Natural Frequencies of Functionally Graded Sandwich Plates Resting on an Elastic Foundation Using Chebyshev Finite Element Method

TON-THAT, Hoang Lan

doi:10.2478/ama-2025-0039

Abstract

In this study, the article uses a finite element method based on Chebyshev polynomials to calculate the natural frequencies of functionally graded sandwich (FGS) plates with hard-core (HC) or soft-core (SC) resting on an elastic foundation. Chebyshev polynomials are a series of orthogonal polynomials defined recursively, and the value of them belongs to the range [-1, 1] as well as vanishes at Gauss points. More clearly, the novelty of this article is to use the high-order shape functions that satisfy the interpolation condition at the points based on Chebyshev polynomials to build the flat quadrilateral element for analysis of FGS plates. On the other hand, these plates are composed of two functionally graded skins and a hard or soft core. The elastic foundation with a two-parameter as a spring stiffness (α₁) and a shear layer stiffness (α₂) are used. Comparative examples are presented to validate the effectiveness of the current approach.

References

Hosseini HS, Taher HRD, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Appl. Math. Model. 2010;34(5):1276-91. Available from: https://doi.org/10.1016/j.apm.2009.08.008
Search in Google Scholar Back to article
Reddy JN. Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 2000;47:663-84.
Search in Google Scholar Back to article
Zenkour AM. Bending analysis of functionally graded sandwich plates using a simple four unknown shear and normal deformations theory. J. Sandw. Struct. Mater. 2013;15:629-56. Available from: https://doi.org/10.1177/1099636213498886
Search in Google Scholar Back to article
Reddy JN. A general nonlinear third-order theory of functionally graded plates. Int. J. Aerosp. Lightweight Struct. 2011;1:1-21. Available from: https://doi.org/10.3850/S201042861100002X
Search in Google Scholar Back to article
Talha M, Singh BN. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl. Math. Model. 2010;34:3991-4011. Available from: https://doi.org/10.1016/j.apm.2010.03.034
Search in Google Scholar Back to article
Zenkour AM. Generalized shear deformation theory for bending analysis of functionally graded materials. Appl. Math. Model. 2006;30(1):67-84. Available from: https://doi.org/10.1016/j.apm.2005.03.009
Search in Google Scholar Back to article
Yaghoobi H, Fereidoon A. 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. B: Eng. 2014;62:54-64. Available from: https://doi.org/10.1016/j.com-positesb.2014.02.014
Search in Google Scholar Back to article
Sobhy M, Al-Mukahal FH. Magnetic control of vibrational behavior of smart FG sandwich plates with honeycomb core via a quasi 3D plate theory. Adv. Eng. Mater. 2023;25(13):2300096. Available from: https://doi.org/10.1002/adem.202300096
Search in Google Scholar Back to article
Zarga D, Tounsi A, Bousahla AA, Bourada F, Mahmoud SR. Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory. Steel Compos. Struct. 2019;32(3):389-410. Available from: https://doi.org/10.12989/scs.2019.32.3.389
Search in Google Scholar Back to article
Alibeigloo A, Alizadeh M. Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method. Eur. J. Mech. A/Solids. 2015;54:252-66. Available from: https://doi.org/10.1016/j.euromechsol.2015.06.011
Search in Google Scholar Back to article
Natarajan S, Manickam G. Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem. Anal. Des. 2012;57:32-42. Available from: https://doi.org/10.1016/j.finel.2012.03.006
Search in Google Scholar Back to article
Li Q, Iu VP, Kou KP. Three-dimensional vibration analysis of functionally graded material sandwich plates. J. Sound Vib. 2008;311:498– 515. Available from: https://doi.org/10.1016/j.jsv.2007.09.018
Search in Google Scholar Back to article
Liu N, Jeffers AE. Isogeometric analysis of laminated composite and functionally graded sandwich plates based on a layerwise displacement theory. Compos. Struct. 2017;176(15):143-53. Available from: https://doi.org/10.1016/j.compstruct.2017.05.037
Search in Google Scholar Back to article
Neves A, Ferreira A, Carrera E, Cinefra M, Roque C, Jorge R, Soares CM. 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. 2013;44:657-74. Available from: https://doi.org/10.1016/j.compositesb.2012.01.089
Search in Google Scholar Back to article
