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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley-ASME Press Series
- Sprache: Englisch
A thorough guide to the fundamental development of linear piezoelectricity for vibrations
Vibrations of Linear Piezostructures is an introductory text that offers a concise examination of the general theory of vibrations of linear piezostructures. This important book brings together in one comprehensive volume the most current information on the theory for modeling and analysis of piezostructures. The authors explore the fundamental principles of piezostructures, review the relevant mathematics, continuum mechanics and elasticity, and continuum electrodynamics as they are applied to electromechanical piezostructures, and include the work that pertains to linear constitutive laws of piezoelectricity.
The book addresses modeling of linear piezostructures via Newton’s approach and Variational Methods. In addition, the authors explore the weak and strong forms of the equations of motion, Galerkin approximation methods for the weak form, Fourier or modal methods, and finite element methods. This important book:
- Covers the fundamental developments to vibrational theory for linear piezostructures
- Provides an introduction to continuum mechanics, elasticity, electrodynamics, variational calculus, and applied mathematics
- Offers in-depth coverage of Newton’s formulation of the equations of motion of vibrations of piezo-structures
- Discusses the variational methods for generation of equations of motion of piezo-structures
Written for students, professionals, and researchers in the field, Vibrations of Linear Piezostructures is an up-to-date volume to the fundamental development of linear piezoelectricity for vibrations from initial development to fully modeled systems using various methods.
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Table of Contents
Cover
Title Page
Copyright
Foreword
Preface
Acknowledgments
List of Symbols
1 Introduction
1.1 The Piezoelectric Effect
1.2 Applications
1.3 Outline of the Book
2 Mathematical Background
2.1 Vectors, Bases, and Frames
2.2 Tensors
2.3 Symmetry, Crystals, and Tensor Invariance
2.4 Problems
3 Review of Continuum Mechanics
3.1 Stress
3.2 Displacement and Strain
3.3 Strain Energy
3.4 Constitutive Laws for Linear Elastic Materials
3.5 The Initial‐Boundary Value Problem of Linear Elasticity
3.6 Problems
4 Review of Continuum Electrodynamics
4.1 Charge and Current
4.2 The Electric and Magnetic Fields
4.3 Maxwell's Equations
4.4 Problems
5 Linear Piezoelectricity
5.1 Constitutive Laws of Linear Piezoelectricity
5.2 The Initial‐Value Boundary Problem of Linear Piezoelectricity
5.3 Thermodynamics of Constitutive Laws
5.4 Symmetry of Constitutive Laws for Linear Piezoelectricity
5.5 Problems
6 Newton's Method for Piezoelectric Systems
6.1 An Axial Actuator Model
6.2 An Axial, Linear Potential, Actuator Model
6.3 A Linear Potential, Beam Actuator
6.4 Composite Plate Bending
6.5 Problems
7 Variational Methods
7.1 A Review of Variational Calculus
7.2 Hamilton's Principle
7.3 Hamilton's Principle for Piezoelectricity
7.4 Bernoulli–Euler Beam with a Shunt Circuit
7.5 Relationship to other Variational Principles
7.6 Lagrangian Densities
7.7 Problems
8 Approximations
8.1 Classical, Strong, and Weak Formulations
8.2 Modeling Damping and Dissipation
8.3 Galerkin Approximations
8.4 Problems
Supplementary Material
S.1 A Review of Vibrations
S.2 Tensor Analysis
S.3 Distributional and Weak Derivatives
Bibliography
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 The crystal systems and their point groups. See [13] Table A.2 or [...
Chapter 6
Table 6.1 Boundary conditions for piezoelectric composite beam driven by two vol...
Chapter 8
Table 8.1 Axial composite piezoelectric actuator, system properties.
Table 8.2 A piezoelectric composite beam material and system properties.
Table 8.3 Material and system properties for the piezoelectric composite beam wi...
List of Illustrations
Chapter 1
Figure 1.1 Barium titanate and lead zirconate titanate. (Left) Barium titana...
Figure 1.2 Polarization versus applied electrical field for ferroelectric ab...
Figure 1.3
and
domains in
, [31].
Figure 1.4 Polarization versus electrical field hysteresis below the Curie t...
Figure 1.5 The direct piezoelectric effect.
Figure 1.6 The converse piezoelectric effect.
Figure 1.7 Piezoelectrically based microphones from PCB®,
Figure 1.8 Piezoelectric stack actuators available from PI ceramic®, Source:...
