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Advanced finite element formulations for modeling thin piezoelectric structures
Author(s) -
Klinkel Sven,
Legner Dieter,
Wagner Werner
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110009
Subject(s) - finite element method , piezoelectricity , transversal (combinatorics) , mixed finite element method , mathematical analysis , timoshenko beam theory , displacement field , beam (structure) , interpolation (computer graphics) , bending , field (mathematics) , mathematics , physics , classical mechanics , structural engineering , engineering , acoustics , motion (physics) , pure mathematics
This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu‐Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner‐Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)