On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates

Abstract

This paper presents a novel C0 higher-order layerwise finite element model for static and free vibration analysis of functionally graded materials (FGM) sandwich plates. The proposed layerwise model, which is developed for multilayer composite plates, supposes higher-order displacement field for the core and first-order displacement field for the face sheets maintaining a continuity of displacement at layer. Unlike the conventional layerwise models, the present one has an important feature that the number of variables is fixed and does not increase when increasing the number of layers. Thus, based on the suggested model, a computationally efficient C0 eight-node quadrilateral element is developed. Indeed, the new element is free of shear locking phenomenon without requiring any shear correction factors. Three common types of FGM plates, namely, (i) isotropic FGM plates; (ii) sandwich plates with FGM face sheets and homogeneous core and (iii) sandwich plates with homogeneous face sheets and FGM core, are considered in the present work. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume power laws of the constituents. The equations of motion of the FGM sandwich plate are obtained via the classical Hamilton’s principle. Numerical results of present model are compared with 2D, quasi-3D, and 3D analytical solutions and other predicted by advanced finite element models reported in the literature. The results indicate that the developed finite element model is promising in terms of accuracy and fast rate of convergence for both thin and thick FGM sandwich plates. Finally, it can be concluded that the proposed model is accurate and efficient in predicting the bending and free vibration responses of FGM sandwich plates.

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References

  1. 1.

    Vinson JR (2001) Sandwich structures. Appl Mech Rev 54(3):201–214

    MathSciNet  Article  Google Scholar 

  2. 2.

    Wang Z-X, Shen H-S (2012) Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Compos B Eng 43(2):411–421

    Article  Google Scholar 

  3. 3.

    Li Q, Iu V, Kou K (2008) Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib 311(1–2):498–515

    Article  Google Scholar 

  4. 4.

    Sofiyev A, Osmancelebioglu E (2017) The free vibration of sandwich truncated conical shells containing functionally graded layers within the shear deformation theory. Compos B Eng 120:197–211

    Article  Google Scholar 

  5. 5.

    Anderson TA (2003) A 3-D elasticity solution for a sandwich composite with functionally graded core subjected to transverse loading by a rigid sphere. Compos Struct 60(3):265–274

    Article  Google Scholar 

  6. 6.

    Vel SS, Batra R (2004) Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vib 272(3–5):703–730

    Article  Google Scholar 

  7. 7.

    Kashtalyan M, Menshykova M (2009) Three-dimensional elasticity solution for sandwich panels with a functionally graded core. Compos Struct 87(1):36–43

    Article  Google Scholar 

  8. 8.

    Swaminathan K, Naveenkumar D, Zenkour A et al (2015) Stress, vibration and buckling analyses of FGM plates. A state-of-the-art review. Compos Struct 120:10–31

    Article  Google Scholar 

  9. 9.

    Ebrahimi MJ, Najafizadeh MM (2014) Free vibration analysis of two-dimensional functionally graded cylindrical shells. Appl Math Model 38(1):308–324

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Chi S-H, Chung Y-L (2006) Mechanical behavior of functionally graded material plates under transverse load. Part I: Analysis. Int J Solids Struct 43(13):3657–3674

    MATH  Article  Google Scholar 

  11. 11.

    Zhang D-G, Zhou Y-H (2008) A theoretical analysis of FGM thin plates based on physical neutral surface. Comput Mater Sci 44(2):716–720

    Article  Google Scholar 

  12. 12.

    Abrate S (2008) Functionally graded plates behave like homogeneous plates. Compos B Eng 39(1):151–158

    Article  Google Scholar 

  13. 13.

    Avcar M, Mohammed WKM (2018) Free vibration of functionally graded beams resting on Winkler-Pasternak foundation. Arab J Geosci 11(10):232

    Article  Google Scholar 

  14. 14.

    Civalek Ö, Uzun B, Yaylı MÖ et al (2020) Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. Eur Phys J Plus 135(4):381

    Article  Google Scholar 

  15. 15.

