#### Vol. 2, No. 3, 2007

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Mixed piezoelectric plate elements with continuous transverse electric displacements

### Erasmo Carrera and Christian Fagiano

Vol. 2 (2007), No. 3, 421–438
##### Abstract

This paper proposes mixed finite elements, FEs, with an a priori continuous transverse electric displacement component ${\mathsc{D}}_{z}$. The Reissner Mixed Variational Theorem (RMVT) and the Unified Formulation (UF) are applied to the analysis of multilayered anisotropic plates with embedded piezoelectric layers. Two forms of RMVT are compared. In a first, partial, form (P-RMVT), the field variables are displacements $u$, electric potential $\Phi$ and transverse stresses ${\sigma }_{n}$. The second, full, form (F-RMVT) adds ${\mathsc{D}}_{z}$ as an independent variable. F-RMVT allows the a priori and complete fulfillment of interlaminar continuity of both mechanical and electrical variables.

We treat both equivalent single-layer models (ESLM), where the number of variables is kept independent of the number of layers, an layerwise models (LWM), in which the number of variables depends in each layer. According to the UF the order $N$ of the expansions assumed for the $u$, $\varphi$, ${\sigma }_{n}$ and ${\mathsc{D}}_{z}$ fields in the plate thickness direction $z$ as well as the number of the element nodes ${N}_{n}$ have been taken as free parameters.

In most cases the results of the classical formulation which are based on Principle of Virtual Displacements (PVD) are given for comparison purposes. The superiority of the F-RMVT results, with respect to the P-RMVT and to PVD ones, is shown by few examples for which three-dimensional solution is available. In particular, the F-RMVT results to be very effective for the evaluation of interlaminar continuous ${\mathsc{D}}_{z}$ fields.

##### Keywords
piezoelectric plates, finite elements, mixed method, transverse continuity, unified formulation