Figure 1.9 Piezoelectric bender actuators available from PI ceramic®, Source...
Chapter 2
Figure 2.1 Frames generated by basis vectors
and
,
, and their cyclic pe...
Figure 2.2 Unit cell and lattice parameters.
Figure 2.3 The unit cell of the triclinic unit cell.
Figure 2.4 Crystallographic coordinates, directions, and planes.
Figure 2.5 The fourteen Bravais lattices, seven crystal systems, and associa...
Figure 2.6 Rotoinversion
of the point
. The rotation of
is followed by ...
Figure 2.7 Examples of objects with two, three, and fourfold symmetry about ...
Figure 2.8 Unit cell of monoclinic crystal system.
Figure 2.9 Invariance of the monoclinic lattice with respect to reflection a...
Figure 2.10 Invariance of the monoclinic lattice with respect to rotation th...
Chapter 3
Figure 3.1 (Left) Continuum body
having surface
and the stress vector
...
Figure 3.2 Tetrahedron with surface normal
used to derive Cauchy's formula...
Figure 3.3 (Left) Differential cube with surface stresses, (Right) All stres...
Figure 3.4 Undeformed configuration
, deformed configuration
, and the def...
Figure 3.5 Axial rod geometry, coordinate alignment, and displacement
Figure 3.6 Beam geometry, coordinate alignment, and displacement
Figure 3.7 Geometry of the thin, rectangular, Kirchoff plate
Figure 3.8 Linearly elastic body
, applied external stress field
, boundar...
Chapter 4
Figure 4.1 Point charges
located at points
, respectively, position vecto...
Figure 4.2 Wire loops carrying the currents
and
, vectors
and
, and di...
Figure 4.3 A typical crystal lattice, an asymmetric unit cell, the centers o...
Figure 4.4 A dielectric parallel plate capacitor.
Figure 4.5 A volume
that straddles the top electrodes surface of the volum...
Figure 4.6 A planar loop of wire carrying current
, having area
, and norm...
Figure 4.7 A lattice with point charges at each corner.
Figure 4.8 A finite wire with uniform linear charge density.
Figure 4.9 An infinite duct.
Figure 4.10 A nonplanar current carrying wire.
Chapter 5
Figure 5.1 (Left) Uniaxial test to measure electric displacement, (Right) El...
Figure 5.2 (Left) Uniaxial test to measure stress, (Right) Stress versus ele...
Figure 5.3 Decomposition of surface
into complementary surfaces
and
Chapter 6
Figure 6.1 Displacement assumed mode
.
Figure 6.2 Potential assumed mode
.
Figure 6.3 Piezoelectric uniaxial rod with the top and bottom surfaces elect...
Figure 6.4 Piezoelectrically actuated composite beam.
Figure 6.5 Shear and bending moment acting on a typical beam section.
Figure 6.6 Piezoelectrically actuated composite plate driven by two voltage ...
Figure 6.7 Composite piezoelectric beam actuated by two patches.
Figure 6.8 Axial piezoelectric specimen with tip mass.
Figure 6.9 Axial piezoelectric specimen with prescribed base motion
.
Figure 6.10 Piezoelectric composite beam specimen with rigidly attached tip ...
Figure 6.11 Piezoelectric composite beam specimen prescribed base motion
....
Chapter 7
Figure 7.1 Piezoelectric composite beam connected to a passive resistive shu...
Figure 7.2 Schematic diagram of piezoelectric element with shunt resistor
Figure 7.3 Piezoelectric composite beam connected to a passive capacitive sh...
Figure 7.4 Piezoelectric element with capacitive shunt
Figure 7.5 Stack actuator, orientation of layers, and electroding pattern
Figure 7.6 Coordinate systems for odd and even layers
Chapter 8
Figure 8.1 Piecewise linear
finite elements
Figure 8.2 Comparison of analytical natural frequencies and numerical estima...
Figure 8.3 Transient response of axial piezoelectric specimen
Figure 8.4 Comparison of Bode plots for models where the number of degrees o...
Figure 8.5 Conventional beam finite element functions over element
with
...
Figure 8.6 Nodes, elements, and beam finite element basis defined over a bea...
Figure 8.7 Comparison of numerically computed frequencies
and analytic fre...
Figure 8.8 Transient response of composite piezoelectric beam specimen
Figure 8.9 Comparison of Bode plot for
and
degrees of freedom in the Gal...
