Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Abstract

The purpose of the paper is to study stability and free vibrations of laminated plates and shallow shells composed of functionally graded materials. The approach proposed incorporates the Ritz method and the R-functions theory. It is assumed that the shell consists of three layers and is loaded in the middle plane. The both cases of uniform as well as non-uniform load are possible. The power-law distribution in terms of volume fractions is applied to get effective material properties for the layers. These properties are calculated for different arrangements and thicknesses of the layers by the analytical formulae obtained in the paper. The mathematical formulation is carried out in framework of the first-order shear deformation theory. The proposed approach consists of two steps. The first step is to define the pre-buckling state by solving the respective elasticity problem. The critical buckling load and frequencies of functionally graded material shallow shells are determined in the second step. The highlight of the method proposed is that it can be used for vibration and buckling analysis of plates and shallow shells of complex shape. The numerical results for frequencies and buckling load of plates and shallow shells of complex shape and different curvatures are presented to demonstrate the potential of the method developed. Different functionally graded material plates and shallow shells composed of a mixture of metal and ceramics are studied. The effects of the power law index, boundary conditions, thickness of the core, and face sheet layers on the fundamental frequencies and critical loads are discussed in this paper. The main advantage of the method is that it provides an analytical representation of the unknown solution, which is important when solving nonlinear problems.

Keywords

Functionally graded material sandwich plates shallow shells complex planform Ritz method R-functions theory

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References

Reddy

. Analysis of functionally graded plates. J Num Method Eng 2000; 47: 663–684.

Shen

. Functionally graded materials of plates and shells, Boca Raton, FL: CRC Press, 2009.

Kou

. Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib 2008; 311: 498–515.

Meiche

NEI

Tounsi

Ziane

, et al. A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. J Mech Sci 2011; 53: 237–247.

Neves

AMA

Ferreira

AJM

Carrera

, et al. 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. J Compos Part B: Eng 2013; 44: 657–674.

Dozio

. Natural frequencies of sandwich plates with FGM core via variable-kinematic 2D Ritz models. J Compos Struct 2013; 96: 561–568.

Thai

Nguye

, et al. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. Euro J Mech A/Solids 2014; 45: 211–225.

Nguyen

Thai

, et al. A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. J Compos Part B: Eng 2014; 66: 233–246.

Matsunaga

. Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory. J Compos Struct 2008; 84: 132–146.

10.

Fazzolari

Carrera

. Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core. J Sound Vib 2014; 333: 1485–1508.

11.

Bennoun

Houari

MSA

Tounsi

. A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. J Mech Adv Mater Struct 2016; 23: 423–431.

12.

Pandey

Pradyumna

. Analysis of functionally graded sandwich plates using a higher-order layerwise theory. J Compos Part B: Eng 2018; 153: 325–336.

13.

Zenkour

. A comprehensive analysis of functionally graded sandwich plates: part 2 - buckling and free vibration. J Solid Struct 2005; 42: 5243–5258.

14.

Focacci

Foraboschi

De Stefano

. Composite beam generally connected: analytical model. Compos Struct 2015; 133: 1237–1248.

15.

Jha

Kant

Singh

. A critical review of recent research on functionally graded plates. J Comp Struct 2013; 96: 833–849.

16.

Swaminathan

Naveenkumar

Zenkour

Carrera

. Stress, vibration and buckling analyses of FGM plates –state-of-the-art review. J Compos Struct 2015; 120: 10–31.

17.

Thai

H-T

Kim

S-E

. A review of theories for the modeling and analysis of functionally graded plates and shells. J Compos Struct 2015; 128: 70–86.

18.

BirmanV Kardomateas

. Review of current trends in research and applications of sandwich structures. J Compos Part B: Eng 2018; 142: 221–240.

19.

Kumar

. The Rayleigh-Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: a literature review. J Vib Control 2018; 24: 1205–1227.

20.

Monterrubio

. Free vibration of shallow shells using the Rayleigh-Ritz method and penalty parameters. J Mech Eng Sci 2009; 223: 2263–2272.

21.

Naeem

Khan

Arshad

, et al. Vibration of three-layered FGM cylindrical shells with middle layer of isotropic material for various boundary conditions. World J Mech 2014; 4: 315–331.

22.

Lee

Kwak

. Free vibration analysis of a circular cylindrical shell using the Rayleigh-Ritz method and comparison of different shell theories. J Sound Vib 2015; 353: 344–377.

23.

Rvachev

. The R-functions theory and its some application, Kiev: Naukova Dumka, 1982, pp. 552–552. (in Russian).

24.

Kurpa

Mazur

. Method of R-functions for investigating parametric vibrations of orthotropic plates with complex shape. J Math Sci 2010; 171: 1–14.

25.

Awrejcewicz

Kurpa

Mazur

. On the parametric vibrations and meshless discretization of orthotropic plates with complex shape. J Nonlin Sci Numer Simul 2010; 11: 371–386.

26.

Kurpa

Mazur

Tkachenko

. Dynamical stability and parametrical vibrations of the laminated plates with complex shape. Latin Am J Solids Struct 2013; 10: 175–188.

27.

Kurpa

Mazur

Tkachenko

. Parametric vibrations of multilayer plates of complex shape. J Math Sci 2014; 174: 101–114.

28.

Kurpa

Shmatko

Timchenko

. Free vibration analysis of laminated shallow shells with complex shape using the R-functions method. J Compos Struct 2010; 93: 225–233.

29.

Kurpa

Pilgun

Amabili

. Nonlinear vibrations of shallow shells with complex boundary: R-functions method and experiments. J Sound Vib 2007; 306: 580–600.

30.

Kurpa

. Nonlinear free vibrations of multilayer shallow shells with a symmetric structure and with a complicated form of the plan. J Math Sci 2009; 162: 85–98.

31.

Awrejcewicz

Kurpa

Shmatko

. Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness. Latin Am J Solids Struct 2013; 10: 147–160.

32.

Kurpa

Shmatko

. Nonlinear vibrations of laminated shells with layers of variable thickness. Shell Struct: Theory Appl 2014; 3: 305–308.

33.

Awrejcewicz

Kurpa

Shmatko

. Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory. J Compos Struct 2015; 125: 575–585.

34.

Kurpa

Timchenko

Osetrov

, et al. Nonlinear vibration analysis of laminated shallow shells with clamped cutouts by the R-functions method. J Nonlin Dyn 2018; 93: 133–147.

35.

Awrejcewicz

Kurpa

Shmatko

. Linear and nonlinear free vibration analysis of laminated functionally graded shallow shells with complex plan form and different boundary conditions. J Nonlin Mech 2018; 107: 161–169.

36.

Shmatko

Kurpa

Awrejcewicz

. Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory. Latin Am J Solids Struct 2019; 16: e95–e95.

37.

Awrejcewicz

Kurpa

Shmatko

. Analysis of geometrically nonlinear vibrations of functionally graded shallow shells of a complex shape. Latin Am J Sol Struct 2017; 14: 1648–1668.

38.

Delale

Erdogan

. The crack problem for nonhomogeneous plane. J Appl Mech 1983; 50: 609–609.

39.

Jin

Shi

, et al. Modified Fourier- Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions. J Mech Sci 2015; 93: 256–269.