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Proceedings Paper

A finite element formulation for piezoelectric shells with well balanced approximation functions
Author(s): Sven Klinkel; Dieter Legner
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Paper Abstract

Smart materials and structures play an important role for sensor and actuator applications. For the simulation of such systems it is essential to predict the material and system behavior as precisely as possible. A reliable simulation may provide an easier, faster and cheaper development of such devices. In a wide range of technical applications piezoelectric sensors and actuators typically have a shell-like structure. This motivates the present contribution to deal with the consistent approximation of a piezoelectric shell formulation. A physical description leads to a system of electromechanical differential equations. Due to the constitutive relations the strains and the electric field are coupled. In case of bending dominated problems incompatible approximation functions of these fields cause incorrect results. This effect occurs in standard finite element formulations, where the mechanical and electrical degrees of freedom are interpolated with lowest order functions. The formulation presented in this paper is based on the classical Reissner-Mindlin shell theory extended by a piezoelectric part. The shell element has four nodes and bilinear interpolation functions. The eight degrees of freedom per node are three displacements, three rotations and the electric potential on top and bottom of the shell. The finite shell element incorporates a 3D-material law and is able to model arbitrary curved shell geometries of piezoelectric devices. In order to overcome the described problem of incompatible approximation spaces a mixed multi-field variational approach is introduced. Six independent fields are employed. These are the displacement, strain, stress, electric potential, dielectric displacement and the electric field. It allows for approximations of the electric field and the strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the corresponding electric field is proposed through the shell thickness. This leads to well balanced approximation functions regarding coupling of electrical and mechanical fields. The numerical results are confirmed by analytical considerations and an example illustrates the more precise results of the present formulation in contrast to standard elements.

Paper Details

Date Published: 1 April 2009
PDF: 9 pages
Proc. SPIE 7289, Behavior and Mechanics of Multifunctional Materials and Composites 2009, 72890F (1 April 2009); doi: 10.1117/12.816368
Show Author Affiliations
Sven Klinkel, Univ. Karlsruhe (Germany)
Dieter Legner, Univ. Karlsruhe (Germany)


Published in SPIE Proceedings Vol. 7289:
Behavior and Mechanics of Multifunctional Materials and Composites 2009
Zoubeida Ounaies; Jiangyu Li, Editor(s)

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