Figure 8.10 Experimental Setup, [27] Source: Vijaya Venkata Malladi / https:...
Figure 8.11 Clamped‐clamped traveling waves, (a) analytical, (b) experimenta...
Figure 8.12 Clamped‐free traveling waves, (a) analytical, (b) experimental [...
Figure 8.13 Free‐free traveling waves, (a) analytical, (b) experimental [27]...
Figure 8.14 Analytical estimates of traveling waves, (a) clamped‐clamped, (b...
Figure 8.15 Envelopes of traveling waves, (a) clamped free, 285 Hz, (b) free...
Supplementary Material
Figure 8.16 First four fixed‐fixed modes
for the axial element
Figure 8.17 First four fixed‐free modes
for the axial element
Figure 8.18 First four cantilever modes
of the Bernoulli–Euler beam
Guide
Cover Page
Table of Contents
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Vibrations of Linear Piezostructures
Andrew J. Kurdila and Pablo A. Tarazaga
Virginia Polytechnic Institute and State University
This edition first published 2021
© 2021 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data
Names: Kurdila, Andrew, author. | Tarazaga, Pablo (Pablo A.), author.
Title: Vibrations of linear piezostructures / Andrew J. Kurdila and Pablo A. Tarazaga.
Description: First edition. | Hoboken, NJ : John Wiley & Sons, 2021. | Series: Wiley-ASME Press series | Includes bibliographical references and index.
Identifiers: LCCN 2020027699 (print) | LCCN 2020027700 (ebook) | ISBN 9781119393405 (cloth) | ISBN 9781119393504 (adobe pdf) | ISBN 9781119393528 (epub) | ISBN 9781119393382 (obook)
Subjects: LCSH: Piezoelectricity. | Vibration.
Classification: LCC QC595 .K78 2021 (print) | LCC QC595 (ebook) | DDC 537/.2446--dc23
LC record available at https://lccn.loc.gov/2020027699
LC ebook record available at https://lccn.loc.gov/2020027700
Cover Design: Wiley
Cover Image: Pablo A. Tarazaga
Foreword
The rise of piezoelectric materials as sensors and actuators in engineering systems got started around 1980 and began to make an impact in the world of vibrations about five years after that. Subsequently, it started to explode into the 90s with topics such shunt damping, active control, structural health monitoring and energy harvesting. As a result, the need to document the fundamentals and intricacies of modeling piezoelectric materials in the context of vibrations in book form will well serve a variation of communities. The presentation here puts the topic on a firm mathematical footing.
The authors are uniquely qualified to provide a sophisticated analytical framework with an eye for applications. Professor Kurdila has nearly four decades of experience in modeling of multi‐physics systems. He authored two other books, one on structural dynamics, and several research monographs. Professor Tarazaga is an experienced creator of piezoelectric solutions to vibration and control problems. Both are well published in their respective research areas of research. Their combined expertise in researching vibratory systems integrated with piezoelectric materials enables this complete and detailed book on the topic. This allows for a formal theoretical background which will enable future research.
Daniel J. Inman
Ann Arbor, Michigan
Preface
The goal of this book is to provide a self‐contained, comprehensive, and introductory account of the modern theory of vibrations of linearly piezoelectric structural systems. While the piezoelectric effect was first investigated by the Curies in the , and systematically investigated in the field of acoustics and the development of sonar during the First World War, it is only much more recently that we have seen the widespread interest in mechatronic systems that feature piezoelectric sensors and actuators. Many of the early, now classical, texts present piezoelectricity from the viewpoint of a material scientist such as in [22] or [53]. Others are difficult, if not impossible, to obtain since they are out of print. Older editions of the excellent text [20] are currently selling for prices in excess of $600 on sites such as Amazon.com. Moreover, it is also quite difficult to find treatments of piezoelectricity that systematically cover all the relevant background material from first principles in continuum mechanics, continuum electrodynamics, or variational calculus that are necessary for a comprehensive introduction to vibrations of piezoelectric structures. The authors know of no text that assimilates all this requisite supporting material into one source. One text may give an excellent overview of piezoelectric constitutive laws, but neglect to discuss variational methods. Another may cover variational methods for piezoelectric systems, but fail to review the first principles of electrodynamics, and so forth. A large, substantive literature on various aspects of piezoelectricity has evolved over the past few years in archival journal articles, but much of this material has never been systematically represented in a single text.
