Analysis of Piezoelectric Structures and Devices 9783110297997, 9783110297881

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Analysis of Piezoelectric Structures and Devices

Analysis of Piezoelectric Structures and Devices Edited by

Daining Fang Ji Wang Weiqiu Chen

De Gruyter

Physics and Astronomy Classification Scheme 2010: 77.65.-j, 85.50.-n, 77.84.-s, 77.65.Dq, 77.84.Lf

ISBN 978-3-11-029788-1 e-ISBN 978-3-11-029799-7 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

” 2013 Higher Education Press and Walter de Gruyter GmbH, Berlin/Boston Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen 앝 Printed on acid-free paper 앪 Printed in Germany www.degruyter.com

Preface

Typical piezoelectric structures of devices such as resonators, actuators, and transducers have been known as targets for analysis with the consideration of coupled fields including mechanical, electrical, and thermal, to name a few, in applications concerning electronic functions for frequency control and detection and sensors for data collection. Indeed, utilization of piezoelectric structures with primary objective of accurate vibration frequency has been found in a wide arrange of practical problems and in-depth studies are required particularly in the strongly demanded biological and chemical sensor technology as sensing elements. To meet application needs, research work focusing on issues in the analysis of piezoelectric elements in devices and structures for sensor and control applications has been conducted for refined predictions of characteristics and behavior towards optimal design and improvement. Such methods and solutions are widely presented in conferences and publications of applied mechanics, which is not well communicated with application engineers of electronic devices for many reasons. The core requirements for the analysis of piezoelectric structures are deformation, vibration frequency, mode shapes, electric potential, and electric charge distribution, among others. These results can be used for the precise design of device structures for sensor and actuator applications with properties in terms of both mechanical and electrical variables. It is clear that the precise analysis is the key to good design which can be better achieved with analytical techniques rather than empirical approaches through the combination of experiments and experiences of good and acceptable designs. To this goal, many works have been done with analytical solutions based on approximate theories with the consideration of configuration, materials, and complications. Such theories have also been expanded to consider the coupled fields required in the analysis of devices with primary considerations of thermal and electrical variables, which are also needed in the formulation and estimation of electrical parameters. Indeed, such efforts have been successful in certain applications with sophisticated methods and techniques for the analysis and design of devices and structures meeting the precision needs of product development such as quartz crystal resonators based on the Mindlin plate equations and follow-up expansion for the thermal considerations. In addition, the Mindlin plate equations have been extended to the finite element analysis for accurate and practical analysis. The advantage of approximate theory has been well demonstrated with the accuracy and simplicity of such analysis with both analytical techniques and numerical methods, showing benefits of approximate methods as one of the needed proofs for further research on the methods. It

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Preface

is certainly well accepted in device and structure analysis because similar efforts have been made in related manners and the results in accelerated analysis have been embraced through improvements. To summarize recent progress on the analysis of piezoelectric structures in engineering applications, we have invited a few active researchers on this subject matter to provide state-of-art accounts on a few topics with broad interests. From these contributions, we can find original research work closely related to structural analysis, device design, fundamental theory, complication factors, and performance evaluation through both theoretical and experimental approaches. Such theory and methods are important in many applications such as actuators, energy harvesters, sensors, resonators with the consideration of novel materials (photostrictive, multiferroic, functionally graded, layered) and various configurations (plates, shells, composites). The wave modes involved in the analysis cover the commonly utilized surface acoustic waves (SAW), bulk acoustic waves (BAW) as the key functioning modes for extensive studies. In addition, there are novel approaches to establish new methods and techniques for the analysis of traditional device structures for possible fast and accurate predictions of essential vibration properties such as the frequency, mode shapes, charge distribution, and effects of complication factors such as surface, thermal, acceleration, stress, electric field and drive-level, and so on. These results can be used not only for the validation and optimization of designs, but also in the calculation of electrical parameters of devices which are commonly functioning as electrical elements in modern electronic circuits. The importance of such analytical techniques and methods is increasingly apparent not only in the computer-based product development process with advantages over time and cost but also international collaboration in the manufacturing and product conceiving process. Clearly, major efforts pioneered through such research should be appreciated and further advances with emphasis on the experimental validation of analytical models and results to improve the product development cycle without any gap left should also be encouraged. The overall research on the acoustic wave devices and structures involving materials, physical acoustics, and electrical parameters has been active in core groups meeting the growing need of new types of electronic and intelligent products, and many of the contributors of this volume have been playing leading roles in research and teaching. More importantly, these active researchers are also heading different directions in diversified communities and groups spanning geologically in the electronic, materials, physical acoustic, and mechanics fields. Such activities will bring the urgent technical challenges to be known by more engineers and scientists with the expected outcome of enriched know ledge and increased involvement in research and studying. Undoubtedly, this will make the subject matters more appealing to generations of students and practical engineers as we have been hoping. This book present the frontiers of piezoelectric structures and devices research in a unified and grouped collection, and will certainly help students, engineers, professors, and technologists to find the information and methods needed to guide their participation and anticipation in the field and industry. This is the latest addition to our dedication of a broader professional and technical exchange

Preface

vii

through conferences such as the IEEE Frequency Control and Ultrasonics Symposia, the SPAWDA, and other workshops and meetings we have been organizing in last decades. Editors

Daining Fang Peking University

Ji Wang Ningbo University

Weiqiu Chen Zhejiang University August, 2012

Contents

Chapter 1 Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou Non-uniform Actuations of Plates and Shells with Piezoelectric and Photostrictive Skew-quad Actuator Designs 1 Introduction 2 New SQ actuator system 3 1.2.1 The distribution profile of induced non-uniform forces and moments 4 1.2.2 Design of an SQ actuator system 7 1.3 Plate control with a piezoelectric SQ actuator system 7 1.3.1 Non-uniform forces and moments induced by the SQ actuator system 8 1.3.2 Modal control 9 1.3.3 Case studies: control of plates 10 1.3.4 Closed-loop actuation with collocated sensors and actuators 16 1.3.5 Summary of non-uniform piezoelectric actuations of plates 20 1.4 Cylindrical shell control with photostrictive SQ actuator systems 21 1.4.1 Uniform and non-uniform photostrictive actuation 22 1.4.2 Photostrictive SQ actuator 24 1.4.3 Modal control 26 1.4.4 Case studies: photostrictive actuation of shells 27 1.4.5 Closed-loop actuation with paired SQ actuator systems 34 1.4.6 Summary of non-uniform photostrictive actuations of shells 36 1.5 Summary 37 References 39 Appendix 41 1.1 1.2

Chapter 2 ChunLi Zhang and WeiQiu Chen Structural Theories of Multiferroic Plates and Shells 2.1 Introduction 44 2.2 Basic formulations 45 2.3 Laminated multiferroic shell equations in orthogonal curvilinear coordinates 46

43

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Contents

2.4 2.5

Equations of first-order theory for laminated multiferroic shells 50 Equations for flat plates and cylindrical/spherical shells 52 2.5.1 Flat plates 52 2.5.2 Cylindrical shells 53 2.5.3 Spherical shells 55 2.6 Applications: evaluation of magnetoelectric effects 57 2.6.1 Magnetoelectric effect in multiferroic bilayers 58 2.6.2 Magnetoelectric effect of multiferroic spherical shell laminates 2.7 Summary 66 References 66 Appendix 67

62

Chapter 3 YuanTai Hu, Huan Xue, and HongPing Hu Piezoelectric Power/Energy Harvesters 71 Introduction 72 Basic structure of a piezoelectric power/energy harvester 74 Piezoelectric power harvesters 75 3.3.1 The related researches on piezoelectric power harvesters 75 3.3.2 Coupling analysis of piezoelectric power harvesters 76 3.3.3 Numerical results 80 3.3.4 Investigation on frequency shift of piezoelectric power harvesters 3.3.5 Broadband design of piezoelectric power harvesters 88 3.4 Piezoelectric energy harvesters 91 3.4.1 Component portions of piezoelectric energy harvesters 91 3.4.2 Integrated analysis of piezoelectric energy harvesters 94 3.4.3 Numerical results 101 References 108 3.1 3.2 3.3

83

Chapter 4 Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates 113 Introduction 114 A two-dimensional analysis of surface acoustic waves in finite isotropic elastic plates 115 4.2.1 Surface acoustic waves in an infinite isotropic elastic plate 115 4.2.2 Two-dimensional equations for finite isotropic elastic plate 120 4.2.3 Conclusions 126 4.3 A two-dimensional analysis of surface acoustic waves in finite anisotropic elastic plates 126 4.3.1 Surface acoustic waves in semi-infinite anisotropic solids 126 4.3.2 Two-dimensional equations for finite anisotropic elastic plates 129 4.1 4.2

Contents

4.3.3 Conclusions 132 A two-dimensional analysis of surface acoustic waves in finite piezoelectric plates 133 4.4.1 Surface acoustic waves in semi-infinite piezoelectric solids 133 4.4.2 Two-dimensional equations for finite piezoelectric elastic plates 135 4.4.3 Conclusions 138 4.5 Summary 138 References 139 4.4

Chapter 5 JianGong Yu Wave Characteristics in the Functionally Graded Piezoelectric Waveguides: Legendre Polynomial Approach 143 Introduction 144 Wave propagation in the FGPM plate 146 5.2.1 Mathematics and formulation 146 5.2.2 Numerical results 151 5.3 Circumferential wave in the FGPM cylindrical curved plate 153 5.3.1 Mathematics and formulation 153 5.3.2 Numerical results and discussion 159 5.3.3 Brief summary 172 5.4 Axial wave in the FGPM hollow cylinders 172 5.4.1 Mathematics and formulation 173 5.4.2 Numerical results and discussion 175 5.4.3 Brief summary 185 5.5 Wave propagation in FGPM spherical curved plates 185 5.5.1 Mathematics and formulation 185 5.5.2 Numerical results and discussion 190 5.6 Summary 196 References 197 5.1 5.2

Chapter 6 HuiMing Wang Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures as Sensors and Actuators 201 Introduction 202 Basic equations for piezoelectric media 203 6.2.1 Basic equations for three-dimensional problems 203 6.2.2 Basic equations for radial motion 204 6.3 Layered model and solution 205 6.3.1 Layered model and governing equations 205 6.3.2 The method of superposition 208 6.3.3 The quasi-static part 209 6.1 6.2

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6.3.4 The dynamic solution 6.4 Numerical results and analysis 6.5 Summary 219 References 220 Appendix 223

211 212

Chapter 7 Zheng-Hua Qian and Feng Jin One Type of Transverse Surface Waves in Piezoelectric Layered Solids for Electro-acoustic Devices 227 Introduction 228 Transverse surface waves in piezoelectric layered solids 230 7.2.1 Piezoelectric layer and metal/dielectric substrate 230 7.2.2 Piezoelectric layered solids with a hard metal interlayer 235 7.3 Transverse surface waves in prestressed piezoelectric layered solids 240 7.4 Transverse surface waves in graded piezoelectric layered solids 243 7.4.1 Functionally graded covering layer 243 7.4.2 Functionally graded substrate 248 7.5 Summary 251 References 252 7.1 7.2

Chapter 8 HaiFeng Zhang Theoretical Investigation of Force-frequency and Electroelastic Effects of Thickness Mode Langasite Resonators 255 Introduction 256 Perturbation integral for resonator frequency shift calculations 257 Force-frequency effect of thickness mode langasite resonators 259 8.3.1 Background 259 8.3.2 Perturbation theory 261 8.3.3 Unperturbed mode 261 8.3.4 Diametrical force 263 8.3.5 Results 265 8.3.6 Conclusions 274 8.4 Electroelastic effect of thickness mode langasite resonators 274 8.4.1 Introduction 275 8.4.2 Unperturbed modes 276 8.4.3 Biasing electric field 276 8.4.4 Results 278 8.4.5 Conclusions 282 References 283 8.1 8.2 8.3

Contents

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Chapter 9 WeiQiu Chen Wave Propagation in a Piezoelectric Plate with Surface Effect Introduction 286 Isotropic elastic material surface 287 Surface piezoelectricity 291 Waves in a piezoelectric plate with surface effect 293 9.4.1 Decomposition 295 9.4.2 Two independent classes of motion 297 9.4.3 Love type or SH waves 299 9.4.4 Rayleigh-Lamb type waves 304 9.5 Summary 309 References 310 9.1 9.2 9.3 9.4

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Chapter 1 Non-uniform Actuations of Plates and Shells with Piezoelectric and Photostrictive Skew-quad Actuator Designs

Jing Jiang1 , HongHao Yue1 , ZongQuan Deng1 , and HornSen Tzou2 1 School of Mechatronic Engineering, Harbin Institute of Technology, Harbin, 150001, China 2 Institute of Applied Mechanics, StrucTronics and Control Lab., School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, 310027, China

Abstract Conventional distributed actuators laminated on shells and plates usually only introduce uniform control forces and moments. Structural actuation and control based on uniform control forces and moments have been investigated for over two decades. This study is to exploit a new actuator design, i.e., a skew-quad (SQ) actuator system made of piezoelectric and photostrictive materials. This new actuator system composed of four regions can induce non-uniform control forces and moments owing to the uneven boundary conditions of each region. The non-uniform distribution of actuator induced forces and moments are defined based on the variation method and validated by ANSYS. The coupling equation of a simply supported plate laminated with the piezoelectric SQ actuator is derived. Distributed control action resulting from the non-uniform control moments is also defined in the modal domain. Control actions of center-placed and corner-placed actuators on a square plate are defined and compared. Furthermore, wireless non-contact actuation of cylindrical shells coupled with the center-placed and corner-placed photostrictive SQ actuator systems are evaluated respectively. The modal control actions change with respect to the modes and the actuator coverage and thus, the actuator size and location are very important to the modal control effectiveness. In order to improve the control actions of the SQ actuator system, control schemes are designed for piezoelectric and photostrictive SQ actuator system respectively to regulate the sign of control forces of each region and to improve actuation effectiveness. Keywords distributed control, plate, cylindrical shell, piezoelectric and photostrictive actuator design, non-uniform control action

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

1.1 Introduction Spatially distributed vibration control of flexible structures with distributed actuators has been extensively studied for over two decades [1-5]. Earlier studies indicate that actuator design and placements are crucial to effective actuation and control of distributed parameter systems (DPSs), e.g., shells and plates [6-11]. A number of different design configurations, e.g., segmentation [12-16] and shaping [17-20], have been thoroughly evaluated over the years. Most of these actuator designs usually induce uniform control forces and moments. The inherent uniformly distributed control forces and control moments, however, often limit the control effectiveness and efficacy of structures. For example, a singlepiece centrally and symmetrically placed distributed actuator reveals control deficiencies to anti-symmetrical modes of symmetrical structures [12,13]. This symmetry problem can be resolved by spatially regulating the output characteristics of actuators [2,21], which can be realized by spatially varying either control signals or thickness of actuators. Sullivan et al. [2] proposed an approximation to a continuous non-uniform distribution of control actions with a combination of gain-weighted and shaped transducers. In this study, a new actuator design, with the two inner adjacent edges fixed and the other two edges freed, is proposed and evaluated. The actuator’s boundary conditions are specifically selected to realize the continuous non-uniform distribution of actuator induced control forces and control moments. Based on the variation method [22], the non-uniform distribution functions of induced forces and moments are calculated and validated. A new skew-quad (SQ) actuator design composed of four regions of the above actuator configuration is proposed and its actuation effectiveness to flexible structures is evaluated. Due to the novel design and the boundary conditions of its each region, this new SQ actuator system can induce multi-DOF (degree-of-freedom) non-uniform control forces and moments and consequently lead to stronger control actions at its four corners. New SQ actuator systems composed of piezoelectric and photostrictive actuation mechanisms are respectively discussed next. Control actions of piezoelectric and photostrictive SQ actuator systems with different surface coverage, location and control scheme are respectively evaluated in case studies. Piezoelectric actuation is based on the converse piezoelectric effect. Distributed actuation and control of shells and plates or DPSs have been investigated [21]. Among commonly used piezoelectric materials, flexible polymeric polyvinylidene fluoride (PVDF) film is versatile in shell applications, owing to its flexibility, durability, sensitivity, manufacturability, etc. [23]. The coupling equations of a simply supported plate laminated with the piezoelectric SQ actuator system is derived based on the generic distributed sensing and control theories of thin shells [21,24,25]. Distributed control action is also defined in the modal domain. Control effects of two locations, i.e., center-located and cornerlocated, of the new SQ actuator are evaluated respectively. In each case, control actions with different surface coverage (i.e., actuator sizes) are evaluated in the first part of this chapter. Furthermore, in order to reduce control deficiencies on anti-symmetrical modes of symmetrical plates when the new SQ actuator is centrally and symmetrically placed, four segmented sensors are also used to regulate the sign of control voltages in each re-

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Non-uniform Actuations of Plates and Shells

3

gion and their control effectiveness is also evaluated. Photostrictive actuation of shells is discussed next and detailed actuation characteristics of shells are presented in the second part of the chapter. Conventional actuators, e.g., piezoelectric, require hard-wire connections to transmit energy sources and control commands to activate the actuator mechanisms. The hardwire signal transmission busses can easily attract undesirable electric noises influenced by electric and/or magnetic fields. Accordingly, noises and uncertainties are often involved and control commands may not be accurately executed. Opto-mechanical actuators controlled by high-energy lights represent a new class of non-contact precision actuators based on the photodeformation process [26]. Irradiating high-energy lights, such as lasers or ultraviolet lights, on a certain class of photostrictive materials can trigger the photodeformation and, consequently, the induced photodeformation can be used for non-contact precision actuation and control. Light-driven opto-mechanical actuators have 1 many advantages over conventional hard-wired electromechanical actuators, such as ° 2 non-contact actuation, ° 3 compact and lightweight, ° 4 high electrical output voltage, ° 5 remote control. One-dimensional immune from electric/magnetic disturbances, and ° (beam type) and two-dimensional (plate type) opto-mechanical actuators with applications to distributed vibration control have been investigated [27-31]. Multi-DOF photostrictive actuators are proposed and their performance are evaluated [32,33]. In the second part of this chapter, cylindrical shell control with the photostrictive SQ actuator system made of four single-piece photostrictive slabs is investigated. Uniform and non-uniform micro-photodeformations are defined first. Photostrictive SQ actuator system design and its non-uniform actuation behavior are discussed, followed by modal control effectiveness of a flexible cylindrical panel respectively coupled with the center-placed and cornerplaced new SQ actuator systems. Closed-loop actuation to improve control effectiveness of unsymmetrical shell modes is also evaluated. Again, distributed control actions of piezoelectric and photostrictive SQ actuator systems with different surface coverage, location and control scheme are respectively reported in this chapter.