    Kandasamy R, Dimitri R, Tornabene F (2016) Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments. Compos Struct 157:207–221

    Article  Google Scholar 

  16. 16.

    Mantari J, Ore M (2015) Free vibration of single and sandwich laminated composite plates by using a simplified FSDT. Compos Struct 132:952–959

    Article  Google Scholar 

  17. 17.

    Avcar M (2019) Free vibration of imperfect sigmoid and power law functionally graded beams. Steel Compos Struct 30(6):603–615

    Google Scholar 

  18. 18.

    Civalek Ö (2017) Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method. Compos B Eng 111:45–59

    Article  Google Scholar 

  19. 19.

    Fakher M, Hosseini-Hashemi S (2020) Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution. Eng Comput. https://doi.org/10.1007/s00366-020-01058-z

    Article  Google Scholar 

  20. 20.

    Ferreira A, Castro LM, Bertoluzza S (2009) A high order collocation method for the static and vibration analysis of composite plates using a first-order theory. Compos Struct 89(3):424–432

    Article  Google Scholar 

  21. 21.

    Neves A, Ferreira A, Carrera E et al (2013) Free vibration analysis of functionally graded shells by a higher-order shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. Eur J Mech-A/Solids 37:24–34

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Thai CH, Zenkour A, Wahab MA et al (2016) A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis. Compos Struct 139:77–95

    Article  Google Scholar 

  23. 23.

    Zenkour A (2005) A comprehensive analysis of functionally graded sandwich plates: part 2—buckling and free vibration. Int J Solids Struct 42(18–19):5243–5258

    MATH  Article  Google Scholar 

  24. 24.

    Zenkour AM (2013) Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J Sandwich Struct Mater 15(6):629–656

    Article  Google Scholar 

  25. 25.

    Mehar K, Kumar Panda S, Devarajan Y et al (2019) Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading. Compos Struct 216:406–414

    Article  Google Scholar 

  26. 26.

    Houari MSA, Tounsi A, Bég OA (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

    Article  Google Scholar 

  27. 27.

    Bourada M, Tounsi A, Houari MSA et al (2012) A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates. J Sandwich Struct Mater 14(1):5–33

    Article  Google Scholar 

  28. 28.

    Ebrahimi F, Barati MR, Civalek Ö (2020) Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng Comput 36:953–964

    Article  Google Scholar 

  29. 29.

    Ebrahimi F, Farazmandnia N, Kokaba MR et al (2019) Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Eng Comput. https://doi.org/10.1007/s00366-019-00864-4

    Article  Google Scholar 

  30. 30.

    Qaderi S, Ebrahimi F (2020) Vibration analysis of polymer composite plates reinforced with graphene platelets resting on two-parameter viscoelastic foundation. Eng Comput. https://doi.org/10.1007/s00366-020-01066-z

    Article  Google Scholar 

  31. 31.

    Zenkour A (2005) A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses. Int J Solids Struct 42(18–19):5224–5242

    MATH  Article  Google Scholar 

  32. 32.

    Hadji L, Atmane HA, Tounsi A et al (2011) Free vibration of functionally graded sandwich plates using four-variable refined plate theory. Appl Math Mech 32(7):925–942

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Neves A, Ferreira A, Carrera E et al (2012) A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos B Eng 43(2):711–725

    Article  Google Scholar 

  34. 34.

    Neves A, Ferreira A, Carrera E et al (2012) A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Struct 94(5):1814–1825

    Article  Google Scholar 

  35. 35.

    Neves A, Ferreira A, Carrera E et al (2013) Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos B Eng 44(1):657–674

    Article  Google Scholar 

  36. 36.

    Ye R, Zhao N, Yang D et al (2020) Bending and free vibration analysis of sandwich plates with functionally graded soft core, using the new refined higher-order analysis model. J Sandw Struct Mater. https://doi.org/10.1177/1099636220909763

    Article  Google Scholar 

  37. 37.

