An Introduction to the Theory of Piezoelectricity
This volume is intended to provide researchers and graduate students with the basic aspects of the continuum modeling of electroelastic interactions in solids. A concise treatment of linear, nonlinear, static and dynamic theories and problems is presented
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Advances in Mechanics and Mathematics Volume 9 Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A. Ray W. Ogden University of Glasgow, U.K.
Advisory Editors: I. Ekeland University of British Columbia, Canada K.R. Rajagopal Texas A&M University, U.S.A. T. Ratiu Ecole Polytechnique, Switzerland W. Yang Tsinghua University, P.R. China
AN INTRODUCTION TO THE THEORY OF PIEZOELECTRICITY
by
JIASHI YANG Department of Engineering Mechanics University of Nebraska-Lincoln, U.S.A.
Springer
eBook ISBN: Print ISBN:
0-387-23546-9 0-387-23573-6
©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America
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Contents
Foreword Preface 1
xi
xiii
CHAPTER ONE NONLINEAR ELECTROELASTICITY FOR STRONG FIELDS 1. DEFORMATION AND MOTION OF A CONTINUUM 2. GLOBAL BALANCE LAWS 2.1 Polarization 2.2 Piezoelectric Effects 2.3 Electric Body Force, Couple, and Power 2.4 Balance Laws 3. LOCAL BALANCE LAWS 4. MATERIAL FORM OF LOCAL BALANCE LAWS 5. CONSTITUTIVE RELATIONS 6. INITIAL-BOUNDARY-VALUE PROBLEM 7. VARIATIONAL FORMULATION 8. TOTAL STRESS FORMULATION
1 1 1 1 9 10 10 11 12 13 18 21 24 26 28
CHAPTER TWO LINEAR PIEZOELECTRICITY FOR INFINITESIMAL FIELDS 1. LINEARIZATION 2. BOUNDARY-VALUE PROBLEM 2.1 Displacement-Potential Formulation 2.2 Boundary-Value Problem 2.3 Principle of Superposition 2.4 Compatibility 3. VARIATIONAL PRINCIPLES 3.1 Hamilton’s Principle 3.2 Mixed Variational Principles 3.3 Conservation Laws 4. UNIQUENESS 4.1 Poynting’s Theorem 4.2 Energy Integral
31 31 31 31 35 35 35 37 38 39 39 40 41 42 42 43
2
vi 4.3 Uniqueness 5. OTHER FORMULATIONS 5.1 Four-Vector Formulation 5.2 Vector Potential Formulation 6. CURVILINEAR COORDINATES 6.1 Cylindrical Coordinates 6.2 Spherical Coordinates 7. COMPACT MATRIX NOTATION 8. POLARIZED CERAMICS 9. QUARTZ AND LANGASITE 10. LITHIUM NIOBATE AND LITHIUM TANTALATE 3
CHAPTER THREE STATIC PROBLEMS 1. EXTENSION OF A CERAMIC ROD 1.1 Boundary-Value Problem 1.2 Shorted Electrodes 1.3 Open Electrodes 1.4 Electromechanical Coupling Factor 2. THICKNESS-STRETCH OF A CERAMIC PLATE 2.1 Boundary-Value Problem 2.2 Shorted Electrodes 2.3 Open Electrodes 2.4 Electromechanical Coupling Factor 3. THICKNESS-SHEAR OF A QUARTZ PLATE 3.1 Boundary-Value Problem 3.2 Free Surfaces 3.3 Clamped Surfaces 3.4 Electromechanical Coupling Factor 4. TORSION OF A CERAMIC CIRCULAR CYLINDER 1.1 Boundary-Value Problem 1.2 Shorted Electrodes 1.3 Open Electrodes 5. TANGENTIAL THICKNESS-SHEAR OF A CERAMIC CIRCULAR CYLINDER 6. ANTI-PLANE PROBLEMS OF POLARIZED CERAMICS 7. A SURFACE DISTRIBUTION OF ELECTRIC POTENTIAL 8. A CIRCULAR HOLE UNDER AXISYMMETRIC LOADS 9. AXIAL THICKNESS-S
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