Theoretical Formulations for Finite Element Models of Functionally Graded Beams with Piezoelectric Layers

In this paper an overview of functionally graded materials and constitutive relations of electro elasticity for three-dimensional deformable  solids is presented, and  governing equations of the Bernoulli–Euler and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material and piezoelectric layers are developed  using the  principle  of virtual  displacements. The formulation is based on a power-law variation of the material in the core with piezoelectric layers at the top and bottom. Virtual work statements of the two theories are also developed and their finite element models are presented. The theoretical formulations and finite element models presented herein can be used in the analysis of piezolaminated and adaptive structures such as beams and plates.