# Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

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## Abstract

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton’s principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson’s ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index *n* and aspect ratio *a*/*b* on frequencies.

## Keywords

Functionally graded material Free vibration Rectangular plate Close form solutions Neutral surface## Notes

### Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grants 11172028, 1372021), Research Fund for the Doctoral Program of Higher Education of China (Grant 20131102110039), and the Innovation Foundation of Beihang University for PhD graduates.

## References

- 1.Koizumi, M.: The concept of FGM. Ceram. Trans. Func. Grad. Mater.
**34**, 3–10 (1993)Google Scholar - 2.Koizumi, M.: FGM activities in Japan. Compos. Pt B. Eng.
**28**, 1–4 (1997)Google Scholar - 3.Xing, Y.F., Liu, B.: New exact solutions for free vibrations of rectangular thin plates by symplectic dual method. Acta. Mech. Sin.
**25**, 265–270 (2009)MathSciNetCrossRefMATHGoogle Scholar - 4.Xing, Y.F., Liu, B.: New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos. Struct.
**89**, 567–574 (2009)Google Scholar - 5.Xing, Y.F., Xu, T.F.: Solution methods of exact solutions for free vibration of rectangular orthotropic thin plates with classical boundary conditions. Compos. Struct.
**104**, 187–195 (2013)CrossRefGoogle Scholar - 6.Yang, J., Shen, H.S.: Dynamic response of initially stressed functionally graded rectangular thin plates. Compos. Struct.
**54**, 497–508 (2001)CrossRefGoogle Scholar - 7.Abrate, S.: Free vibration, buckling, and static deflections of functionally graded plates. Compos. Sci. Technol.
**66**, 2383–2394 (2006)CrossRefGoogle Scholar - 8.Abrate, S.: Functionally graded plates behave like homogeneous plates. Compos. Pt B. Eng.
**39**, 151–158 (2008)CrossRefGoogle Scholar - 9.Zhang, D.G., Zhou, Y.H.: A theoretical analysis of FGM thin plates based on physical neutral surface. Comput. Mat. Sci.
**44**, 716–720 (2008)CrossRefGoogle Scholar - 10.Yin, S., Yu, T., Liu, P.: Free vibration analyses of FGM thin plates by isogeometric analysis based on classical plate theory and physical neutral surface. Adv. Mech. Eng.
**5**, 634584 (2013)Google Scholar - 11.Li, S.R., Wang, X., Batra, R.C.: Correspondence relations between deflection, buckling load, and frequencies of thin functionally graded material plates and those of corresponding homogeneous plates. J. Appl. Mech.
**82**, 111006 (2015)CrossRefGoogle Scholar - 12.Thai, H.T., Uy, B.: Levy solution for buckling analysis of functionally graded plates based on a refined plate theory. J. Mech. E. Pt C.
**227**, 2649–2664 (2013)Google Scholar - 13.Reddy, J.N., Wang, C., Kitipornchai, S.: Axisymmetric bending of functionally graded circular and annular plates. Eur. J. Mech.
**18**, 185–199 (1999)CrossRefMATHGoogle Scholar - 14.Hosseini-Hashemi, S., Fadaee, M., Atashipour, S.R.: A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates. Int. J. Mech. Sci.
**53**, 11–22 (2011)CrossRefGoogle Scholar - 15.Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Meth. Eng.
**684**, 663–684 (2000)CrossRefMATHGoogle Scholar - 16.Yang, J., Shen, H.S.: Vibration characteristics and transient response of shear deformable functionally graded plates in thermal environments. J. Sound. Vib.
**255**, 579–602 (2002)CrossRefGoogle Scholar - 17.Kim, Y.W.: Temperature dependent vibration analysis of functionally graded rectangular plates. J. Sound. Vib.
**284**, 531–549 (2005)CrossRefGoogle Scholar - 18.Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M., et al.: Stress, vibration and buckling analyses of FGM plates–a state-of-the art review. Compos. Struct.
**120**, 10–31 (2015)CrossRefGoogle Scholar - 19.Thai, H.T., Kim, S.E.: A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct.
**128**, 70–86 (2015)CrossRefGoogle Scholar - 20.Sayyad, A.S., Ghugal, Y.M.: On the free vibration analysis of laminated composite and sandwich plates: a review of recent literature with some numerical results. Compos. Struct.
**129**, 177–201 (2015)CrossRefGoogle Scholar - 21.Reddy, J.N., Chin, C.D.: Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stresses.
**21**, 593–626 (1998)CrossRefGoogle Scholar