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On Free Vibration of Functionally Graded Mindlin Plate and Effect of In-Plane Displacements

Published online by Cambridge University Press:  29 January 2013

A. Hasani Baferani ,
A.R. Saidi  and
H. Ehteshami
A. Hasani Baferani
Affiliation:
Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., 424, Tehran, Iran
A.R. Saidi*
Affiliation:
Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
H. Ehteshami
Affiliation:
Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
*
*Corresponding author (saidi@mail.uk.ac.ir)
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Abstract

In this paper, free vibration analysis of functionally graded rectangular plate is investigated based on the first order shear deformation theory and the effect of in-plane displacements on the natural frequencies of such plate is studied. The governing equations of motion are obtained, which are five coupled partial differential equations, without any simplification. Some mathematical manipulation leads us to decouple the equations. The decoupled equations are solved by the Levy's method for various boundary conditions. As the results show, in some boundary conditions the in-plane displacements cause a drastic change of frequencies. In other words, neglecting the in-plane displacement, which is assumed in some papers, is not proper for these boundary conditions. However, in the other boundary conditions, the natural frequencies are not significantly affected by the in-plane displacements. The results for various boundary conditions are discussed in detail and some interpretations for these differences are provided. Besides to the comparisons, the accurate natural frequencies of the plate for six different boundary conditions with several aspect ratios, thickness-length ratios and power law indices are presented. The natural frequencies of Mindlin functionally graded rectangular plates with considering the in-plane displacements are reported for the first time and can be used as benchmark.

Keywords

Free vibrationFunctionally gradedIn-Plane displacementsAnalytical solution
Type
Articles
Information
Journal of Mechanics , Volume 29 , Issue 2 , June 2013 , pp. 373 - 384
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2013

