DOI QR코드

DOI QR Code

A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates

Bessaim, Aicha;Houari, Mohammed Sid Ahmed;Bernard, Fabrice;Tounsi, Abdelouahed

  • 투고 : 2015.03.31
  • 심사 : 2015.10.10
  • 발행 : 2015.10.25

초록

In this work, a nonlocal quasi-3D trigonometric plate theory for micro/nanoscale plates is proposed. In order to introduce the size influences, the Eringen's nonlocal elasticity theory is utilized. In addition, the theory considers both shear deformation and thickness stretching effects by a trigonometric variation of all displacements within the thickness, and respects the stress-free boundary conditions on the top and bottom surfaces of the plate without considering the shear correction factor. The advantage of this theory is that, in addition to considering the small scale and thickness stretching effects (${\varepsilon}_z{\neq}0$), the displacement field is modelled with only 5 unknowns as the first order shear deformation theory (FSDT). Analytical solutions for vibration of simply supported micro/nanoscale plates are illustrated, and the computed results are compared with the available solutions in the literature and finite element model using ABAQUS software package. The influences of the nonlocal parameter, shear deformation and thickness stretching on the vibration behaviors of the micro/nanoscale plates are examined.

키워드

trigonometric shear deformation theory;nanoplates;nonlocal elasticity theory;navier solution;stretching effect;vibration

