Qing-Hua Qin. Advanced Mechanics of Piezoelectricity

Transcription

1 Qing-Hua Qin Advanced Mechanics of Piezoelectricity

2 Qing-Hua Qin Advanced Mechanics of Piezoelectricity With 77 figures

3 Author Prof. Qing-Hua Qin Research School of Engineering Australian National University Canberra, Australia ISBN Higher Education Press, Beijing ISBN ISBN Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publishers locations, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publishers can accept any legal responsibility for any errors or omissions that may be made. The publishers make no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Preface This book contains a comprehensive treatment of piezoelectric materials using linear electroelastic theory, the symplectic model, and various special solution methods. The volume summarizes the current state of practice and presents the most recent research outcomes in piezoelectricity. Our hope in preparing this book is to present a stimulating guide and then to attract interested readers and researchers to a new field that continues to provide fascinating and technologically important challenges. You will benefit from the authors thorough coverage of general principles for each topic, followed by detailed mathematical derivations and worked examples as well as tables and figures in appropriate positions. The study of piezoelectricity was initiated by Jacques Curie and Pierre Curie in They found that certain crystalline materials generate an electric charge proportional to a mechanical stress. Since then new theories and applications of the field have been constantly advanced. These advances have resulted in a great many publications including journal papers and monographs. Although many concepts and theories have been included in earlier monographs, numerous new developments in piezoelectricity over the last two decades have made it increasingly necessary to collect significant information and to present a unified treatment of these useful but scattered results. These results should be made available to professional engineers, research scientists, workers and postgraduate students in applied mechanics and material engineering. The objective of this book is to fill this gap, so that readers can obtain a sound knowledge of the solution methods for piezoelectric materials. This volume details the development of solution methods for piezoelectric composites and is written for researchers, postgraduate students, and professional engineers in the areas of solid mechanics, physical science and engineering, applied mathematics, mechanical engineering, and materials science. Little mathematical knowledge besides the usual calculus is required, although conventional matrixes, vectors, and tensor presentations are used throughout the book. Chapter 1 provides a brief description of piezocomposites and the linear theory of piezoelectric materials in order to establish notation and fundamental concepts for reference in later chapters. Chapter 2 presents various solution methods for piezoelectric composites which can be taken as a common source for subsequent chapters. It includes the potential function method, Lekhnitskii formalism, techniques of Fourier transformation, Trefftz finite element method, integral equation approach, shear-lag model, and symplectic method. Chapter 3 deals with problems of fibrous piezoelectric composites, beginning with a discussion of piezoelectric fiber push-out and pull-out, and ending with a brief description of the

5 vi Preface solution for a piezoelectric composite with an elliptic fiber. Chapter 4 is concerned with applications of Trefftz method to piezoelectric materials. Trefftz finite element method, Trefftz boundary element method, and Trefftz boundary-collocation method are presented. Chapter 5 describes some solutions of piezoelectric problems using a symplectic approach. Chapter 6 presents Saint-Venant decay analysis of piezoelectric materials by way of symplectic formulation and the state space method. Chapter 7 reviews solutions for piezoelectric materials containing penny-shaped cracks. Chapter 8 describes solution methods for functionally graded piezoelectric materials. I am indebted to a number of individuals in academic circles and organizations who have contributed in different, but important, ways to the preparation of this book. In particular, I wish to extend appreciation to my postgraduate students for their assistance in preparing this book. Special thanks go to Ms. Jianbo Liu of Higher Education Press for her commitment to the publication of this book. Finally, we wish to acknowledge the individuals and organizations cited in the book for permission to use their materials. I would be grateful if readers would be so kind as to send reports of any typographical and other errors, as well as their more general comments. Qing-Hua Qin Canberra, Australia May 2012

6 Contents Chapter 1 Introduction to Piezoelectricity Background Linear theory of piezoelectricity Basic equations in rectangular coordinate system Boundary conditions Functionally graded piezoelectric materials Types of gradation Basic equations for two-dimensional FGPMs Fibrous piezoelectric composites References Chapter 2 Solution Methods Potential function method Solution with Lekhnitskii formalism Techniques of Fourier transformation Trefftz finite element method Basic equations Assumed fields Element stiffness equation Integral equations Fredholm integral equations Volterra integral equations Abel s integral equation Shear-lag model Hamiltonian method and symplectic mechanics State space formulation References Chapter 3 Fibrous Piezoelectric Composites Introduction Basic formulations for fiber push-out and pull-out tests Piezoelectric fiber pull-out Relationships between matrix stresses and interfacial shear stress Solution for bonded region Solution for debonded region... 62

