Advertisement

Arabian Journal for Science and Engineering

, Volume 44, Issue 10, pp 8727–8745 | Cite as

Nonlinear Hygro-Thermo-Mechanical Analysis of Functionally Graded Plates Using a Fifth-Order Plate Theory

  • Shantaram M. Ghumare
  • Atteshamuddin S. SayyadEmail author
Research Article -Civil Engineering
  • 55 Downloads

Abstract

In the present research, a new fifth-order plate theory is developed and applied to evaluate the static response of functionally graded (FG) plates resting on Winkler–Pasternak elastic foundation under nonlinear hygro-thermo-mechanical loading. This theory involves polynomial shape functions expanded up to fifth order in terms of the thickness coordinates. The effects of shear deformation and normal deformation are accounted; hence, it can be called as quasi-3D plate theory and abbreviated as FOSNDT. The theory involves nine unknowns. Zero transverse shear stress conditions at top and bottom surfaces are satisfied using constitutive relations. Analytical solutions are obtained using the double Fourier series technique suggested by Navier. The non-dimensional displacements and stresses are compared with available results.

Keywords

FG plate Shear deformation Normal deformation Hygro-thermo-mechanical loading Elastic foundation 

References

  1. 1.
    Kirchhoff, G.R.: Uber das gleichgewicht und die bewegung einer elastischen Scheibe. J. Reine. Angew. Math. (Crelle’s J) 40, 51–88 (1850)Google Scholar
  2. 2.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)zbMATHGoogle Scholar
  3. 3.
    Jha, D.K.; Kant, T.; Singh, R.K.: A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013)Google Scholar
  4. 4.
    Swaminathan, K.; Naveenkumar, D.T.; Zenkour, A.M.; Carrera, E.: Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review. Compos. Struct. 120, 10–31 (2015)Google Scholar
  5. 5.
    Swaminathan, K.; Sangeetha, D.M.: Thermal analysis of FGM plates—a critical review of various modeling techniques and solution methods. Compos. Struct. 160, 43–60 (2017)Google Scholar
  6. 6.
    Sayyad, A.S.; Ghugal, Y.M.: On the free vibration analysis of laminated composite and sandwich plates: a review of recent literature with some numerical results. Compos. Struct. 129, 177–201 (2015)Google Scholar
  7. 7.
    Sayyad, A.S.; Ghugal, Y.M.: Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature. Compos. Struct. 171, 486–504 (2017)Google Scholar
  8. 8.
    Sayyad, A.S.; Ghugal, Y.M.: Modeling and analysis of functionally graded sandwich: a review. Mech. Adv. Mater. Struct. (2018).  https://doi.org/10.1080/15376494.2018.1447178 Google Scholar
  9. 9.
    Zenkour, A.M.: Generalized shear deformation theory for bending analysis of functionally graded plates. Appl. Math. Model. 30, 67–84 (2006)zbMATHGoogle Scholar
  10. 10.
    Ameur, M.; Tounsi, V.; Mechab, I.; Bedia, E.A.: A trigonometric shear deformation theory for bending analysis of functionally graded plates resting on elastic foundations. KSCE J. Civ. Eng. 15, 1405–1414 (2011)Google Scholar
  11. 11.
    Neves, A.M.A.; Ferreira, A.J.M.; Carrera, E.; Cinefra, M.; Roque, C.M.C.; Jorge, R.M.N.; Soares, C.M.M.: A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos. Struct. 94, 1814–1825 (2012)Google Scholar
  12. 12.
    Neves, A.M.A.; Ferreira, A.J.M.; Carrera, E.; Cinefra, M.; Roque, C.M.C.; Jorge, R.M.N.; Soares, C.M.M.: A quasi-3D sinusoidal shear deformation and free vibration analysis of functionally graded plates. Compos. Part B-Eng. 43, 711–725 (2012)Google Scholar
  13. 13.
    Thai, H.T.; Choi, D.H.: A simple refined theory for bending, buckling and vibration of thick plates resting on elastic foundation. Int. J. Mech. Sci. 73, 40–52 (2013)Google Scholar
  14. 14.
