Vibration Characteristics of Functionally Graded Material Skew Plate in Thermal Environment

Authors: Gulshan Taj M. N. A., Anupam Chakrabarti, Vipul Prakash

Abstract:

In the present investigation, free vibration of functionally graded material (FGM) skew plates under thermal environment is studied. Kinematics equations are based on the Reddy’s higher order shear deformation theory and a nine noded isoparametric Lagrangian element is adopted to mesh the plate geometry. The issue of C₁ continuity requirement related to the assumed displacement field has been circumvented effectively to develop C₀ finite element formulation. Effective mechanical properties of the constituents of the plate are considered to be as position and temperature dependent and assumed to vary in the thickness direction according to a simple power law distribution. The displacement components of a rectangular plate are mapped into skew plate geometry by means of suitable transformation rule. One dimensional Fourier heat conduction equation is used to ascertain the temperature profile of the plate along thickness direction. Influence of different parameters such as volume fraction index, boundary condition, aspect ratio, thickness ratio and temperature field on frequency parameter of the FGM skew plate is demonstrated by performing various examples and the related findings are discussed briefly. New results are generated for vibration of the FGM skew plate under thermal environment, for the first time, which may be implemented in the future research involving similar kind of problems.

Keywords: Functionally graded material, finite element method, higher order shear deformation theory, skew plate, thermal vibration.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1090741

References:

[1] J.N. Reddy, and C.D. Chin,"Thermo mechanical analysis of functionally graded cylinders and plates,” J. of Ther. Stress., vol.21, pp. 593-626, 1998.
[2] J.Suresh Kumar, B. Sidda Reddy, and C. Eswara Reddy, "Higher order theory for free vibration analysis of functionally graded material plates,” ARPN J. of Eng. and App. Sci., vol.6, pp. 105-111, 2011.
[3] Hiroyuki Matsunaga, "Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory,” Comp. Struct., vol. 82, pp. 499-512, 2008.
[4] N. Noda, "Thermal stresses in materials with temperature-dependent properties,” App. Mech. Revision vol.44, pp. 383-97, 1991.
[5] Mohammed Talha, and B.N.Singh, "Static response and free vibration analysis of FGM plates using higher order shear deformation theory,” App. Math. Model., vol.34, pp. 3991-4011, 2010.
[6] M. Talha, and B.N. Singh, "Thermo-mechanical deformation behavior of functionally graded rectangular plates subjected to various boundary conditions and loadings,” Int. J. Aerospace and Mech. Eng., vol. 6, no.1, pp. 14-25, 2012.
[7] L. Hadji, H.A.Atmane, and A.Tounsi, "Free vibration of functionally graded sandwich plates using four-variable refined plate theory,” App. Math. and mech., vol.32, pp. 925-942, 2011.
[8] A.J.M. Ferreria, R.C. Batra, and C.M.C.Roque, "Natural frequencies of functionally graded plates by a meshless method” Comp. Struct., vol. 75, pp.593-600, 2006.
[9] Bahar uymaz and Metin aydogdu, "Three-Dimensional Vibration Analyses of Functionally Graded Plates under Various Boundary Conditions,” J. Rein. Plas. Comp., vol. 26, pp. 1847-1863, 2007.
[10] Serge Abrate, "Free vibration, buckling, and static deflection of functionally graded plates,” Comp. Sci. and Tech., vol.66, pp.2383–2394, 2006.
[11] Senthil S. Vel, and R.C.Batra, "Three dimensional exact solution for the vibration of functionally graded rectangular plates,” J. of Sound and Vibr., vol.272, pp. 703-730, 2004.
[12] Xiao-Lin Huang, and Hui-Shen Shen, "Non-linear vibration and dynamic response of functionally graded plates in thermal environments,” Int. J. of Solids and Struct., vol.41, pp. 2403-2427, 2004.
[13] J. Yang, and H.S.Shen, "Vibration characteristics and transient response of shear deformable functionally graded plates in thermal environments,” J. of Sound and Vib., vol.255, pp. 579-602, 2002.
[14] M. Talha, and B.N. Singh, "Thermo-mechanical induced vibration characteristics of shear deformable functionally graded ceramic-metal plates using finite element method,” Proc. of the Institution of Mech. Engineers Part C J. of Mech. Eng. Sci., vol. 225, pp. 50-65, 2011.
[15] N. Soundararajan, T.Prakash, and M.Ganapathi, "Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments,” Finite Ele. in Ana. and Design vol.42, pp.152-168, 2005.
[16] L.F.Qian, R.C. Batra, and L.M.Chen, " Free and forced vibrations of thick rectangular plates using higher order shear and normal deformable plate theory and meshless Petrov-Galerkin method,” CMES vol.4, pp. 519-534, 2003.
[17] Jae-Sang Park and Ji-Hwan Kim, "Thermal postbuckling and vibration analyses of functionally graded plates,” J. of Sound and Vib., vol.289, pp.77–93, 2006.
[18] Jae-Sang Park and Ji-Hwan Kim, "Thermal postbuckling and vibration analyses of functionally graded plates,” J. of Sound and Vib., vol.289, pp.77–93, 2006.
[19] D.P.Makhecha, B.P. Patel, and M.Ganapathi, "Transient dynamics of thick skew sandwich laminates under thermal/mechanical loads,” Journal of Plastics and Composites 20: 1524-1545, 2001.
[20] Ali Shahrjerdi, F.Mustapha, and S.M.Sapuan, "Thermal free vibration analysis of temperature-dependent functionally graded plates using second order shear deformation,” Key Eng., Mat., vol.471-472, pp.133-139, 2011.
[21] W.U. Lei, and M.A.Liansheng, "Thermal vibration of functionally graded circular plates,” Key Eng. Mat., vol.353-358, pp. 1777-1780, 2007.
[22] Maziar Janghorban, and Amin Zare, "Thermal effect on free vibration analysis of functionally graded arbitrary straight-sided plates with different cutouts,” Latin Americal J. Solids and Struct., vol.8, pp. 245 – 257, 2011.
[23] Y.S. Touloukian, "Thermophysical properties of high temperature solid materials,” MacMillan, New York, 1967.
[24] R.D., Mindlin, "Influence of rotator inertia and shear on flexural vibration of isotropic elastic plates,” J. of App. Mech., vol.73, pp. 31-38, 1951.
[25] J.N.Reddy, "A simple higher-order theory for laminated composite plate,” J. of App. Mech., vol.51, pp.745-752, 1984.
[26] M.N.A. Gulshan Taj, Anupam chakrabarti, "Static and dynamic analysis of functionally graded skew plates,” ASCE J. Eng. Mech. Vol. 139, no. (7), pp. 848-857, 2013.
[27] M.N.A. Gulshan Taj, Anupam Chakrabarti, and Abdul Hamid Sheikh, "Analysis of functionally graded plates using higher order shear deformation theory (Periodical style- accepted for publication),” App. Math. Modeling, to be published.
[28] M.N.A. Gulshan Taj, and Anupam Chakrabarti, "Dynamic analysis of functionally graded skew shell panel,” Latin American J. of Solids and Struct., vol.10, no.6, pp., 1243-1266, 2013.
[29] M.N.A. Gulshan Taj, and Anupam Chakrabarti, "An Efficient Finite Element Approach for Bending Analysis of Functionally Graded Ceramic-Metal Skew Shell Panels (Periodical style- submitted for publication),” submitted for publication.
[30] M.N.A. Gulshan Taj, and Anupam Chakrabarti, "Bending analysis of functionally graded skew sandwich plates with through-the thickness displacement variations (Periodical style- submitted for publication),” submitted for publication.