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On the effect of the micromechanical models on the free vibration of rectangular FGM plate resting on elastic foundation

  • Mahmoudi, Abdelkader (Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Benyoucef, Samir (Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Tounsi, Abdelouahed (Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Benachour, Abdelkader (Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Bedia, El Abbas Adda (Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology)
  • Received : 2017.01.11
  • Accepted : 2018.01.29
  • Published : 2018.02.25
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Abstract

In this research work, free vibrations of simply supported functionally graded plate resting on a Winkler-Pasternak elastic foundation are investigated by a new shear deformation theory. The influence of alternative micromechanical models on the macroscopic behavior of a functionally graded plate based on shear-deformation plate theories is examined. Several micromechanical models are tested to obtain the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which continuously varies through the thickness of a functionally graded plate. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The energy functional of the system is obtained using Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Finally, the numerical results are provided to reveal the effect of explicit micromechanical models on natural fundamental frequencies.

Keywords

References

  1. Abrate, S. (2006), "Free vibration, buckling, and static deflections of functionally graded plates", Compos. Sci. Technol., 66, 2383-94. https://doi.org/10.1016/j.compscitech.2006.02.032
  2. Abualnour, M., Houari, M.S.A., Tounsi, A., Adda Bedia, E.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
  3. 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
  4. 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
  5. Akbarzadeh, A.H., Abedini, A. and Chen, Z.T. (2015), "Effect of micromechanical models on structural responses of functionally graded plates", Compos. Struct., 119, 598-609. https://doi.org/10.1016/j.compstruct.2014.09.031
  6. Akhavan, H., Hosseini Hashemi, Sh., Rokni Damavandi Taher, H., Alibeigloo, A. and Vahabi, Sh. (2009), "Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation. Part II: Frequency analysis", Comput. Mater. Sci., 44, 951-961. https://doi.org/10.1016/j.commatsci.2008.07.001
  7. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006
  8. Atmane, H.A., Tounsi, A. and Mechab, I. (2010), "Free vibration analysis of functionally graded plates resting on Winkler-Pasternak elastic foundations using a new shear deformation theory", Int. J Mech. Mater. Des., 6, 113-121. https://doi.org/10.1007/s10999-010-9110-x
  9. Attia, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187
  10. Baferani, A.H., Saidi, A.R. and Ehteshami, H. (2011), "Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation", Compos. Struct., 93(7), 1842-53. https://doi.org/10.1016/j.compstruct.2011.01.020
  11. 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
  12. Bellifa, H., 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
  13. Benachour, A., Daouadji, H.T., 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 Eng., 42, 1386-1394. https://doi.org/10.1016/j.compositesb.2011.05.032
  14. 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
  15. Benveniste, Y. (1987), "A new approach to the application of Mori-Tanaka's theory in composite materials", Mech. Mater., 6, 147-157. https://doi.org/10.1016/0167-6636(87)90005-6
  16. Benyoucef, S., Mechab, I., Tounsi, A., Fekrar, A., Ait Atmane, H. and Adda Bedia, E.A. (2010), "Bending of thick functionally graded plates resting on Winkler-Pasternak elastic foundations", Mech. Compos. Mater., 46(4), 425-434. https://doi.org/10.1007/s11029-010-9159-5
  17. Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Adda Bedia, E.A. (2013), "A new 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
  18. 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