1.2 New SQ actuator system As discussed previously, uniform actuation and control of shells and plates have been studied over the years. Non-uniform actuation and control are emphasized in this chapter. An actuator element with uneven boundary conditions is introduced first and its nonuniform actuation behavior is evaluated. A new SQ actuator design based on the actuator element is proposed and its actuation effectiveness to plates and shells are presented later.

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

1.2.1 The distribution profile of induced non-uniform forces and moments When a control voltage is applied to an unconstrained piezoelectric actuator, the actuator usually induces uniform strains or actuations. However, an actuator can induce nonuniform strains when its boundary conditions are carefully manipulated. As shown in Fig.1.1, an actuator element, with the two inner adjacent edges fixed and the other two edges freed, can induce non-uniform strains or actuations. In this section, the non-uniform deformation and actuation behavior is evaluated.

Fig. 1.1 An actuator element with fixed-fixed-free-free boundary conditions.

When a control voltage is applied to a single mono-axial piezoelectric piece fixed at one end, the actuator element induces a uniform strain S=

d31 φ a ha

(1.1)

where d31 is the piezoelectric constant, φ a is a control voltage applied to the actuator, a exerted by the and ha is the actuator thickness. The equivalent uniform tension stress T11 actuator can be expressed as d31Ya φ a a T11 = = SYa (1.2) ha where Ya is Young’s modulus of the piezoelectric actuator. However, with the boundary conditions shown in Fig.1.1, the deformation is obviously zero at the fixed side and the deformation is the maximum at the free corner. This deformation profile (or an equivalent force profile when fixed) can be calculated using the variation method. Because of a , the the linear relationship between the actuator deformation and the tension stress T11 a magnitude of T11 is normalized, i.e., 1, to simplify the profile calculation. Thus, the true a , once the deformation is the calculated value multiplied by the true actuator stress T11 actuator material, control signal and dimensions are specified. Since the actuator is very

1

Non-uniform Actuations of Plates and Shells

5

thin, this is a plane problem in elasticity. According to the displacement boundary conditions, this problem can be solved by the planar displacement variation method. The displacements in the x and y directions can be respectively set as u = u0 + ∑ Am um

(1.3)

m

=

0+

∑ Bm

(1.4)

m

m

where u and are the displacements in the x and y directions respectively; Am and Bm are 2m independent coefficients; and u0 , 0 , um and m are functions set to satisfy the given boundary conditions. The boundary values of u0 and 0 are equal to the known boundary displacements; the values of um and m are zero on this boundary in this case. The boundary conditions is illustrated in Fig.1.1, i.e., u(x, y)(x = 0, y = 0) = 0, (x, y)(x = 0, y = 0) = 0, and hence u0 = 0, 0 = 0. The x and y displacement functions are assumed to be u(x, y) = xy(A1 + A2 y + A3 xy + A4 x + A5 xy2 + A6 x2 y + A7 x2 y2 + A8 x2 y3 +A9 x3 y2 + A10 x3 y3 + A11 x3 y4 + · · · ) 2

(1.5) 2

2 2

3 2

(x, y) = xy(B1 + B2 x + B3 xy + B4 y + B5 x y + B6 xy + B7 x y + B8 x y +B9 x2 y3 + B10 x3 y3 + B11 x4 y3 + · · · )

(1.6)

The potential energy for a plane stress problem can be expressed as [22] " ¶ µ ¶2 µ ¶ # ZZ µ Ya ∂u 2 ∂ ∂u ∂ (1 − µ ) ∂ ∂u 2 Vε = + + 2µ + + dxdy 2(1 − µ 2 ) ∂x ∂y ∂x ∂y 2 ∂x ∂y (1.7) where Vε is the deformation potential energy, µ is actuator’s Poisson’s ratio. The 2m coefficients can be derived by ZZ

Z

∂ Vε = fx um dxdy + fx um ds ∂ Am ZZ Z ∂ Vε = fy m dxdy + fy m ds ∂ Bm

(1.8) (1.9)

where fx and fy are body forces, fx and fy are surface forces acted on the actuator respectively in the x and y directions. As shown in Fig.1.1, the body forces fx = fy = 0 and a , f = 0. The 2m independent coefficient A and B can be the surface forces fy = T11 x m m determined by solving Eqs.(1.5)-(1.9). The calculated displacement changes with respect to the number of coefficients m. The parameters of this single-piece actuator element are La = 0.05 m, W a = 0.05 m, Ya = 2.0 × 109 N/m2 . The induced deformation in the y direction of the corner point (W a , La ) with respect to coefficients m are summarized in Table 1.1. Analysis results indicate that when the number of coefficients m is set to 6, 7, 8, 9 and 10, solutions of the corner displacement is consistent with each other, i.e., the dis-

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

placement solution is converged. In later calculations, the number of coefficients is set as 10. Table 1.1 Solutions obtained by the variation method. Number of coefficients

6

7

8

9

10

Corner displacement /(×10−11 m)

2.638

2.661

2.635

2.634

2.637

In order to validate the results, the actuator deformation profile is also calculated by the finite element software ANSYS. The two displacement curves calculated respectively by ANSYS and by the variation method are plotted in Fig.1.2 and they do compare very favorably. Thus, the deformation profile of one mono-axial actuator element with the boundary condition of two inner adjacent edges fixed and the other two edges freed is defined and the actuation force can also be derived accordingly.

Fig. 1.2 Comparison of actuator deformations.

When m is set to 10, the actuator deformation in the y direction shown in Fig.1.2 can be expressed as (x, y) = xy(B1 + B2 x + B3 xy + B4 y + B5 x2 y + B6 xy2 +B7 x2 y2 + B8 x3 y2 + B9 x2 y3 + B10 x3 y3 )

(1.10)

The total actuator deformation in the y direction can be defined by substituting y = La into Eq.(1.10). Thus, the induced equivalent actuator strain can be written as 0

S =

a (x, La ) T11 SYa (x, La ) = a L La

(1.11)

The non-uniform actuation force of the actuator element can be defined accordingly. Design of a new SQ actuator system and its control forces and moments are discussed next.

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Non-uniform Actuations of Plates and Shells

7

1.2.2 Design of an SQ actuator system A new SQ actuator system is made of four regions (Fig.1.3). Two inner adjacent edges of each region of the SQ system are fixed to a stable cross fixture and the other two outer adjacent edges are free. With these boundary conditions, each region can induce nonuniform deformation when a control signal is applied to the actuator. Consequently, it can induce non-uniform forces and moments when attached to plate or shell structures. The deformation function of each region of this new actuator system is defined next. With the deformation profile, the equivalent control forces and control moments are derived. Actuation and control characteristics of piezoelectric and photostrictive SQ actuator systems applied to plate and shell structures are respectively investigated later.

Fig. 1.3 A new SQ actuator system.

1.3 Plate control with a piezoelectric SQ actuator system Piezoelectric SQ actuator design consists of four flexible mono-axial piezoelectric pieces or actuator regions. Based on the above calculations, distributions of induced non-uniform control forces and moments of each region are defined. Independent modal control actions of a square simply supported plate attached with the center-located and the corner-located SQ actuator are studied respectively in this section. In each case, control actions with different surface coverage (i.e., actuator sizes) are evaluated. Control effectiveness and modal actuations of these two locations are compared. Closed-loop control characteristics with four segmented sensors are used to regulate the sign of control voltages in each region and their control effectiveness is also evaluated in case studies.

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

1.3.1 Non-uniform forces and moments induced by the SQ actuator system A new piezoelectric SQ actuator design consists of four flexible mono-axial piezoelectric elements respectively fixed to a center cross fixture. Figure 1.4 illustrates an SQ actuator system (with length La and width W a ) centrally placed on a plate (with length a and width b). When an electric control signal is applied to this actuator system, it induces non-uniform control forces and moments to the plate. The directions and distributions of induced control forces of each region are also shown in Fig.1.4. Non-uniform control forces and moments are defined in this section; modal control effects of rectangular plates are presented in the next section.

Fig. 1.4 A plate attached with a center-located SQ actuator system (not to scale).

According to the above variation procedures, the distribution profile of induced nonuniform forces and moments of each region can be written as f1 (x) = B1 xb1 + B2 x2 b1 + B3 x2 b21 + B4 xb21 + B5 x3 b21 +B6 x2 b31 + B7 x3 b31 + B8 x4 b31 + B9 x3 b41 + B10 x4 b41 f2 (y) = f3 (x) = f4 (y) =

−(B1 ya1 + B2 y a1 + B3 y2 a21 + B4 ya21 + B5 y3 a21 +B6 y2 a31 + B7 y3 a31 + B8 y4 a31 + B9 y3 a41 + B10 y4 a41 ) B1 xb1 − B2 x2 b1 − B3 x2 b21 + B4 xb21 + B5 x3 b21 −B6 x2 b31 + B7 x3 b31 − B8 x4 b31 + B9 x3 b41 − B10 x4 b41 −B1 ya1 + B2 y2 a1 + B3 y2 a21 − B4 ya21 − B5 y3 a21 +B6 y2 a31 − B7 y3 a31 + B8 y4 a31 − B9 y3 a41 + B10 y4 a41

(1.12)

2

Thus, the actuator induced control forces and moments (i.e., Niia and Miia ) are

(1.13) (1.14) (1.15)

1 a Nxx =

a Nyy =

Non-uniform Actuations of Plates and Shells

9

SYa Ya ha · { f4 (y) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (y − y∗1 ) − us (y − y∗2 )] La /2 − f2 (y) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (y − y∗2 ) − us (y − y∗3 )]} (1.16) SYa Ya ha · { f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (y − y∗2 ) − us (y − y∗3 )] W a /2 − f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (y − y∗1 ) − us (y − y∗2 )]} (1.17)

where us (·) is a step function. The two sets of step functions define the location of the actuator induced forces. The actuator induced control moments are (h + ha ) 2 (h + ha ) a a Myy = Nyy · 2

a a Mxx = Nxx ·

(1.18) (1.19)

Non-uniform control forces and moments induced by the new SQ actuator system are used to control the rectangular plate. Modal control effectiveness of plates is evaluated next.

1.3.2 Modal control It is assumed that the transverse bending oscillation dominates the plate motion, i.e., the in-plane membrane oscillations are neglected. The transverse governing equation of the plate with the distributed actuator can be expressed as [21] µ D

∂ 4 u3 ∂ 4 u3 ∂ 4 u3 +2 2 2 + 4 ∂x ∂x ∂y ∂ y4

¶ + ρ hu¨3 + cu˙3 =

a a ∂ 2 Myy ∂ 2 Mxx + 2 2 ∂x ∂y

(1.20)

where D is the bending stiffness, D = Y h3 /(1 − µ 2 ), Y is the plate Young’s modulus, µ is the plate Poisson’s ratio; ρ is the plate mass density; u3 is the plate transverse displacement; h is the plate thickness; c is the damping constant; and u¨3 and u˙3 are the plate transverse acceleration and velocity respectively. Note that the membrane control force does not contribute any control action to the transversely oscillating plate. It is also assumed that the plate is simply supported on all four edges. Based on the modal expansion technique, the dynamic response of the system can be represented by a sum of the responses of all participating modes, i.e., ∞

u3 (x, y,t) =

∞

∑ ∑ η3mn (t)U3mn (x, y)

(1.21)

m=1 n=1

where U3mn is the mnth transverse mode shape function and η3mn is the modal participation factor. Using the modal expansion and imposing the modal orthogonality of natural modes yields the mnth transverse modal equation of the plate [21].

10

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou c 2 η¨ mn + 2ζmn ωmn η˙ mn + ωmn ηmn = Fmn

(1.22)

where ζmn is the modal damping ratio, ωmn is the natural frequency of the mnth mode, and c is the modal control force Fmn à ! Z +a Z +b a a ∂ 2 Myy 2 2 1 ∂ 2 Mxx c Fmn = + U3mn (x, y)dxdy (1.23) ρ hNmn − a2 − b2 ∂ x2 ∂ y2 where Nmn is defined by the squared mode shape functions: Z Z

Nmn =

x y

2 U3mn dxdy

(1.24)

In case studies, control actions and comparison of a center-located and a corner-located piezoelectric SQ actuator system on a square plate are respectively evaluated next.

1.3.3 Case studies: control of plates As mentioned above, distributed modal actuations and control of a center-located and a corner-located piezoelectric SQ actuator system on a square plate are respectively investigated. Their modal actuation characteristics and effectiveness of these two cases are also compared. 1.3.3.1 Center-located piezoelectric SQ actuator on square plates In this case, the SQ actuator is symmetrically located at the plate center and both the actuator and the plexiglas plate are square, that is a = b and La = W a . Dimensions of the square plate are 0.2 m × 0.2 m × 0.0016 m (a × b × h). Material properties of the plate and the actuator are listed in Appendix: Table 1.A1 and Table 1.A2. Since the structure and the actuator system are symmetrical, the origin is assumed at the center of the plate (Fig.1.4). Thus, the transverse mode shape function of a simply supported plate (with dimensions a × b) in this coordinate system is ¶ ¶  µ  µ a b mπ x + nπ y +   2  2   · sin   U3mn = sin  (1.25)     a b where a and b are respectively the length and the width of the plate. Effects of actuator sizes defined by the length ratio ∆ , i.e., ∆ = La /a (actuator length/plate length), are also evaluated. Substituting Eqs.(1.18), (1.24) and (1.25) into the first term of the right side of the modal control force in Eq.(1.23) yields

1 Non-uniform Actuations of Plates and Shells

4 ρ hab =

11

Z +a Z +b µ 2 a ¶ 2 2 ∂ Mxx − a2

− b2

∂ x2 Z

U3mn (x, y)dxdy

a

Z

b

4 SYa (h + ha ) + 2 + 2 ∂ 2 Ya ha { f4 (y) · [us (x − x2∗ ) a 2 ρ hab L 2 − a2 − b2 ∂ x 2 −us (x − x3∗ )] · [us (y − y∗1 ) − us (y − y∗2 )]

− f2 (y) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (y − y∗2 ) − us (y − y∗3 )]}U3mn (x, y)dxdy = SM˜ xmn (1.26) Similarly, substituting Eqs.(1.18), (1.19), (1.24) and (1.25) into the second term of the modal control force in Eq.(1.23) gives à ! Z +a Z +b a ∂ 2 Myy 2 2 4 U3mn (x, y)dxdy ρ hab − a2 − b2 ∂ y2 Z

=

a

Z

b

4 SYa (h + ha ) + 2 + 2 ∂ 2 Ya ha { f1 (x) · [us (x − x2∗ ) a 2 ρ hab W 2 − a2 − b2 ∂ y 2 −us (x − x3∗ )] · [us (y − y∗2 ) − us (y − y∗3 )]

− f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (y − y∗1 ) − us (y − y∗2 )]}U3mn (x, y)dxdy = SM˜ ymn (1.27) The actuator is located from coordinates x1∗ to x3∗ and y∗1 to y∗3 . As shown in Fig.1.4, x1∗ = −La /2, x2∗ = 0, x3∗ = La /2, y∗1 = −W a /2, y∗2 = 0, and y∗3 = +W a /2. Substituting Eqs.(1.26) and (1.27) into Eq.(1.23), one obtains c c Fmn = S · (M˜ xmn + M˜ ymn ) = S · F˜mn

(1.28)

c (i.e., the magnitude of control moments) becomes Thus, the total control action F˜mn c F˜mn = M˜ xmn + M˜ ymn

(1.29)

where control actions M˜ xmn and M˜ ymn denote “actuation magnitudes” which are used as comparison indices in future comparisons. Note that for an unbiased comparison, the magnitude and sign of control voltages applied to the actuator system remains unchanged in all cases. The modal control effectiveness of the actuator can be evaluated with these control actions. Recall that the control action is determined by the material properties, actuator and plate dimensions, the mode number and the location of the actuator. Modal control actions of the new SQ actuator with different length ratios (∆ = 1/4, 1/3, 1/2, 2/3, 3/4, 1) or actuator coverage are calculated and summarized in Table 1.2. This table suggests that the new SQ actuator has control effects on square plates only when both m and n are odd wave numbers. Because the location of the actuator is centerlocated and the mode shape function of the simply supported plate, the symmetrical

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

modes, such as (1,1), (1,3), (3,3), etc., are effectively controlled. But the anti-symmetrical modes, such as (2,2), (2,4), etc., are not controllable, because the positive and negative control actions cancel out each other. To further evaluate the SQ actuator, plot this actuators’ control actions of the square plate with respect to the actuator/plate length ratio (∆ = La /L) in Fig.1.5. Because it has no control effects on even modes when m = 2, e.g., (2,1), (2,2), (2,3), only the control actions of odd mode groups of m = 1 (Fig.1.5(a)) and m = 3 (Fig.1.5(b)), e.g., (1,1), (1,2), (1,3), (3,1), (3,2), and (3,3), are plotted.

Fig. 1.5 Control actions of the center-located SQ actuator. (a) (1,1), (1,2), (1,3) modes; (b) (3,1), (3,2), (3,3) modes.

From these two figures, it can be observed that the actuator has identical control actions on modes (1,3) and (3,1), due to symmetry. The control actions on all controllable modes fluctuate with respect to the length ratio, i.e., the actuator coverage. Thus, the actuator size is very important to the overall modal control effectiveness. Figure 1.5(b) shows that the new SQ actuator performs best on mode (3,3) when the length ratio approaches to 1. This is contributed by the actuator configurations and the mode shape function of simply supported plates. Note that the (3,3) actuation magnitude of the SQ actuator is much larger (about 48%) than that of the multi-DOF actuator [34], due to its non-uniform

1 Non-uniform Actuations of Plates and Shells

13

boundary conditions, when ∆ = 1. As shown in Fig.1.6, the plate nodal lines of mode (3,3) and the actuator area are respectively drawn in dashed lines and solid lines, and the mode shapes are represented by “+” and “–” regions. Because in this case the sign and magnitude of control voltage do not change, the real control action of mode (3,3) is the difference between the control actions of “+” and “–” regions. Since the new SQ

Fig. 1.6 Control effectiveness of mode (3,3).

actuator induces non-uniform control forces and moments, it performs better at its four corners. When the length ratio approaches to 1, the induced control actions on the “+” regions would be much more significant than those on the “–” regions. Thus, this center-located SQ actuator is only effective to odd modes and ineffective to even modes. Accordingly, control effectiveness of a corner-located SQ actuator system is evaluated next. 1.3.3.2 Corner-located piezoelectric SQ actuator on square plates In this case, the SQ actuator is located at one quadrant of the square plate. The dimensional and material properties of the square plate and those of the actuator are the same as before. In order to simplify the calculation, the origin of the coordinate system is assumed at the center of the actuator, as shown in Fig.1.7. Following the previous procedures, one can define the modal control force of the corner-located actuator system. Ã ! Z + 3a Z + 3b 2Ma 2Ma ∂ 4 4 1 ∂ yy xx c Fmn = + U3mn (x, y)dxdy (1.30) ρ hNmn − a4 − b4 ∂ x2 ∂ y2 where the modal shape function in this new coordinate system can be rewritten as ¶  µ ³ b  a´ nπ y + mπ x + 4  4  sin    U3mn = sin  (1.31)   a b and

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

Fig. 1.7 The SQ actuator system is located at one corner of the plate.

¶  µ ³ ´ b a nπ y + Z Z Z + 3a Z + 3b mπ x + 4 4 4  2 2 4  sin2    dxdy = ab Nmn = U3mn dxdy = a sin   b a b 4 x y −4 −4 

(1.32) Substituting Eqs.(1.31) and (1.32) into Eq.(1.30) yields the two control moments induced by the corner-placed actuator system: 4 ρ hab =

Z + 3a Z + 3b µ 2 a ¶ 4 4 ∂ Mxx − 4a

− b4

∂ x2 Z

U3mn (x, y)dxdy

3a

Z

3b

4 SYa (h + ha ) + 4 + 4 ∂ 2 Ya ha { f4 (y) · [us (x − x2∗ ) − us (x − x3∗ )] 2 ρ hab 1 a 2 − a4 − b4 ∂ x L 2 ·[us (y − y∗1 ) − us (y − y∗2 )] − f2 (y) · [us (x − x1∗ ) − us (x − x2∗ )]

·[us (y − y∗2 ) − us (y − y∗3 )]}U3mn (x, y)dxdy = SM˜ xmn à ! Z + 3a Z + 3b a ∂ 2 Myy 4 4 4 U3mn (x, y)dxdy ρ hab − 4a − b4 ∂ y2 Z

=

3a

Z

(1.33)

3b

4 SYa (h + ha ) + 4 + 4 ∂ 2 Ya ha { f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] 2 ρ hab 1 a 2 − a4 − b4 ∂ y W 2 ·[us (y − y∗2 ) − us (y − y∗3 )] − f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )]

·[us (y − y∗1 ) − us (y − y∗2 )]}U3mn (x, y)dxdy = SM˜ ymn

(1.34)

1 Non-uniform Actuations of Plates and Shells

15

c (i.e., the magnitude of control moments) becomes Similarly, the total control action F˜mn c F˜mn = M˜ xmn + M˜ ymn

(1.35)

Assume the length ratio is set as its maximal value, i.e., ∆ = 1/2, comparisons of mode control actions between the center-located actuator and the corner-located actuator are summarized in Table 1.3. This table indicates that the corner-located actuator leads to control effectiveness on all of the first nine modes. It provides the same control actions on modes (1,2) and (2,1), (2,3) and (3,2), and less control actions on modes (1,1), (1,3) and (3,3), as compared with the center-located actuator when the length ratio is 1/2. Table 1.3 Comparison of mode control actions (∆ = 1/2).

(unit: N/kg)

Mode

Corner-located

Center-located

(1,1)

–2323

–4646

(1,2)

–5641

0

(1,3)

314

7230

(2,1)

–5641

0

(2,2)

–7963

0

(2,3)

–3509

0

(3,1)

–7544

7230

(3,2)

–3509

0

(3,3)

–784

–1566

Magnitudes of control actions on all controllable modes also fluctuate with respect to the length ratio or actuator size, as shown in Figs.1.8(a)-(c). These figures indicate that when the length ratio is ∆ = 1/2, the actuator introduce the maximal control action on modes (1,1), (1,2), (2,1), and (3,1). However, it does not provide the best control action on modes (1,3), (2,2), (2,3), (3,2), (3,3). Unlike the center-located actuator, the maximal coverage size of corner-located actuator is only 1/4, i.e., ∆ = 1/2. The comparison between Fig.1.8 and Fig.1.5 suggests that the center-located actuator performs better on symmetrical modes, such as (1,1), (1,3), (3,3), etc. The corner-located actuator improves the control actions on anti-symmetrical modes, but degrades the control effects on symmetrical modes. Another method to improve the modal control effectiveness of the center-located SQ actuator is discussed next.

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

Fig. 1.8 Modal control actions and size relationship of the corner-located SQ actuator system. (a) (1,1), (1,2), (1,3) modes; (b) (2,1), (2,2), (2,3) modes; (c) (3,1), (3,2), (3,3) modes.

1.3.4 Closed-loop actuation with collocated sensors and actuators As discussed previously, the corner-located actuator can control more modes than the center-located actuator. However, it is less effective than the center-located actuator in symmetrical plate modes, such as the (1,1) mode. To improve the control effectiveness, the sensor segmentation technique and closed-loop feedback [21] are used to enable the center-located SQ actuator to control all of the first nine modes. As shown in Fig.1.9,

Fig. 1.9 A plate attached with four sensor segments and a center-located SQ actuator system.

1 Non-uniform Actuations of Plates and Shells

17

four segmented biaxial piezoelectric sensors are located symmetrically to the plate’s center. Each sensor segment responds to the local motion state and generates a signal output which is fed back to its collocated region of the SQ actuator. In order to prevent the sensors from short-circuiting, it is assumed that a small gap is left open between the two adjacent sensor segments, but it is ignored in the mathematical model due to its smallness. Because the sensing signals are primarily contributed by bending strains of the plate in the transverse oscillation, membrane strains due to in-plane oscillation are ignored. The s can be expressed as mnth unit modal sensing signal of distributed piezoelectric sensor φmn a function of mode shape functions [13]: · µ 2 ¶ µ 2 ¶¸ ∂ U3mn s s ∂ U3mn φmn = −hs h31 r1s + h r (1.36a) 32 2 ∂ x2 ∂ y2 µ 2 ¶ µ 2 ¶¸ Z · ∂ U3mn s s ∂ U3mn φmn = −(hs /As ) h31 r1s + h r dAs (1.36b) 32 2 ∂ x2 ∂ y2 As where hs is the thickness of the distributed piezoelectric sensor layer, As is the effective electrode sensor area, h31 and h32 are piezoelectric constants, ris denotes the distance measured from the neutral surface to the mid-plane of the sensor layer, and ris = (h + hsi )/2. Note that r1s = r2s = (h + hs )/2 for uniform-thickness plates and sensor segments. Equation (1.36a) denotes the spatial distribution and Eq.(1.36b) denotes the averaged signal output of the sensor segment. It is assumed that four distributed piezoelectric sensors cover the whole surface of the plate. In order to simplify the calculation, the origin of the coordinate system is set at the center of the plate. For the mnth mode, the s of four sensor segments are respectively [12] output signals φmn s1 = (hs /As1 )[h31 r1s (mπ/a)2 + h32 r2s (nπ/b)2 ] φmn

×

Z a/2 Z b/2 0

0

sin[mπ(x + a/2)/a] sin[nπ(y + b/2)/b]dxdy

= 4Smn [cos(mπ/2) − cos(mπ)] · [cos(nπ/2) − cos nπ] s2 = 4Smn [1 − cos(mπ/2)] · [cos(nπ/2) − cos nπ] φmn s3 φmn = 4Smn [1 − cos(mπ/2)] · [1 − cos(nπ/2)] s4 = 4Smn [cos(mπ/2) − cos mπ][1 − cos(nπ/2)] φmn

(1.37) (1.38) (1.39) (1.40)

where Smn = (hs /mn)[h31 r1s (m/a)2 + h32 r2s (n/b)2 ] is the mnth modal sensitivity. With the above four signal equations, the output signals exist for most modes, except for either m or n is multiples of 4. The sign of feedback voltages to each region of the SQ actuator is regs3 s3 s1 s2 ulated for different plate modes, and φmn = (−1)m · (−1)n φmn , φmn = (−1) · (−1)n φmn , s s4 φmn = (−1) · (−1)m φmn3 . Again, it is assumed that the amplitude of feedback voltage of each sensor is kept at a constant maximum |φ a |, and only the sign of each sensor’s s1 voltage is regulated with respect to the wave number, i.e., φmn = (−1)m · (−1)n |φ a |, s3 s2 s4 a a a n m φmn = (−1) · (−1) |φ |, φmn = |φ |, φmn = (−1) · (−1) |φ |, for both m and n are odd number modes, such as modes (1,1), (1,3), (3,1), (3,3). Thus, this SQ actuator would induce identical control actions as that without four sensors. However, for either m or n is

18

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

even mode, such as (1,2), (2,1), (2,2), (2,3), (3,2), regulating the sign of feedback voltages can introduce control effects to even modes and this is different from the earlier case with a single uniform control voltage. When this SQ actuator is laminated with a plate, the desired induced control forces/moments on even modes of its four regions are illustrated in Fig.1.10.

Fig. 1.10 A plate attached with a center-located SQ actuator system and collocated sensor segments. (a) (2,1), (2,3) modes; (b) (2,2) mode; (c) (3,2) mode.

Based on these signal output, the actuator induced modal control actions are à ! Z +a Z +b a 2 2 4 ∂ 2 Mxx U3mn (x, y)dxdy ρ hab − a2 − b2 ∂ x2 =

4 SYa (h + ha ) Ya ha ρ hab 1 a 2 L 2 Z +a Z +b 2 2 2 ∂ × a {−(−1)m · f4 (y) · [us (x − x2∗ ) − us (x − x3∗ )] 2 b −2 −2 ∂x

1 Non-uniform Actuations of Plates and Shells

19

·[us (y − y∗1 ) − us (y − y∗2 )] + (−1)n · f2 (y) · [us (x − x1∗ ) − us (x − x2∗ )] ·[us (y − y∗2 ) − us (y − y∗3 )]}U3mn (x, y)dxdy = SM˜ xmn à ! Z +a Z +b a ∂ 2 Myy 2 2 4 U3mn (x, y)dxdy ρ hab − a2 − b2 ∂ y2 =

(1.41)

4 SYa (h + ha ) Ya ha ρ hab 1 a 2 W 2 Z +a Z +b 2 2 2 ∂ × a {(−1)m · (−1)n · f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] 2 b −2 −2 ∂y ·[us (y − y∗2 ) − us (y − y∗3 )] − f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )]

·[us (y − y∗1 ) − us (y − y∗2 )]}U3mn (x, y)dxdy = SM˜ ymn

(1.42)

S = (d31 · |φ a |)/ha

(1.43)

where Closed-loop control actions of the new SQ actuator with different length ratios (∆ = 1/4, 1/3, 1/2, 2/3, 3/4, 1) are summarized in Table 1.4.

Table 1.4 suggests that the actuator induces identical actuation magnitudes on modes (1,2) and (2,1), (1,3) and (3,1), (2,3) and (3,2). Comparing Table 1.2 with Table 1.4 suggests that when both n and m are odd numbers, such as (1,1), (1,3), (3,1), (3,3) mode, either with or without segmented sensors, the control actions of the SQ actuator system are identical. Thus, with the collocated sensors, the control actions for even modes can be improved without degrading the control actions for odd modes. The control actions and the size relationship of the first nine modes are plotted in Fig.1.11 to further illustrate

20

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

the actuation magnitudes changing with respect to the actuator coverage. Note that larger actuator doesn’t always guarantee better control effects.

Fig. 1.11 Control actions of the center-located SQ actuator system with four collocated sensor segments. (a) (1,1), (1,2), (1,3) modes; (b) (2,1), (2,2), (2,3) modes; (c) (3,1), (3,2), (3,3) modes.

1.3.5 Summary of non-uniform piezoelectric actuations of plates Piezoelectric SQ actuator system composed of four pieces of mono-axial piezoelectric actuators was proposed. Due to the uneven boundary conditions of each region, this actuator system can induce non-uniform control forces and moments. Analyses, comparisons and parametric evaluations suggest that: (1) When the piezoelectric new SQ actuator system is symmetrically placed at the plate center (center-located), it provides control effects only to modes when both m or n are odd wave numbers, such as (1,1), (1,3), (3,1) and (3,3), and the actuator generates identical control actions on modes (1,3) and (3,1). It is ineffective to even modes or antisymmetrical modes. (2) The modal control actions fluctuate with respect to the length ratio, i.e., the actuator coverage or size. Larger size doesn’t always guarantee better modal control effects. Thus, the actuator size is very important to the individual modal control effectiveness of various

1 Non-uniform Actuations of Plates and Shells

21

modes. The center-located SQ actuator performs best on mode (3,3) when the length ratio approaches to 1. (3) When the SQ actuator is located at one of the corners of the plate (corner-located), it generates control effectiveness on all of the first nine modes, and it delivers the same control actions on modes (1,2) and (2,1), also (2,3) and (3,2). However, actuation magnitudes in some modes, such as (1,1), are less than those of the center-located actuator. (4) With four collocated sensor segments and regulating signs of control voltages, the center-located SQ actuator system can generate control actions for even modes without degrading the control actions for odd modes. Thus, with the uneven control moments, independent modal control of square plexiglas plates attached with the center-located and the corner-located SQ actuators is feasible. In order to improve the control effectiveness of the SQ actuator system when symmetrically placed at the plate center, closed-loop feedbacks using four collocated sensor segments to independently regulate the sign of control voltages to each actuator region also demonstrate the improvement of modal control effectiveness. Wireless and non-contact actuation of shells with photostrictive SQ actuator systems is presented next.

1.4 Cylindrical shell control with photostrictive SQ actuator systems Light-driven non-contact actuation and control exhibits many advantages over conventional hard-wired actuator and control systems, especially in high electrical- and electromagnetic-noise environments. Photostrictive (Pb,La)(Zr,Ti)O3 ceramic doped with WO3 (PLZT) materials respond to a narrow-band ultraviolet light and, with specific configuration, can generate strains which can be used for precision actuation and control [26,35]. Precision actuation and vibration control of distributed structures laminated with PLZT actuators have been evaluated over the years. These structures include beams [36,37], plates [28,38], cylindrical shells [39,40], spherical and parabolic shells [41,42], etc. However, photostrictive actuators used in these structures were generally uniform thickness and surfaced laminated. Thus, under the uniform illumination of high-energy lights, they generated uniform control forces/moments, due to the photodeformation effect, a combined photovoltaic and converse piezoelectric effect. Actuation and control characteristics of distributed structures subjected to non-uniform control forces and moments were seldom investigated. The modal actuation effectiveness of a cylindrical shell panel laminated with this new photostrictive SQ actuator system is investigated in this section. Uniform and non-uniform actuation behaviors are defined first. New actuator design and its non-uniform actuation behavior of this new photostrictive SQ actuator system are discussed, followed by modal control effectiveness of a flexible cylindrical panel respectively coupled with the center-placed and corner-placed new actuator systems. Closedloop actuation to improve control effectiveness of unsymmetrical shell modes is also evaluated.

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

1.4.1 Uniform and non-uniform photostrictive actuation Figure 1.12 shows a PLZT photostrictive actuator polarized in the z-direction and its two electrodes are on the x-y surfaces. The actuator can induce strains along the polarization direction when subjected to high-intensity light illuminations. The strains are generated 1 the photo-electromechanical (photodeformation) from three light induced effects, i.e., ° 2 the photo-thermo-electromechanical effect and, ° 3 the photo-thermoelastic efeffect, ° fect. When the photostrictive actuator is laminated with flexible structures, it can induce control actions from these strains. In this section, all of the three effects are respectively discussed next.

Fig. 1.12 Unconstrained photostrictive actuator.

