Coupled three-layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three-layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed- or open-circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a threelayered smart beam with different EBC is proposed considering a fully coupled electro-mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi-static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC.