Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation
Keywords:FG sandwich plate, Pasternak foundation, Mori-Tanaka scheme, quasi-3D theory, free vibration, finite element formulation
AbstractFree vibration of functionally graded (FG) sandwich plates partially supported by a Pasternak elastic foundation is studied. The plates consist of three layers, namely a pure ceramic hardcore and two functionally graded skin layers. The effective material properties of the skin layers are considered to vary in the plate thickness by a power gradation law, and they are estimated by Mori--Tanaka scheme. The quasi-3D shear deformation theory, which takes the thickness stretching effect into account, is adopted to formulate a finite element formulation for computing vibration characteristics. The accuracy of the derived formulation is confirmed through a comparison study. The numerical result reveals that the foundation supporting area plays an important role on the vibration behavior of the plates, and the effect of the layer thickness ratio on the frequencies is governed by the supporting area. A parametric study is carried out to highlight the effects of material distribution, layer thickness ratio, foundation stiffness and area of the foundation support on the frequencies and mode shapes of the plates. The influence of the side-to-thickness ratio on the frequencies of the plates is also examined and discussed.
Y. Fukui. Fundamental investigation of functionally gradient material manufacturing system using centrifugal force. Japan Society of Mechanical Engineering International Journal, Series III, 34, (1), (1991), pp. 144–148. https://doi.org/10.1299/jsmec1988.34.144.
D. K. Jha, T. Kant, and R. K. Singh. A critical review of recent research on functionally graded plates. Composite Structures, 96, (2013), pp. 833–849. https://doi.org/10.1016/j.compstruct.2012.09.001.
K. Swaminathan, D. T. Naveenkumar, A. M. Zenkour, and E. Carrera. Stress, vibration and buckling analyses of FGM plates—A state-of-the-art review. Composite Structures, 120, (2015), pp. 10–31. https://doi.org/10.1016/j.compstruct.2014.09.070.
G. N. Praveen and J. N. Reddy. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. International Journal of Solids and Structures, 35, (33), (1998), pp. 4457–4476. https://doi.org/10.1016/s0020-7683(97)00253-9.
A. M. Zenkour. A comprehensive analysis of functionally graded sandwich plates: Part 1–Deflection and stresses. International journal of solids and structures, 42, (18-19), (2005), pp. 5224–5242. https://doi.org/10.1016/j.ijsolstr.2005.02.015.
A. M. Zenkour. A comprehensive analysis of functionally graded sandwich plates: Part 2–Buckling and free vibration. International Journal of Solids and Structures, 42, (18-19), (2005), pp. 5243–5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016.
A. M. Zenkour and M. Sobhy. Thermal buckling of various types of FGM sandwich plates. Composite Structures, 93, (1), (2010), pp. 93–102. https://doi.org/10.1016/j.compstruct.2010.06.012.
S. Xiang, Y. Jin, Z. Bi, S. Jiang, and M. Yang. A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates. Composite Structures, 93, (11), (2011), pp. 2826–2832. https://doi.org/10.1016/j.compstruct.2011.05.022.
S. Xiang, G. Kang, M. Yang, and Y. Zhao. Natural frequencies of sandwich plate with functionally graded face and homogeneous core. Composite Structures, 96, (2013), pp. 226–231. https://doi.org/10.1016/j.compstruct.2012.09.003.
A. M. A. Neves, A. J. M. Ferreira, E. Carrera, M. Cinefra, C. M. C. Roque, R. M. N. Jorge, and C. M. M. Soares. 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: Engineering, 44, (1), (2013), pp. 657–674. https://doi.org/10.1016/j.compositesb.2012.01.089.
H.-T. Thai and D.-H. Choi. Finite element formulation of various four unknown shear deformation theories for functionally graded plates. Finite Elements in Analysis and Design, 75, (2013), pp. 50–61. https://doi.org/10.1016/j.finel.2013.07.003.
H.-T. Thai and S.-E. Kim. A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Composite Structures, 96, (2013), pp. 165–173. https://doi.org/10.1016/j.compstruct.2012.08.025.
H.-T. Thai and S.-E. Kim. A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates. Composite Structures, 99, (2013), pp. 172–180. https://doi.org/10.1016/j.compstruct.2012.11.030.
H.-T. Thai, T.-K. Nguyen, T. P. Vo, and J. Lee. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. European Journal of Mechanics-A/Solids, 45, (2014), pp. 211–225. https://doi.org/10.1016/j.euromechsol.2013.12.008.