1.4.1.1 Photo-electromechanical effect The light irradiated on the photostrictive actuator first induces a current which generates a voltage between the two surface electrodes. This phenomenon is known as the photovoltaic effect. It is assumed that the light intensity I(t) is regulated with respect to time, i.e., I(t) is a time function. The induced photovoltaic electric field El (t j ) at t j = t j−1 + ∆t, including the voltage leakage effect, can be approximated as [28] α α I(t j−1 )e(− as I(t j−1 )∆t ) · ∆t − El (t j−1 )βo e−βo ∆t · ∆t as (1.44) where Es is the saturated photovoltaic voltage, as is the aspect ratio of actuator’s length and width and as = La /W a , I(t j ) is the light intensity at time t j , ∆t is the time step, βo is a constant related to the leakage time, and α is the optical actuator constant.

El (t j ) = El (t j−1 ) + [Es − El (t j−1 )]

1.4.1.2 Photo-thermo-electromechanical effect The high photonic energy can also heat up the photostrictive actuator. An additional electric field Eθ (t) resulting from actuator’s temperature rise is generated owing to the pyro-

1 Non-uniform Actuations of Plates and Shells

23

electric effect:

pn θ (t) (1.45) ε where Pn is the pyroelectric constant, ε is the permittivity. The temperature response θ (t) is [28] [I(t j )P − γθ (t j−1 )]∆t θ (t j ) = θ (t j−1 ) + (1.46) H + γ ∆t Eθ (t) =

where P is the power of absorbed heat, H is the heat capacity of the optical actuator, and γ is the heat transfer rate. 1.4.1.3 Photo-thermoelastic effect The high-energy light also induces thermal expansion λ θ (t)/Ya of the actuator, where λ is a thermal stress coefficient, and Ya is actuator’s Young’s modulus.This thermal expansion effect reduces the overall actuation effectiveness. All these three effects contribute the resultant photostrictive actuation effectiveness. Thus, based on the converse piezoelectric effect and the thermal expansion, the overall magnitude of the actuator induced strains becomes S(t) = d33 [El (t) + Eθ (t)] − λ θ (t)/Ya

(1.47)

where d33 is the piezoelectric constant of the PLZT. Accordingly the induced actuation a (t) in the z-direction is force per unit width N11 ª © a N11 (t) = ha Ya d33 [El (t) + Eθ (t)] − λ θ (t) (1.48) where ha is the actuator thickness. As discussed earlier, an unconstrained PLZT photostrictive actuator usually only induces uniform strains under uniform irradiation of highenergy lights. However, the actuator can induce non-uniform strains when its boundary conditions are modified. As shown in Fig.1.13, a PLZT actuator, with the two inner adjacent edges fixed and the other two edges freed, can induce non-uniform strains. In this section, the non-uniform photodeformation is evaluated.

Fig. 1.13 Photostrictive actuator with fixed-fixed-free-free boundary conditions.

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Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

The photostrictive actuator deformation is obviously zero at the fixed edges and the maximal at the corner. This deformation profile is equivalent to the deformation profile a (t) which can be written as when the actuator is exerted by a uniform tensional stress T11 a T11 (t) = Ya d33 [El (t) + Eθ (t)] − λ θ (t) = Ya · S(t)

(1.49)

Thus, with the specified boundary conditions, the actuator induced non-uniform actuation force per unit width along its deformation direction becomes a (t) = ha [Ya d33 [El (t) + Eθ (t)] − λ θ (t)] · (x, La )/La N11

(1.50)

where (x, y) = xy(B1 + B2 x + B3 xy + B4 y + B5 x2 y + B6 xy2 +B7 x2 y2 + B8 x3 y2 + B9 x2 y3 + B10 x3 y3 + · · · )

(1.51)

Bm are the coefficients determined to fit the deformation profile and m = 10 in this case. This will be used to calculate actuation profiles when the actuator coordinates are specified later. An SQ photostrictive actuator system based on this component is presented next. Actuation characteristics of the new design applied to cylindrical shells are also evaluated later.

1.4.2 Photostrictive SQ actuator The photostrictive SQ actuator system can be used to induce multi-DOF non-uniform control actions, and its induced control forces and moments are defined in this section. As shown in Fig.1.14, the photostrictive SQ actuator system is made of four single-piece

Fig. 1.14 A new photostrictive SQ actuator system.

1 Non-uniform Actuations of Plates and Shells

25

PLZT slabs whose polarities respectively pointing outwards. Two inner adjacent edges of each region of the system are fixed to a stable cross fixture and the other two outer adjacent edges are free. With these boundary conditions, each region can induce nonuniform deformation under the irradiation of high-energy lights. This SQ actuator system can induce four individual non-uniform control actions independently or coordinately when attached to plate or shell structures. When the SQ actuator is irradiated with a high-energy light, it can introduce in-plane membrane control forces and these forces multiplied by their respective moment arms resulting in control moments to the shell structure. Now assume a curved SQ actuator is applied to a simply supported cylindrical shell defined by (x, ψ , α3 ), where x, ψ and α3 denote the longitudinal, circumferential, and transverse directions respectively. Figure 1.15 shows the cylindrical shell attached with an SQ actuator system and each actuator region induces a non-uniform control membrane force.

Fig. 1.15 A cylindrical shell attached with an SQ actuator system.

The control forces and moments generated by the SQ actuator system are defined by their respective regions and coordinates. As shown in Fig.1.15, regions 1 and 3 can induce control actions in the ψ direction; regions 2 and 4 can induce control actions in the x direction. Thus, the total control forces and control moments induced by the SQ actuator respectively in the x and ψ directions are  −SYa f2 (ψ ) a   Ya h [us (x − x1∗ ) − us (x − x2∗ )][us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]  La /2 a Nxx =    SYa f4 (ψ ) Ya ha [us (x − x2∗ ) − us (x − x3∗ )][us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] La /2

(1.52)

 −SYa f2 (ψ ) a (h + ha )   Ya h [us (x − x1∗ ) − us (x − x2∗ )][us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]  La /2 2 a Mxx =  SYa f4 (ψ ) a (h + ha )   Ya h [us (x − x2∗ ) − us (x − x3∗ )][us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] La /2 2 (1.53)

26

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

 SYa f1 (x) a   Y h [us (x − x2∗ ) − us (x − x3∗ )][us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]  a /2 a R ψ a Nψψ = (1.54)    −SYa f3 (x) Ya ha [us (x − x1∗ ) − us (x − x2∗ )][us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] Rψ a /2  SYa f1 (x) a (h + ha )   Ya h [us (x − x2∗ ) − us (x − x3∗ )][us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]  Rψ a /2 2 a Mψψ = a    −SYa f3 (x) Ya ha (h + h ) [us (x − x1∗ ) − us (x − x2∗ )][us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] Rψ a /2 2 (1.55) where R is the radius of the cylindrical shell; La , ψ a and ha are the length, circumferential angle and thickness of the actuator respectively; us (·) is a step function and the two sets of step functions define the location of the actuator induced forces. As shown in Fig.1.15, the deformation profile of regions 1 and 3 is the function of x, i.e., f1 (x) and f3 (x) and the other two regions’ deformation profile is the function of ψ , i.e., f2 (ψ ) and f4 (ψ ). Note that the functions are changed when the origin of coordinate system is changed, i.e., they are functions of actuator locations.

1.4.3 Modal control In this section, control actions induced by the SQ actuator system to cylindrical shells are evaluated based on the modal expansion technique. It is assumed that the added mass and stiffness of the actuator are neglected. The closed-loop system equations of the cylindrical shell laminated with the SQ actuator are [21] −

−

a ∂ Nxx 1 ∂ Nxψ ∂ Nxx + cu˙x + ρ hu¨x = − ∂x R ∂ψ ∂x

(1.56)

a a ∂ Nxψ 1 ∂ Nψψ 1 ∂ Mxψ 1 ∂ Mψψ 1 ∂ Nψψ 1 ∂ Mψψ − − − 2 + cu˙ψ + ρ hu¨ψ = + 2 ∂x R ∂ψ R ∂x R ∂ψ R ∂ψ R ∂ψ (1.57)

∂ 2 Mxx 2 ∂ 2 Mxψ 1 ∂ 2 Mψψ Nψψ − − 2 + + cu˙3 + ρ hu¨3 2 ∂x R ∂ x∂ ψ R ∂ ψ2 R a a a Nψψ ∂ 2 Mxx 1 ∂ 2 Mψψ = − + ∂ x2 R R2 ∂ ψ 2 −

(1.58)

where Ni j and Mi j are the “elastic” forces and moments of the shell, the superscript “a” indicates the photostrictive induced control forces and moments, R is the shell radius, ρ is the shell density, h is the shell thickness, c is an equivalent viscous damping factor, and u¨ i is the acceleration. Based on the modal expansion technique, the dynamic response of the shell system can be represented by a sum of the response of all participating modes,

1 Non-uniform Actuations of Plates and Shells

i.e.,

∞

ui (x, ψ ,t) =

27

∞

∑ ∑ ηimn (t)Uimn (x, ψ )

(1.59)

m=1 n=1

where i = x, ψ , α3 ; Uimn is the mnth mode shape function; and ηimn is the modal participation factor. Assume the transverse vibration dominates in shell oscillations and the transverse modal function of a simply supported shell panel is U3mn = sin(mπx/L) sin(nπψ /ψ ∗ )

(1.60)

where L and ψ ∗ are respectively the length and the curvature angle of the shell panel. Substituting Eq.(1.59) into Eq.(1.58), integrating over the whole shell surface, and using the modal orthogonality, one can define the mnth transverse modal equation of the cylindrical shell: 2 c η¨ mn + 2ξmn ωmn η˙ mn + ωmn ηmn = Fmn (1.61) where ξmn = c/(2ρ hωmn ) is the mnth modal damping ratio, ωmn is the mnth natural frec is quency, and the modal control force Fmn à ! Z L Z ψ∗ a a a Nψψ 4 ∂ 2 Mxx 1 ∂ 2 Mψψ c Fmn = − + 2 Umn (x, ψ )Rdxdψ (1.62) ρ hLRψ ∗ 0 0 ∂ x2 R R ∂ ψ2 a , N a and M a are defined previously. They are determined by material Recall that Mxx ψψ ψψ properties, actuator and shell dimensions, modes and actuator locations. By comparing these control forces, one can study the modal control effectiveness of the new photostrictive SQ actuator system. Actuation characteristics and effectiveness of the photostrictive SQ actuator system to shells are presented next.

1.4.4 Case studies: photostrictive actuation of shells Control characteristics and effectiveness of the SQ actuator system on cylindrical shell panels are evaluated respectively when the actuator system is center-located or cornerlocated on the shell structure. Closed-loop modal control behaviors of paired SQ actuators respectively placed on the top and bottom shell surfaces are also investigated later. 1.4.4.1 Center-located photostrictive SQ actuator In this case, the SQ actuator is located at the cylindrical shell center. Figure 1.16 illustrates the top view of the shell/actuator system. Both the actuator and the plexiglas cylindrical shell are “square” in prospective, i.e., L = R × ψ ∗ and La = R × ψ a . Dimensions π of the cylindrical shell are 0.4 m × m × 0.0016 m(L × ψ ∗ × h), the actuator thickness is 3 0.0001 m. Material properties of the plate and the actuator are listed in Table 1.A3 and Table 1.A4 of Appendix.

28

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

Fig. 1.16 Center-located SQ actuator system.

In order to simplify the calculation, the origin is assumed at the center of the actuator. Thus, the transverse mode shape function of a simply supported cylindrical shell panel in this coordinate system is · µ ¶ ¸ · µ ¶ ¸ L ψ∗ ∗ U3mn = sin mπ x + /L sin nπ ψ + /ψ (1.63) 2 2 According to the variation method [22], with b1 = Rψ a /2, a1 = La /2, the distribution profile of induced non-uniform forces and moments of each actuator region can be defined as f1 (x) = B1 xb1 + B2 x2 b1 + B3 x2 b21 + B4 xb21 + B5 x3 b21 +B6 x2 b31 + B7 x3 b31 + B8 x4 b31 + B9 x3 b41 + B10 x4 b41 f2 (ψ ) = f3 (x) = f4 (ψ ) =

2

2

−(B1 Rψ a1 + B2 R ψ a1 + B3 R ψ 2 a21 + B4 Rψ a21 + B5 R3 ψ 3 a21 +B6 R2 ψ 2 a31 + B7 R3 ψ 3 a31 + B8 R4 ψ 4 a31 + B9 R3 ψ 3 a41 + B10 R4 ψ 4 a41 ) B1 xb1 − B2 x2 b1 − B3 x2 b21 + B4 xb21 + B5 x3 b21 −B6 x2 b31 + B7 x3 b31 − B8 x4 b31 + B9 x3 b41 − B10 x4 b41 −B1 Rψ a1 + B2 R2 ψ 2 a1 + B3 R2 ψ 2 a21 − B4 Rψ a21 − B5 R3 ψ 3 a21 +B6 R2 ψ 2 a31 − B7 R3 ψ 3 a31 + B8 R4 ψ 4 a31 − B9 R3 ψ 3 a41 + B10 R4 ψ 4 a41

(1.64)

2

(1.65) (1.66) (1.67)

With these profiles, one can define exact actuation forces and moments in Eqs.(1.52)(1.55). Substituting Eqs.(1.52)-(1.55) respectively into each term of Eq.(1.62) yields the three contributing actuation components: 4 ρ hLRψ ∗

Z + L Z + ψ∗ µ 2 a ¶ 2 2 ∂ Mxx − L2

∗

− ψ2

∂ x2

U3mn (x, ψ )Rdxdψ

1 Non-uniform Actuations of Plates and Shells

=

29

4 SYa (h + ha ) Ya ha ∗ a ρ hLRψ L /2 2 Z + L Z + ψ∗ 2 2 2 ∂

{ f4 (ψ ) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] ∂ x2 − f2 (ψ ) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]}U3mn (x, ψ )Rdxdψ = SM˜ xmn (1.68) Ã ! ∗ Z +L Z +ψ a 2 2 4 1 ∂ 2 Mψψ U3mn (x, ψ )Rdxdψ ρ hLRψ ∗ − L2 − ψ2∗ R2 ∂ ψ2 ×

=

− L2

∗

− ψ2

a 4 SYa a (h + h ) Y h a ρ hLRψ ∗ Rψ a /2 2

×

Z + L Z + ψ∗ 2 2 1 ∂2 − L2

∗

− ψ2

R2 ∂ ψ 2

{ f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]

− f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )Rdxdψ = SM˜ ymn (1.69) 4 ρ hLRψ ∗ =− ×

Z + L Z + ψ∗ µ 2 2 − L2

∗

− ψ2

¶ 1 a − Nψψ U3mn (x, ψ )Rdxdψ R

4 SYa Ya ha ρ hLRψ ∗ Rψ a /2 Z + L Z + ψ∗ 2 2 − L2

∗

− ψ2

{ f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]

− f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )dxdψ = SN˜ ψ mn (1.70) Thus, the mnth modal control force can be rewritten as c Fmn = S(M˜ xmn + N˜ ψ mn + M˜ ymn )

(1.71)

c is defined as and the total control action F˜mn c F˜mn = M˜ xmn + N˜ ψ mn + M˜ ymn

(1.72)

c . Substituting Eqs.(1.64)-(1.67) into Eqs.(1.68)-(1.70) gives the total control action F˜mn Thus, modal control actions of the new SQ actuator with different length ratios (∆ = 1/4, 1/2, 2/3, 1) or actuator coverage are calculated and summarized in Table 1.5. This table suggests that the new SQ actuator has no control effects on shell modes (1,2), (2,1), (2,3) and (3,2). Because the location of the actuator is center-located and the mode shape function of the simply supported cylindrical shell, the positive and negative control actions cancel out each other on these modes. To further evaluate the SQ actuator,

30

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

plot actuators’ control actions with respect to the actuator/shell length ratio (∆ = La /L) of the first nine modes (m = 1, 2, 3; n = 1, 2, 3) in Fig.1.17 where “Max.” indicates the maximal actuation amplitude at a specific length ratio.

Fig. 1.17 Control action vs. length ratio of the center-located SQ actuator system. (a) (1,1), (1,2), (1,3) modes; (b) (2,1), (2,2), (2,3) modes; (c) (3,1), (3,2), (3,3) modes.

These figures indicate that the control actions on all controllable modes fluctuate with respect to the length ratio, i.e., the actuator coverage. These figures suggest that larger

1 Non-uniform Actuations of Plates and Shells

31

size doesn’t always guarantee better control effects, i.e., the 100% coverage is not the most effective size to all nine shell modes. Thus, the actuator size is very important to the overall modal control effectiveness. Control effectiveness of each mode and its optimal size need to be carefully evaluated. 1.4.4.2 Corner-located SQ actuator system In this case, the SQ actuator is located at one quadrant of the shell panel. Dimensions and material properties of the shell and those of the actuator are the same as before. In order to simplify the calculation, the origin of the coordinate system is assumed at the center of the actuator (Fig.1.18).

Fig. 1.18 Corner-located photostrictive SQ actuator system.