    Meziane MAA, Abdelaziz HH, 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

    Article  Google Scholar 

  38. 38.

    Nguyen V-H, Nguyen T-K, Thai H-T et al (2014) A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos B Eng 66:233–246

    Article  Google Scholar 

  39. 39.

    Bennoun M, Houari MSA, 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

    Article  Google Scholar 

  40. 40.

    Meksi R, Benyoucef S, Mahmoudi A et al (2019) An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J Sandw Struct Mater 21(2):727–757

    Article  Google Scholar 

  41. 41.

    Sayyad AS, Ghugal YM (2019) 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. https://doi.org/10.1177/1099636219840980

    Article  Google Scholar 

  42. 42.

    Belkhodja Y, Ouinas D, Zaoui FZ et al (2020) An exponential-trigonometric higher order shear deformation theory (HSDT) for bending, free vibration, and buckling analysis of functionally graded materials (FGMs) plates. Adv Compos Lett 29:0963693519875739

    Google Scholar 

  43. 43.

    Saini R, Lal R (2020) Axisymmetric vibrations of temperature-dependent functionally graded moderately thick circular plates with two-dimensional material and temperature distribution. Eng Comput. https://doi.org/10.1007/s00366-020-01056-1

    Article  Google Scholar 

  44. 44.

    Sharma R, Jadon V, Singh B (2015) A review on the finite element methods for heat conduction in functionally graded materials. J Inst Eng 96(1):73–81

    Google Scholar 

  45. 45.

    Chareonsuk J, Vessakosol P (2011) Numerical solutions for functionally graded solids under thermal and mechanical loads using a high-order control volume finite element method. Appl Therm Eng 31(2–3):213–227

    Article  Google Scholar 

  46. 46.

    Pandey S, Pradyumna S (2018) Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Compos B Eng 153:325–336

    Article  Google Scholar 

  47. 47.

    Belarbi M-O, Tati A, Ounis H et al (2017) On the free vibration analysis of laminated composite and sandwich plates: a layerwise finite element formulation. Latin Am J Solids Struct 14(12):2265–2290

    Article  Google Scholar 

  48. 48.

    Das M, Barut A, Madenci E et al (2006) A triangular plate element for thermo-elastic analysis of sandwich panels with a functionally graded core. Int J Numer Meth Eng 68(9):940–966

    MATH  Article  Google Scholar 

  49. 49.

    Talha M, Singh B (2010) Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl Math Model 34(12):3991–4011

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Natarajan S, Manickam G (2012) Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem Anal Des 57:32–42

    Article  Google Scholar 

  51. 51.

    Nguyen T-K, Nguyen V-H, Chau-Dinh T et al (2016) Static and vibration analysis of isotropic and functionally graded sandwich plates using an edge-based MITC3 finite elements. Compos B Eng 107:162–173

    Article  Google Scholar 

  52. 52.

    Gupta A, Talha M, Singh B (2016) Vibration characteristics of functionally graded material plate with various boundary constraints using higher order shear deformation theory. Compos B Eng 94:64–74

    Article  Google Scholar 

  53. 53.

    Kulikov G, Plotnikova S, Carrera E (2018) A robust, four-node, quadrilateral element for stress analysis of functionally graded plates through higher-order theories. Mech Adv Mater Struct 25(15–16):1383–1402

    Article  Google Scholar 

  54. 54.

    Carrera E (2003) Historical review of Zig-Zag theories for multilayered plates and shells. Appl Mech Rev 56(3):287–308

    Article  Google Scholar 

  55. 55.

    Li D (2020) Layerwise theories of laminated composite structures and their applications: a review. Arch Comput Methods Eng. https://doi.org/10.1007/s11831-019-09392-2

    Article  Google Scholar 

  56. 56.

