Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment

Free vibration of statically thermal post-buckled functionally graded material beams with surface-bonded piezoelectric layers subjected to both temperature rise and voltage is studied. By accurately considering the axial extension and based on Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surfacebonded piezoelectric layers subjected to thermo-electro-mechanical loadings were formulated. It was assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate and that the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of beam. s vibration is small and its response harmonic, the above mentioned non-linear partial differential equations were reduced to two sets of coupled ordinary differential equations; the one for the postbuckling, and the other for linear vibration of the beam superimposed upon the post buckled configuration. By using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subjected to transversely non-uniform heating and uniform electric field were 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 were plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with an increase in 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.