Keywords: finite element, multilayered plates, piezoelectric, potential function approximation, static and modal analysis.

This paper deals with the problem of multilayered plates with piezoelectric actuators and sensors. Development of new numerical tools for analysing non-homogeneous structures is very important since few years and a survey of beam, plate and shell finite elements available for this kind of coupled field problems are given in [1,2]. The aim of this work is about the development of a new computational tool without any classical numerical pathologies in the field of finite element, simple to use and very efficient for both convergence velocity and accuracy for a very low cost. Two different plate finite elements have been previously presented for mechanical analysis :

• a C eight nodes with 40 degrees of freedom (dof), based on the first order model, very accurate for displacement values for a low cost, see [3,4];
• a C six nodes with 81 dof, based on a refined plate model, very efficient for both displacements and stresses ; a quasi-exact three dimensional stress field can be obtained for multilayered structures, see [5].

The purpose of this paper is to present a new finite element, simple to use and accurate on both displacement and potential function used later for active control of plate structures. Optimal number and location of actuators and sensors will then be studied using theoretical developments presented in [6] for beams. From this point of view, accurate transverse shear stress distribution is not needed and the choice of the C finite element is obvious.

Therefore, a new plate finite element is defined in order to analyse this kind of coupled problems for both moderately thick and thin plates without any pathologies of classical plate finite elements (shear locking, membrane locking, spurious modes, ...). It is based on the papers [3,4]. The potential function is approximated using the layers-wise approach and is quadratic with respect to the in-plane coordinates : one degree of freedom at each node. Two kinds of finite element approximations for the potential with respect to the thickness co-ordinate are studied in this work :

• a linear variation in each layer;
• a quadractic variation in each layer.

The major advantage of this choice is to obtain a total independence between the middle plane mesh and the discretisation of the potential function in the thickness direction.

In order to evaluate the efficiency of these finite elements, some static and dynamic tests have been conducted about the cylindrical bending of laminated plates and compared with reference solutions, given in [7]. The numerical simulations are about a monomorph plate (PZT4 material) and two sandwich plates with piezoelectric skins and elastic core (Zn0+/SI/Zn0- ; PZT4/Epoxy/PZT4 materials). Simply supported opposite ends with mechanical and/or electric loads are used in all the static tests and free vibrations analysis are also conducted. Results obtained are in very good agreement and show the efficiency of these new finite elements for both displacements and potential function. Furthermore, linear and quadratic variations of the potential function in each layer are also evaluated. For the monomorph tests, the results obtained using the quadratic variation are more accurate with less dof than using the linear variation. In the case of the sandwich tests, results are of same accuracy for both approximations.

1

D.A. Saravanos and P.R. Heyliger. Mechanics and computational models for laminated piezoelectric beams, plates and shells. App. Mech. Rev., 52(10):305�-320, 1999. doi:10.1115/1.3098918

2

A. Benjeddou. Advances in piezoelectric finite element modeling of adaptive structural elements : a survey. Comp. and Struc., 37(3):378�-383, 2000. doi:10.1016/S0045-7949(99)00151-0

3

O. Polit, M. Touratier, and P. Lory. A new eight-node quadrilateral shear-bending plate finite element. Int. Jour. Num. Meth. Eng., 37:387�-411, 1994. doi:10.1002/nme.1620370303

4

M. Ganapathi, O. Polit, and M. Touratier. A C eight-node membrane-shearbending element for geometrically non-linear (static and dynamic) analysis of laminates. Int. Jour. Num. Meth. Eng., 39:3453�-3474, 1996. doi:10.1002/(SICI)1097-0207(19961030)39:20<3453::AID-NME9>3.0.CO;2-7

5

O. Polit and M. Touratier. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comp. Meth. Applied Mech. and Eng., 185:305�-324, 2000. doi:10.1016/S0045-7825(99)00264-9

6

I. Bruant, G. Coffignal, F. Léné, and M. Vergé. A methodology for determination of piezoelectric actuator and sensor location on beam structures. J. Sound Vibr., 243(5), 2001. doi:10.1006/jsvi.2000.3448

7

A. Fernandez. Modèle et étude de composants piézoéelectriques : applications aux structures multifonctionnelles. PhD thesis, Université Paris VI, 2000. in French.

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