Keywords: finite element, multilayered plates, piezoelectric, electric potential approximations, static and modal analysis.

The development of multilayered plates with piezoelectric actuators and sensors is a very active trend in the research community, as a result of different kinds of attractive applications such as vibration control and noise attenuation. Development of new numerical tools for analysing non-homogeneous structures is very important for the last 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]. This work is about the development of a 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 at a very low cost. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account of the transverse shear stress effects and using only five unknown generalized displacements : the C finite element approximation based on first order shear deformation theories (FSDT) [3] and the C finite element approximations using a high order shear deformation theory (HSDT) [4].

In this work, a simple eight node plate finite element is presented, based on the FSDT for the displacement field. This 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 (for example shear locking, membrane locking, spurious modes). It is based on previous work [3].

The potential function is approximated using the layerwise approach and two kinds of finite element approximations for the electric potential with respect to the thickness co-ordinate are presented in this work :

• a linear variation in each layer ;

• a quadratic variation in each layer.

The major advantage of this choice is that the discretization in the thickness direction is independent of the mid-plane mesh. Furthermore, a better description of the electric variation is obtained defining more numerical layers in one physical layer. For the in-plane variation, a quadratic variation using one degree of freedom at each node is used, according to an isoparametric approach.

Furthermore and at the post-processing level, the transverse shear stresses and the normal electric displacement vector component are deduced using the equilibrium equations.

In order to evaluate the efficiency of these finite elements, some static and dynamic tests have been previously conducted for the cylindrical bending of laminated plates [5]. In this work, laminated plates are evaluated in static and modal analysis and comparison are conducted with respect to elasticity solutions available in the literature from [6,7]. Some evaluations with respect to HSDT from [8] are also presented. Finally, some results using embedded piezoelectric shear actuator are given with respect to elasticity solutions from [9]. Results obtained are in good agreement and show the efficiency of these new finite elements.

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

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

5

I. Bruant and O. Polit. "A new piezoelectric 8 node finite element for active control", In Hellenic European Research on Computer Mathematics and its Applications (HERCMA 2003), 2003.

6

D.A. Saravanos, P.R. Heyliger, and D.A. Hopkins. "Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates", Int. J. Solids Struc., 34(3):359-378, 1997. doi:10.1016/S0020-7683(96)00012-1

7

P. Heyliger and D.A. Saravanos. "Exact free-vibration analysis of laminated plates with embedded piezoelectric layers", J. Acoust. Soc. Am., 98(3):1547-1556, 1995. doi:10.1121/1.413420

8

V.M.Franco Correia, M.A.Aguiar Gomes, A. Suleman, Cr.M Mota Soares, and Ca.M Mota Soares. "Modelling and design of adaptative composite structures", Comp. Meth. Applied Mech. and Eng., (185):325-346, 2000. doi:10.1016/S0045-7825(99)00265-0

9

A. Benjeddou and J.F. Deu. "Piezoelectric transverse shear actuation and sensing of plates, part 2 : Application and analysis", J. of Intel. Mat. Syst. and Struc., 12:451-467, 2001. doi:10.1106/J8DX-R3AT-EA6G-85K7

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