Keywords: vibrations of shells, multilayered structures, piezoelectric materials, closed-form solution, higher-order formulations.

This paper presents a closed-form solution for the free vibration of multilayered shells with embedded piezoelectric materials. The modelling of the through-thickness behavior is based on the "Unified Formulation" (UF) developed by Carrera (a comprehensive description may be found e.g. in [1]) and extended to piezoelectric materials by Ballhause et al. [2]. The UF allows to account in a compact notation for a large number of axiomatic two-dimensional theories for multilayered structures: well known Equivalent Single Layer (ESL) formulations - like the First-Order Shear Deformation Theory (FSDT) - as well as layerwise (LW), higher-order descriptions of the unknowns are presented and compared. Irrespective of the description method (LW or ESL) employed for the passive layers, the electric unknowns, i.e. the electric potential, has been always considered at layer level.
The UF can be compactly described by the following expansion for the unknown along the local thickness coordinate ( is the layer index):

where the usual summation convention is employed, denotes the order of the expansion and are the curvilinear coordinates defined on the shell reference surface. The functions can be defined at layer level (LW description) or on the reference surface of the shell (ESL description). Assembly at multilayered level is performed accounting for interlaminar continuity. In this work an analytic approach has been used with functions expressed as a combination of trigonometric functions, e.g.

and are the lengths of the shell edges along the coordinates and , respectively, while and indicate the number of half-waves in these two directions. This closed-form solution satisfies exactly all differential equations and the boundary conditions of a simply supported shell. The considered geometries consist of arbitrarily thick and doubly curved shells with constant Lamé coefficients. All curvature terms have been retained, i.e. no simplifying Love-type or Donnell-type assumptions have been made. A linear material behavior has been used; piezoelectric layers satisfy the linearly coupled electro-mechanical constitutive equations for an orthorombic mm2 crystal polarized in thickness direction.
The main results of this work confirm the necessity of layerwise, higher-order descriptions for an accurate evaluation of the mechanical characteristics of moderately thick and thick multilayered components. Introduction of fictitious mathematical layers improves the resolution of the curvature variation in thickness direction. Additionally, the primary role played by the so-called "electric stiffness" in the free-vibration response of piezoelectric laminates has been pointed out. The capability of piezoelectric layers to convert in electric energy part of the mechanical energy induces a stiffness increase which is clearly observable in the values of the fundamental frequencies. In order to capture this crucial effect, at least quadratic assumptions for the electric potential are required.
The present formulation has been employed for estimating the influence of the electro-mechanical coupling on resonant frequencies. It is shown that this important design parameter of laminated piezoelectric plates depends on the laminate's thickness: For thin plates, the frequencies computed by considering the electric stiffness result to be up to 5% larger than those obtained by neglecting the piezoelectric coupling; for thick plates, an increase of about 2% due to the electric stiffness can be seen. For hollow cylinders, the difference produced on the frequency associated to the first axisymmetric mode is of about 2%-3%, depending on the curvature radius and on the wall thickness; in this case, the influence of these two geometric parameters is less than in the case of plates, leading to differences of up to 1%.
In general, the present implementation has shown to recover exact 3D solutions in the case of passive shells and piezoelectric plates, thus ensuring the capability of accurately resolve both the geometry and the physical fields. With this in mind and since only few works are available concerning exact three-dimensional free vibration analysis for piezoelectric shells, the presented results obtained with higher-order approaches could be taken as reference results for further studies.

1

E. Carrera, "Theories and Finite Elements for Multilayered Plates and Shells: A Unified Compact Formulation with Numerical Assessment and Benchmarking", Arch. Comp. Meth. Eng., 10, pp. 215-296, 2003. doi:10.1007/BF02736224

2

D. Ballhause, M. D'Ottavio, B. Kröplin, E. Carrera, "A Unified Formulation to Assess Multilayered Theories for Piezoelectric Plates", Comp. and Struct., to be published, 2004. doi:10.1016/j.compstruc.2004.09.015

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