands, Balkema Publishers, Broken Sound Pkwy NW, Boca Raton, Florida, U.S.A.
  94. Yao, X.H. and Han, Q. (2007), "The thermal effect on axially compressed buckling of a double-walled carbon nanotube", Eur. J. Mech. A/Sol., 26(2), 298-312. https://doi.org/10.1016/j.euromechsol.2006.05.009
  95. 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
  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. 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
  99. 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.
  100. Zidi, M., Tounsi, A., Houari, M.S.A. and Beg, O.A. (2014), "Bending analysis of FGM plates under hygrothermomechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001
  101. Zidour, M., Benrahou, K.H., Semmah, A., Naceri, M., Belhadj, H.A., Bakhti, K. and Tounsi, A. (2012), "The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory", Comput. Mater. Sci., 51(1), 252-260. https://doi.org/10.1016/j.commatsci.2011.07.021
  102. Zidour, M., Hadji, L., Bouazza, M., Tounsi A. and Adda Bedia, E.A. (2015), "The mechanical properties of zigzag carbon nanotube using the energy-equivalent model", J. Chem. Mater. Res., 3, 9-14.
  103. Zine, A., Tounsi, A., Draiche, K., Sekkal, M. and Mahmoud, S.R. (2018), "A novel higher-order shear deformation theory for bending and free vibration analysis of isotropic and multilayered plates and shells", Steel Compos. Struct., 26(2), 125-137. https://doi.org/10.12989/SCS.2018.26.2.125

Cited by

  1. Free vibration analysis of angle-ply laminated composite and soft core sandwich plates vol.33, pp.5, 2019, https://doi.org/10.12989/scs.2019.33.5.663
  2. Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model vol.34, pp.5, 2018, https://doi.org/10.12989/scs.2020.34.5.643
  3. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory vol.25, pp.3, 2020, https://doi.org/10.12989/cac.2020.25.3.225
  4. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations vol.25, pp.4, 2018, https://doi.org/10.12989/cac.2020.25.4.311
  5. Porosity-dependent mechanical behaviors of FG plate using refined trigonometric shear deformation theory vol.26, pp.5, 2020, https://doi.org/10.12989/cac.2020.26.5.439
  6. Buckling analysis of functionally graded plates using HSDT in conjunction with the stress function method vol.27, pp.1, 2018, https://doi.org/10.12989/cac.2021.27.1.073