Keywords: laminated shells, actuators, sensors, piezoelectric materials, finite element method.

This chapter deals with our recent developments on the analysis of laminated shells with embedded and/or surface bonded piezoelectric actuators or sensors. It is briefly reviewed the most relevant works related with this theme outlining the corresponding theories. Two shell finite element models are presented for the structural analysis of composite laminated piezoelectric shells. One is a semi-analytical axisymmetric conical frustum with two nodal rings using Fourier series expansion to represent the displacement field, the electric potential and applied loads. The other is a conical shell quadrilateral finite element with eight nodes using conforming type shape functions for the transversal displacement. Both models are based in a mixed laminated theory that combines a higher order shear deformation mechanical displacement field with a layerwise representation with linear functions for the electric potential through each piezoelectric layer.

The use of adaptive structures is currently receiving wide attention owing to its potential applications in several branches of engineering. Piezoelectric materials have the advantage that they can be used as actuators and sensors and are available in two broad classes: ceramics and polymers. Shells made of laminated composites with layers made of piezoelectric material are currently being used in complex technological applications. Such applications increase the demand for numerical tools for the analyses of these structures with high accuracy and to use these tools to achieve efficient optimised designs. Numerous analytical and numerical solutions for the structural analysis of laminated shells with piezoelectric layers had been continuously developed, which are described in survey papers by Benjeddou [1,2].

The two finite elements here described are specifically formulated to be applied in laminated shells with piezoelectric layers. The strain displacement relations used are obtained by specializing the general three-dimensional strain displacement relations from the Green´s strain tensor expressed in arbitrary orthogonal curvilinear coordinates as in Palazotto and Dennis [3]. A high order shear deformation displacement field is used with the imposed condition of zero transversal shear stresses at the surfaces of the shell. The electric potential is considered as a degree of freedom in the models developed allowing that sensor and actuator effects can be analyzed. A mixed laminate theory approach is used, Saravanos [4], which combines the equivalent single layer higher order shear deformation theory, to represent the mechanical behaviour with a layerwise discretization in the thickness direction to represent the distribution of the electric potential in each piezoelectric layer. Due to the imposition of the condition of vanishing of the shear transverse stresses at the shell surfaces a C1 continuity type function must be used for the shape function of the transversal displacement.

The first model was constructed to be applied to shell conical frustums, Pinto Correia et al [5], and takes advantage of the axisymmetric geometry as the dependence in the circumferential coordinate is taken in account by Fourier series expansions of the displacements, electrical potential and applied loads. This formulation allows the development of a semi-analytic finite element in the circumferential direction. The other finite element applies to conical shells which do not have to be axisymmetric and is a numerical quadrilateral bi-dimensional shell finite element , Pinto Correia et al [6], of the conforming type since the transversal displacement and its derivatives are continuous between adjacent elements.

1

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

2

Benjedou A., "Advances in Hybrid Active-Passive Vibration and Noise Control via Piezoelectric and Viscoelastic Constrained Layer Treatments Materials: Review and Assessment", Journal of Vibration and Control, 7, 565-602, 2001. doi:10.1177/107754630100700406

3

Palazotto A.N. and Dennis S.T., "Nonlinear Analysis of Shell Structures", AIAA Education Series, Washington D.C., 1992.

4

Saravanos D.A., "Mixed Laminated Theory and Finite Element for Smart Piezoelectric Composite Shell Structures", AIAA Journal, 35(8), 1327-1333, 1997. doi:10.2514/2.264

5

Pinto Correia I.F., Mota Soares C.M., Mota Soares C.A. and Herskovits J., "Development of semianalytical axisymmetric shell models with embedded sensors and actuators", Composite Structures, 47, 531-541, 1999. doi:10.1016/S0263-8223(00)00017-9

6

Pinto Correia I.F., Mota Soares C.M., Mota Soares C.A. and Herskovits J., "Analysis of adaptive shell structures using a refined laminated model", Composite Structures, (to appear 2004). doi:10.1016/j.compstruct.2004.04.047

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