참고문헌

  1. Aagesen, M. and Sorensen, C.B. (2008), "Nanoplates and their suitability for use as solar cells", Proceedings of Clean Technology, 109-112.
  2. Adda Bedia, W., Benzair, A. Semmah, A., Tounsi, A. and Mahmoud, S.R. (2015), "On the thermal buckling characteristics of armchair single-walled carbon nanotube embedded in an elastic medium based on nonlocal continuum elasticity", Braz. J. Phys., 45, 225-233. https://doi.org/10.1007/s13538-015-0306-2
  3. Aghababaei, R. and Reddy, J.N. (2009), "Non-local third-order shear deformation plate theory with application to bending and vibration of plates", J. Sound Vib., 326, 227-289.
  4. Aissani, K., Bachir Bouiadjra, M., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., 55(4), 743-762. https://doi.org/10.12989/sem.2015.55.4.743
  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. Aksencer, T. and Aydogdu, M. (2011), "Levy type solution for vibration and buckling of nanoplates using nonlocal elasticity theory", Physica E, 43(4), 954959.
  7. 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
  8. 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
  9. Alibeigloo, A. (2011), "Free vibration analysis of nano-plate using three-dimensional theory of elasticity", Acta Mechanica, 222(1-2), 149-159. https://doi.org/10.1007/s00707-011-0518-7
  10. Amara, K., Tounsi, A., Mechab, I. and Adda-Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34, 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  11. Babaei, H. and Shahidi, A.R. (2010), "Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method", Arch. Appl. Mech., 81(8), 1051-1062.
  12. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B. 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  13. 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
  14. Benachour, A., Daouadji, H.I., Ait Atmane, H., Tounsi, A. and Meftah, S.A. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Compos. Part B. 42, 1386-1394. https://doi.org/10.1016/j.compositesb.2011.05.032
  15. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B. 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  16. Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "Anew higher-order shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521
  17. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys. 41, 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  18. Berrabah, H.M., Tounsi, A., Semmah, A., and Adda Bedia, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  19. Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., and Adda Bedia, E.A. (2013), "Anew higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater., 15(6), 671-703. https://doi.org/10.1177/1099636213498888
  20. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nono Res. 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029
  21. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "Anew 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
  22. 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
  23. Draiche, K., Tounsi, A. and Khalfi, Y. (2014), "A trigonometric four variable plate theory for free vibration of rectangular composite plates with patch mass", Steel Compos. Struct. 17(1), 69-81. https://doi.org/10.12989/scs.2014.17.1.069
  24. El Meiche, N., Tounsi, A., Ziane, N., Mechab, I. and Adda Bedia, E.A. (2011), "A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate", Int. J. Mech. Sci. 53, 237-247. https://doi.org/10.1016/j.ijmecsci.2011.01.004
  25. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int J. Eng Sci., 10, 233-48. https://doi.org/10.1016/0020-7225(72)90039-0
  26. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-10. https://doi.org/10.1063/1.332803
  27. Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates", Meccanica, 49, 795-810. https://doi.org/10.1007/s11012-013-9827-3
  28. Fritz, J., Baller, M.K., Lang, H.P., Rothuizen, H., Vettiger, P., Meyer, E., Guntherodt, H.J., Gerber, C. and and Gimzewski, J.K. (2000), "Translating biomolecular recognition into nanomechanics", Science, 288(5464),316-318. https://doi.org/10.1126/science.288.5464.316
  29. 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
  30. 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", ASCE J. Eng. Mech., 140, 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  31. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008a), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E, 40, 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  32. Houari, M.S.A., Tounsi, A. and Anwar Beg, O. (2013), "Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory", Int. J. Mech. Sci., 76, 102-111. https://doi.org/10.1016/j.ijmecsci.2013.09.004
  33. Huang, D.W. (2008), "Size-dependent response of ultra-thin films with surface effects", Int. J. Solid Struct., 45(2), 568-579. https://doi.org/10.1016/j.ijsolstr.2007.08.006
  34. Janghorban, M. and Zare, A. (2011), "Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method", Physica E, 43, 1602-1604. https://doi.org/10.1016/j.physe.2011.05.002
  35. Janghorban, M. (2012), "Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment", Arch. Appl. Mech., 82, 669-675. https://doi.org/10.1007/s00419-011-0582-4
  36. Karimi, M., Haddad, H.A., Shahidi, A.R. (2015), "Combining surface effects and non local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates", Micro Nano Lett., 10(6),276-281. https://doi.org/10.1049/mnl.2014.0651
  37. Khalfi, Y., Houari, M.S.A. and Tounsi, A. (2014), "A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation", Int. J. Comput. Meth., 11(5), 135007.
  38. Kiani, K. (2011a), "Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory", J. Sound Vib., 330(20), 4896-4914. https://doi.org/10.1016/j.jsv.2011.03.033
  39. Kiani, K. (2011b), "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: theoretical formulations", Physica E, 44(1), 229-248. https://doi.org/10.1016/j.physe.2011.08.020
  40. Kiani, K. (2011c), "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part II: parametric studies", Physica E, 44(1), 249-269. https://doi.org/10.1016/j.physe.2011.08.021
  41. Kiani, K. (2013a), "Free vibration of conducting nanoplates exposed to unidirectional in-plane magnetic fields using nonlocal shear deformable plate theories", Physica E: Low-dimens. Syst. Nanostruct., 57, 179-192.
  42. Kiani, K. (2013b), "Vibrations of biaxially tensioned embedded nanoplates for nanoparticle delivery", Indi. J. Sci. Tech., 6(7), 48944902.