    Thai, H.T.; Choi, D.H.: Finite element formulation of various four unknown shear deformation theories for functionally graded plates. Finite Elem. Anal. Des. 75, 50–61 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sayyad, A.S.; Ghugal, Y.M.: A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich plates. Int. J. Appl. Mech. 9(1), 1–36 (2017)Google Scholar
  16. 16.
    Carrera, E.; Brischetto, S.; Robaldo, A.: Variable kinematic model for the analysis of functionally graded material plates. AIAA J. 46, 194–203 (2008)Google Scholar
  17. 17.
    Carrera, E.; Brischetto, S.; Cinefra, M.; Soave, M.: Effect of thickness stretching in functionally graded plates and shells. Compos. Part B-Eng. 42(2), 123–133 (2011)Google Scholar
  18. 18.
    Sayyad, A.S.; Ghugal, Y.M.: A four-variable plate theory for thermoelastic bending analysis of laminated composite plates. J. Therm. Stresses 38, 904–925 (2015)Google Scholar
  19. 19.
    Sayyad, A.S.; Ghugal, Y.M.: Thermoelastic bending analysis of laminated composite plates according to various shear deformation theories. Open Engg. 5, 18–30 (2015)Google Scholar
  20. 20.
    Sayyad, A.S.; Ghugal, Y.M.: Thermal stress analysis of laminated composite plates using exponential shear deformation theory. Int. J. Auto. Compos. 2(1), 23–40 (2016)Google Scholar
  21. 21.
    Zenkour, A.M.; Sobhy, M.: Thermal buckling of various types of FGM sandwich plates. Compos. Struct. 93, 93–102 (2010)Google Scholar
  22. 22.
    Zenkour, A.M.; Sobhy, M.: Thermal buckling of functionally graded plates resting on elastic foundations using the trigonometric theory. J. Therm. Stresses 93, 1119–1138 (2011)Google Scholar
  23. 23.
    Tounsi, A.; Houari, M.S.A.; Benyoucef, S.; Bedia, E.A.A.: A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plate. Aerosp. Sci. Technol. 24, 209–220 (2013)Google Scholar
  24. 24.
    Zhao, X.; Lee, Y.Y.; Liew, K.M.: Mechanical and thermal buckling analysis of functionally graded plates. Compos. Struct. 90, 161–171 (2009)Google Scholar
  25. 25.
    Leetsch, R.; Wallmersperger, T.; Kroplin, K.: Thermo-mechanical modeling of functionally graded plates. J. Intell. Mater. Syst. Struct. 20, 1799–1813 (2009)Google Scholar
  26. 26.
    Akavci, S.S.: Mechanical behavior of functionally graded sandwich plates on elastic foundation. Compos. Part B-Eng. 96, 136–152 (2016)Google Scholar
  27. 27.
    Akavci, S.S.: An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation. Compos. Struct. 108, 667–676 (2014)Google Scholar
  28. 28.
    Thai, H.T.; Choi, D.H.: A refined plate theory for functionally graded plates resting on elastic foundation. Compos. Sci. Technol. 71, 1850–1858 (2011)Google Scholar
  29. 29.
    Thai, H.T.; Choi, D.H.: A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Compos. Part B-Eng. 43, 2335–2347 (2012)Google Scholar
  30. 30.
    Gao, K.; Gao, W.; Chen, D.; Yang, J.: Nonlinear free vibration of functionally graded graphene platelets reinforced porous nano composite plates resting on elastic foundation. Compos. Struct. 204, 831–846 (2018)Google Scholar
  31. 31.
    Sheikholeslami, S.A.; Saidi, A.R.: Vibration analysis of functionally graded rectangular plates resting on elastic foundation using higher-order shear and normal deformable plate theory. Compos. Struct. 106, 350–361 (2013)Google Scholar
  32. 32.
    Zhou, K.; Huang, X.; Tian, J.; Hua, H.: Vibration and flutter analysis of supersonic porous functionally graded material plates with temperature gradient and resting on elastic foundation. Compos. Struct. 204, 63–79 (2018)Google Scholar
  33. 33.
    Gupta, A.; Talha, M.; Chaudhari, V.K.: Natural frequency of functionally graded plates resting on elastic foundation using finite element method. Procedia Technol. 23, 163–170 (2016)Google Scholar
  34. 34.
