Archive of Applied Mechanics

, Volume 81, Issue 3, pp 361–383 | Cite as

Nonlinear vibration analysis of piezo-thermo-electrically actuated functionally graded circular plates

  • F. EbrahimiEmail author
  • A. Rastgoo


A theoretical model for geometrically nonlinear vibration analysis of thermo-piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s–Love hypothesis with von-Karman type geometrical large nonlinear deformations. The material properties of the FG core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. Dynamic equations and boundary conditions including thermal, elastic and piezoelectric couplings are formulated and solutions are derived. An exact series expansion method combined with perturbation approach is used to model the nonlinear thermo-electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixtures of ceramic and metal are presented in dimensionless forms. A parametric study is also undertaken to highlight the effects of the thermal environment, applied actuator voltage and material composition of the FG core plate on the nonlinear vibration characteristics of the composite structure.


Functionally graded plate Piezo-thermo-electrically actuated plate Circular plate vibration 

List of symbols


Plate radius

\({\hat {a}}\)

Dimensionless vibration amplitude


Permeability constant of piezoelectric material


Young’s modulus


Temporal function

hf, hp

FG plate and Piezoelectric layers thickness


FGM volume fraction index


Non-dimensional force


Transverse shear component

Rd, Sd

Eigen-functions (mode shape function)

r, θ, z

Radial, circumferential and transverse direction




Non-dimensional temperature load

ur, uθ, w

Radial, circumferential, transverse displacements


Non-dimensional voltage

\({V_z^t, V_z^b }\)

Applied control voltages to the top and bottom piezoelectric layers


Non-dimensional transverse deflection

x, y

Non-dimensional radical distances

\({{X}_{\rm s},Y_{\rm s}^m}\)

Non-dimensional slope and force


Coefficient of thermal expansion

\({\overline{\varepsilon }_{ij}, \kappa _{ij}}\)