  43. Kiani, K. (2015), "Free vibrations of elastically embedded stocky single-walled carbon nanotubes acted upon by a longitudinally varying magnetic field", Meccanica. (accepted paper)
  44. Kitipornchai, S., He, X.Q. and Liew K.M. (2005), "Continuum model for the vibration of multilayered graphene sheets", Phys. Rev. B, 72, 075443. https://doi.org/10.1103/PhysRevB.72.075443
  45. 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
  46. Lee, W.H., Han, S.C. and Park, W.T. (2012), "Nonlocal elasticity theory for bending and free vibration analysis of nano plates", J. Korea Acad. Indus. Coop. Soc., 13(7), 3207-3215.
  47. Liew, K.M., Hung. K.C. and Lim, M.K. (1993), "A continuum three-dimensional vibration analysis of thick rectangular plates", Int. J. Solid Struct., 30(24), 3357-3379. https://doi.org/10.1016/0020-7683(93)90089-P
  48. Liu, C. and Rajapakse, R.K.N.D. (2010), "Continuum models incorporating surface energy for static and dynamic response of nanoscale beams", IEEE Tran. Nanotechnol., 9(4), 422-431. https://doi.org/10.1109/TNANO.2009.2034142
  49. Lu, P., He. L.H., Lee, H.P. and Lu, C. (2006), "Thin plate theory including surface effects", Int. J. Solid Struct., 43(16), 4631-4647. https://doi.org/10.1016/j.ijsolstr.2005.07.036
  50. Ma, Q. and Clarke, D.R. (1995), "Size dependent hardness of silver single crystals", J. Mater. Res., 10, 853-863. https://doi.org/10.1557/JMR.1995.0853
  51. Ma, M., Tu, J.P., Yuan, Y.F., Wang, X.L., Li, K.F., Mao, F. and Zeng, Z.Y. (2008), "Electrochemical performance of ZnO nanoplates as anode materials for Ni/Zn secondary batteries", J. Power Sour., 179, 395-400. https://doi.org/10.1016/j.jpowsour.2008.01.026
  52. 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. 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  53. Malekzadeh, P., Setoodeh, A.R. and Beni, A.A. (2011), "Small scale effect on the free vibration of orthotropic arbitrary straight straightsided quadrilateral nanoplates", Compos. Struct., 93(7), 16311639. https://doi.org/10.1016/j.compstruct.2011.01.008
  54. Mohammadi, M., Ghayour, M. and Farajpour, A. (2013), "Free transverse vibration analysis of circular and annular grapheme sheets with various boundary conditions using the nonlocal continuum plate model", Compos. Part E: Eng., 45(1), 32-42. https://doi.org/10.1016/j.compositesb.2012.09.011
  55. Murmu, T., and Pradhan, S.C. (2009), "Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity", J. Appl. Phys., 106, 104301. https://doi.org/10.1063/1.3233914
  56. Murmu, T. and Pradhan, S.C. (2010), "Small scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model", Physica E, 41(8), 1628-1633.
  57. Murmu, T. and Adhikari, S. (2011), "Nonlocal vibration of bonded double-nanoplate-systems", Compos. Part E: Eng., 42(7), 1901-1911. https://doi.org/10.1016/j.compositesb.2011.06.009
  58. Nami, M.R. and Janghorban, M. (2013), "Static analysis of rectangular nanoplates using trigonometric shear deformation theory based on nonlocal elasticity theory", Beil. J. Nanotech., 4, 968-973. https://doi.org/10.3762/bjnano.4.109
  59. Nami, M.R. and Janghorban, M. (2014), "Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant", Compos. Struct., 111, 349-353. https://doi.org/10.1016/j.compstruct.2014.01.012
  60. Nami, M.R. and Janghorban, M. (2015), "Free vibration of functionally graded size dependent nanoplates based on second order shear deformation theory using nonlocal elasticity theory", Iran. J. Sci. Tech., 39, 15-28.
  61. Ould Larbi, L., Kaci, A., Houari, M.S.A. and Tounsi, A. (2013), "An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams", Mech. Bas. Des. Struct. Mach., 41, 421-433. https://doi.org/10.1080/15397734.2013.763713
  62. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103,. 023511. https://doi.org/10.1063/1.2833431
  63. Phadikar, J.K. and Pradhan, S.C. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Computat. Mater. Sci., 49(3), 492-499. https://doi.org/10.1016/j.commatsci.2010.05.040
  64. Pradhan, S.C. and Murmu, I. (2009), "Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics", Comput. Mater. Sci., 47, 268-274. https://doi.org/10.1016/j.commatsci.2009.08.001
  65. Samaei, A.T., Aliha, M.R.M. and Mirsayar, M.M. (2015), "Frequency analysis of a graphene sheet embedded in an elastic medium with consideration of small scale", Mater. Phys. Mech., 22, 125-135.
  66. Sheng, H.Y., Li. H.P., Lu, P. and Xu, H.Y. (2010), "Free vibration analysis for micro-structures used in MEMS considering surface effects", J. Sound Vib., 329(2), 236-246. https://doi.org/10.1016/j.jsv.2009.08.035
  67. Sobhy, M. (2014), "Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions", Acta Mechanica, 225(9), 2521-2538. https://doi.org/10.1007/s00707-014-1093-5
  68. Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013a), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  69. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013b), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12, 1350025. https://doi.org/10.1142/S0219581X13500257
  70. Tounsi, A., Semmah, A. and Bousahla, A.A. (2013c), "Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory", ASCE J. Nanomech. Micromech., 3, 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057
  71. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013d), A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Techn., 24, 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  72. Wang, G.F. and Feng, X.Q. (2009), "Timoshenko beam model for buckling and vibration of nan ow ires with surface effects", J. Phys. D. Appl. Phys., 42,155411. https://doi.org/10.1088/0022-3727/42/15/155411
  73. Yguerabide, J. and Yguerabide, E.E. (2001), "Resonance light scattering particles as ultrasensitive labels for detection of analytes in a wide range of applications", J. Cell. Biochem. Suppl., 37, 71-81.
  74. 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
  75. Zhang, Z., Wang, C. and Challamel, N. (2015), "Eringen's length-scale coefficients for vibration and buckling of nonlocal rectangular plates with simply supported edges", ASCE J. Eng. Mech., 141(2), 04014117. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000838
  76. Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Tech., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001

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