This book has evolved from the course notes that the authors have generated while offering courses in active materials, smart systems, and piezoelectric materials over the past decade at various research universities. Most recently, the authors have taught active materials and smart structures courses that feature piezoelectricity at Virginia Tech to a diverse collection of first year graduate students. So much time was dedicated to the particular systems that include piezoelectric components that this textbook emerged. The backgrounds of the students in our classes have varied dramatically. Many students have not had a graduate class in vibrations, continuum mechanics, advanced strength of materials, nor electrodynamics. For this reason, the notes that evolved into this book make every effort to be self‐contained. Admittedly, this text covers in one chapter what other courses may cover over one or two semesters of dedicated study. As an example, Chapter 3 reviews the fundamentals of continuum mechanics for this text, a topic that is covered in other graduate classes at an introductory level during a full semester. So, while the presentation attempts to be comprehensive, the pace is sometimes brisk.
While preparing this text, we have tried to structure the material so that it is presented at the senior undergraduate or first year graduate student level. It is intended that this text provide the student with a good introduction to the topic, one that will serve them well when they seek to pursue more advanced topics in other texts or in their research. For example, this text can serve as a introduction to the fundamentals of modeling piezoelectric systems, and it can prepare the student specializing in energy harvesting when they consult a more advanced text such as [21].
This text begins in Chapter 2 with a review of the essential mathematical tools that are used frequently throughout the book. Topics covered include frames, coordinate systems, bases, vectors, tensors, introductory crystallography, and symmetry. Chapter 3 then gives a fundamental summary of topics from continuum mechanics. The stress vector and tensor is defined, Cauchy's Principle and the equilibrium equations are derived. The strain tensor is defined, and an introduction to constitutive laws for linearly elastic materials is also covered in this chapter. Chapter 4 provides the student the required introduction to continuum electrodynamics that is essential in building the theory of linear piezoelectricity in subsequent chapters. The definitions of charge, current, electric field, electric displacement, and magnetic field are introduced, and then Maxwell's equations of electromagnetism are studied.
Linear piezoelectricity is covered in Chapter 5. The discussion begins by introducing a physical example of the piezoelectric effect in one spatial example, and subsequently giving a generalization of the phenomenon in terms of piezoelectric constitutive laws. The initial‐boundary value problem of linear piezoelectricity is then derived from the analysis of Maxwell's equations and principles of continuum mechanics. While the equations governing any particular piezoelectric structure can be derived in principle from the initial‐boundary value problem of linear piezoelectricity, it is often possible and convenient to derive them directly for a problem at hand. Chapter 6 discusses the application of Newton's equations of motion for several prototypical piezoelectric composite structural systems. Chapter 7 provides a detailed account of how variational techniques can be used, instead of Newton's method, for many linearly piezoelectric structures. In some cases the variational approach can be much more expedient in deriving the governing equations. This chapter starts with a review of variational methods and Hamilton's Principle for linearly elastic structures. The approach is then extended by formulating Hamilton's Principle for Piezoelectric Systems and Hamilton's Principle for Electromechanical Systems. Several examples are considered, including the piezoelectrically actuated rod and Bernoulli–Euler beam, as well as the electromechanical systems that result when these structures are connected to ideal passive electrical networks. The book finishes in Chapter 8 with a discussion of approximation methods. Both modal approximations and finite element methods are discussed. Numerous example simulations are described in the final chapter, both for the actuator equation alone and for systems that couple the actuator and sensor equations.
June, 2017
Andrew J. Kurdila
Pablo A. Tarazaga
Acknowledgments
This book is the culmination of research carried out and courses taught by the authors over the years at a variety of institutions. The authors would like to thank the various research laboratories and sponsors that have supported their efforts over the years in areas related to active materials, smart structures, linearly piezoelectric systems, vibrations, control theory, and structural dynamics. These sponsors most notably include the Army Research Office, Air Force Office of Scientific Research, Office of Naval Research, and the National Science Foundation. We likewise extend our appreciation to the institutes of higher learning that have enabled and supported our efforts in teaching, research, and in disseminating the fruits of teaching and research: this volume would not have been possible without the infrastructure that makes such a sustained effort possible. In particular, we extend our gratitude to the Aerospace Engineering Department at Texas A&M University, the Department of Mechanical and Aerospace Engineering at the University of Florida, and most importantly, the Department of Mechanical Engineering at Virginia Tech. We extend our appreciation to the many colleagues that have worked with us over the years in areas related to active materials and smart structures. In particular, we thank Dr. Dan Inman for his support and for being a source of inspiration.