Membrane and bending strains


Thermal conductivity




Non-dimensional eigenvalue


Nonlinear coefficient functions


Poisson’s ratio


Non-dimensional time


Nonlinear frequency


Natural frequency



Electric, mechanical and temperature


Linear and nonlinear force


The variable in FGM and Piezo layer



Metal and ceramic

s, d

Static and dynamic condition


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  1. 1.
    Bailey T., Hubbard J.E.: Distributed piezoelectric polymer active vibration control of a cantilever beam. J. Guid. Control Dyn. 8, 605–611 (1985)zbMATHCrossRefGoogle Scholar
  2. 2.
    Miller, S.E., Hubbard, J.E.: Observability of a Bernoulli–Euler beam using PVF2 as a distributed sensor MIT Draper Laboratory Report (1987)Google Scholar
  3. 3.
    Spencer W.J., Corbett W.T., Dominguez L.R., Shafer B.D.: An electronically controlled piezoelectric insulin pump and valves. IEEE Trans. Sonics Ultrason. 25, 153–156 (1978)Google Scholar
  4. 4.
    Dong S., Du X., Bouchilloux P., Uchino K.: Piezoelectric ring-morph actuation for valve application. J. Electroceram. 8, 155–161 (2002)CrossRefGoogle Scholar
  5. 5.
    Chee C.Y.K., Tong L., Steve G.P.: A review on the modeling of piezoelectric sensors and actuators incorporated in intelligent structures. J. Intell. Mater. Syst. Struct. 9, 3–19 (1998)CrossRefGoogle Scholar
  6. 6.
    Cao L., Mantell S., Polla D.: Design and simulation of an implantable medical drug delivery system using microelectromechanical systems technology. Sens. Actuators A 94, 117–125 (2001)CrossRefGoogle Scholar
  7. 7.
    Chen X., Fox C.H.J., McWilliam S.: Optimization of a cantilever microswitch with piezoelectric actuation. J. Intell.Mater. Syst. Struct. 15, 823–834 (2004)CrossRefGoogle Scholar
  8. 8.
    Dobrucki A.B., Pruchnicki P.: Theory of piezoelectric axisymmetric bimorph. Sens. Actuators A 58, 203–212 (1997)CrossRefGoogle Scholar
  9. 9.
    Morris C.J., Forster F.K.: Optimization of a circular piezoelectric bimorph for a micropump driver. J. Micromech. Microeng. 10, 459–465 (2000)CrossRefGoogle Scholar
  10. 10.
    Li S., Chen S.: Analytical analysis of a circular PZT actuator for valveless micropump. Sens. Actuators A 104, 151–161 (2003)CrossRefGoogle Scholar
  11. 11.
    Crawley E.F., Anderson E.H.: Detailed models of piezoceramic actuation of beams. J. Intell. Mater. Syst. Struct. 1, 4–25 (1990)CrossRefGoogle Scholar
  12. 12.
    Tzou, H.S.: Piezoelectric Shells. Distributed Sensing and Control of Continua. Kluwer, Dordrecht (1993)Google Scholar
  13. 13.
    Tzou H.S., Zhong J.P.: Electromechanics and vibrations of piezoelectric shell distributed systems theory and applications. ASME J. Dyn. Syst. Meas. Control 115(3), 506–517 (1993)CrossRefGoogle Scholar
  14. 14.
    Koizumi M.: The concept of FGM Ceram. Trans. Funct. Grad. Mater. 34, 3–10 (1993)Google Scholar
  15. 15.
    Reddy J.N., Cheng Z.Q.: Three-dimensional solutions of smart functionally graded plates. ASME J. Appl. Mech. 68, 234–241 (2001)zbMATHCrossRefGoogle Scholar
  16. 16.
    Wang B.L., Noda N.: Design of smart functionally graded thermo-piezoelectric composite structure. Smart Mater. Struct. 10, 189–193 (2001)CrossRefGoogle Scholar
  17. 17.
    He X.Q., Ng T.Y., Sivashanker S., Liew K.M.: Active control of FGM plates with integrated piezoelectric sensors and actuators. Int. J. Solids Struct. 38, 1641–1655 (2001)zbMATHCrossRefGoogle Scholar
  18. 18.
    Liew K.M., Yang J., Kitipornchai S.: Post buckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int. J. Solids Struct. 40, 3869–3892 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Shen H.S.: Post buckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings. Int. J. Solids Struct. 42, 6101–6121 (2005)zbMATHGoogle Scholar
  20. 20.
    Liew K.M., He X.Q., Ng T.Y., Sivashanker S.: Active control of FGM plates subjected to a temperature gradient: modeling via finite element method based on FSDT. Int. J. Numer. Meth. Eng. 52, 1253–1271 (2001)zbMATHCrossRefGoogle Scholar
  21. 21.
    Liew K.M., He X.Q., Ng T.Y., Kitipornchai S.: Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computat. Mech. 31, 350–358 (2003)zbMATHGoogle Scholar
  22. 22.
    Yang J., Kitipornchai S., Liew K.M.: Nonlinear analysis of thermo-electro-mechanical behavior of shear deformable FGM plates with piezoelectric actuators. Int. J. Numer. Methods Eng. 59, 1605–1632 (2004)zbMATHCrossRefGoogle Scholar
  23. 23.
    Yang J., Kitipornchai S., Liew K.M.: Large amplitude vibration of thermo-electric-mechanically stressed FGM laminated plates. Comput. Methods Appl. Mech. Eng. 192, 3861–3885 (2003)zbMATHCrossRefGoogle Scholar
  24. 24.
    Huang X.L., Shen H.S.: Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. J. Sound Vib. 289, 25–53 (2006)CrossRefGoogle Scholar
  25. 25.
    Ebrahimi, F., Rastgoo, A.: Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Mater. Struct. vol. 17, No. 015044, 13 (2008). doi: 10.1088/0964-1726/17/1/015044
  26. 26.
    Ebrahimi F., Rastgoo A., Kargarnovin M.H.: Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers. J. Mech. Sci. Technol. 22(16), 1056–1072 (2008)Google Scholar
  27. 27.
    Ebrahimi, F., Rastgoo, A., Atai, A.A.: Theoretical analysis of smart moderately thick shear deformable annular functionally graded plate. Eur. J. Mech. A Solids 28, 962–973 (2009)Google Scholar
  28. 28.
    Ebrahimi F., Rastgoo A.: An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Struct. 46, 1402–1408 (2008)CrossRefGoogle Scholar
  29. 29.
    Ebrahimi, F., Rastgoo, A.: Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers. Int. J. Mech. Mater. Des. 5, 157–165 (2009). doi: 10.1007/s10999-008-9091-1 Google Scholar
  30. 30.
    Reddy J.N., Praveen G.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic–metal plate. Int. J. Solids Struct. 35, 4457–4476 (1998)zbMATHCrossRefGoogle Scholar
  31. 31.
    Wetherhold R.C., Seelman S., Wang S.: The use of functionally graded materials to eliminate or control thermal deformation. Compos. Sci. Technol. 56, 1099–1104 (1996)CrossRefGoogle Scholar
  32. 32.
    Tanigawa Y., Morishita H., Ogaki S.: Derivation of system of fundamental equations for a three dimensional thermoelastic field with nonhomogeneous material properties and its application to a semi infinite body. J. Therm. Stresses 22, 689–711 (1999)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Reddy, J.N.: Theory and Analysis of Elastic Plates. Taylor and Francis, Philadelphia (1999)Google Scholar
  34. 34.
    Song G., Sethi V., Lic H.N.: Vibration control of civil structures using piezoceramic smart materials: a review. Eng. Struct. 28, 1513–1524 (2006)CrossRefGoogle Scholar
  35. 35.
    Zheng X.J., Zhou Y.H.: Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells. Acta Mech. Sin. 6(1), 69–81 (1990)zbMATHCrossRefGoogle Scholar
  36. 36.
    William H.P., Brain P.F., Sau A.T.: Numerical Recipes—the Art of Scientific Computing. Cambridge University Press, New York (1986)zbMATHGoogle Scholar
  37. 37.
    Nayfe A.H., Mook D.T.: Nonlinear Oscillations. Wiley, New York (1979)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentTehran UniversityTehranIran

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