Vibrations of Linear Piezostructures. Andrew J. Kurdila

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

1 Introduction

      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