We also would like to specifically thank Dr. Vijaya V. N. Sriram Malladi and Dr. Sai Tej Paruchuri for their tireless efforts in editing and correcting the draft manuscript. Their meticulous attention to detail, suggestions and tireless effort has made this book a better version from its original draft. Additionally, we would like to thank our students Dr. Sheyda Davaria, Dr. Mohammad Albakri, Manu Krishnan, Mostafa Motaharibidgoli who have worked through the manuscript in order to improve its clarity. We would also like to also thank Sourabh Sangle, Murat Ambarkutuk, Lucas Tarazaga and Vanessa Tarazaga for their help in proofreading the last draft of the document. Finally, we would like to acknowledge anyone else not mentioned that contributed to the manuscript, including the students in our classes who provided valuable input throughout the years.
And, of course, we thank our families for their continued support and encouragement in efforts just like this one over the years.
Andrew J. Kurdila
Pablo A. Tarazaga
Blacksburg, VA
February, 2021
List of Symbols
Symbol
Description
Vectors and Tensors
Kronecker delta function
Levi‐Civita permutation tensor
generic basis vector
rotation matrix and its components
vector space of
order tensors
tensor product of
and
characteristic function of
lattice parameters
unit cell or lattice angles
domain
boundary of
Electrodynamics
speed of light
electric permitivity of free space
magnetic permeability of free space
current
,
total, free, bound, and polarization current density
total, free, and bound charge density
electric potential
magnetic vector potential
electric field vector and its components
dipole moment and its components
polarization and its components
electric displacement vector and its components
magnetic field and its components
magnetic dipole moment and its components
magnetization or magnetic polarization
magnetic field intensity and its components
“external” electric field induced by free charge
“internal” electric field induced by polarization charge
Elasticity
mass density
body force and its components
displacement field and its components
second order stress tensor and its components
second order linear strain tensor and its components
fourth order material stiffness tensor and its components
boundary of
on which
is prescribed
boundary of
on which
is prescribed
prescribed displacements on
prescribed stress vector on
initial condition on
in
initial condition on
in
strain energy density
strain energy or potential energy
Kinetic energy
Work
Virtual work
Beam shear force and bending moment
Plate bending moment per unit length
Plate shear force per unit length
beam area moment
Beam bending stiffness
Piezoelectricity
boundary of
on which
is prescribed
boundary of
on which
is prescribed
prescribed potential
on
prescribed charge distribution on
heat
1Introduction
1.1 The Piezoelectric Effect
In the most general terms, a material is piezoelectric if it transforms electrical into mechanical energy, and vice versa, in a reversible or lossless process. This transformation is evident at a macroscopic scale in what are commonly known as the direct and converse piezoelectric effects. The direct piezoelectric effect refers to the ability of a material to transform mechanical deformations into electrical charge. Equivalently, application of mechanical stress to a piezoelectric specimen induces flow of electricity in the direct piezoelectric effect. The converse piezoelectric effect describes the process by which the application of an electrical potential difference across a specimen results in its deformation. The converse effect can also be viewed as how the application of an external electric field induces mechanical stress in the specimen.
While the brothers Pierre and Jacques Curie discovered piezoelectricity in 1880, much the early impetus motivating its study can be attributed to the demands for submarine countermeasures that evolved during World War I. An excellent and concise history, before, during, and after World War I, can be found in [43]. With the increasing military interest in detecting submarines by their acoustic signatures during World War I, early research often studied naval applications, and specifically sonar. Paul Langevin and Walter Cady had pivotal roles during these early years. Langevin constructed ultrasonic transducers with quartz and steel composites. Shortly thereafter, the use of piezoelectric quartz oscillators became prevalent in ultrasound applications and broadcasting. The research by W.G. Cady was crucial in determining how to employ quartz resonators to stabilize high frequency electrical circuits.
A number of naturally occurring crystalline materials including Rochelle salt, quartz, topaz, tourmaline, and cane sugar exhibit piezoelectric effects. These materials were studied methodically in the early investigations of piezoelectricity. Following World War II, with its high demand for quartz plates, research and development of techniques to synthesize piezoelectric crystalline materials flourished. These efforts have resulted in a wide variety of synthetic piezoelectrics, and materials science research into specialized piezoelectrics continues to this day.
