Open Access

ARTICLE

Variable Kinematics and Advanced Variational Statements for Free Vibrations Analysis of Piezoelectric Plates and Shells

E. Carrera, S. Brischetto¹, M. Cinefra²

1 Corresponding author: Salvatore Brischetto, Department of Aeronautics and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. Tel: +39.011.564.6869, Fax: +39.011.564.6899, Email: salvatore.brischetto@polito.it.
2 Department of Aeronautics and Space Engineering, Politecnico di Torino, Italy

Computer Modeling in Engineering & Sciences 2010, 65(3), 259-342. https://doi.org/10.3970/cmes.2010.065.259

Abstract

This paper investigates the problem of free vibrations of multilayered plates and shells embedding anisotropic and thickness polarized piezoelectric layers. Carrera's Unified Formulation (CUF) has been employed to implement a large variety of electro-mechanical plate/shell theories. So-called Equivalent Single Layer and Layer Wise variable descriptions are employed for mechanical and electrical variables;linear to fourth order expansions are used in the thickness direction z in terms of power of z or Legendre polynomials. Various forms are considered for the Principle of Virtual Displacements (PVD) and Reissner's Mixed Variational Theorem (RMVT) to derive consistent differential electro-mechanical governing equations. The effect of electro-mechanical stiffness has been evaluated in both PVD and RMVT frameworks, while the effect of continuity of transverse variables (transverse shear and normal stresses and transverse normal electric displacement) has been addressed by comparing various forms of RMVT. According to CUF, governing equations related to a given variational statement have been written in terms of fundamental nuclei whose form is independent of the order of expansion and of the adopted variable description. The numerical results have been restricted to simply supported orthotropic plates and shells, for which exact three-dimensional solutions are available. A large numerical investigation has been conducted to compute fundamental and higher vibrations modes. An exhaustive numerical evaluation of assumptions, related to the various PVD and RMVT forms, is given. Classical, higher-order, layer-wise and mixed assumptions have been compared to available three-dimensional solutions. The convenience of hierarchical approaches based on CUF is shown, along with the suitability of the implemented RMVT forms to accurately trace the free vibration response of piezoelectric plates and shells. RMVT applications permit the vibration modes of transverse electro-mechanical variables to be accurately evaluated in the thickness plate/shell direction.

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Keywords

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