Mixed piezoelectric plate elements with continuous transverse electric displacements

Abstract

This paper proposes mixed finite elements, FEs, with an a priori continuous transverse electric displacement component Ᏸ 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 Φ and transverse stresses σ n . The second, full, form (F-RMVT) adds Ᏸ 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, φ, σ n and Ᏸ 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 Ᏸ z fields.

Refined Multilayered Plate Elements for Coupled Magneto‐Electro‐Elastic Analysis

“…The present work proposes a variable kinematic finite element model which is based on Unified Formulation (UF) by extending previous authors' findings [9], [23], [24], [25], [26] and [27]. What at points 1, 2 and 4, they are fully addressed in the present work.…”

“…Such a continuity is not enforced in usual models that make use of only displacement variables [7]. The same could be said for transverse electrical displacements, heat flux or transverse magnetical inductance [8], [9].…”

Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement

“…Cartesian x, y, z reference system is considered and notation already used in previous work [19] is referred to: subscript ' p' denotes in-plane unknowns and subscript 'n' denotes out-of-plane unknowns; subscript 'z' indicates through-the-thickness z-direction, while subscripts 'x' and 'y' are for the two in-plane directions. If electrical contributions are included, the PVD in Equation (4) becomes…”

“…Upon substitution of Equations (17), (18), (19), (23) and (31), the variational statement in Equation (10) leads to a set of equilibrium equations which can be formally put in the following compact form:…”

“…In this case, the transverse electric displacement D z was calculated by post-processing. More accurate results for the evaluation of the D z have recently been provided in [19], where the transverse stresses, together with the D z , have been modeled by RMVT. The employed variational statement has been called F-RMVT, where the 'F' stands for 'Full'.…”