Wang Y, Tham L, Cheung Y. Beams and plates on elastic foundations: a review. Prog Struct Eng Mat. 2005;7:174–82. Available from: https://doi.org/10.1002/pse.202
Search in Google Scholar Back to article
Winkler E. Die Lehre von der Elastizität und Festigkeit (The Theory of Elasticity and Stiffness), H. Dominicus, Prague, Czechoslovakia. 1867.
Search in Google Scholar Back to article
Kolahchi R, Safari M, Esmailpour M. Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium, Compos. Struct. 2016;150:255–65. Available from: https://doi.org/10.1016/j.compstruct.2016.05.023
Search in Google Scholar Back to article
Pasternak P. 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.
Search in Google Scholar Back to article
Kneifati MC. Analysis of plates on a Kerr foundation model. J. Eng. Mech. 1985;111:1325–42.
Search in Google Scholar Back to article
Mohamed M, Samir B, Abdelkader M, Abdelouahed T, Abdelmoumen AB, Mahmoud SR. Thermodynamic behavior of functionally graded sandwich plates resting on different elastic foundation and with various boundary conditions. J. Sandw. Struct. Mater. 2021;23(3):1028-57. Available from: https://doi.org/10.1177/1099636219851281
Search in Google Scholar Back to article
Guerroudj HZ, Yeghnem R, Kaci A, Zaoui FZ, Benyoucef S. Eigenfre-quencies of advanced composite plates using an efficient hybrid quasi-3D shear deformation theory. Smart Struct. Syst. 2018;22(1):121-32. Available from: https://doi.org/10.12989/sss.2018.22.1.121
Search in Google Scholar Back to article
Lazreg H, Mehmet A, Nafissa Z. Natural frequency analysis of imperfect FG sandwich plates resting on Winkler-Pasternak foundation. Mater. Today: Proc. 2022;53(1):153-60. Available from: https://doi.org/10.1016/j.matpr.2021.12.485
Search in Google Scholar Back to article
Kurpa L, Shmatko T, Linnik A. Buckling Analysis of Functionally Graded Sandwich Plates Resting on an Elastic Foundation and Subjected to a Nonuniform Loading. Mech. Compos. Mater. 2023;59:645-58. Available from: https://doi.org/10.1007/s11029-023-10122-w
Search in Google Scholar Back to article
Pham VV, Le QH. Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory. Def. Technol. 2022;18(3):490-508. Available from: https://doi.org/10.1016/j.dt.2021.03.006
Search in Google Scholar Back to article
Dang TH, Yang DJ, Liu Y. Improvements in shear locking and spurious zero energy modes using Chebyshev finite element method. J. Comput. Inf. Sci. Eng. 2019;19:011006. Available from: https://doi.org/10.1115/1.4041829
Search in Google Scholar Back to article
Ton-That HL, Nguyen-Van H, Chau-Dinh T. A novel quadrilateral element for analysis of functionally graded porous plates/shells reinforced by graphene platelets. Arch. Appl. Mech. 2021;91:2435-66. Available from: https://doi.org/10.1007/s00419-021-01893-6
Search in Google Scholar Back to article
Fornberg B, Zuev J. The Runge Phenomenon and Spatially Variable Shape Parameters in RBF Interpolation. Comput. Math. Appl. 2007;54(3):379–98. Available from: https://doi.org/10.1016/j.camwa.2007.01.028
Search in Google Scholar Back to article
Lee SJ. Free Vibration Analysis of Plates by Using a Four-Node Finite Element Formulated With Assumed Natural Transverse Shear Strain. J. Sound Vib. 2004;278(3):657–84. Available from: https://doi.org/10.1016/j.jsv.2003.10.018
Search in Google Scholar Back to article
Abassian F, Haswell DJ, Knowles NC. Free Vibration Benchmarks. National Agency for Finite Element Methods and Standards, Glasgow, UK. 1987.
Search in Google Scholar Back to article
Shahsavari D, Shahsavari M, Li L, Karami B. A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp. Sci. Technol. 2018;72:134-49. Available from: https://doi.org/10.1016/j.ast.2017.11.004
Search in Google Scholar Back to article
Baferani AH, Saidi A, Ehteshami H. Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Compos. Struct. 2011;93(7):1842-53. Available from: https://doi.org/10.1016/j.compstruct.2011.01.020
Search in Google Scholar Back to article
Akavci S. Mechanical behaviour of functionally graded sandwich plates on elastic foundation. Compos. B: Eng. 2016;96:136-52. Available from: https://doi.org/10.1016/j.compositesb.2016.04.035
Search in Google Scholar Back to article

Natural Frequencies of Functionally Graded Sandwich Plates Resting on an Elastic Foundation Using Chebyshev Finite Element Method

Abstract

Paradigm

My account