Finite Element Analysis of Functionally Graded‎ ‎Piezoelectric Beams

In this paper‎, ‎the static bending‎, ‎free vibration‎, ‎and dynamic response of functionally graded‎ ‎piezoelectric beams have been carried out by finite element method‎‎under different sets of mechanical‎, ‎thermal‎, ‎and electrical‎ ‎loadings‎. ‎The beam with functionally graded piezoelectric material‎ ‎(FGPM) is assumed to be graded across the thickness with a simple‎ ‎power law distribution in terms of the volume fractions of the‎ ‎constituents‎. ‎The electric potential is assumed linear across the‎ ‎FGPM beam thickness‎. ‎The temperature field is assumed to be of‎ ‎uniform distribution over the beam surface and through the beam‎ ‎thickness‎. ‎The governing equations are obtained using potential‎ ‎energy and Hamilton's principle based on the Euler-Bernoulli beam‎ ‎theory that includes thermo-piezoelectric effects‎. ‎The finite‎ ‎element model is derived based on the constitutive equation of‎ ‎piezoelectric material accounting for coupling between the‎ ‎elasticity and the electric effect by two nodes Hermitian beam‎ ‎element‎. ‎The present finite element is modelled with displacement‎ ‎components and electric potential as nodal degrees of freedom‎. ‎The‎ ‎temperature field is calculated by post-computation through the‎ ‎constitutive equation‎. ‎Results are presented for two-constituent‎ ‎FGPM beam under different mechanical boundary conditions‎. ‎Numerical results include the influence of the different power law‎ ‎indexes‎, ‎the effect of mechanical‎, ‎thermal‎, ‎and electrical‎ ‎loadings and the type of in-plane boundary conditions on the‎ ‎deflection‎, ‎stress‎, ‎natural frequencies‎, ‎and dynamic response‎. ‎The‎ ‎numerical results obtained by the present model are in good‎ ‎agreement with the available solutions reported in the‎ ‎literature.