Thus, the distribution profile of induced non-uniform forces and moments of each region is described in Eqs.(1.64)-(1.67). Modal control force in this coordinate system is rewritten as à ! Z + 3L Z + 3ψ ∗ a a a Nψψ 4 4 4 ∂ 2 Mxx 1 ∂ 2 Mψψ c Fmn = U3mn (x, ψ )Rdxdψ − + 2 ρ hLRψ ∗ − L4 − ψ4∗ ∂ x2 R R ∂ ψ2 (1.73) where the mode shape function is · µ ¶ ¸ · µ ¶ ¸ L ψ∗ ∗ U3mn = sin mπ x + /L sin nπ ψ + /ψ (1.74) 4 4

32

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

With the corner location, the three contributing actuation components can be rewritten as 4 ρ hLRψ ∗ =

Z + 3L Z + 3ψ ∗ µ 2 a ¶ 4 4 ∂ Mxx − L4

∗

∂ x2

− ψ4

U3mn (x, ψ )Rdxdψ

4 SYa (h + ha ) Ya ha ∗ a ρ hLRψ L /2 2 Z + 3L Z + 3ψ ∗ 2 4 4 ∂

{ f4 (ψ ) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] ∂ x2 − f2 (ψ ) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]}U3mn (x, ψ )Rdxdψ = SM˜ xmn (1.75) Ã ! ∗ Z + 3L Z + 3ψ a 4 4 4 1 ∂ 2 Mψψ U3mn (x, ψ )Rdxdψ ρ hLRψ ∗ − L4 − ψ4∗ R2 ∂ ψ2 ×

=

− L4

∗

− ψ4

a 4 SYa a (h + h ) Y h a ρ hLRψ ∗ Rψ a /2 2

×

Z + 3L Z + 3ψ ∗ 4 4 1 ∂2 − L4

∗

− ψ4

R2 ∂ ψ 2

{ f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]

− f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )Rdxdψ = SM˜ ymn (1.76) 4 ρ hLRψ ∗ =− ×

Z + 3L Z + 3ψ ∗ µ 4 4 − L4

∗

− ψ4

¶ 1 a − Nψψ U3mn (x, ψ )Rdxdψ R

4 SYa Ya ha ρ hLRψ ∗ Rψ a /2 Z + 3L Z + 3ψ ∗ 4 4 − L4

∗

− ψ4

{ f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] · [us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]

− f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] · [us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )dxdψ = SN˜ ψ mn (1.77) Accordingly, magnitudes of control actions on all controllable modes are calculated and plotted in Fig.1.19, which also fluctuate with respect to the length ratio or actuator size. Unlike the center-located actuator, the maximal coverage size of corner-located actuator is only 1/4 of the total shell surface area, i.e., ∆ = 1/2. These figures indicate that when the length ratio is ∆ = 1/2, the actuator introduces the maximal control action on modes (1,1), (1,2) and (3,3). However, it does not provide the best control action on modes (1,3), (2,1), (2,2), (2,3), (3,1), (3,2). Again, “Max.” indicates the maximal actuation amplitude at a specific length ratio. Observe that there is an optimal size for each shell modal control. Thus, one needs to fully understand the shell/actuator coupling and its modal characteristics in order to select the optimal size for various shell modes.

1 Non-uniform Actuations of Plates and Shells

33

Fig. 1.19 Control action vs. length ratio of the corner-located SQ actuator system. (a) (1,1), (1,2), (1,3) modes; (b) (2,1), (2,2), (2,3) modes; (c) (3,1), (3,2), (3,3) modes.

Assume the length ratio is set at its maximal value, i.e., ∆ = 1/2, comparisons of modal control actions between the center-located actuator and the corner-located actuator are summarized in Table 1.6. This table indicates that the corner-located actuator leads to control effects on all of the first nine modes. However, it provides less control actions on modes (1,1), (1,3), (3,1) and (3,3), as compared with the center-located actuator when the length ratio is 1/2.

34

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

1.4.5 Closed-loop actuation with paired SQ actuator systems As discussed previously, the corner-located actuator can control more modes than the center-located actuator. However, it is less effective than the center-located actuator in some cylindrical modes, such as the (1,1), (1,3), (3,1) and (3,3) modes. In this section, one method to improve the modal control effectiveness of the center-located SQ actuator is proposed. When the SQ actuator is located at the center of a “square” cylindrical shell panel, some modes, such as (1,2), (2,1), (2,3), (3,2) can not be controlled, because regions 1 and 3 of the SQ actuator induce identical positive and negative control actions respectively, and regions 2 and 4 also respectively induce identical positive and negative control actions. Thus, they cancel out each other. In order to alter the sign of each regional induced control forces, one pair of photostrictive SQ actuators are respectively placed on the top and bottom surfaces of the cylindrical shell, as shown in Fig.1.20.

Fig. 1.20 A cylindrical shell attached with one pair of SQ actuator systems.

It is assumed that the light induces positive actions to the shell when it is applied on the upper actuator, and the light induces negative actions to the shell when it is applied on the bottom actuator. By closed-loop actuations, the light intensity applied to each region is identical to |Imax | and only the light irradiations to various regions of one pair of top and bottom placed SQ actuator systems is regulated with respect to the shell modes. The sign of control action of each region can be changed due to the regulation, thus the total control action can be rewritten as c0 c1 c2 c3 c4 F˜mn = (−1)m (−1)n F˜mn + (−1)(−1)n F˜mn + F˜mn + (−1)(−1)m F˜mn

(1.78)

ci is the modal control action of region i. Thus, the three contributing actuation where F˜mn components can be rewritten as Z + L Z + ψ∗ µ 2 a ¶ 2 2 4 ∂ Mxx U3mn (x, ψ )Rdxdψ ∗ ∗ ψ L ρ hLRψ − 2 − 2 ∂ x2

=

a 4 SYa a (h + h ) Y h a ρ hLRψ ∗ La /2 2

1 Non-uniform Actuations of Plates and Shells

35

Z + L Z + ψ∗ 2 2 2 ∂

{(−1)(−1)m f4 (ψ ) · [us (x − x2∗ ) − us (x − x3∗ )] ∗ 2 − ψ2 ∂ x ·[us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )] + (−1)n f2 (ψ ) · [us (x − x1∗ ) − us (x − x2∗ )] ·[us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )]}U3mn (x, ψ )Rdxdψ ×

− L2

= SM˜ xmn 4 ρ hLRψ ∗ =

(1.79) Z + L Z + ψ∗ 2 2 1 − L2

∗ − ψ2

Ã

R2

a ∂ 2 Mψψ ∂ ψ2

! U3mn (x, ψ )Rdxdψ

a 4 SYa a (h + h ) Y h a ρ hLRψ ∗ Rψ a /2 2

Z + L Z + ψ∗ 2 2 1 ∂2

{(−1)m (−1)n f1 (x) · [us (x − x2∗ ) − us (x − x3∗ )] ∗ 2 2 − ψ2 R ∂ ψ ·[us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )] − f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] ·[us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )Rdxdψ ×

− L2

= SM˜ ymn 4 ρ hLRψ ∗ =−

(1.80) Z + L Z + ψ∗ µ 2 2 − L2

∗

− ψ2

−

¶ 1 a Nψψ U3mn (x, ψ )Rdxdψ R

4 SYa Ya ha ∗ ρ hLRψ Rψ a /2 Z + L Z + ψ∗ 2 2

m n ∗ ∗ ∗ {(−1) (−1) f 1 (x) · [us (x − x2 ) − us (x − x3 )] − ψ2 ·[us (ψ − ψ2∗ ) − us (ψ − ψ3∗ )] − f3 (x) · [us (x − x1∗ ) − us (x − x2∗ )] ·[us (ψ − ψ1∗ ) − us (ψ − ψ2∗ )]}U3mn (x, ψ )dxdψ

×

− L2

= SN˜ ψ mn

(1.81)

Closed-loop control actions of the new SQ actuator with different length ratios (∆ = 1/4, 1/2, 2/3, 1) are calculated and summarized in Table 1.7. Comparing Table 1.7 with Table 1.5 suggests that for both m and n are odd modes, such as modes (1,1), (1,3), (3,1), (3,3), this SQ actuator would induce identical control actions as the single SQ actuator. However, for either m or n is even mode, such as (1,2), (2,1), (2,2), (2,3), (3,2), regulating the lights on different regions of the SQ system on the top and bottom surfaces independently can significantly improve the control effectiveness of the center-located SQ actuator. In order to further evaluate the actuation magnitudes with respect to the actuator coverage, the control actions and size relationship of the first nine modes are plotted in Fig.1.21. As discussed previously, note that larger actuator doesn’t always guarantee better control effects.

36

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

Fig. 1.21 Control action vs. length ratio of the paired center-located SQ actuator system. (a) (1,1), (1,2), (1,3) modes; (b) (2,1), (2,2), (2,3) modes; (c) (3,1), (3,2), (3,3) modes.

1.4.6 Summary of non-uniform photostrictive actuations of shells Under the irradiation of high-energy lights, photostrictive actuators can induce noncontact control actions. Thus, they can be used for non-contact actuation and active vi-

1 Non-uniform Actuations of Plates and Shells

37

bration control of flexible structures. A number of photostrictive actuator configurations with uniform control actions have been designed and evaluated over the years. A new photostrictive SQ actuator system consisting of four single PLZT pieces was designed to induce multi-DOF non-uniform control actions. Independent modal control actions of a plexiglas cylindrical shell panel attached with the center-located and the corner-located SQ actuator system was studied respectively. In order to improve the control effectiveness of the center-located actuator system, one pair of SQ actuator systems were respectively placed on the top and bottom surfaces of the shell panel. Regulating the lights in different regions of the SQ system on the top and bottom surfaces independently yielded both positive and negative control actions on the shell panel and thus closed-loop actuations of the shell were achieved. By analyses, comparisons and parametric evaluations, this study suggests that (1) When the new SQ actuator system is symmetrically placed at the shell center (center-located), it provides control effects only to modes (1,1), (1,3), (2,2), (3,1) and (3,3). It is ineffective to modes (1,2), (2,1), (2,3) and (3,2), due to cancellation of control actions. (2) The maximal modal control actions fluctuate with respect to the length ratio, i.e., the actuator coverage or size. Larger size doesn’t always guarantee better modal control effects. Thus, not only the actuator location, but also the actuator size is very important to the individual modal control effectiveness of various shell modes. (3) When the SQ actuator is located at one quadrants of the cylindrical shell (cornerlocated), it generates control effectiveness on all of the first nine modes. However, actuation magnitudes in some modes, such as (1,1), are less than those of the center-located actuator system. (4) Regulating the light irradiations to various regions of one pair of SQ actuator systems with respect to the shell modes proves to be an effective closed-loop control technique. The center-located SQ actuator system now can induce control actions for all of the first nine modes, including both odd and even modes. Thus, it is evident that the new photostrictive SQ actuator system is effective to generate non-uniform control forces and moments. With regulating light illuminations to various regions of the actuator system, both odd and even shell modal oscillations can be controlled. However, there is an optimal size for each shell modal control. Thus, one needs to fully understand the shell/actuator coupling and its modal characteristics in order to select the optimal size for different shell modes.

1.5 Summary Actively induced control forces and moments of distributed actuators, resulting from imposed control stimuli e.g., voltages, high-energy lights, etc., can be used to control static shapes and/or undesirable oscillations of shells and plates. Conventional distributed actuators usually induce uniform control forces and moments. In this chapter, boundary

38

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

conditions of actuator elements are specifically manipulated to realize a continuous nonuniform distribution of actuator induced control forces and control moments. An actuator element, with the two inner adjacent edges fixed and the other two edges freed was the fundamental element used in a new SQ actuator system. Distribution functions of induced forces and moments were calculated based on the variation method and validated by ANSYS. A new SQ actuator system composed of four regions of fix-fix/free-free actuator elements was configured to induce distributed multi-DOF non-uniform control forces and moments applied to control of plate or shell structures. Two SQ actuator systems made of piezoelectric and photostrictive materials were respectively applied to distributed modal actuation and control of plates and shells. The piezoelectric SQ actuator system was made of four flexible mono-axial piezoelectric elements. The coupling equations of a simply supported plate laminated with this piezoelectric SQ actuator system were defined and distributed control actions were also derived in the modal domain. The modal control actions change with respect to the plate modes and the actuator coverage and locations. Thus, modal control effects of centerlocated and corner-located SQ actuators were evaluated respectively. Furthermore, in order to reduce control deficiencies on anti-symmetrical modes of symmetrical plates when the new SQ actuator is centrally and symmetrically placed, closed-loop feedbacks using four collocated sensor segments to independently regulate the sign of control voltages to each actuator region were investigated. The closed-loop configuration of the centerlocated SQ actuator system can control all of the first nine plate modes, while the openloop configuration only control odd modes, i.e., (1,1), (1,3), (3,1), (3,3), of the first nine modes. Photostrictive actuation, which can directly turn light energy into mechanical energy, represents a new promising non-contact photoactuation technique for active vibration control of flexible structures. By considering the photovoltaic effect, the pyroelectric effect and the thermal strain effect, the total magnitude of photostrictive actuator induced control strains was defined first. A new photostrictive SQ actuator system was also designed and evaluated. Photonic control of a simply supported cylindrical shell using the centerlocated and corner-located photostrictive SQ actuator systems were investigated respectively. Control effectiveness with different actuator coverage (i.e., actuator sizes) were evaluated. In order to improve modal control effectiveness of the center-located photostrictive SQ actuator system, one pair of photostrictive SQ actuator systems were respectively placed on the top and bottom surfaces of the shell. By regulating the lights on different regions of the SQ system on the top and bottom surfaces independently, the sign of each region induced control action can be manipulated and control can be effectively enhanced. This study suggests that the proposed SQ actuator system does provide non-uniform control forces and moments. Actuator location and coverage of this SQ actuator system strongly influence the modal controllability of plates and shells. In general, control moments dominate the plate control, while membrane control forces dominate shell control, especially deep shells. Laboratory experiments to evaluate the design and control effectiveness are underway and results will be reported later.

1 Non-uniform Actuations of Plates and Shells

39

Acknowledgments This research was supported, in part, by two grants from the National Natural Science Foundation of China (NSFC Nos. 51105095, 51175103) and the Natural Scientific Research Innovation Foundation at Harbin Institute of Technology (HIT NSRIF 2011111). Professor Tzou would like to thank the “111 Project” (B07018) program sponsored by the Chinese Ministry of Education at Harbin Institute of Technology, the 2ndphase “985” project at Zhejiang University and also NSFC No. 11172262.

References [1] Tzou H S, Gadre M. Piezoelectric electromechanical dynamics of axially excited polyvinylidene — fluoride polymer. In: Soedel W, Hamilton J F, ed. Developments in Mechanics, 1987, 14B, 716-721. [2] Sullivan J M, Hubbard J E, Burke S E. Distributed sensor/actuator design for plates: spatial shape and shading as design parameters. J. Sound Vib., 1997, 203(3), 473-493. [3] Baz A, Poh S. Performance of an active control system with piezoelectric actuators. J. Sound Vib., 1998, 126(2), 327-343. [4] Costa L, Figueiredo I, Leal R, et al. Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate. Comput. Struct., 2007, 85, 385-403. [5] Lin J C, Nien M H. Adaptive modeling and shape control of laminated plates using piezoelectric actuators. J. Mater. Process. Technol., 2007, 189, 231-236. [6] Kang Y K, Park H C, Han K S. Optimum placement of piezoelectric sensor/actuator for vibration control of laminated beams. AIAA J., 1996, 34(9), 1921-1926. [7] Choe K, Baruh H. Actuator placement in structural control. J. Guid. Control Dynam., 1991, 15(1), 40-46. [8] Sadri A M, Wright J R, Wyne R J. Modeling and optimal placement of piezoelectric actuators in isotropic plates using genetic algorithms. Smart Mater. Struct., 1999, 8, 490-498. [9] Lee A C, Chen S T. Collocated sensor/actuator positioning and feedback design in the control of flexible structure system. ASME J. Vib. Acoust., 1994, 116, 146-154. [10] Lim K B. Method for optimal actuator and sensor placement for large flexible structures. J. Guid. Control Dynam., 1992, 15(1), 49-57. [11] Yue H H, Deng Z Q, Tzou H S. Optimal actuator locations and precision micro-control actions on free paraboloidal membrane shells. Commun. Nonlinear Sci. Numer. Simulat., 2008, 13(10), 2298-2307. [12] Tzou H S, Fu H Q. A study of segmentation of distributed piezoelectric sensors and actuators. Part I: Theoretical analysis. J. Sound Vib., 1994, 172(2), 247-259. [13] Tzou H S, Fu H Q. A study of segmentation of distributed piezoelectric sensors and actuators. Part II: Parametric study and active vibration controls. J. Sound Vib., 1994, 172(2), 261-275. [14] Tzou H S, Bao Y. Parametric study of segmented transducers laminated on cylindrical shells. Part 1: Sensor patches. J. Sound Vib., 1996, 197(2), 207-224. [15] Tzou H S, Bao Y. Parametric study of segmented transducers laminated on cylindrical shells. Part 2: Actuator patches. J. Sound Vib., 1996, 197(2), 225-249. [16] Tzou H S, Wang D W, Chai W K. Dynamics and distributed control of conical shells laminated with full and diagonal actuators. J. Sound Vib., 2002, 256(1), 65-79.