    Raissi H, Shishehsaz M, Moradi S (2019) Stress distribution in a five-layer sandwich plate with FG face sheets using layerwise method. Mech Adv Mater Struct 26(14):1234–1244

    Article  Google Scholar 

  57. 57.

    Belarbi MO, Tati A (2015) A new C0 finite element model for the analysis of sandwich plates using combined theories. Int J Struct Eng 6(3):212–239

    Article  Google Scholar 

  58. 58.

    Belarbi M-O, Tati A, Ounis H et al (2016) Development of a 2D isoparametric finite element model based on the layerwise approach for the bending analysis of sandwich plates. Struct Eng Mech 57(3):473–506

    Article  Google Scholar 

  59. 59.

    Nikbakht S, Salami SJ, Shakeri M (2019) A 3D full layer-wise method for yield achievement in Functionally Graded Sandwich Plates with general boundary conditions. Eur J Mech-A/Solids 75:330–347

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Methods Eng 9(2):87–140

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Pluciński P, Jaśkowiec J (2020) Three-dimensional analysis of laminated plates with functionally graded layers by two-dimensional numerical model. Eng Trans 68(1):21–45. https://doi.org/10.24423/EngTrans.1063.20200102

  62. 62.

    Iurlaro L, Gherlone M, Di Sciuva M (2014) Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory. J Sandw Struct Mater 16(6):669–699

    Article  Google Scholar 

  63. 63.

    Belarbi MO, Tati A (2016) Bending analysis of composite sandwich plates with laminated face sheets: new finite element formulation. J Solid Mech 8(2):280–299

    Google Scholar 

  64. 64.

    Reddy JN (1993) An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Compos Struct 25(1–4):21–35

    Article  Google Scholar 

  65. 65.

    Liew K, Pan Z, Zhang L (2019) An overview of layerwise theories for composite laminates and structures: development, numerical implementation and application. Compos Struct 316:240–259

    Article  Google Scholar 

  66. 66.

    Liu M, Cheng Y, Liu J (2015) High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core. Compos B Eng 72:97–107

    Article  Google Scholar 

  67. 67.

    Zenkour AM (2006) Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Model 30(1):67–84

    MATH  Article  Google Scholar 

  68. 68.

    Reddy J (2000) Analysis of functionally graded plates. Int J Numer Meth Eng 47(1–3):663–684

    MATH  Article  Google Scholar 

  69. 69.

    Delale F, Erdogan F (1983) The crack problem for a nonhomogeneous plane. J Appl Mech 50:609–614

    MATH  Article  Google Scholar 

  70. 70.

    Nguyen TN, Thai CH, Nguyen-Xuan H (2016) A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method. Thin-Walled Struct 107:473–488

    Article  Google Scholar 

  71. 71.

    Gilhooley D, Batra R, Xiao J et al (2007) Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions. Compos Struct 80(4):539–552

    MATH  Article  Google Scholar 

  72. 72.

    Uymaz B, Aydogdu M (2007) Three-dimensional vibration analyses of functionally graded plates under various boundary conditions. J Reinf Plast Compos 26(18):1847–1863

    Article  Google Scholar 

  73. 73.

    Jin G, Su Z, Shi S et al (2014) Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Compos Struct 108:565–577

    Article  Google Scholar 

  74. 74.

    Xiang S, Kang G-W, Yang M-S et al (2013) Natural frequencies of sandwich plate with functionally graded face and homogeneous core. Compos Struct 96:226–231

    Article  Google Scholar 

  75. 75.

    Alibeigloo A, Alizadeh M (2015) Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method. Eur J Mech-A/Solids 54:252–266

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Mohamed-Ouejdi Belarbi.

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Hirane, H., Belarbi, MO., Houari, M.S.A. et al. On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates. Engineering with Computers (2021). https://doi.org/10.1007/s00366-020-01250-1

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Keywords

  • Functionally graded materials
  • Sandwich plates
  • Layerwise
  • Bending analysis
  • Free vibration
  • Finite element method