Electric potential approximations for an eight node plate finite element

Abstract

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C 0 finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387-411] and C 1 finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305-24].In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.

High-order plate finite elements for smart structure analysis

“…This table presents results obtained using Ansys, different versions of the present high-order model (P9ZZ, P9Z and P9) as well as FSDT. Furthermore, the model (P9ZZ w/o Du 2 3 ) is deduced from P9ZZ neglecting the contribution of the derivative of u 2 3 in the transverse shear strain, see Eq. (7).…”

“…While most developments employ an ESL description for the mechanical behavior, and particularly the First order Shear Deformation Theory (FSDT), a Layer-Wise description is necessary for the piezoelectric approximation to impose electric boundary conditions at each piezoelectric layer interfaces, i.e., the electrodes, within the stack. Inside each piezo-electric layer, the electric potential can be linear, quadratic or higher and a comparison has been proposed in [2].…”

A Refined Sinus Finite Element Model for the Analysis of Piezoelectric-Laminated Beams

“…3 in the corresponding transverse shear strain components due to bending, and enables to avoid transverse shear locking phenomenon using the field compatibility approach. Finally, u and u 1 31 are interpolated by Lagrange quadratic functions (Polit et al, 1994;Polit and Bruant, 2006;Vidal and Polit, 2006;Vidal and Polit, 2008). Mechanical representation of the laminated beam FE is shown in Figure 3.…”

An Efficient Spectral Element Model with Electric DOFs for the Static and Dynamic Analysis of a Piezoelectric Bimorph

“…The ESL model is simple and capable of predicting the global responses of the bimorph, but it does not account for the nonlinear distribution of the electric potential across piezoelectric layers as observed from 3D solutions [4]. This shortcoming inspired the researchers to develop layer-wise theory [4,[14][15][16] or the sublayer theory [2,7,[17][18][19] for approximation of the electric potential. In the latter case, the piezoelectric layer is divided into appropriate number of sublayers along the thickness direction and a linear through-the-thickness electric potential distribution for each sublayer is assumed.…”