40

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

[17] Lee C K, Moon F C. Modal sensors/actuator. ASME J. Appl. Mech., 1990, 57, 434-441. [18] Lee C K, Moon F C. Laminated piezopolymer plates for torsion and bending sensors and actuators. J. Acoust. Soc. Am., 1989, 85(6), 2432-2439. [19] Tzou H S. Spatially distributed orthogonal piezoelectric shell actuators: theory and applications. J. Sound Vib., 1994, 177(3), 363-378. [20] Ryou J K, Park K Y, Kim S J. Electrode pattern design of piezoelectric sensors and actuators using genetic algorithms. AIAA J., 1998, 36(2), 227-233. [21] Tzou H S. Piezoelectric Shells: Distributed Sensing and Control of Continua. Kluwer Academic Publishers, Boston, Dordrecht, 1993, 18-55. [22] Xu Z L. Elasticity. Higher Education Press, Beijing, 2006. [23] Tzou H S, Grade M. Active vibration isolation by piezoelectric polymer with variable feedback gain. AIAA J., 1998, 26(8), 1014-1017. [24] Soedel. Vibrations of Shells and Plates. Marcel Dekker, Inc., 1981, 8-54. [25] Lee C K. Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part I: Governing equations and reciprocal relationships. J. Acoust. Soc. Am., 1990, 87(3), 1144-1158. [26] Brody P S. Optomechanical bimorph actuator. Ferroelectrics, 1983, 50, 27-32. [27] Liu B, Tzou H S. Photodeformation and light-temperature-electric coupling of optical actuators. Part-1: Parameter calibration; Part-2: Vibration control. Proceedings of the ASME Aerospace Division, 1996, AD-52, 663-678. ASME 1996 International Congress, Atlanta, GA, November 17-22. [28] Liu B, Tzou H S. Distributed photostrictive actuation and opto-piezothermoelasticity applied to vibration control of plates. ASME J. Vib. Acoust., 1998, 120, 937-943. [29] Fukuda T, Hattori S, Arai F, et al. Characteristics of optical actuator-servomechanisms using bimorph optical piezoelectric actuator. Proceedings of 1993 IEEE Robotics and Automation Conference, 1993, 618-623. [30] Fukuda T, Hattori S, Arai F, et al. Performance improvement of optical actuator by double side irradiation. IEEE T. Ind. Electron., 1995, 42(5), 455-461. [31] Shih H R, Tzou H S. Opto-piezothermoelastic constitutive modeling of a new 2-d photostrictive composite plate actuator. Proceeding of ASME IMECE 2000, 61, 1-8. [32] Tzou H S, Chou C S. Nonlinear optoelectromechanics and photodeformation of optical actuators. Smart Mater. Struct., 1996, 5, 230-235. [33] Yue H H, Sun G L, Deng Z Q, et al. Distributed shell control with a new multi-DOF photostrictive actuator design. J. Sound Vib., 2010, 329, 3647-3659. [34] Jiang J, Yue H H, Deng Z Q, et al. Non-uniform control moments induced by a new skewquad actuator design applied to plate controls. IMECE2009 Proceeding, IMECE2009-11102. Lake Buena Vista, Florida, November 13-19, 2009. [35] Chu S Y, Uchino K. Photoacoustic devices using PLZT ceramics. Proceedings of 1994 IEEE Intel Symposium of Applied Ferroelectrics. [36] Sun D C, Tong L Y. Theoretical investigation on wireless vibration control of thin beams using photostrictive actuators. J. Sound Vib., 2008, 312, 182-194. [37] Yue H H, Deng Z Q, Tzou H S. Non-contact precision actuation and optimal actuator placement of hybrid photostrictive beam structure systems. IDETC2007 Proceeding, DETC200734995. Las Vegas, Vevada, 2007. [38] Shih H R, Tzou H S, Saypuri M. Structural vibration control using spatially configured optoelectromechanical actuator. J. Sound Vib., 2005, 284, 1-2, 361-378. [39] Shih H R, Smith R, Tzou H S. Photonic control of cylindrical shells with electro-optic photostrictive actuators. AIAA J., 2004, 42(2), 341-347.

1 Non-uniform Actuations of Plates and Shells

41

[40] Tzou H S, Liu B J, Cseledy D. Optopiezothermoelastic actions and micro-control sensitivity analysis of cylindrical opto-mechanical shell actuators. J. Theor. App. Mech. 2002, 40(3), 775-796. [41] Shih H R, Tzou H S. Photostrictive actuators for photonic control of shallow spherical shells. Smart Mater. Struct., 2007, 16, 1712-1717. [42] Shih H R, Tzou H S. Wireless control of parabolic shells using photostrictive actuators. Proceedings of IMECE2006, IMECE2006-13064. Chicago, Illinois.

Appendix Table 1.A1 Material properties of plexiglas plate. Property

Value

Mass density ρ

1.19×103 kg/m3

Young’s modulus Y

3.10×109 N/m2

Thickness h

0.0016 m

Poisson’s ratio µ

0.35

Table 1.A2 Material properties of PVDF. Property

Value

Mass density ρ

1.8×103 kg/m3

Young’s modulus Ya

2×109 N/m2

Piezoelectric constant d31

1.0×10−11 m/V

Thickness ha

0.00004 m

Poisson’s ratio µ

0.2

Table 1.A3 Material properties of plexiglas cylindrical shell. Property

Value

Mass density ρ

1.19×103 kg/m3

Young’s modulus Y

3.10×109 N/m2

Thickness h

0.0016 m

Poisson’s ratio µ

0.35

Table 1.A4 Material properties of PLZT. Variable

Value

Notes

Es

2.43×105 V/m

Saturated electric field

Ya

6.3×1010 N/m2

Young’s modulus

α

0.002772 m2 /(W·s)

Optical actuator constant

β

0.01 V/s

Voltage leakage constant

P

0.23×103 cm2 /s

Power of absorbed heat

42

Jing Jiang, HongHao Yue, ZongQuan Deng, and HornSen Tzou

Continued Variable

Value

Notes

d33

1.79×10−10 m/V

Piezoelectric strain constant

H

16 W/◦ C

Heat capacity

γ

0.915 W/(◦ C·s)

Heat transfer rate

λ

6.8086×104 N/(m2 ·◦ C)

Stress-temperature constant

Pn

0.25×10−4 C/(m2 ·◦ C)

Pyroelectric constant

ε

1.65×10−8

F/m

Electric permittivity

Chapter 2 Structural Theories of Multiferroic Plates and Shells

ChunLi Zhang and WeiQiu Chen Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

Abstract Analogical to that for elastic plates as proposed by R. D. Mindlin, the linear structural theories of multiferroic plates and shells are derived using the series expansion method. In an orthogonal curvilinear coordinate system, the displacement field is expanded in an infinite series of powers of the thickness-coordinate of the shell. The local coordinate of the thickness is introduced in the expansions of the electric and magnetic potentials, which are different from the expansion of displacement. Two-dimensional governing equations in general form and those for cylindrical and spherical shells in particular forms are presented. By setting the Lam´e coefficients corresponding to the coordinates at the middle surface of the shell to unit and the two principal radii of the middle surface of the shell to infinity, the shell theory can directly be degenerated into the plate theory. In numerical examples, the magnetoelectric effects of laminated multiferroic structures are evaluated using the established structural theories. Keywords structural theory, multiferroic materials, plates, shells, series expansion

44

ChunLi Zhang and WeiQiu Chen

2.1 Introduction Multiferroic materials, sometimes also known as magnetoelectric materials, exhibit simultaneously several ferroic orders such as ferroelectricity and ferromagnetism [1-3]. They have recently stimulated a sharply increasing number of research activities for their scientific interests and significant technological promise in novel multi-functional devices [1-7]. Due to multi-field coupling and material anisotropy, the governing equations of multiferroic materials are rather complicated. For certain relatively simple and special problems, solutions from three-dimensional (3D) equations can be obtained [8-10]. To analyze device problems in general, one-dimensional (1D) and two-dimensional (2D) structural theories of beams, plates and shells are needed. Plates and shells are common structures for multi-functional devices. Various 2D theories for thin piezoelectric plates and shells [11-13] have been developed and proved to be very effective in device modeling. Recently, the authors and their coworkers have extended the structural theories to accommodate multiferroic structures (such as rod-or beam-shaped and laminated structures) [14, 15] and successfully applied them to the analysis of magnetoelectric coupling effects of multiferroic structures [16-18], as well as modeling magnetic energy harvesters [19-21]. Tensor equations in mathematical form describing the laws of mechanics (or physical events) have the most important property [22] that if a tensor equation can be established in one coordinate system, then it must hold in all coordinate systems. This statement affords a powerful method for establishing equations in mathematical physics. For example, if a certain tensor equation is true in Cartesian coordinates, then it is also true in general curvilinear coordinates in Euclidean space. It should be remembered that the tensor components of a physical quantity which is referred to a particular curvilinear coordinate system may or may not have the same physical dimensions. For physical understanding, it is desirable to employ the physical components in practical applications. With the use of Lam´e coefficients, the physical components of vectors and tensors are conveniently displayed in a certain coordinate system. In this chapter, we present the general tensor equations in general orthogonal curvilinear coordinate systems for laminated multiferroic plates and shells. Then the detailed equations in physical component form for plates, cylindrical shells and spherical shells are derived by substituting the individual tensor components by their physical counterpart. In developing structural theories, the 3D equations of multiferroic structures are converted to a series of 2D equations. This is done by expanding the displacement in a series of powers of the thickness-coordinate, as well as expanding the electric and magnetic potentials of the I-th layer in a series of powers of the local coordinate of the thickness. Substituting them into the 3D equations and then integrating these equations over the thickness of the structure, the 2D equations are thus obtained for multiferroic structures. In a theoretical sense, the result will be exact and the same as that of 3D equations when the number of terms in the series approaches infinite. However, the infinite series of equations are no easier to treat than the original 3D equations. Actually, a few terms properly retained in the series will be enough accurate in practice as already demonstrated for elastic and

2 Structural Theories of Multiferroic Plates and Shells

45

piezoelectric plates [12, 13, 23]. Following the same idea, we obtain in this chapter the first-order theory which includes the coupled extensional, flexural and thickness-shear motions of multiferroic structures and show its applications in the analysis of magnetoelectric effects of composite multiferroic structures.

2.2 Basic formulations In orthogonal curvilinear coordinates, the equations of motion and the Gaussian equations of electrostatics and magnetostatics can be written in tensor form as T ji | j = ρ u¨i

(2.1)

Di |i = 0

(2.2)

Bi |i = 0

(2.3)

where Ti j is the stress tensor, ρ is the mass density, ui is the displacement vector, Di is the electric displacement vector, and Bi is the magnetic induction. In this chapter, the summation convention over repeated indices is employed, and a superimposed dot represents time derivative. A tensor or vector followed by the symbol “|i ” denotes the covariant derivative. The detailed expansions of the covariant derivative are given in the Appendix. Note that in Cartesian coordinates they have the same form as the common derivative which is often indicated by a comma followed by an index ( ,i = ∂ /∂ xi ). The constitutive relations describing the linear material behavior are given by T ji = c jikl Skl − ek ji Ek − hk ji Hk

(2.4)

Di = eikl Skl + εik Ek + αik Hk

(2.5)

Bi = hikl Skl + αki Ek + µik Hk

(2.6)

where Si j is the strain tensor, Ei is the electric field, and Hi is the magnetic field. c jikl , εi j , ei jk , hi jk and αi j are the elastic, dielectric, piezoelectric, piezomagnetic and magnetoelectric constants, respectively, and µi j are the magnetic permeability. The gradient relations of respective fields are given by Si j = (ui | j + u j |i )/2

(2.7)

Ei = −ϕ |i

(2.8)

Hi = −ψ |i

(2.9)

where ϕ is the electric potential and ψ is the magnetic potential. On the boundary surface of a body, mechanical displacement or traction vector, electric potential or surface electric charge, and magnetic potential or surface magnetic charge may be prescribed as boundary data.

46

ChunLi Zhang and WeiQiu Chen

2.3 Laminated multiferroic shell equations in orthogonal curvilinear coordinates In Section 2.2, the 3D equations for multiferroic materials have been given in a general tensor form. On the basis of these equations, we establish here the general structural theories for laminated multiferroic shell structures in orthogonal curvilinear coordinates. Consider a differential element of an N-layered multiferroic shell (see Fig. 2.1). (γ1 , γ2 , γ3 ) is an orthogonal curvilinear coordinate system with the corresponding Lam´e coefficients (ξ1 , ξ2 , and ξ3 = 1). Namely, the coordinate line γ3 is a straight line. γ1 and γ2 are the middle surface principal coordinates, and γ3 is the thickness coordinate. The thickness of the shell, 2h, is much smaller than the principal radii of curvature R1 and R2 of the middle surface. The two major faces and the N − 1 interfaces are sequentially determined by γ3 = −h = h0 , h1 , · · · , hN−1 and hN = h. A1 and A2 are Lam´e coefficients of the middle surface corresponding to γ1 and γ2 . There exist the following relations:

ξ1 = A1 (1 + γ3 /R1 ),

ξ2 = A2 (1 + γ3 /R2 )

(2.10)

Fig. 2.1 A shell element and coordinate system.

We expand the mechanical displacement u j in a series of powers of the thickness coordinate γ3 by g

uj =

(n)

∑ γ3n u j

(γ1 , γ2 ,t)

(2.11)

n=0

For the electric and magnetic potentials of the I-th layer (hI−1 < γ3 < hI ), we introduce a local thickness coordinate, γ3I = γ3 −(hI +hI−1 )/2, whose origin is at the middle surface of the I-th layer. The expansions of the electric and magnetic potentials of the I-th layer are

ϕI = ψI =

g

∑ (γ3I )n ϕ I(n) (γ1 , γ2 ,t)

(2.12)

∑ (γ3I )n ψ I(n) (γ1 , γ2 ,t)

(2.13)

n=0 g

n=0

(n)

In Eqs.(2.11)-(2.13), g is a positive integer. u j , ϕ I(n) and ψ I(n) vanish for n > g. Succes-

2 Structural Theories of Multiferroic Plates and Shells

47

sively higher values of g lead to higher-order theories. With the expansions in the forms of Eqs.(2.11)-(2.13), the strains, electric and magnetic fields can be written as g

Si j =

(n)

∑ (γ3 )n Si j , EiI =

n=0

g

I(n)

∑ (γ3I )n Ei

, HiI =

n=0

g

I(n)

∑ (γ3I )n Hi

(2.14)

n=0

where 1 (n) 1 (n) (n) (n+1) (n+1) Si j = (ui | j + u j |i ) + (n + 1)(δ3 j ui + δ3i u j ) 2 2 I(n) Ei = −ϕ I(n) |i − (n + 1)δ3i ϕ (n+1) I(n) Hi

(2.15) (2.16)

= −ψ I(n) |i − (n + 1)δ3i ψ (n+1)

(2.17)

Here δi j is the Kronecker delta. Then we multiply Eq.(2.1) for the I-th layer by γ3n , integrate the resulting equations over the layer thickness from hI−1 to hI , sum the resulting equations over I, and make use of the interface continuity conditions of the traction vector to obtain the following n-th order governing equations: Ã ! (n)

(n−1)

Ta j |a − nT3 j

(n)

+ Fj

N

=

∞

ρI

∑

I=1

(m)

∑ Gmn u¨ j

(2.18)

m=0

where the index a = 1, 2, and  m+n+1  2h (m + n is even) Gmn = m + n + 1  0 (m + n is odd)

(2.19)

Multiplying Eqs.(2.2) and (2.3) for the I-th layer and integrating the resulting equation over the layer thickness from hI−1 to hI , we obtain the following equations of electrostatics and magnetostatics: I(n)

I(n−1)

Da |a − nD3

+ d I(n) = 0

I(n) I(n−1) Ba |a − nB3 + bI(n)

(2.20)

=0

(2.21)

In Eqs.(2.18), (2.20) and (2.21), the shell resultants of various orders are (n)

Ti j =

Z h −h

I(n)

Ti j γ3n dγ3 , Di

=

Z h¯ I −h¯ I

I(n)

DIi (γ3I )n dγ3I , Bi

=

Z h¯ I −h¯ I

BIi (γ3I )n dγ3I

(2.22)

where h¯ I = (hI − hI−1 )/2 and the surface loads are (n)

Fj

¯

¯

= [T3 j γ3n ]h−h , d I(n) = [DI3 (γ3I )n ]h−Ih¯ , bI(n) = [BI3 (γ3I )n ]h−Ih¯ I

I

(2.23)

Next, we derive shell constitutive equations for thin shells. Since the thickness of the shell is very small, we adopt the usual stress relaxation approximation of vanishing normal

48

ChunLi Zhang and WeiQiu Chen

stress T33 = 0. Through Eq.(2.4) by setting i = j = 3, the strain component S33 is given by c33kl Skl − c3333 S33 − ek33 Ek − hk33 Hk S33 = − (2.24) c3333 It should be noted that in the above S33 has been eliminated on the right-hand side. Then substituting Eq.(2.24) back into Eqs.(2.4)-(2.6), we can obtain the following constitutive relations relaxed for thin shells: T ji = c¯ jikl Skl − e¯k ji Ek − h¯ k ji Hk

(2.25)

Di = e¯ikl Skl + ε¯ik Ek + α¯ ik Hk Bi = h¯ ikl Skl + α¯ ki Ek + µ¯ ik Hk

(2.26) (2.27)

where the effective material constants are defined by ci j33 c33kl ek33 c33i j , e¯ki j = eki j − , c3333 c3333 hk33 c33i j ei33 e j33 h¯ ki j = hki j − , ε¯i j = εi j + , c3333 c3333 ei33 h j33 hi33 h j33 α¯ i j = αi j + , µ¯ i j = µi j + c3333 c3333 c¯i jkl = ci jkl −

(2.28)

Substitution of Eqs.(2.25)-(2.27) into Eq.(2.22), respectively, gives the n-th order stress constitutive equations for thin shells and the electric and magnetic constitutive equations of n-th order for the I-th layer as N

(n)

Ti j =

∑∑

I=1 m=0 Z hI

− I(n) Di

I(n)

Bi

N

=

hI−1 g

·Z

hI

hI−1

(m)

c¯Ii jkl γ3n γ3m dγ3 Skl

¸

I(m) I(m) (e¯Iki j Ek + h¯ Iki j Hk )γ3n (γ3I )m dγ3

·Z

(2.29)