    Li, Q.; Wu, D.; Chen, X.; Liu, L.; Yu, Y.; Gao, W.: Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler-Pasternak elastic foundation. Int. J. Mech. Sci. 148, 596–610 (2018)Google Scholar
  35. 35.
    Bahmyaria, E.; Banatehranib, M.M.; Ahmadic, M.; Bahmyari, M.: Vibration analysis of thin plates resting on Pasternak foundations by element free Galerkin method. Shock Vib. 20, 309–326 (2013)Google Scholar
  36. 36.
    Zaoui, F.Z.; Ouinas, D.; Tounsi, A.: New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Compos. Part B-Eng. 159, 231–247 (2019)Google Scholar
  37. 37.
    Huang, C.S.; Lee, H.T.; Li, P.Y.; Hu, K.C.; Lan, C.W.; Chang, M.J.: Three-dimensional buckling analyses of cracked functionally graded material plates via the MLS-Ritz method. Thin Wall. Struct. 134, 189–202 (2019)Google Scholar
  38. 38.
    Taczała, M.; Buczkowski, R.; Kleiber, M.: Post buckling analysis of functionally graded plates on an elastic foundation. Compos. Struct. 132, 842–847 (2015)Google Scholar
  39. 39.
    Thai, H.T.; Kim, S.E.: Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation. Int. J. Mech. Sci. 75, 34–44 (2013)Google Scholar
  40. 40.
    Giunta, G.; Crisafulli, D.; Belouettar, S.; Carrera, E.: Hierarchical theories for the free vibration analysis of functionally graded beams. Compos. Struct. 94, 68–74 (2011)Google Scholar
  41. 41.
    Jung, W.Y.; Han, S.C.; Park, W.T.: Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation. Int. J. Mech. Sci. 111(112), 73–87 (2016)Google Scholar
  42. 42.
    Zenkour, A.M.; Radwan, A.F.: Compressive study of functionally graded plates resting on Winkler-Pasternak foundations under various boundary conditions using hyperbolic shear deformation theory. Arch. Civ. Mech. Eng. 18, 645–658 (2018)Google Scholar
  43. 43.
    Sayyad, A.S.; Ghugal, Y.M.: A unified five-degree-of freedom theory for the bending analysis of softcore and hardcore functionally graded sandwich beams and plates. J. Sandw. Struct. Mater. (2019).  https://doi.org/10.1177/1099636219840980. In press Google Scholar
  44. 44.
    Sayyad, A.S.; Ghugal, Y.M.: Effects of non-linear hygro-thermo-mechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory. J. Therm. Stresses. (2018).  https://doi.org/10.1080/01495739.2018.1469962 Google Scholar
  45. 45.
    Zenkour, A.M.; Allam, M.L.; Radwan, A.F.: Effects of transverse shear and normal strains on FG plates resting on elastic foundations under hygro-thermo-mechanical loading. Int. J. Appl. Mech. 6(5), 1–26 (2014)Google Scholar
  46. 46.
    Zenkour, A.M.; Radwan, A.F.: Hygro-thermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model. Int. J. Comput. Meth. Eng. Sci. Mech. (2019).  https://doi.org/10.1080/15502287.2019.1568618 Google Scholar
  47. 47.
    Zidi, M.; Taunsi, A.; Hauari, M.; Bedia, E.; Beg, O.A.: Bending analysis of an FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory. Aerosp. Sci. Technol. 14, 24–34 (2014)Google Scholar
  48. 48.
    Daouadji, T.H.; Adim, B.; Benferhat, R.: Bending analysis of an imperfect FGM plates under hygro-thermo-mechanical loading with analytical validation. Adv. Mater. Res. 5(1), 35–53 (2016)Google Scholar
  49. 49.
    Han, J.B.; Liew, K.M.: Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations. Int. J. Mech. Sci. 39, 977–989 (1997)zbMATHGoogle Scholar
  50. 50.
    Carrera, E.; Brischetto, S.: Modeling and analysis of functionally graded beams, plates and shells Part-II. Mech. Adv. Mater. Struct. 18(1), 1–2 (2011)zbMATHGoogle Scholar
  51. 51.
    Neves, A.M.A.; Ferreira, A.J.M.; Carrera, E.; Cinefra, M.; Roque, C.M.C.; Jorge, R.M.N.; Soares, C.M.M.: Bending of FGM plate by a sinusoidal plate formulation and collocation with radial basis functions. Mech. Res. Commun. 38, 368–371 (2011)zbMATHGoogle Scholar
  52. 52.
