Keywords: piezoelectric, multilayered plates, finite elements, higher-order theories, smart structures, vibration analysis, unified formulation.

As a result of potential applications, during the last decades, significant efforts have been devoted to the research on the so-called smart structures. Such structures differ from the conventional ones by the presence of elements able to perform as actuators and/or sensors, allowing the structures itself to adapt and/or sense to the external environment. This capability leads to a wide range of applications, in particular in the aerospace field such as vibration suppression, shape adaption of aerodynamic surfaces, noise reduction, precision positioning of antennas, aeroelastic control of lifting surfaces and shape control of optical devices. Even if a variety of different materials can be utilized in smart structures, only piezoelectric materials have shown the capability to perform effectively both as actuators and sensors elements. Another advantage of piezoelectric materials is their simple integration within multilayered composite structures combining low density and superior mechanical and thermal properties along with sensing, actuation and control. However, multilayered structures embedding piezoelectric layers require appropriate electromechanical modelling, see [1,2,3,4]. On the other hand, the solution of practical problems demand the use of computational methods such as the finite element method.

This work presents some finite elements for the analysis of laminated plates embedding piezoelectric layers based on the Principle of Virtual Displacements (PVD) and the unified formulation introduced by Carrera [5,6]. One of the most interesting features of this element is the use of the unified formulation for the description of the unknowns which allows to keep the order of the expansion of the variables along the thickness direction as a parameter of the model and at the same time to perform both Equivalent Single Layer (ESL) and Layer-Wise (LW) descriptions of the state mechanical variables. The displacement unknowns are expanded up to the fourth order through a set of functions that depend only on the thickness coordinate. These functions can be simple Taylor polynomials or a manipulation of the Legendre ones respectively for an ESL or a LW description of the laminate. In case of a ESL model the zig-zag form has been recovered introducing the ZZ-function of Murakami [5]. The electric potential assumption has been limited to a LW description and the order of the expansion goes from 1 up to 4 as the displacement field. This feature is particularly suitable since it has been demonstrated by Benjeddou [1] that the widely used first-order expansion of the electric potential neglects the contribution of the induced potential, which in actuation problems increases the structure stiffness, and leads to a partial electromechanical coupling. With the present formulation the errors induced by the linear assumption of the electric potential can be easily pointed out. Combining all the possible parameters, up to thirteen theories are addressed in this work.

The full coupling between the electric and mechanical fields is considered; thus, the electric potential is taken as a state variable of the problem. The governing equations have been derived substituting the constitutive relations expressed in terms of the displacements and electric potential into the PVD statement. The discretized governing equations have then been given in the form of matrices that can be easily assembled starting from simpler arrays whose dimensions are in general 3x3 called Fundamental Nuclei. Both four-noded and nine-noded quadrilateral isoparametric elements have been formulated. Results for the free vibration frequencies in the case for a single layer piezoelectric plate and of a hybrid sandwich plate have been reported for various finite elements. Simply-supported square plates have been considered, while the lamination angle have been limited to cross-ply in order to compare the finite element results with those obtained by a Navier-type solution based on the same formulation [2].

The proposed numerical solution has shown a very good agreement with the analytical one [2]. Higher order LW elements lead to results very close to the 3D exact solution.

1

Benjeddou A., "Advances in piezoelectric finite element modelling of adaptive structural elements: a survey", Computer & Structures, 76, pp. 347-363, 2000. doi:10.1016/S0045-7949(99)00151-0

2

Ballhause D., "Assessment of multilayered theories for piezoelectric plates using a unified formulation", Master Thesis, Universität Stuttgart, Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, 2003.

3

Touratier M. Ossadzow-David C., "Multilayered Piezoelectric Refined Plate Theory", American Institute of Aeronautics and Astronautics Journal, vol. 41, no 1, pp. 90-99, 2003.

4

Benjeddou A. and Deü J.-F., "A two-dimensional closed-form solution for the free-vibrations analysis of piezoelectric sandwich plates", International Journal of Solids and Structures, vol.39, pp. 1463-1486, 2001. doi:10.1016/S0020-7683(01)00287-6

5

Carrera E., "An assessment of mixed and classical theories for thermal stress analysis of orthotropic plates", Journal of Thermal Stresses, vol. 23, pp. 797-831, 2000. doi:10.1080/014957300750040096

6

Carrera E., Robaldo A., "Formulazione unificata di elementi finiti piani per analisi termoelastica di strutture multistrato anisotrope", XVII Congresso Nazionale AIDAA, Rome, 2003.

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