(hI −hI−1 )/2

(m) e¯Iikl (γ3I )n γ3m dγ3I Skl (hI−1 −hI )/2 I=1 m=0 ¸ Z (hI −hI−1 )/2 I(n) I(n) + (ε¯ikI Ek + α¯ ikI Hk )(γ3I )n (γ3I )m dγ3I (hI−1 −hI )/2

∑∑

N

=

g

g

∑∑

·Z

(hI −hI−1 )/2

(hI−1 −hI )/2

I=1 m=0 Z (hI −hI−1 )/2

+

(hI−1 −hI )/2

(2.30)

(m) h¯ Iikl (γ3I )n γ3m dγ3I Skl

¸ I(n)

(α¯ ikI Ek

I(n)

+ µ¯ ikI Hk

)(γ3I )n (γ3I )m dγ3I

(2.31)

Hereto, we have obtained a complete system of general 2D equations of n-th order in tensor form for laminated multiferroic shells in orthogonal curvilinear coordinates. Especially, for a single-layered multiferroic shell, the general equations of n-th order can

2 Structural Theories of Multiferroic Plates and Shells

49

be derived by omitting the superscript “ I” and setting N = 1, hI = h and hI−1 = −h in the above equations for laminated multiferroic shells. For practical applications, we now transform the basic tensor equations, i.e., Eqs.(2.15)(2.18), (2.20) and (2.21), into the basic equations with the forms of physical components (n) I(n) according to the Appendix . The expressions of physical components of Si j , Ei and I(n)

Hi

are, respectively, ! (n) ∂ u1 1 ∂ ξ1 (n) ∂ ξ1 (n) + u + u , ∂ γ1 ξ2 ∂ γ 2 2 ∂ γ3 3 Ã (n) ! 1 ∂ u2 1 ∂ h2 (n) ∂ ξ2 (n) (n) S22 = + u + u , ξ2 ∂ γ2 ξ1 ∂ γ 1 1 ∂ γ3 3 (n) S11

1 = ξ1

Ã

(n)

(n+1)

S33 = (n + 1)u3

(n)

(n)

2S23 =

(n)

1 ∂ u1 1 ∂ u2 1 ∂ ξ1 (n) 1 ∂ ξ2 (n) + − u − u , ξ2 ∂ γ2 ξ1 ∂ γ 1 ξ1 ξ2 ∂ γ2 1 ξ1 ξ2 ∂ γ 1 2 1 ∂ u3 1 ∂ ξ1 (n) (n+1) + (n + 1)u1 − u ξ1 ∂ γ1 ξ1 ∂ γ3 1

1 ∂ ϕ I(n) , ξ1 ∂ γ 1

E2

1 ∂ ψ I(n) , ξ1 ∂ γ1

H2

I(n)

=−

I(n)

=−

(2.32)

(n)

(n)

2S13 =

H1

1 ∂ ξ2 (n) 1 ∂ u3 (n+1) + (n + 1)u2 − u , ξ2 ∂ γ2 ξ2 ∂ γ3 2 (n)

(n)

2S12 =

E1

,

I(n)

=−

1 ∂ ϕ I(n) , ξ2 ∂ γ2

E3

I(n)

=−

1 ∂ ψ I(n) , ξ2 ∂ γ 2

H3

I(n)

= −(n + 1)ϕ I(n+1)

(2.33)

I(n)

= −(n + 1)ψ I(n+1)

(2.34)

The n-th order governing equations in physical components from Eqs.(2.18), (2.20), and (2.21) are, µ ¶ 2 2 1 2 ∂ ξ1 ξ2 (n) 1 ∂ ξ j (n) 1 ∂ ξ j (n) Ta j + ∑ Ta j − ∑ Tj j ∑ ξ1 ξ2 a=1 ∂ xa ξa j6=a,a=1 ξa ξ j ∂ xa j6=a,a=1 ξa ξ j ∂ xa ! Ã (n−1)

−nT3 j 2

1 ∂ ∑ ξ1 ξ2 ∂ xa a=1 2

1 ∂ ∑ ξ1 ξ2 ∂ xa a=1 (n)

µ

(n)

+ Fj

N

=

∑

I=1

ρI

∞

(m)

∑ Gmn u¨ j

¶ 2 3 ξ1 ξ2 I(n) 1 ∂ ξa I(n) I(n−1) Da +∑ ∑ D j − nD3 + d I(n) = 0 ξa ξ ∂ x j a=1 j=1 j

µ

(2.35)

m=0

¶ 2 3 ξ1 ξ2 I(n) 1 ∂ ξa I(n) I(n−1) Ba +∑ ∑ B j − nB3 + bI(n) = 0 ξa ξ ∂ x j a=1 j=1 j

(2.36)

(2.37)

where Fj , d I(n) and bI(n) are the same as Eq.(2.23), and the Einstein summation convention for repeated indexes is not adopted in Eqs.(2.35)-(2.37). It should be noted that the

50

ChunLi Zhang and WeiQiu Chen

Lam´e coefficient ξ3 is unit in our coordinate system, which has been made use of in the above equations.

2.4 Equations of first-order theory for laminated multiferroic shells Generally the first-order theory, which is able to describe the common modes of extension, flexure, and thickness-shear, is enough for practical device modeling. We set (n)

(n)

u1 = 0,

u2 = 0 (n > 1),

(n)

u3 = 0 (0)

(2.38)

(n > 2) (0)

(1)

(1)

where ua (a = 1, 2) is the extension, u3 is the flexure, ua is the thickness-shear, and u3 is the thickness-stretch. In order to accommodate the thickness-strain which accompanies (2) the low-frequency flexure, the retention of u3 is necessary. Although we are mainly (0)

interested in ui

(1)

(1)

(2)

and ua , we have included u3 and u3 in Eq.(2.38). The 2D equations (0)

(1)

that we will obtain are for ui and ua only. The electric and magnetic potentials of the I-th layer are approximated by

ϕ I(n) = 0,

ψ I(n) = 0

(n > 1)

(2.39)

Now, the thin shell equations of the first-order theory can be derived by setting n to 0 and 1, respectively, in Eqs.(2.32)-(2.37). For thin shells, the Lam´e coefficients ξ1 = A1 , ξ2 = A2 from Eq.(2.10). Letting n = 0, 1 in Eqs.(2.32)-(2.34), we can obtain the zero-and first-order strains of thin shells and the zero-and first-order electric and magnetic fields. Their detailed explicit expressions in physical components are (0)

1 ∂ u1 1 (0) + u3 , A1 ∂ γ1 R1

(0)

S11 =

(0)

(0)

2S12 =

(1)

(0)

1 ∂ u1 1 ∂ u2 + , A2 ∂ γ2 A1 ∂ γ1

(2.40a)

1 ∂ u3 1 (0) (1) + u1 − u1 A1 ∂ γ1 R1 (1)

1 ∂ u1 1 (1) + u3 , A1 ∂ γ1 R1

(1)

2S23 = (1)

1 ∂ u3 1 (0) (1) + u2 − u2 , A2 ∂ γ2 R2

(0)

S33 = u3 ,

(0)

(0)

2S13 = (1)

(0)

1 ∂ u2 1 (0) + u3 , A2 ∂ γ 2 R2

(0)

(0)

2S23 =

S11 =

(0)

S22 =

2S13 =

(1)

1 ∂ u3 1 (1) − u2 , A2 ∂ γ2 R2 (1)

1 ∂ u3 1 (1) − u1 A1 ∂ γ1 R1

(1)

(1)

S22 = (1)

1 ∂ u2 1 (1) + u3 , A2 ∂ γ2 R2

2S12 =

(1)

(1)

(2)

S33 = 2u3 , (1)

1 ∂ u1 1 ∂ u2 + , A2 ∂ γ 2 A1 ∂ γ1

(2.40b)

2 Structural Theories of Multiferroic Plates and Shells I(0)

E1

=−

1 ∂ ϕ I(0) , A1 ∂ γ1

I(1)

E1 I(0)

H1

=−

1 ∂ ϕ I(1) , A1 ∂ γ1

1 ∂ ψ I(0) , A1 ∂ γ1

I(1)

H1

=−

=−

I(0)

E2

I(0)

H2

1 ∂ ψ I(1) , A1 ∂ γ1

=−

1 ∂ ϕ I(0) , A2 ∂ γ2

I(1)

E2

=−

1 ∂ ϕ I(1) , A2 ∂ γ2

1 ∂ ψ I(0) , A2 ∂ γ 2

I(1)

H2

=−

=−

I(0)

E3

I(0)

H3

1 ∂ ψ I(1) , A2 ∂ γ2

51

= −ϕ I(1)

(2.41a)

I(n)

=0

(2.41b)

= −ψ I(1)

(2.42a)

I(1)

(2.42b)

E3

H3

=0

Let n = 0, 1 in Eq.(2.35), the shell equations of motion in physical components are

∂ (N11 A2 ) ∂ (N21 A1 ) N12 ∂ A1 N22 ∂ A2 + + − ∂ γ1 ∂ γ2 ∂ γ2 ∂ γ1 Ã ! N ˆ I u¨(1) h A1 A2 Q13 (0) (0) 1 + + F1 = ∑ A1 A2 ρ I h˜ I u¨1 + R1 2 I=1 ∂ (N12 A2 ) ∂ (N22 A1 ) N21 ∂ A2 N11 ∂ A1 + + − ∂ γ1 ∂ γ2 ∂ γ1 ∂ γ2 Ã ! (1) N hˆ I u¨2 A1 A2 Q23 (0) I ˜ (0) + + F2 = ∑ A1 A2 ρ hI u¨2 + R2 2 I=1 N ∂ (Q13 A2 ) ∂ (Q23 A1 ) N11 A1 A2 N22 A1 A2 (0) (0) + − − + F3 = ∑ A1 A2 ρ I (h˜ I u¨3 ) ∂ γ1 ∂ γ2 R1 R2 I=1

∂ (M11 A2 ) ∂ (M21 A1 ) M12 ∂ A1 M22 ∂ A2 + + − ∂ γ1 ∂ γ2 ∂ γ2 ∂ γ1 Ã (0) ! (1) N ó ˆ hI u¨1 (1) I hI u¨1 −Q31 A1 A2 + F1 = ∑ A1 A2 ρ + 2 3 I=1 ∂ (M12 A2 ) ∂ (M22 A1 ) M21 ∂ A2 M11 ∂ A1 + + − ∂ γ1 ∂ γ2 ∂ γ1 ∂ γ2 Ã (0) ! (1) N ó ˆ hI u¨2 (1) I hI u¨2 −Q32 A1 A2 + F2 = ∑ A1 A2 ρ + 2 3 I=1

(2.43a)

(2.43b) (2.43c)

(2.43d)

(2.43e)

(0) (0) where h˜ I = hI − hI−1 , hˆ I = h2I − h2I−1 , ó hI = h3I − h3I−1 , and Nab = Tab , Q3c = T3c , Qc3 = (0)

(1)

Tc3 , Mab = Tab , (a, b = 1, 2), which are the often-used symbols for shell theories. Similarly, the Gaussian equations of electric and magnetic fields are µ ¶ I(0) I(0) ∂ (D1 A2 ) ∂ (D2 A1 ) 1 1 I(0) + A1 A2 D3 + d I(0) = 0 + + ∂ γ1 ∂ γ2 R1 R2

(2.44a)

52

ChunLi Zhang and WeiQiu Chen

µ ¶ I(1) I(1) ∂ (D1 A2 ) ∂ (D2 A1 ) 1 1 I(1) I(0) + A1 A2 D3 − A1 A2 D3 + d I(1) = 0 + + ∂ γ1 ∂ γ2 R1 R2 µ ¶ I(0) I(0) ∂ (B1 A2 ) ∂ (B2 A1 ) 1 1 I(0) + + + A1 A2 B3 + bI(0) = 0 ∂ γ1 ∂ γ2 R1 R2 µ ¶ I(1) I(1) ∂ (B1 A2 ) ∂ (B2 A1 ) 1 1 I(1) I(0) + A1 A2 B3 − A1 A2 B3 + bI(1) = 0 + + ∂ γ1 ∂ γ2 R1 R2

(2.44b) (2.45a) (2.45b)

Thus, the basic equations of the first-order theory of laminated multiferroic shells are established. For particular plate or shell structures, the principal radii of curvature and the Lam´e coefficients of the middle surface are readily known, and hence the basic equations of the structural theory can be easily obtained from the previous equations.

2.5 Equations for flat plates and cylindrical/spherical shells The most frequently used structures in practice such as flat plates and cylindrical and spherical shells are special cases of the general shell structures. Therefore, the equations for laminated multiferroic plates and cylindrical and spherical shells can be directly derived from the general equations of laminated multiferroic shells presented in the previous section. Now we present the basic equations for these special shell structures. When we consider special shell structures such as cylindrical and spherical shells and plates, the basic equations can be easily derived by assigning in the equations of general shells the principal radii of curvature R1 and R2 of the middle surface and the Lam´e coefficients A1 and A2 with appropriate values.

2.5.1 Flat plates For flat plates, the principal radii of curvature are infinite and the Lam´e coefficients of the middle surface are unit, namely, R1 = R2 = ∞,

A1 = A2 = 1

(2.46)

Substituting Eq.(2.46) in the shell equations of the first-order theory, we can obtain the basic plate equations for multiferroic plates as follows. The strain-displacement relations: (0)

(0)

S11 = u1,1 , (0)

(0)

(0)

(0)

(1)

2S31 = u3,1 + u1 ,

S22 = u2,2 ,

(0)

(0)

2S23 = u3,2 + u2 , (1)

(1)

S11 = u1,1 , (1)

(1)

2S23 = u3,2 ,

(1)

S33 = u3 , (0)

(1)

(1)

(1)

S22 = u2,2 , (1)

(1)

2S31 = u3,1 ,

(0)

(0)

(0)

2S12 = u1,2 + u2,1

(1)

(2.47a)

(2)

S33 = 2u3 , (1)

(1)

(1)

2S12 = u1,2 + u2,1

(2.47b)

2 Structural Theories of Multiferroic Plates and Shells

53

The relations between electric field and potential: I(0)

E1

I(0)

I(0)

= −ϕ,1 ,

I(1)

E1

E2

I(1)

I(0)

= −ϕ,2 ,

I(1)

= −ϕ,1 ,

I(0)

E3

I(1)

I(1)

= −ϕ,2 ,

E2

= −ϕ I(1)

E3

=0

(2.48a) (2.48b)

The relations between magnetic field and potential: I(0)

I(0)

I(0)

I(0)

= −ψ,1 , H2

H1

I(1)

H1

I(1)

I(0)

= −ψ,2 , H3

I(1)

I(1)

= −ψ,1 , H2

= −ψ I(1)

I(1)

= −ψ,2 , H3

=0

"

#

(2.49a) (2.49b)

The equations of motion: (0) (0) Tab,a + Fb

N

=

∑ρ

hˆ I (1) (0) h˜ I u¨b + u¨b 2

I

I=1

(0)

(0)

Ta3,a + F3 (1) (0) (1) Tab,a − T3b + Fb

N

=

(0)

∑ ρ I h˜ I u¨3

I=1

Ã

N

=

∑ρ

I=1

(2.50a)

I

hˆ I (0) ó hI (1) u¨b + u¨b 2 3

(2.50b) ! (2.50c)

And the Gaussian equations for quasi-static electromagnetic field: I(0)

Da,a + d I(0) = 0 I(1)

I(0)

Da,a − D3

+ d I(1) = 0

I(0)

Ba,a + bI(0) = 0 I(1)

I(0)

Ba,a − B3

+ bI(1) = 0

(2.51a) (2.51b) (2.52a) (2.52b)

The above basic equations for flat plates are the same as those presented in Ref. [15] which were established directly in a Cartesian coordinate system.

2.5.2 Cylindrical shells For cylindrical shells with the mean radius R and cylindrical coordinates (θ , z, r) corresponding to (γ1 , γ2 , γ3 ), we have R1 = R, A1 = R,

R2 = ∞, A2 = 1

So, the basic equations for multiferroic cylindrical shells are obtained as follows.

