Sensitivity Analysis of Thickness Assumptions for Piezoelectric Plate Models

This article compares different axiomatic 2D theories for linear homogeneous piezoelectric plates. Simple actuator and sensor configurations are considered of thin and thick piezoelectric ceramics working either in transverse extension (31 mode) or in shear mode (15 mode). By generalizing a previously established unified formulation, a large number of different through-thickness approximations for the in-plane displacement, the transverse displacement and the electrostatic potential are introduced. Additionally, either full 3D constitutive law or reduced constitutive equations accounting for a vanishing transverse normal stress are employed in these plate theories. By referring to an analytical Naviertype closed-form solution, a systematic assessment of various 2D models is performed. The proposed sensitivity analysis of plate theories with respect to their electromechanical response can serve as a useful guide for choosing appropriate models depending on the piezoelectric polarization scheme, the use as sensor or actuator, and the plate thickness.

[1]  H. F. Tiersten,et al.  Linear Piezoelectric Plate Vibrations , 1969 .

[2]  N. Rogacheva.  The Theory of Piezoelectric Shells and Plates , 1994 .

[3]  Paul R. Heyliger,et al.  Exact Solutions for Laminated Piezoelectric Plates in Cylindrical Bending , 1996 .

[4]  I. Chopra,et al.  The Effect of Laminate Kinematic Assumptions on the Global Response of Actuated Plates , 2006 .

[5]  Gaetano Giunta,et al.  Hierarchical closed form solutions for plates bent by localized transverse loadings , 2007 .

[6]  Roger Ohayon,et al.  A Unified Beam Finite Element Model for Extension and Shear Piezoelectric Actuation Mechanisms , 1997 .

[7]  D. Ballhause,et al.  A unified formulation to assess multilayered theories for piezoelectric plates , 2005 .

[8]  Erasmo Carrera,et al.  Analysis of thickness locking in classical, refined and mixed multilayered plate theories , 2008 .

[9]  Roger Ohayon,et al.  Parametric Analysis of the Vibration Control of Sandwich Beams Through Shear-Based Piezoelectric Actuation , 1999 .

[10]  Paolo Bisegna,et al.  An Exact Three-Dimensional Solution for Simply Supported Rectangular Piezoelectric Plates , 1996 .

[11]  R Sreedeep,et al.  Bending Behavior of Hybrid-Actuated Piezoelectric Sandwich Beams , 2004 .

[12]  Luciano Demasi,et al.  ∞3 Hierarchy plate theories for thick and thin composite plates: The generalized unified formulation , 2008 .

[13]  Amâncio Fernandes,et al.  Accurate modelling of piezoelectric plates: single-layered plate , 2001 .

[14]  Jiashi Yang,et al.  Higher-Order Theories of Piezoelectric Plates and Applications , 2000 .

[15]  J. N. Reddy,et al.  On laminated composite plates with integrated sensors and actuators , 1999 .

[16]  Erasmo Carrera,et al.  Analysis of thickness locking in classical, refined and mixed theories for layered shells , 2008 .

[17]  Paolo Gaudenzi,et al.  A finite element evaluation of single-layer and multi-layer theories for the analysis of laminated plates , 1995 .

[18]  M. D'Ottavio,et al.  An Extension of Reissner Mixed Variational Theorem to Piezoelectric Laminates , 2006 .

[19]  Inderjit Chopra,et al.  Review of State of Art of Smart Structures and Integrated Systems , 2002 .

[20]  T. K. Varadan,et al.  The Contradicting Assumptions of Zero Transverse Normal Stress and Strain in the Thin Plate Theory: A Justification , 2001 .

[21]  Brian P. Baillargeon,et al.  Active Vibration Suppression of Sandwich Beams using Piezoelectric Shear Actuators: Experiments and Numerical Simulations , 2005 .

[22]  Inderjit Chopra,et al.  State of the art of smart structures and integrated systems , 1998, Other Conferences.

[23]  Erasmo Carrera,et al.  Reissner Mixed Theorem Applied to Static Analysis of Piezoelectric Shells† , 2007 .

[24]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[25]  Ayech Benjeddou,et al.  Advances in piezoelectric finite element modeling of adaptive structural elements: a survey , 2000 .

[26]  R. Batra,et al.  Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories , 2002 .

[27]  D. Saravanos,et al.  Mechanics and Computational Models for Laminated Piezoelectric Beams, Plates, and Shells , 1999 .

[28]  K. Y. Sze,et al.  Electric Assumptions for Piezoelectric Laminate Analysis , 2004 .

[29]  Paul C-P Chao,et al.  Dynamic modeling and experimental verification of a piezoelectric part feeder in a structure with parallel bimorph beams. , 2007, Ultrasonics.

