Skip to main content
Log in

An efficient shear deformation theory for vibration of functionally graded plates

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Koizumi M.: FGM activities in Japan. Compos. Part B Eng. 28(1–2), 1–4 (1997)

    Article  Google Scholar 

  2. Leissa A.W.: The free vibration of rectangular plates. J. Sound Vib. 31(3), 257–293 (1973)

    Article  MATH  Google Scholar 

  3. Chen C.S., Chen T.J., Chien R.D.: Nonlinear vibration of initially stressed functionally graded plates. Thin-Walled Struct. 44(8), 844–851 (2006)

    Article  Google Scholar 

  4. Ebrahimi F., Rastgo A.: An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Struct. 46(12), 1402–1408 (2008)

    Article  Google Scholar 

  5. Ebrahimi F., Rastgoo A.: Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers. Int. J. Mech. Mater. Des. 5(2), 157–165 (2009)

    Article  Google Scholar 

  6. Baferani A.H., Saidi A.R., Jomehzadeh E.: An exact solution for free vibration of thin functionally graded rectangular plates. Proc. Inst. of Mech. Eng. Part C J. Mech. Eng. Sci. 225(3), 526–536 (2011)

    Article  Google Scholar 

  7. 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(1), 11–22 (2011)

    Article  Google Scholar 

  8. Zhao X., Lee Y.Y., Liew K.M.: Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J. Sound Vib. 319(3–5), 918–939 (2009)

    Article  Google Scholar 

  9. Della Croce L., Venini P.: Finite elements for functionally graded Reissner–Mindlin plates. Comput. Methods Appl. Mech. Eng. 193(9–11), 705–725 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hosseini-Hashemi S., Rokni Damavandi Taher H., Akhavan H., Omidi M.: Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Appl. Math. Model. 34(5), 1276–1291 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nelson R.B., Lorch D.R.: A refined theory for laminated orthotropic plates. J. Appl. Mech. 41, 177 (1974)

    Article  Google Scholar 

  12. Lo K.H., Christensen R.M., Wu E.M.: A high-order theory of plate deformation-Part 2: Laminated plates. J. Appl. Mech. 44, 669 (1977)

    Article  MATH  Google Scholar 

  13. Bhimaraddi A., Stevens L.: A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. J.Appl. Mech. 51(1), 195–198 (1984)

    Article  Google Scholar 

  14. Reddy J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51(4), 745 (1984)

    Article  MATH  Google Scholar 

  15. Kant T., Pandya B.: A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates. Compos. Struct. 9(3), 215–246 (1988)

    Article  Google Scholar 

  16. Kant T., Khare R.K.: A higher-order facet quadrilateral composite shell element. Int. J. Numer. Methods Eng. 40(24), 4477–4499 (1997)

    Article  MATH  Google Scholar 

  17. Talha M., Singh B.N.: Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl. Math. Model. 34(12), 3991–4011 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang J., Liew K.M., Kitipornchai S.: Dynamic stability of laminated FGM plates based on higher-order shear deformation theory. Comput. Mech. 33(4), 305–315 (2004)

    Article  MATH  Google Scholar 

  19. Reddy J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47(1–3), 663–684 (2000)

    Article  MATH  Google Scholar 

  20. Cheng Z.Q., Batra R.C.: Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. J. Sound Vib. 229(4), 879–895 (2000)

    Article  MATH  Google Scholar 

  21. Ferreira A.J.M., Batra R.C., Roque C.M.C., Qian L.F., Martins P.A.L.S.: Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Compos. Struct. 69(4), 449–457 (2005)

    Article  Google Scholar 

  22. Ferreira A.J.M., Batra R.C., Roque C.M.C., Qian L.F., Jorge R.M.N.: Natural frequencies of functionally graded plates by a meshless method. Compos. Struct. 75(1–4), 593–600 (2006)

    Article  Google Scholar 

  23. Abrate S.: Functionally graded plates behave like homogeneous plates. Compos. Part B Eng. 39(1), 151–158 (2008)

    Article  Google Scholar 

  24. Hosseini-Hashemi S., Fadaee M., RokniDamavandi Taher H.: Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory. Appl. Math. Model. 35(2), 708–727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hosseini-Hashemi S., Fadaee M., Atashipour S.R.: Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure. Compos. Struct. 93(2), 722–735 (2011)

    Article  Google Scholar 

  26. Kitipornchai S., Yang J., Liew K.M.: Random vibration of the functionally graded laminates in thermal environments. Comput. Methods Appl. Mech. Eng. 195(9–12), 1075–1095 (2006)

    Article  MATH  Google Scholar 

  27. Allahverdizadeh A., Naei M.H., Nikkhah Bahrami M.: Nonlinear free and forced vibration analysis of thin circular functionally graded plates. J. Sound Vib. 310(4–5), 966–984 (2008)

    Article  Google Scholar 

  28. Reddy J.N.: Energy Principles and Variational Methods in Applied Mechanics. Wiley, New York (2002)

    Google Scholar 

  29. Vel S.S., Batra R.C.: Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J. Sound Vib. 272(3–5), 703–730 (2004)

    Article  Google Scholar 

  30. Matsunaga H.: Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos. Struct. 82(4), 499–512 (2008)

    Article  Google Scholar 

  31. Pradyumna S., Bandyopadhyay J.N.: Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation. J. Sound Vib. 318(1–2), 176–192 (2008)

    Article  Google Scholar 

  32. Mori T., Tanaka K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21(5), 571–574 (1973)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dong-Ho Choi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thai, HT., Park, T. & Choi, DH. An efficient shear deformation theory for vibration of functionally graded plates. Arch Appl Mech 83, 137–149 (2013). https://doi.org/10.1007/s00419-012-0642-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-012-0642-4

Keywords

Navigation