Applied Mathematics and Mechanics

, Volume 30, Issue 8, pp 969–982 | Cite as

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

  • Shi-rong Li (李世荣)
  • Hou-de Su (苏厚德)
  • Chang-jun Cheng (程昌钧)


Free vibration of statically thermal postbuckled functionally graded material (FGM) beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electromechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Key words

functionally graded material laminated beams with piezoelectric layers thermal buckling free vibration natural frequency 

Chinese Library Classification


2000 Mathematics Subject Classification

74H45 74F05 74H15 


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Copyright information

© Shanghai University and Springer-Verlag GmbH 2009

Authors and Affiliations

  • Shi-rong Li (李世荣)
    • 1
  • Hou-de Su (苏厚德)
    • 1
  • Chang-jun Cheng (程昌钧)
    • 2
  1. 1.Department of Engineering MechanicsLanzhou University of TechnologyLanzhouP. R. China
  2. 2.Department of MechanicsShanghai UniversityShanghaiP. R. China

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