[30]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[31]  E. Carrera,et al.  Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells , 2006 .

[32]  Vijay K. Varadan,et al.  A review and critique of theories for piezoelectric laminates , 1999 .

[33]  Vijay K. Varadan,et al.  A review and critique of theories for piezoelectric laminates , 1999 .

[34]  A. Benjeddou,et al.  Piezoelectric Transverse Shear Actuation and Sensing of Plates, Part 1: A Three-Dimensional Mixed State Space Formulation , 2001 .

[35]  Osama J. Aldraihem,et al.  Exact deflection solutions of beams with shear piezoelectric actuators , 2003 .

[36]  Ayech Benjeddou,et al.  On Higher-Order Modelling of Smart Beams with Embedded Shear-Mode Piezoceramic Actuators and Sensors , 2006 .

[37]  Erasmo Carrera,et al.  An assessment of Mixed and Classical Theories on Global and Local Response of Multilayered, Orthotropic Plates , 2000 .

[38]  Olivier Polit,et al.  Electric potential approximations for an eight node plate finite element , 2006 .

[39]  Amâncio Fernandes,et al.  Analytical and numerical approaches to piezoelectric bimorph , 2003 .

[40]  C. Sun,et al.  Formulation of an adaptive sandwich beam , 1996 .

[41]  C. Sun,et al.  Use of thickness-shear mode in adaptive sandwich structures , 1995 .

[42]  Paolo Gaudenzi,et al.  On the Formulation of a Piezoelectric Plate Model , 2005 .

[43]  Marek Pietrzakowski,et al.  Piezoelectric control of composite plate vibration: Effect of electric potential distribution , 2008 .

[44]  Gérard A. Maugin,et al.  AN ASYMPTOTIC THEORY OF THIN PIEZOELECTRIC PLATES , 1990 .

[45]  P. M. Naghdi,et al.  The Theory of Shells and Plates , 1973 .

[46]  C. Sun,et al.  Analysis of a sandwich plate containing a piezoelectric core , 1999 .

[47]  R. Batra,et al.  Missing frequencies in previous exact solutions of free vibrations of simply supported rectangular plates , 2003 .

[48]  Romesh C. Batra,et al.  Exact solution for the cylindrical bending of laminated plates with embedded piezoelectric shear actuators , 2001 .

[49]  J. Paumier.  Asymptotic Consistency Of The Polynomial Approximation In The Linearized Plate Theory. Application T , 1997 .

[50]  Erasmo Carrera,et al.  Piezoelectric shell theories with "a priori" continuous transverse electro-mechanical variables , 2007 .

[51]  Paul R. Heyliger,et al.  Exact Solutions for Simply Supported Laminated Piezoelectric Plates , 1997 .

[52]  Maenghyo Cho,et al.  Higher order zig-zag plate theory under thermo-electric-mechanical loads combined , 2003 .

[53]  S. Kitipornchai,et al.  THREE-DIMENSIONAL ASYMPTOTIC APPROACH TO INHOMOGENEOUS AND LAMINATED PIEZOELECTRIC PLATES , 2000 .

[54]  Paolo Gaudenzi,et al.  Multi-layer higher-order finite elements for the analysis of free-edge stresses in piezoelectric actuated laminates , 2003 .

[55]  S. M. Shiyekar,et al.  Cylindrical bending of piezoelectric laminates with a higher order shear and normal deformation theory , 2008 .

[56]  E. Ramm,et al.  Three‐dimensional extension of non‐linear shell formulation based on the enhanced assumed strain concept , 1994 .

[57]  J. Reddy,et al.  Deformations of Piezothermoelastic Laminates with Internal Electrodes , 2001 .

[58]  M. D'Ottavio,et al.  Classical and Advanced Models for Laminated Plates with Piezoelectric Layers Actuated in Shear Mode , 2008 .

[59]  R. Batra,et al.  Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle , 2002 .

[60]  Ayech Benjeddou,et al.  Refined sandwich model for the vibration of beams with embedded shear piezoelectric actuators and sensors , 2008 .

[61]  Paolo Gaudenzi,et al.  MULTI-LAYER HIGHER-ORDER FINITE ELEMENTS FOR THE ANALYSIS OF FREE-EDGE STRESSES IN COMPOSITE LAMINATES , 1998 .

[62]  Paolo Bisegna,et al.  A Consistent Theory of Thin Piezoelectric Plates , 1996 .

[63]  Amâncio Fernandes,et al.  An accurate modelling of piezoelectric multi-layer plates , 2002 .

[64]  Romesh C. Batra,et al.  Analysis of piezoelectric bimorphs and plates with segmented actuators , 2001 .

[65]  Luciano Demasi,et al.  Three-dimensional closed form solutions and exact thin plate theories for isotropic plates , 2007 .