L. Iurlaro, M. Gherlone, and M. Di Sciuva. Bending and free vibration analysis of functionally graded sandwich plates using the refined zigzag theory. Journal of Sandwich Structures & Materials, 16, (6), (2014), pp. 669–699. https://doi.org/10.1177/1099636214548618.
S. Pandey and S. Pradyumna. Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Composites Part B: Engineering, 153, (2018), pp. 325–336. https://doi.org/10.1016/j.compositesb.2018.08.121.
Z. Belabed, A. A. Bousahla, M. S. A. Houari, A. Tounsi, and S. Mahmoud. A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate. Earthquakes and Structures, 14, (2), (2018), pp. 103–115. https://doi.org/10.12989/eas.2018.14.2.103.
A. A. Daikh and A. M. Zenkour. Effect of porosity on the bending analysis of various functionally graded sandwich plates. Materials Research Express, 6, (6), (2019), p. 065703. https://doi.org/10.1088/2053-1591/ab0971.
C. F. L¨ u, C.W. Lim, andW. Q. Chen. Exact solutions for free vibrations of functionally graded thick plates on elastic foundations. Mechanics of Advanced Materials and Structures, 16, (8), (2009), pp. 576–584. https://doi.org/10.1080/15376490903138888.
S. Benyoucef, I. Mechab, A. Tounsi, A. Fekrar, H. A. Atmane, E. A. A. Bedia, Bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations. Mechanics of Composite Materials, 46, (4), (2010), pp. 425–434. https://doi.org/10.1007/s11029-010-9159-5.
M. Sobhy. Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Composite Structures, 99, (2013), pp. 76–87. https://doi.org/10.1016/j.compstruct.2012.11.018.
S. A. Al Khateeb and A. M. Zenkour. A refined four-unknown plate theory for advanced plates resting on elastic foundations in hygrothermal environment. Composite Structures, 111, (2014), pp. 240–248. https://doi.org/10.1016/j.compstruct.2013.12.033.
H. V. Tung. Thermal and thermomechanical postbuckling of FGM sandwich plates resting on elastic foundations with tangential edge constraints and temperature dependent properties. Composite Structures, 131, (2015), pp. 1028–1039. https://doi.org/10.1016/j.compstruct.2015.06.043.
N. M. Khoa and H. V. Tung. Nonlinear thermo-mechanical stability of shear deformable FGM sandwich shallow spherical shells with tangential edge constraints. Vietnam Journal of Mechanics, 39, (4), (2017), pp. 351–364. https://doi.org/10.15625/0866-7136/9810.
S. S. Akavci. Mechanical behavior of functionally graded sandwich plates on elastic foundation. Composites Part B: Engineering, 96, (2016), pp. 136–152. https://doi.org/10.1016/j.compositesb.2016.04.035.
R. Benferhat, T. H. Daouadji, and M. S. Mansour. Free vibration analysis of FG plates resting on an elastic foundation and based on the neutral surface concept using higher-order shear deformation theory. Comptes Rendus Mecanique, 344, (9), (2016), pp. 631–641. https://doi.org/10.12989/eas.2016.10.5.1033.
A. Benahmed, M. S. A. Houari, S. Benyoucef, K. Belakhdar, and A. Tounsi. A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation. Geomechanics and Engineering, 12, (1), (2017), pp. 9–34. https://doi.org/10.12989/gae.2017.12.1.009.
F. Z. Zaoui, D. Ouinas, and A. Tounsi. New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Composites Part B: Engineering, 159, (2019), pp. 231–247. https://doi.org/10.1016/j.compositesb.2018.09.051.
M. Eisenberger, D. Z. Yankelevsky, and M. A. Adin. Vibrations of beams fully or partially supported on elastic foundations. Earthquake Engineering & Structural Dynamics, 13, (5), (1985), pp. 651–660. https://doi.org/10.1002/eqe.4290130507.
T. Yokoyama. Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations. Computers & Structures, 61, (6), (1996), pp. 995–1007. https://doi.org/10.1016/0045-7949(96)00107-1.
S. Motaghian, M. Mofid, and J. E. Akin. On the free vibration response of rectangular plates, partially supported on elastic foundation. Applied Mathematical Modelling, 36, (9), (2012), pp. 4473–4482. https://doi.org/10.1016/j.apm.2011.11.076.
H. S. Shen. Functionally graded materials: nonlinear analysis of plates and shells. CRC Press, Taylor & Francis Group, Boca Raton, (2009).
R. D. Cook, D. S. Malkus, and M. E. Plesha. Concepts and applications of finite element analysis. JohnWilley & Sons, New York, 3rd edition, (1989).
S. S. Rao. The finite element method in engineering. Elsevier, Amsterdam, 4th edition, (2005).