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References

REFERENCES

1.Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I., Proceedings of First International Symposium on Functionally Gradient Materials, Japan, Sendai, (1990).Google Scholar
2.Reddy, J. N., “Analysis of Functionally Graded Plates,” International Journal for Numerical Methods in Engineering, 47, pp. 663–684 (2000).Google Scholar
3.Birman, V. and Byrd, L. W., “Modeling and Analysis of Functionally Graded Materials and Structures,” Journal of Applied Mechanics, ASME, 60, pp. 195–216 (2007).Google Scholar
4.Fukui, Y., “Fundamental Investigation of Functionally Gradient Material Manufacturing System Using Centrifugal Force,” International Journal of the Japan Society for Mechanics Engineering, 4, pp. 144–148 (1991).Google Scholar
5.Yang, J. and Shen, H. S., “Dynamic Response of Initially Stressed Functionally Graded Rectangular Thin Plates,” Composite Structures, 54, pp. 497–508 (2001).Google Scholar
6.He, X. Q., Ng, T. Y., Sivashanker, S. and Liew, K. M., “Active Control of FGM Plates with Integrated Piezoelectric Sensors and Actuators,” International Journal of Solids and Structures, 38, pp. 1641–1655 (2001).Google Scholar
7.Qian, L. F., Batra, R. C. and Chen, L. M., “Static and Dynamic Deformations of Thick Functionally Graded Elastic Plates by Using Higher-Order Shear and Normal Deformable Plate Theory and Meshless Local Petrov–Galerkin Method,” Composites: Part B, 35, pp. 685–697 (2004).Google Scholar
8.Vel, S. S. and Batra, R. C., “Three-Dimensional Exact Solution for the Vibration of Functionally Graded Rectangular Plates,” Journal of Sound and Vibration, 272, pp. 703–730 (2004).CrossRefGoogle Scholar
9.Kim, Y. W., “Temperature Dependent Vibration Analysis of Functionally Graded Rectangular Plates,” Journal of Sound and Vibration, 284, pp. 531–549 (2005).Google Scholar
10.Shiau, L. C. and Zeng, J. Y., “Free Vibration of Rectangular Plate with Delamination,” Journal of Mechanics, 26, pp. 87–93 (2010).Google Scholar
11.Ferreira, A. J. M, Batra, R. C, Roque, C. M. C., Qian, L. F. and Jorge, R. M. N., “Natural Frequencies of Functionally Graded Plates by a Mesh Less Method,” Composite Structures, 75, pp. 593–600 (2006).Google Scholar
12.Sundararajan, N., Prakash, T. and Ganapathi, M., “Nonlinear Free Flexural Vibrations of Functionally Graded Rectangular and Skew Plates Under Thermal Environments,” Finite Elements in Analysis and Design, 42, pp. 152–168 (2005).Google Scholar
13.Woo, J., Meguid, S. A. and Ong, L. S., “Nonlinear Free Vibration Behavior of Functionally Graded Plates,” Journal of Sound and Vibration, 289, pp. 595–611 (2006).CrossRefGoogle Scholar
14.Gilhooley, D. F., Batra, R. C., Xiao, J. R., McCarthy, M. A. and Gillespie, J. W., “Analysis of Thick Functionally Graded Plates by Using Higher-Order Shear and Normal Deformable Plate Theory and MLPG Method with Radial Basis Functions,” Composite Structures, 80, pp. 539–552 (2007).Google Scholar
15.Matsunaga, H., “Free Vibration and Stability of Functionally Graded Plates According to a 2-D Higher-Order Deformation Theory,” Composite Structures, 82, pp. 499–512 (2008).CrossRefGoogle Scholar
16.Zhang, D. G. and Zhou, Y. H., “A Theoretical Analysis of FGM Thin Plates Based on Physical Neutral Surface,” Computational Materials Science, 44, pp. 716–720 (2008).Google Scholar
17.Fares, M. E., Elmarghany, M. K. and Atta, D., “An Efficient and Simple Refined Theory for Bending and Vibration of Functionally Graded Plates,” Composite Structures, 91, pp. 296–305 (2009).CrossRefGoogle Scholar
18.Malekzadeh, P., “Three-Dimensional Free Vibration Analysis of Thick Functionally Graded Plates on Elastic Foundations,” Composite Structures, 89, pp. 367–373 (2009).Google Scholar
19.Wu, C. P. and Lu, Y. C., “A Modified Pagano Method for the 3D Dynamic Responses of Functionally Graded Magneto-Electro-Elastic Plates,” Composite Structures, 90, pp. 363–372 (2009).CrossRefGoogle Scholar
20.Li, Q., Iu, V. P. and Kou, K. P., “Three-Dimensional Vibration Analysis of Functionally Graded Material Plates in Thermal Environment,” Journal of Sound and Vibration, 324, pp. 733–750 (2009).CrossRefGoogle Scholar
21.Liu, D. Y., Wang, C. Y. and Chen, W. Q., “Free Vibration of FGM Plates with In-Plane Material Inhomogeneity,” Composite Structures, 92, pp. 1047–1051 (2010).Google Scholar
22.Zhao, X., Lee, Y. Y. and Liew, K. M., “Free Vibration Analysis of Functionally Graded Plates Using the Element-Free Kp-Ritz Method,” Journal of Sound and Vibration, 319, pp. 918–939 (2009).CrossRefGoogle Scholar
23.Hosseini-Hashemi, Sh., Rokni Damavandi Taher, H., Akhavan, H. and Omidi, M., “Free Vibration of Functionally Graded Rectangular Plates Using First-Order Shear Deformation Plate Theory,” Applied Mathematical Modelling, 34, pp. 276–129 (2010).Google Scholar
24.Hosseini-Hashemi, Sh., Fadaee, M. and Atashipour, S. R., “A New Exact Analytical Approach for Free Vibration of Reissner–Mindlin Functionally Graded Rectangular Plates,” International Journal of Mechanical Sciences, 53, pp. 11–22 (2011).Google Scholar
25.Hosseini-Hashemi, Sh., Fadaee, M. and Atashipour, S. R., “A Study on the Free Vibration of Thick Functionally Graded Rectangular Plates According to a New Exact Closed-Form Procedure,” Composite Structures, 93, pp. 722–735 (2011).Google Scholar
26.Reddy, J. N., Energy Principles and Variational Methods in Applied Mechanics, 2nd Edition, John Wiley and Sons Inc, New York (2002).Google Scholar
27.Reddy, J. N., Theory and Analysis of Elastic Plates, Taylor & Francis, Philadelphia (1999).Google Scholar