1.1.1 Ferroelectric Piezoelectrics
Perhaps one of the most important classes of piezoelectric materials that have become popular over the past few decades are the ferroelectric dielectrics. A ferroelectric can have coupling between the mechanical and electrical response that is several times a large as that in natural piezoelectrics. Ferroelectrics include materials such as barium titanate and lead zirconate titanate, and their unit cells are depicted in Figure 1.1. When the centers of positive and negative charge in a unit cell of a crystalline material do not coincide, the material is said to be polar or dielectric. An electric dipole moment is a vector that points from the center of negative charge to the center of positive charge, and its magnitude is equal to where is the magnitude of the charge at the centers and is the separation between the centers. The limiting volumetric density of dipole moments is the polarization vector . Intuitively we think of the polarization vector as measuring the asymmetry of the internal electric field of the piezoelectric crystal lattice. Ferroelectrics exhibit spontaneous electric polarization that can be reversed by the application of an external electric field. In other words, the polarization of the material is evident during a spontaneous process, one that evolves to a state that is thermodynamically more stable. Understanding this process requires a discussion of the micromechanics of a ferroelectric.
Figure 1.1 Barium titanate and lead zirconate titanate. (Left) Barium titanate with cation at the center, anions on the faces, and cations at the corners of the unit cell. (Right) Lead zircanate titanate with or cation at the center, anions on the faces, and cations at the corners of the unit cell.
The micromechanics of ferroelectric dielectrics is subtle and interesting. Above a critical temperature , the Curie temperature, the crystal structure of a ferroelectric is usually symmetric, and a plot of the polarization versus applied electric charge is generally nonlinear and single‐valued as shown in Figure 1.2.
However, with cooling below the Curie temperature , a thermodynamic process drives a structural phase transition so that the final crystalline phase has a lower symmetry. At the lower temperature it can be shown [18] that the lower symmetry crystal phase has at least two energetically equivalent configurations or variants. Furthermore, with the application of an external electric field, it must be the case that it is possible switch among these crystalline variants in a reversible process. The ferroelectric material forms domains that consist of these energetically equivalent crystalline variants. Figure 1.3 depicts schematically the and domains [31] that can appear in single crystal barium titanate [31]. Note in the figure that the polarization vectors are opposite from one domain to the next, and their average polarization over a macroscale can have zero effective polarization. Because of the presence of these domains, below the Curie temperature the polarization versus applied electric field takes the form of a hysteresis loop as shown in Figure 1.4. Initially, the domains cancel their effects over the macroscopic specimen and at . The polarization increases as in Figure 1.2 for a range of electric field . When a critical value , the coercive electric field strength, is reached, the domains abruptly switch so that they are approximately well‐aligned with the external electric field. With all domains having approximately aligned polarization vectors, the polarization again follows a nonlinear single valued curve until saturation is achieved. When the electric field is reversed, and reaches the opposite coercive electric field strength , the domains switch again so their polarization vectors are approximately aligned with the second variant. The result of this cyclic process is that after the transient response there is a nonzero polarization, the spontaneous polarization, for an electric field strength . At a macroscopic scale, then, the effective or average polarization can switch with the application of the external electric field.
Figure 1.2 Polarization versus applied electrical field for ferroelectric above the Curie temperature .
Figure 1.3 and domains in , [31].
Source: Walter J. Merz, Domain Formation and Domain Wall Motion in Ferro‐electric BaTiO3 Single Crystals, em Physical Review, Volume 95, Number 3, August 1, 1954, pp. 690–698.
Figure 1.4 Polarization versus electrical field hysteresis below the Curie temperature .
1.1.2 One Dimensional Direct and Converse Piezoelectric Effect
In view of these observations, at a fundamental level, the micromechanics of piezoelectricity is understood in terms of crystalline asymmetry. While the most general theory of linear piezoelectricity of material continua in three dimensions is discussed in Chapter 5, intuition can be built by considering a one dimensional example. Figure 1.5 depicts the direct piezoelectric effect graphically, while the converse effect is shown in Figure 1.6. For the specimens shown, the mechanical variables are the stress and strain , and the electrical variables include the electric field , electric displacement , voltage , and the electrical potential . In Figure 1.5 we suppose that the top and bottom of the specimen are free to displace. A thin film electrode, one that does not alter the mechanical properties of the specimen, is applied to the top and bottom surfaces by a deposition or sputtering process. An ideal current meter, over which the potential difference is approximately zero, is attached to the top and bottom electroded surfaces. A positive stress is applied as shown. As we discuss in Chapter 5