    Winkler, E.: Die Lehre von der Elasticitaet und Festigkeit. Prag, Dominicus (1867)Google Scholar
  53. 53.
    Pasternak, P.L.: On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow (1954)Google Scholar
  54. 54.
    Kerr, A. D.: Elastic and viscoelastic foundation models. J. Appl. Mech. 491–498 (1964)Google Scholar
  55. 55.
    Kerr, A.D.: A study of a new foundation model. 135–136 (1964)Google Scholar
  56. 56.
    Kutlu, A.; Ugurlu, B.; Omurtag, M.H.; Ergin, A.: Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid. Ocean Eng. 42, 112–125 (2012)Google Scholar
  57. 57.
    Abdeen, M.A.M.; Bichir, S.M.: Analysis of simply supported thin FGM rectangular plate resting on fluid layer. Arab. J. Sci. Eng. 38, 3267–3273 (2013)Google Scholar
  58. 58.
    Rahman, A.A.A.: Closed formed solution for thick plates resting on Kerr foundation. Int. J. Appl. Eng. Res. 12(22), 12133–12143 (2017)Google Scholar
  59. 59.
    Ta, H.D.; Noh, H.C.: Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory. Appl. Math. Model. 39, 6243–6257 (2015)MathSciNetGoogle Scholar
  60. 60.
    Vallabhan, C.V.G.; Daloglu, A.T.: Consistent FEM-Vlasov model for plates on layered soil. J. Strut. Eng. 125, 113–118 (1999)Google Scholar
  61. 61.
    Kamiya, N.; Sawaki, Y.: An alternative boundary element analysis of plates resting on elastic foundation. Boundary Elements Conf. 561–562 (1986)Google Scholar
  62. 62.
    Hetenyi, M.: A general solution for the bending of beams on an elastic foundation of arbitrary continuity. J. Appl. Phys. 21, 55–58 (1950)zbMATHGoogle Scholar
  63. 63.
    Chilton, D.S.; Wekezer, J.W.: Plates on elastic foundation. J. Struct. Eng. 116(11), 3236–3241 (1990)Google Scholar
  64. 64.
    Sheng, C.X.: A free rectangular plate on elastic foundation. J. Appl. Math. Mech. 13(10), 977–982 (1992)Google Scholar
  65. 65.
    Shen, H.S.; Xiang, Y.; Lin, F.: Nonlinear bending of functionally graded grapheme reinforced composite laminated plates resting on elastic foundations in thermal environments. Compos. Struct. 170, 80–90 (2017)Google Scholar
  66. 66.
    Najafi, F.; Shojaeefard, M.H.; Googarchin, H.S.: Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field. Compos. Part B-Eng. 107, 123–140 (2016)Google Scholar
  67. 67.
    Adineh, M.; Kadkhodayan, M.: Three-dimensional thermo-elastic analysis and dynamic response of a multi-directional functionally graded skew plate on elastic foundation. Compos. Part B-Eng. 125, 227–240 (2017)Google Scholar
  68. 68.
    Alimirzaei, S.; Sadighi, B.M.; Nikbakht, A.: Wave propagation analysis in viscoelastic thick composite plates resting on Visco-Pasternak foundation by means of quasi-3D sinusoidal shear deformation theory. Eur. J. Mech. A Solids. 74, 1–15 (2019)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Dogruoglu, A.N.; Omurtag, M.H.: Stability analysis of composite-plate foundation interaction by mixed FEM. J. Eng. Mech. 126, 928–936 (2000)Google Scholar
  70. 70.
    Ayvaz, Y.; Daloglu, A.; Dogangun, A.: Application of a modified Vlasov model to earthquake analysis of plates resting on elastic foundations. J. Sound Vib. 212(3), 388–498 (1998)Google Scholar
  71. 71.
    Daloglu, A.T.; Ozgan, K.: The effective depth of soil stratum for plates resting on elastic foundation. Struct. Eng. Mech. 18(2), 1–12 (2004)Google Scholar
  72. 72.
    Bodaghi, M.; Saidi, A.R.: Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation. Arch. Appl. Mech. 81, 765–780 (2011)zbMATHGoogle Scholar
  73. 73.
    Li, Z.D.; Yang, T.Q.; Luo, W.B.: An improved model for bending of thin viscoelastic plate on elastic foundation. Sci. Res. 1(2), 120–123 (2009)Google Scholar
  74. 74.
    Zenkour, A.M.; Sobhy, M.: Elastic foundation analysis of uniformly loaded functionally graded viscoelastic sandwich plates. J. Mech. 28(03), 439–452 (2012)Google Scholar