  19. 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
  20. Bourada, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., 21(6), 1287-1306. https://doi.org/10.12989/scs.2016.21.6.1287
  21. Bourada, M., Tounsi, A., Houari, M.S.A. and Adda Bedia, E.A. (2012), "A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates", J. Sandw. Struct. Mater., 14, 5-33. https://doi.org/10.1177/1099636211426386
  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. Dehghan, H. and Baradaran, G.H. (2011), "Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method", Appl. Math. Comput., 218, 2772-2784.
  24. 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
  25. Fang, X.Q., Hu, C. and Huang, W.H. (2007), "Determination of dynamic effective properties in functionally graded materials", Acta Mechanica, 192, 49-63. https://doi.org/10.1007/s00707-006-0440-6
  26. Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F. and Jorge, R.M.N. (2006), "Natural frequencies of functionally graded plates by a meshless method", Compos. Struct., 75, 593-600. https://doi.org/10.1016/j.compstruct.2006.04.018
  27. Gasik, M. (1995), "Scand. Ch226", Acta Polytech, 72.
  28. Gasik, M.M. (1998), "Micromechanical modeling of functionally graded materials", Comput. Mater. Sci., 13, 42-55. https://doi.org/10.1016/S0927-0256(98)00044-5
  29. Hadji, L., Atmane, H.A., Tounsi, A., Mechab, I. and Adda Bedia, E.A. (2011), "Free vibration of functionally graded sandwich plates using four variable refined plate theory", Appl. Math. Mech., 32, 925-942. https://doi.org/10.1007/s10483-011-1470-9
  30. 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
  31. Hebali, H., Tounsi, A., Houari, M.S.A, Bessaim, A. and Adda Bedia, E.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., ASCE, 140, 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  32. 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
  33. Jha, D.K., Kant, T. and Singh, R.K. (2013), "Critical review of recent research on functionally graded plates", Compos. Struct., 96, 833-849. https://doi.org/10.1016/j.compstruct.2012.09.001
  34. Ju, J. and Chen, T.M. (1994), "Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities", Acta. Mech., 103, 103-121. https://doi.org/10.1007/BF01180221
  35. Kar, V.R. and Panda, S.K. (2015), "Free vibration responses of temperature dependent functionally graded curved panels under thermal environment", Lat. Am. J. Solid. Struct., 12(11), 2006-2024. https://doi.org/10.1590/1679-78251691
  36. Kerr, A.D. (1964), "Elastic and viscoelastic foundation models", ASME J. Appl. Mech., 31(3), 491-498. https://doi.org/10.1115/1.3629667
  37. Kettaf, F.Z., Houari, M.S.A., Benguediab, M. and Tounsi, A. (2013), "Thermal buckling of functionally graded sandwich plates using a new hyperbolic shear displacement model", Steel Compos. Struct., 15(4), 399-423. https://doi.org/10.12989/scs.2013.15.4.399
  38. 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.
  39. Koizumi, M. (1993), "The concept of FGM", Ceram. Tran. Funct. Grad. Mater., 34, 3-10.
  40. 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
  41. Matsunaga, H. (2008), "Free vibration and stability of functionally graded plates according to a 2D higher-order deformation theory", Compos. Struct., 82, 499-512. https://doi.org/10.1016/j.compstruct.2007.01.030
  42. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2018), "An analytical solution for bending, buckling and vibration responses of FGM sandwich plates", J. Sandw . Struct. Mater., 1099636217698443.
  43. Mishnaevsky, Jr. L. (2007), Computational Mesomechanics of Composites, John Wiley & Sons, England.
  44. Mori, T. and Tanaka, K. (1973), "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta. Metall., 21, 571-574. https://doi.org/10.1016/0001-6160(73)90064-3
  45. Nedri, K., El Meiche, N. and Tounsi, A. (2014), "Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory", Mech. Compos. Mater., 49(6), 641-650. https://doi.org/10.1007/s11029-013-9380-0
  46. 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
