Mechanic of Advanced and Smart Materials
Quarterly

Natural Frequency analysis of Multi-directional Functionally Graded Rectangular Plates on elastic Foundation using Three-dimensional Elasticity Theory

Document Type : Original Article

Author

Mahdi Adineh

Assistant Professor . University of Gonabad

https://doi.org/10.61186/masm.4.1.40
Abstract
In the present study, the natural frequencies of rectangular plates made of multi-directional functionally graded materials on an elastic substrate was investigated for the first time. The mechanical properties of the material in the examined plate can be changed in all three coordinate directions according to a power law function. Equations of motion are written according to the three-dimensional theory of elasticity and then discretized using the method of Generalized Differential Quadratics. By comparing the results of several examples in the published articles, the validity of the method and the solution was examined, indicating the high accuracy of the method used. The influence of a change in direction on the mechanical properties is examined using several examples and the results are examined. In addition, the effects of plate thickness, plate dimension ratio and the effects of elastic foundation parameters for different boundary conditions were investigated and presented in the form of diagrams. The results show that the direction of change in material properties can have a significant effect on the natural frequency of the plate.

Keywords

Natural frequency
Functionally graded plate
Three dimensional elasticity
[1] Khorshidi K, Karimi M, Rezaeisaray M. Piezoelectric Energy Harvesting from Functionally Graded Beams Using Modified Shear Deformation Theories. Mechanics of Advanced and Smart Materials journal. 2022;1:136-54.
[2] Vel SS, Batra R. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration. 2004;272:703-30.
[3] Li Q, Iu V, Kou K. Three-dimensional vibration analysis of functionally graded material rectangular plates by Chebyshev polynomials. Proceedings of the Tenth International Conference on Enhancement and Promotion of Computational Methods in Engineering and Science, Sanya, China, 2006.
[4] Zhou F, Li S. Three-dimensional analysis for static bending and free vibration of functionally graded rectangular plate. 2010;42:325-31.
[5] Hosseini-Hashemi S, Taher HRD, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Applied Mathematical Modelling. 2010;34:1276-91.
[6] Hosseini-Hashemi S, Fadaee M, Atashipour SR. A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates. International Journal of Mechanical Sciences. 2011;53:11-22.
[7] Hosseini-Hashemi S, Fadaee M, Atashipour SR. Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure. Composite Structures. 2011;93:722-35.
[8] Baferani AH, Saidi A, Ehteshami H. Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Composite Structures. 2011;93:1842-53.
[9] Chakraverty S, Pradhan K. Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions. Aerospace Science and Technology. 2014;36:132-56.
[10] Jin G, Su Z, Shi S, Ye T, Gao S. Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Composite Structures. 2014;108:565-77.
[11] Akavci S. An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation. Composite Structures. 2014;108:667-76.
[12] Chakraverty S, Pradhan K. Free vibration of functionally graded thin rectangular plates resting on Winkler elastic foundation with general boundary conditions using Rayleigh–Ritz method. International Journal of Applied Mechanics. 2014;6:1450043.
[13] Kumar S, Ranjan V, Jana P. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Composite Structures. 2018;197:39-53.
[14] Xing Y, Wang Z, Xu T. Closed-form analytical solutions for free vibration of rectangular functionally graded thin plates in thermal environment. International Journal of Applied Mechanics. 2018;10:1850025.
[15] Talley M, Khorshidvand AR. Free vibration analysis of functionally graded rectangular plates via differential quadrature method. Iranian Journal of Mechanical Engineering Transactions of the ISME. 2019;20:46-70.
[16] Xu T, Xing Y. Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation. Acta Mechanica Sinica. 2016;32:1088-103.
[17] Kang R, Xin F, Shen C, Lu TJ. 3D free vibration analysis of functionally graded plates with arbitrary boundary conditions in thermal environment. Advanced Engineering Materials. 2022;24:2100636.
[18] Huang C, Chung W. Analytical solution for the three-dimensional vibration of a rectangular functionally graded material plate with two simply supported opposite faces. International Journal of Structural Stability and Dynamics. 2023;23:2350014.
[19] Thai S, Nguyen VX, Lieu QX. Bending and free vibration analyses of multi-directional functionally graded plates in thermal environment: A three-dimensional Isogeometric Analysis approach. Composite Structures. 2022;295:115797.
[20] Adineh M, Kadkhodayan M. Three-dimensional thermo-elastic analysis of multi-directional functionally graded rectangular plates on elastic foundation. Acta Mechanica. 2017;228:881-99.
[21] Tahouneh V, Naei M. A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation. Meccanica. 2014;49:91-109.
[22] Khorshidi K, Bakhsheshy A, Ghadirian H. The study of the effects of thermal environment on Free Vibration analysis of two-dimensional Functionally Graded Rectangular plates on Pasternak elastic foundation. Journal of Solid and Fluid Mechanics. 2016;6:137-47.
[23] Adineh M, Kadkhodayan M. Three-dimensional thermo-elastic analysis and dynamic response of a multidirectional functionally graded skew plate on elastic foundation. Composites Part B: Engineering. 2017;125:227-40.
[24] Yin S, Yu T, Bui TQ, Zheng X, Tanaka S. In-plane material inhomogeneity of functionally graded plates: A higher-order shear deformation plate isogeometric analysis. Composites Part B: Engineering. 2016;106:273- 84.
[25] Xiang T, Natarajan S, Man H, Song C, Gao W. Free vibration and mechanical buckling of plates with inplane material inhomogeneity–A three dimensional consistent approach. Composite Structures. 2014;118:634-42.
[26] Yousefi P, Khodadadi M. Solution of out-of-Plane vibration of moderately thick rectangular nano-plate using nonlocal sinusoidal shear deformation theory. Mechanics of Advanced and Smart Materials journal. 2022;1:231-46.
[27] Mehrabadi SJ, Nezamabadi A, Moayeedi E. Deflection analysis of composite micro-beam on elastic foundation by strain gradient theory. Mechanics of Advanced and Smart Materials journal. 2022;2:94-107.
[28] Yang J, Shen H-S. Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. Journal of Sound and Vibration. 2002;255:579-602.
[29] Kim Y-W. Temperature dependent vibration analysis of functionally graded rectangular plates. Journal of Sound and Vibration. 2005;284:531-49.
[30] Li Q, Iu V, Kou K. Three-dimensional vibration analysis of functionally graded material plates in thermal environment. Journal of Sound and Vibration. 2009;324:733-50.
[31] Zhou D, Cheung Y, Lo S, Au F. Three‐dimensional vibration analysis of rectangular thick plates on Pasternak foundation. International journal for numerical methods in engineering. 2004;59:131334.
[32] Matsunaga H. Vibration and stability of thick plates on elastic foundations. Journal of engineering mechanics. 2000;126:27-34.
[33] Yas M, Aragh BS. Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation. International Journal of Engineering Science. 2010;48:1881-95.
Mechanic of Advanced and Smart Materials
Volume 4, Issue 1
Spring 2024
Pages 40-63

  • Receive Date 13 April 2024
  • Revise Date 13 May 2024
  • Accept Date 19 May 2024

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