7 viii Contents Numerical results Piezoelectric fiber push-out Stress transfer in the bonded region Frictional sliding PFC push-out driven by electrical and mechanical loading Numerical assessment Interfacial debonding criterion Micromechanics of fibrous piezoelectric composites Overall elastoelectric properties of FPCs Extension to include magnetic and thermal effects Solution of composite with elliptic fiber Conformal mapping Solutions for thermal loading applied outside an elliptic fiber Solutions for holes and rigid fibers References Chapter 4 Trefftz Method for Piezoelectricity Introduction Trefftz FEM for generalized plane problems Basic field equations and boundary conditions Assumed fields Modified variational principle Generation of the element stiffness equation Numerical results Trefftz FEM for anti-plane problems Basic equations for deriving Trefftz FEM Trefftz functions Assumed fields Special element containing a singular corner Generation of element matrix Numerical examples Trefftz boundary element method for anti-plane problems Indirect formulation The point-collocation formulations of Trefftz boundary element method Direct formulation Numerical examples Trefftz boundary-collocation method for plane piezoelectricity General Trefftz solution sets Special Trefftz solution set for a problem with elliptic holes Special Trefftz solution set for impermeable crack problems

8 Contents ix Special Trefftz solution set for permeable crack problems Boundary collocation formulation References Chapter 5 Symplectic Solutions for Piezoelectric Materials Introduction A symplectic solution for piezoelectric wedges Hamiltonian system by differential equation approach Hamiltonian system by variational principle approach Basic eigenvalues and singularity of stress and electric fields Piezoelectric bimaterial wedge Multi-piezoelectric material wedge Extension to include magnetic effect Basic equations and their Hamiltonian system Eigenvalues and eigenfunctions Particular solutions Symplectic solution for a magnetoelectroelastic strip Basic equations Hamiltonian principle The zero-eigenvalue solutions Nonzero-eigenvalue solutions Three-dimensional symplectic formulation for piezoelectricity Basic formulations Hamiltonian dual equations The zero-eigenvalue solutions Sub-symplectic system Nonzero-eigenvalue solutions Symplectic solution for FGPMs Basic formulations Eigenvalue properties of the Hamiltonian matrix H Eigensolutions corresponding to =0 and Extension to the case of magnetoelectroelastic materials References Chapter 6 Saint-Venant Decay Problems in Piezoelectricity Introduction Saint-Venant end effects of piezoelectric strips Hamiltonian system for a piezoelectric strip Decay rate analysis Numerical illustration

9 x Contents 6.3 Saint-Venant decay in anti-plane dissimilar laminates Basic equations for anti-plane piezoelectric problem Mixed-variable state space formulation Decay rate of FGPM strip Two-layered FGPM laminates and dissimilar piezoelectric laminates Saint-Venant decay in multilayered piezoelectric laminates State space formulation Eigensolution and decay rate equation Decay rate of piezoelectric-piezomagnetic sandwich structures Basic equations and notations in multilayered structures Space state differential equations for analyzing decay rate Solutions to the space state differential equations References Chapter 7 Penny-Shaped Cracks Introduction An infinite piezoelectric material with a penny-shaped crack A penny-shaped crack in a piezoelectric strip A fiber with a penny-shaped crack embedded in a matrix Fundamental solution for penny-shaped crack problem Potential approach Solution for crack problem Fundamental solution for penny-shaped crack problem A penny-shaped crack in a piezoelectric cylinder Problem statement and basic equation Derivation of integral equations and their solution Numerical results and discussion A fiber with a penny-shaped crack and an elastic coating Formulation of the problem Fredholm integral equation of the problem Numerical results and discussion References Chapter 8 Solution Methods for Functionally Graded Piezoelectric Materials Introduction Singularity analysis of angularly graded piezoelectric wedge Basic formulations and the state space equation Two AGPM wedges AGPM-EM-AGPM wedge system

10 Contents xi Numerical results and discussion Solution to FGPM beams Basic formulation Solution procedure Parallel cracks in an FGPM strip Basic formulation Singular integral equations and field intensity factors Mode cracks in two bonded FGPMs Basic formulation of the problem Impermeable crack problem Permeable crack problem References Index

11 Notation English symbols a ij, b ij reduced material constants defined in Eq. (1.26) B i magnetic flux c i unknown coefficients in Eq. (4.9) and elastic stiffness constants in Chapter 3 c ijkl, c ij elastic stiffness constants d ij piezoelectric charge constants D i electric displacements e ijk, e ij piezoelectric constants e piezomagnetic coefficient ij E i f i f ij g ij H i electric field mechanical body forces elastic compliances piezoelectric voltage constants magnetic field intensity m ij reduced material constants defined in Eq. (6.5) q s surface charge Q electric charge density t i surface tractions u, v, w displacement in x, y, z directions, respectively u i displacements Greek symbols ij magnetoelectric coupling coefficient 2 c e n ( ) ( ) ( ) c n n e n 2 * ij ij ij ij for n = 0,1,2,, defined in Eq. (5.73) defined in Eq. (5.119) elastic strains temperature change dielectric constants magnetic permeability stresses Poisson s ratio electric potential magnetic potential

12 xiv Notation Other symbols / x partial derivative of a variable with respect to x [ ] denotes a rectangular or a square matrix { } denotes a column vector [ ] 1 denotes the inverse of a matrix [ ] T denotes the transpose of a matrix ( ) a bar over a variable represents the variable being prescribed or complex conjugate = 2 / x / y 2