  47. Pasternak, P.L. (1954), "On a new method of analysis of an elastic foundation by means of two foundation constants", Cosudarstrennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, USSR, Moscow. (in Russian)
  48. Pindera, M., Aboudi, J. and Arnold, S. (1995), "Limitations of the uncoupled, RVE-based micromechanical approach in the analysis of functionally graded composites", Mech. Mater., 20, 77-94. https://doi.org/10.1016/0167-6636(94)00052-2
  49. Qian, L.F., Batra, R.C. and Chen, L.M. (2004), "Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method", Compos. Part B, 35, 685-697. https://doi.org/10.1016/j.compositesb.2004.02.004
  50. Rahman, S. and Chakraborty, A. (2007), "A stochastic micromechanical model for elastic properties of functionally graded materials", Mech. Mater., 39, 548-563. https://doi.org/10.1016/j.mechmat.2006.08.006
  51. Reddy, J.N. (2000), "Analysis of functionally graded plates", Int. J. Numer. Meth. Eng., 47, 663-684. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8
  52. Reiter, T. and Dvorak, G.J. (1997), "Micromechanical models for graded composite materials", J. Mech. Phys. Solid., 46, 1655-1673.
  53. Sadoune, M., Tounsi, A., Houari, M.S.A. and Adda Bedia, E.A. (2014), "A novel first-order shear deformation theory for laminated composite plates", Steel Compos. Struct., 17(3), 321-338. https://doi.org/10.12989/scs.2014.17.3.321
  54. Sekkal, M., Fahsi, B., Tounsi, A. and Mahmoud, S.R. (2017), "A new quasi-3D HSDT for buckling and vibration of FG plate", Struct. Eng. Mech., 64(6), 737-749. https://doi.org/10.12989/SEM.2017.64.6.737
  55. Shen, H.S. and Wang, Z.X. (2012), "Assessment of Voigt and Mori-Tanaka models for vibration analysis of functionally graded plates", Compos. Struct., 94(7), 2197-2208. https://doi.org/10.1016/j.compstruct.2012.02.018
  56. Talha, M. and Singh, B.N. (2010), "Static response and free vibration analysis of FGM plates using higher order shear deformation theory", Appl. Math. Modell., 34, 3991-4011. https://doi.org/10.1016/j.apm.2010.03.034
  57. Thai, H. and Choi, D. (2011), "A refined plate theory for functionally graded plates resting on elastic foundation", Compos. Sci. Technol., 71, 1850-8. https://doi.org/10.1016/j.compscitech.2011.08.016
  58. 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
  59. 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
  60. Vel, S.S. and Batra, R.C. (2004), "Three-dimensional exact solution for the vibration of functionally graded rectangular plates", J. Sound Vib., 272(3), 703-730. https://doi.org/10.1016/S0022-460X(03)00412-7
  61. Weng, G.J. (2003), "Effective bulk moduli of two functionally graded composites", Acta Mechanica, 166, 57-67. https://doi.org/10.1007/s00707-003-0063-0
  62. Winkler, E. (1867), Die Lehre von der Elasticitaet und Festigkeit, Dominicus, Prag.
  63. Yaghoobi, H. and Fereidoon, A. (2014), "Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: an assessment of a simple refined nth-order shear deformation theory", Compos. Part B, 62, 54-64. https://doi.org/10.1016/j.compositesb.2014.02.014
  64. Yaghoobi, H. and Torabi, M. (2013), "Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation", Appl. Math. Model., 37(18-19), 8324-8340. https://doi.org/10.1016/j.apm.2013.03.037
  65. Yaghoobi, H. and Yaghoobi, P. (2013), "Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: An analytical approach", Meccanica, 48, 2019-2035. https://doi.org/10.1007/s11012-013-9720-0
  66. Yu, J. and Kidane, A. (2014), "Modeling functionally graded materials containing multiple heterogeneities", Acta. Mech., 225, 1931-1943. https://doi.org/10.1007/s00707-013-1033-9
  67. Zimmerman, R.W. (1994), "Behavior of the poisson ratio of a twophase composite material in the high-concentration limit", Appl. Mech. Rev., 47(1), 38-44. https://doi.org/10.1115/1.3122819
  68. Zine, A., Tounsi, A., Draiche, K., Mohamed 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
  69. Zuiker, J.R. (1995), "Functionally graded materials-choice of micromechanics model and limitations in property variation", Compos. Eng., 5(7), 807-819. https://doi.org/10.1016/0961-9526(95)00031-H

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