Journal of Applied and Computational Mechanics

Bending and Free Vibration Analysis of Functionally Graded Plates via Optimized Non-polynomial Higher Order Theories

Document Type : Research Paper

Authors

1 Faculty of Mechanical Engineering, National University of Engineering (UNI), Av. Túpac Amaru 210, Rimac, Lima, Peru

2 Department of Civil Engineering, Universidad Peruana de Ciencias Aplicadas (UPC), Lima, Peru

3 Faculty of Mechanical Engineering, Universidad de Ingeniería y Tecnología (UTEC), Jr. Medrano Silva 165, Barranco, Lima, Peru

Abstract

Optimization concept in the context of shear deformation theories was born for the development of accurate models to study the bending problem of structures. The present study seeks to extend such an approach to the dynamic analysis of plates. A compact and unified formulation with non-polynomial shear strain shape functions (SSSFs) is employed to develop a static and free vibration analysis of simply supported functionally graded plates. In this context, three new non-polynomial displacement fields are proposed using trigonometric and hyperbolic SSSFs. Then, the non-polynomial SSSFs are optimized by varying the arguments of the trigonometric and hyperbolic functions. Additionally, the Mori-Tanaka approach is used to estimate the effective properties of the functionally graded plates. The Principle of Virtual Displacement (PVD) and the Hamilton’s Principle along with the Navier closed-form solution technique are used to obtain exact results. The obtained numerical results are in a good agreement with 3D and 2D higher order shear deformation theory solutions available in the literature.

Keywords

Main Subjects

[1] Bever MB, Duwez PE. Gradients in composite materials. Mater Sci Eng 10 (1972) 1–8.
[2] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally graded materials: design, processing and applications. Kluwer Academic Publishers; 1999.
[3] Kieback B, Neubrand A, Riedel H. Processing techniques for functionally graded materials. Materials Science and Engineering A 362 (2003) 81–105.
[4] Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E. Stress, vibration and buckling analyses of FGM plates - A state-of-the-art review. Composite Structures 120 (2015) 10–31.
[5] Reddy JN, Cheng ZQ. Frequency of functionally graded plates with three-dimensional asymptotic approach. J. Eng. Mech. 129(18) (2003) 896–900.
[6] Vel SS, Batra RC. Vel SS, Batra RC. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J. Sound Vib. 272 (2004) 703–30.
[7] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21 (1973) 571–4.
[8] Benveniste Y. A new approach to the application of Mori–Tanaka’s theory of composite materials, Mechanics of Materials 6 (1987) 147–157.
[9] Zenkour AM. On vibration of functionally graded plates according to a refined trigonometric plate theory. International Journal of Structural Stability and Dynamics 5(2) (2005) 279–297.
[10] Zenkour AM. A comprehensive analysis of functionally graded sandwich plates: part 1 – Deflection and stresses. Int J. Solids Struct. 42 (2005) 5224–42.
[11] Zenkour AM. A comprehensive analysis of functionally graded sandwich plates: part 2 – buckling and free vibration. Int. J. Solids Struct. 42(18–19) (2005) 5243–58.
[12] Matsunaga H. Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos. Struct. 82 (2008) 499–512.
[13] Fares ME, Elmarghany MK, Atta D. An efficient and simple refined theory for bending and vibration of functionally graded plates. Compos. Struct. 91(3) (2009) 296–305.
[14] Cinefra M, Belouettar S, Soave M, Carrera E. Variable kinematic models applied to free-vibration analysis of functionally graded material shells. Eur. J. Mech. – A/Solids 29(6) (2010) 1078–87.
[15] Hadji L, Atmane HA, Tounsi A, Mechab I, Bedia EAA. Free vibration of functionally graded sandwich plates using four-variable refined plate theory. Appl. Math. Mech. – Engl. Ed. 32(7) (2011) 925–42.
[16] Qian LF, Batra RC, Chen LM. Free and forced vibrations of thick rectangular plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin (MLPG) method. Comput. Model. Eng. Sci. 4(5) (2003) 519–34.
[17] Ferreira AJM, Batra RC, Roque CMC, Qian LF, Jorge RMN. Natural frequencies of functionally graded plates by a meshless method. Compos. Struct. 75(1-4) (2006) 593–600.
[18] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC., Jorge RMN, Soares CMM. A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Composites: Part B 43 (2012) 711-725.
[19] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC., Jorge RMN, Soares CMM. 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. Composites: Part B 44 (2013) 657-674.
[20] Carrera E, Brischetto S, Robaldo A. Variable Kinematic Model for the Analysis of Functionally Graded Material Plates. AIAA Journal 46(1) (2008) 194-203.
[21] Brischetto S, Carrera E. Advanced mixed theories for bending analysis of functionally graded plates. Computers & Structures 88 (2010) 1474-1483.
[22] Carrera E, Brischetto S, Cinefra M, Soave M. Effects of thickness stretching in functionally graded plates and shells. Compos. Part B Eng. 42(2) (2011) 123–33.
[23] Mantari JL, Oktem AS, Soares CG. Bending response of functionally graded plates by using a new higher order shear deformation theory. Compos. Struct. 94 (2012) 714–23.
[24] Mantari JL, Soares CG. Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos. Struct. 94 (2012) 1991–2000.
[25] Mantari JL, Soares CG. A novel higher-order shear deformation theory with stretching effect for functionally graded plates. Compos. Part B Eng. 45(1) (2013) 268–81.
[26] Mantari JL, Soares CG. Optimized sinusoidal higher order shear deformation theory for the analysis of functionally graded plates and shells. Compos. Part B Eng. 56 (2014) 126–36.
[27] Ait Amar Meziane, Abdelaziz HH, Tounsi A. 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) (2014) 293-318.
[28] Abdelaziz HH, Ait Amar Meziane, Bousahla AA, Alwabli AS. An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions. Steel and Composite Structures 25(6) (2017) 693-704.
Journal of Applied and Computational Mechanics
Volume 5, Issue 2
April 2019
Pages 281-298