  75. 75.
    Zhang, C.; Zhu, H.; Shi, B.; Liu, L.: Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation. J. Rock Mech. Geotech. Eng. 6, 373–379 (2014)Google Scholar
  76. 76.
    Gupta, U.S.; Sharma, S.; Singhal, P.: Effect of two-parameter foundation on free transverse vibration of non-homogeneous orthotropic rectangular plate of linearly varying thickness. Int. J. Eng. Sci. 6(2), 32–51 (2014)Google Scholar
  77. 77.
    Gupta, U.S.; Sharma, S.; Singhal, P.: DQM modeling of rectangular plate resting on two parameter foundation. Eng. Solid Mech. 4, 33–44 (2016)Google Scholar
  78. 78.
    Ozcelikors, Y.; Omurtag, M.H.; Demir, H.: Analysis of orthotropic plate-foundation interaction by mixed finite element formulation using Gateaux differential. Comput. Struct. 62(1), 93–106 (1997)zbMATHGoogle Scholar
  79. 79.
    Naderi, A.; Saidi, A.R.: Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation. Compos. Struct. 93, 629–638 (2011)Google Scholar
  80. 80.
    Zenkour, A.M.; Allam, M.N.M.; Shaker, M.O.; Radwan, A.F.: On the simple and mixed first-order theories for plates resting on elastic foundations. Acta Mech. 220, 33–46 (2011)zbMATHGoogle Scholar
  81. 81.
    Kiani, Y.; Akbarzadeh, A.H.; Chen, Z.T.; Eslami, M.R.: Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation. Compos. Struct. 94, 2474–2484 (2012)Google Scholar
  82. 82.
    Civalek, O.: Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation. J. Sound Vib. 294, 966–980 (2006)Google Scholar
  83. 83.
    Buczkowski, R.; Taczała, M.; Kleiber, M.: A 16-node locking-free Mindlin plate resting on two-parameter elastic foundation-static and eigenvalue analysis. Comput. Assist. Meth. Eng. Sci. 22, 99–114 (2015)MathSciNetGoogle Scholar
  84. 84.
    Zenkour, A.M.: The refined sinusoidal theory for FGM plates on elastic foundations. Int. J. Mech. Sci. 51, 869–880 (2009)Google Scholar
  85. 85.
    Omurtag, M.H.; Ozutok, A.; Akoz, A.Y.: Free vibration analysis of Kirchhoff plates resting on elastic foundation by mixed finite element formulation based on gateaux differential. Int. J. Numer. Meth. Eng. 40, 295–317 (1997)Google Scholar
  86. 86.
    Sayyad, A.S.; Ghugal, Y.M.: Bending of shear deformable plates resting on Winkler foundations according to trigonometric plate theory. J. Appl. Comput. Mech. 4(3), 187–201 (2018)Google Scholar
  87. 87.
    Sayyad, A.S.; Ghugal, Y.M.: An inverse hyperbolic theory for FG beams resting on Winkler-Pasternak elastic foundation. Adv. Aircr. Spacecr. Sci. 5(6), 671–689 (2018)Google Scholar
  88. 88.
    Ghumare, S.M.; Sayyad, A.S.: A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams. Lat. Am. J. Solids Struct. 14, 1893–1911 (2017)Google Scholar
  89. 89.
    Ghumare, S.M.; Sayyad, A.S.: A new quasi-3D model for functionally graded plates. J. Appl. Comput. Mech. 5(2), 367–380 (2019)Google Scholar
  90. 90.
    Naik, N.S.; Sayyad, A.S.: 2D analysis of laminated composite and sandwich plates using new fifth order theory. Lat. Am. J. Solids Struct. 15(9), 114–125 (2018)Google Scholar
  91. 91.
    Naik, N.S.; Sayyad, A.S.: 1D analysis of laminated composite and sandwich plates using a new fifth-order plate theory. Lat. Am. J. Solids Struct. 15, 1–17 (2018)Google Scholar
  92. 92.
    Naik, N.S.; Sayyad, A.S.: An accurate computational model for thermal analysis of laminated composite and sandwich plates. J. Therm. Stresses. (2019).  https://doi.org/10.1080/01495739.2018.1522986 Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  • Shantaram M. Ghumare
    • 1
  • Atteshamuddin S. Sayyad
    • 1
    Email author
  1. 1.Department of Civil Engineering, Sanjivani College of Engineering, KopargaonSavitribai Phule Pune UniversityPuneIndia

Personalised recommendations