A unified formulation for finite element analysis of piezoelectric adaptive plates

This paper presents some finite elements for the dynamic analysis of laminated plates embedding piezoelectric layers based on the principle of virtual displacements (PVD) and an unified formulation. The description of the unknowns allows to keep the order of the expansion of the variables along the thickness direction as a parameter of the model and at the same time to perform both equivalent single layer (ESL) and layer-wise (LW) descriptions of the state mechanical variables. The full coupling between the electric and mechanical fields is considered; thus, the electric potential is taken as a state variable of the problem and is assumed LW with an order of the expansion that goes from 1 to 4 as the displacement field. In case of a ESL model the ZZ-function of Murakami has been introduced to recover the zig-zag form of the displacement field. Combining all the possible parameters, up to 12 different finite elements are addressed. Numerical results have been given for the free-vibrations frequencies of simply supported plates embedding piezoelectric layers. The lamination of the pure elastic layers has been limited to cross-ply in order to compare numerical results to analytical ones obtained by a Navier-type solution based on the same formulation.