(2.53)

54

ChunLi Zhang and WeiQiu Chen

The strain-displacement relations: Ã (0) ! 1 ∂ u1 (0) (0) S11 = + u3 , R ∂ γ1 (0)

(0)

2S23 =

(0)

(0)

S22 =

∂ u2 , ∂ γ2

(0)

∂ u3 (1) + u2 , ∂ γ2

2S12 =

(0)

(0)

(0)

(1)

(0)

∂ u1 1 ∂ u2 + , ∂ γ2 R ∂ γ1

u 1 ∂ u3 (1) + u1 − 1 R ∂ γ1 R Ã (1) ! (1) ∂u 1 ∂ u1 (1) (1) (1) S11 = + u3 , S22 = 2 , R ∂ γ1 ∂ γ2 (0)

(0)

S33 = u3 , (2.54a)

2S13 =

(1)

(1)

2S23 =

∂ u3 , ∂ γ2 (1)

(1)

2S13 =

(1)

(1)

2S12 =

(1)

(2)

S33 = 2u3 ,

(1)

∂ u1 1 ∂ u2 + , ∂ γ2 R ∂ γ1

(2.54b)

(1)

u 1 ∂ u3 − 1 R ∂ γ1 R

The relations between electric field and potential: I(0)

E1

=−

I(1)

E1

1 ∂ ϕ I(0) , R ∂ γ1I

=−

I(0)

E2

1 ∂ ϕ I(1) , R ∂ γ1I

=−

I(1)

E2

∂ ϕ I(0) , ∂ γ2I

I(0)

∂ ϕ I(1) , ∂ γ2I

=−

= −ϕ I(1)

E3

I(1)

E3

(2.55a)

=0

(2.55b)

= −ψ I(1)

(2.56a)

I(1)

(2.56b)

The relations between magnetic field and potential: I(0)

H1

=−

1 ∂ ψ I(0) , R ∂ γ1I

I(1)

H1

=−

I(0)

H2

1 ∂ ψ I(1) , R ∂ γ1I

=−

∂ ψ I(0) , ∂ γ2I

I(1)

H2

=−

I(0)

H3

∂ ψ I(1) , ∂ γ2I

H3

=0

The equations of motion: Ã ! (0) (1) N hˆ I u¨1 1 ∂ N11 ∂ N21 Q13 F1 I ˜ (0) + + + = ∑ ρ hI u¨1 + R ∂ γ1 ∂ γ2 R R 2 I=1

(2.57a)

à ! (0) (1) N hˆ I u¨2 1 ∂ N12 ∂ N22 F2 I ˜ (0) + + = ∑ ρ hI u¨2 + R ∂ γ1 ∂ γ2 R 2 I=1

(2.57b)

(0)

N 1 ∂ Q13 1 ∂ Q23 N11 F3 (0) + − + = ∑ ρ I h˜ I u¨3 R ∂ γ1 R ∂ γ2 R R I=1

(2.57c)

2 Structural Theories of Multiferroic Plates and Shells (1)

N F 1 ∂ M11 ∂ M21 + − Q31 + 1 = ∑ ρ I R ∂ γ1 ∂ γ2 R I=1 (1)

N F 1 ∂ M12 ∂ M22 + − Q32 + 2 = ∑ ρ I R ∂ γ1 ∂ γ2 R I=1

Ã

Ã

(0) (1) ó hˆ I u¨1 hI u¨1 + 2 3 (0) (1) ó hˆ I u¨2 hI u¨2 + 2 3

55

! (2.57d) ! (2.57e)

And the Gaussian equations for quasi-static electromagnetic field: I(0)

I(0)

∂ D1 ∂D I(0) + R 2 + D3 + d I(0) = 0 ∂ γ1 ∂ γ2 I(1)

(2.58a)

I(1)

∂ D1 ∂D I(1) I(0) + R 2 + D3 − RD3 + d I(1) = 0 ∂ γ1 ∂ γ2 I(0)

(2.58b)

I(0)

∂ B1 ∂B I(0) + R 2 + B3 + bI(0) = 0 ∂ γ1 ∂ γ2 I(1)

(2.59a)

I(1)

∂ B1 ∂B I(1) I(0) + 2 + B3 − RB3 + bI(1) = 0 ∂ γ1 ∂ γ2

(2.59b)

2.5.3 Spherical shells For the spherical shells with mean radius R and spherical coordinates (θ , φ , r) corresponding to (γ1 , γ2 , γ3 ), we have R1 = R, R2 = R, (2.60) A1 = R sin φ , A2 = R Similarly, the basic equations for multiferroic spherical shells are as follows. The strain-displacement relations: (0)

(0) Sθ θ

(0) (0) uφ 1 ∂ uθ ur = + cot φ + , R sin φ ∂ θ R R

(0) Srr

(1) = ur ,

(0)

(0) Sφ φ

(0)

1 ∂ uφ ur = + , R ∂φ R

(0)

(0) 2Sφ r

(1) = uφ +

(0) uφ 1 ∂ ur − , R ∂φ R (0)

(0) 2Sθ φ (0)

(0) (0) u cot φ 1 ∂ uθ 1 ∂ uφ = + − θ , R ∂φ R sin φ ∂ θ R

2Sθ r =

(0)

(0)

u 1 ∂ ur (1) + uθ − θ R sin φ ∂ θ R

(2.61a)

56

ChunLi Zhang and WeiQiu Chen (1)

(1)

Sθ θ =

(1) (1) uφ 1 ∂ uθ ur + cot φ + , R sin φ ∂ θ R R

(1)

(1)

Sφ φ =

(1)

1 ∂ uφ ur + , R ∂φ R

(1)

(1)

(2)

Srr = 2ur ,

(1)

2Sφ r =

(1) uφ 1 ∂ ur − , R ∂φ R

(2.61b)

(1)

(1) (1) u 1 ∂ uθ 1 ∂ uφ (1) 2Sθ φ = + − θ cot φ , R ∂φ R sin φ ∂ θ R (1)

(1) u 1 ∂ ur = − θ R sin φ ∂ θ R

(1) 2Sθ r

The relations between electric field and potential: I(0)

Eθ

=−

I(1)

Eθ

1 ∂ ϕ I(0) , R sin φ ∂ θ

=−

I(0)

Eφ

1 ∂ ϕ I(1) , R sin φ ∂ θ

=−

I(1)

Eφ

1 ∂ ϕ I(0) , R ∂φ

=−

I(0)

Er

1 ∂ ϕ I(1) , R ∂φ

= −ϕ I(1)

I(1)

Er

(2.62a)

=0

(2.62b)

= −ψ I(1)

(2.63a)

The relations between magnetic field and potential: 1 ∂ ψ I(0) , R sin φ ∂ θ

Hφ

1 ∂ ψ I(1) , R sin φ ∂ θ

Hφ

I(0)

=−

I(1)

=−

Hθ

Hθ

1 ∂ ψ I(0) , R ∂φ

I(0)

=−

I(1)

=−

I(0)

Hr

1 ∂ ψ I(1) , R sin φ ∂ φ

I(1)

Hr

=0

(2.63b)

The equations of motion: 1 ∂ Nθ θ 1 ∂ Nφ θ cot φ Qθ r + +2 Nθ φ + R sin φ ∂ θ R ∂φ R R Ã ! (0) (1) N F hˆ I u¨θ (0) + 2θ = ∑ ρ I h˜ I u¨θ + R sin φ I=1 2 ¢ Q23 1 ∂ Nθ φ 1 ∂ Nφ φ cot φ ¡ + + Nφ φ − Nθ θ + R sin φ ∂ θ R ∂φ R R   (0) (1) N Fφ hˆ I u¨φ (0)  + 2 = ∑ ρ I ˜hI u¨φ + R sin φ I=1 2

(2.64a)

(2.64b)

1 ∂ Qθ r 1 ∂ Qφ r cot φ + + Qφ r R sin φ ∂ θ R ∂φ R (0)

N 1 Fr (0) − (Nθ θ + Nφ φ ) + 2 = ∑ ρ I h˜ I u¨r R R sin φ I=1

(2.64c)

2 Structural Theories of Multiferroic Plates and Shells

1 ∂ Mθ θ 1 ∂ Mφ θ cot φ + +2 Mφ θ R sin φ ∂ θ R ∂φ R Ã (0) ! (1) (1) N ó ˆ Fθ hI u¨θ I hI u¨θ −Qrθ + 2 = ∑ρ + R sin φ I=1 2 3

57

(2.64d)

¢ 1 ∂ Mθ φ 1 ∂ Mφ φ cot φ ¡ + + Mφ φ − Mθ θ R sin φ ∂ θ R ∂φ R   (1) (1) ˆ I u¨(0) ó N h h u ¨ I F2 φ φ  −Qrφ + 2 = ∑ ρI  + R sin φ I=1 2 3

(2.64e)

And the Gaussian equations for quasi-static electromagnetic field: I(0)

I(0)

∂ Dθ ∂θ

+ sin φ

∂φ

I(0)

+ cos φ Dφ

I(0)

+ 2 sin φ Dr

+

d I(0) =0 R

I(1)

I(1)

∂ Dθ ∂θ

∂ Dφ

+ sin φ

∂ Dφ

∂φ

I(1)

+ cos φ Dφ

I(1)

+ sin φ (2Dr

I(0)

− RDr ) +

d I(1) =0 R

(2.65b)

I(0)

I(0)

∂ Bφ ∂ Bθ bI(0) I(0) I(0) + sin φ + cos φ Bφ + 2 sin φ Br + =0 ∂θ ∂φ R I(1)

(2.65a)

(2.66a)

I(1)

∂ Bφ ∂ Bθ bI(1) I(1) I(1) I(0) + sin φ + cos φ Bφ + sin φ (2Br − RBr ) + =0 ∂θ ∂φ R

(2.66b)

2.6 Applications: evaluation of magnetoelectric effects In this part, we use the structural theories established above to evaluate the magnetoelectric (ME) effects in laminated multiferroic structures. The magnetoelectric effects in multiferroic bilayers and spherical shell bilayers are analyzed with the conventional definition

Fig. 2.2 Multiferroic bilayers. (a) L-T configuration; (b) T-L configuration.

58

ChunLi Zhang and WeiQiu Chen

of the magnetoelectric coupling coefficient [24] as α = ∆E/∆H or α 0 = v∆E/∆H (that is α 0 = α v), here v is the volume fraction of the piezoelectric phase (Fig.2.2).

2.6.1 Magnetoelectric effect in multiferroic bilayers First, we consider the L-T bilayer [18] as shown in Fig.2.2. The piezoelectric layer is PZT-4 and the piezomagnetic layer is CoFe2 O4 . The coordinate plane Ox1 x2 coincides with the geometric middle plane at equal distance from the top and bottom of the bi(0) layer. Let the extensional displacement at x3 = 0 be u1 (x1 ,t), the flexural displacement (0) (0) be u3 (x1 ,t), the extensional resultant force be N (= T11 ), the transverse shear force (0)

(1)

be Q (= T31 ), the bending moment be M (= T11 ), the electric potential be ϕ (0) (x1 ,t), (0)

the longitudinal electric field be E1 = −∂ ϕ (0) /∂ x1 = −ϕ,1 , and the longitudinal electric displacement resultant be D. The equations of motion and the equation of magnetoelectrostatics for multiferroic laminates have been established in Section 2.5.1. For the special case of bilayers in coupled flexure and extension, these equations become (0)

(0)

N,1 = ρ (0) u¨1 − ρ (1) u¨3,1 , (0)

Q,1 = ρ (0) u¨3 ,

(2.67a)

|x1 | < a

(2.67b)

|x1 | < a

D,1 = 0, where

|x1 | < a

(2.67c)

(0) (0)

(1) (0)

(0)

(0)

11

11

11

31

N = c11 u1,1 − c11 u3,11 − e11 E1 − h31 H3 , (1) (0) (2) (0) (1) (1) M = c u −c u − e¯ E1 − h¯ H3 , 1,1

3,11

(2.68a)

Q = M,1 , (0) (0)

(1) (0)

(0)

D = e11 u1,1 − e¯11 u3,11 + ε11 E1

ρ (0) = ρ p (h0 − hp ) + ρ m (h0 − hp ), ρ (1) = ρ p (h20 − h2p ) + ρ m (h2m − h20 ), (0)

p

c11 = (h0 − hp )c¯11 + (hm − h0 )c¯m 11 , (1)

p

2c11 = h20 − h2p c¯11 + h2m − h20 c¯m 11 , (2)

(2.68b)

p

3c11 = (h30 − h3p )c¯11 + (h3m − h30 )c¯m 11 , (0)

p

e11 = (h0 − hp )e¯11 , (0) p ε11 = (h0 − hp )ε¯11 , (1) 2 2 m ¯ ¯ 2h = (hm − h )h 31

0

(1)

p

2e¯11 = h20 − h2p e¯11 , (0) h = (hm − h0 )h¯ m , 31

31

31

m m 2 m c¯m 11 = c11 − (c13 ) /c33 , p p p p p e¯11 = e33 − e31 c13 /c11 , m m m m h¯ m 31 = h31 − h33 c31 /c33

p

p

p

p

c¯11 = c33 − (c13 )2 /c11 , p p p p ε¯11 = ε33 + (e31 )2 /c11 ,

(2.68c)

2 Structural Theories of Multiferroic Plates and Shells

59

The superscripts “p” and “m” denote the piezoelectric and piezomagnetic layers, respectively. We consider a bilayer with free ends and open end electrodes, namely, N, M, Q and D are all zero at both ends. In the static case, the inertial terms on the right-hand sides of Eqs.(2.67a) and (2.67b) vanish. When subject to an applied magnetic field H3 , the ME effect per unit volume can be readily obtained as

α=

(0) (2) (1) (1) (1) (0) h31 (c11 − c11 h0p ) − h¯ 31 (c11 − c11 h0p ) (1) (1)

(0) (2)

p

(1) (1)

p

(0) (1)

(2) (0)

(c11 c11 − c11 c11 )ε¯11 /e¯11 + (2c11 e¯11 − c11 e¯11 h0p − c11 e11 )

(2.69a)

and

α0 =

(0) (2) (1) (1) (1) (0) h31 (c11 − c11 h0p ) − h¯ 31 (c11 − c11 h0p )v (1) (1)

(0) (2)

p

(1) (1)

p

(0) (1)

(2) (0)

(c11 c11 − c11 c11 )ε¯11 /e¯11 + (2c11 e¯11 − c11 e¯11 h0p − c11 e11 )

(2.69b)

where v = hp /(hm + hp ) is the volume fraction of the piezoelectric phase, and h0p = (h0 − hp )/2. In the case of harmonic motion, the applied magnetic field in the piezomagnetic layer is given by H3 = exp(iω t). For steady-state motions, all fields have the same time dependence which will be dropped for convenience. From the electrostatic equation in Eq.(2.67c), considering open electrodes at both ends, we can obtain the electric potential as (1) (0) e¯11 (0) e (0) ϕ (0) = 11 u − u (2.70) (0) 1 (0) 3,1 ε11 ε11 Substituting Eq.(2.70) into Eqs.(2.67a) and (2.67b), letting u1 = A sin kx1 and u3 = B cos kx1 , we obtain (0)

(1)

(c¯11 k2 − ω 2 ρ (0) )A + (c¯11 k3 − ω 2 ρ (1) k)B = 0, (1)

(2.71)

(2)

c¯11 k3 A + (c¯11 k4 − ω 2 ρ (0) )B = 0 (0)

(0)

(0) (0)

(0)

(1)

(1)

(0) (1)

(0)

(2)

(2)

(1) (1)

(0)

where c¯11 = c11 + e11 e11 /ε11 , c¯11 = c11 + e11 e¯11 /ε11 and c¯11 = c11 + e¯11 e¯11 /ε11 . For nontrivial solutions, the determinant of the coefficient matrix of Eq.(2.71) must vanish, leading to a cubic equation in k2 . For trigonometric solutions, a sign difference in k does not matter. Denoting the three roots of k with positive real parts by k j ( j = 1, 2, 3), we obtain the solution symmetric with respect to x1 = 0 as 3

(0)

u1 = (0) u3

=

∑ B j β j sin k j x1 ,

j=1 3

∑

B j cos k j x1 , j=1 (0) 3 (1) e e¯11 3 ϕ (0) = 11 B β sin k x + B j k j sin k j x1 j j j 1 (0) (0) ε11 j=1 ε11 j=1

∑

∑

(2.72)

60

ChunLi Zhang and WeiQiu Chen (2)

(1)

where β j = −(c¯11 k4j − ω 2 ρ (0) )/c¯11 k3j , and B j ( j = 1, 2, 3) are three undetermined constants. Substituting Eq.(2.72) into the mechanical boundary conditions of free ends yields three equations for determining B j . The electric field is a function of x1 . We define the average of E1 through E=

1 2a

Z a −a

E1 dx1 =

1

3

aε11

j=1

(0)

(1)

B (e β + e11 k j ) sin k j a (0) ∑ j 11 j

(2.73)

and use E to calculate the ME coupling coefficient:

α=

1 (0)

aε11

3

(0)

(1)

∑ B j (e11 β j + e11 k j ) sin k j a

(2.74)

j=1

For the T-L configuration in Fig.2.2, the solution procedure is similar and is not presented here. The ME coupling coefficient under the constant magnetic field is

α=

(2) (1) (0) (0) (1) (1) (c11 − c11 h0p )h11 + (c11 h0p − c11 )h¯ 11 (1) (1)

(2) (0)

(1) (1)

(2) (0)

(0) (1)

I /e¯I + (2c e¯ (c11 c11 − c11 c11 )ε¯33 31 11 31 − c11 e31 − c11 e¯31 h0p )

Under the applied harmonic magnetic field, there is # " 3 3 1 (1) (0) ¯ ¯ α = − (0) e31 ∑ B j β j sin(k j a) + e¯31 ∑ B j k j sin(k j a) aε33 j=1 j=1 For numerical examples, consider bilayers with the total thickness hp + hm = 0.7 mm. The length is 2a = 9.2 mm. The material constants of PZT and CoFe2 O4 can be found in Ref. [25, 26]. In the static case, the ME coupling coefficients for both the T-L and the L-T bilayers are simultaneously given in Fig.2.3 as a function of v, the volume fraction of PZT-4. The peak value is about 0.26 V/A for T-L and 0.138 V/A for L-T when v is around 0.25. This optimal volume fraction is much smaller than that for the extensional mode

Fig. 2.3 Magnetoelectric effect versus PZT-4 volume fraction [18].

2 Structural Theories of Multiferroic Plates and Shells

61

(v = 0.55) as reported in Ref. [17]. Since coupling exists between flexure and extension, each curve in Fig. 2.3 looks like a saddle and has two local peak values. In the case of harmonic motion, the effect of damping is taken into consideration using a complex elastic constant in the numerical calculation, i.e., c is replaced by c = c (1 + 0.01i) where c is a typical elastic constant. Figures 2.4(a) and 2.4(b) show the ME coupling coefficients versus driving frequency for T-L and L-T bilayers, respectively. The peak at resonance takes place at v = 0.45. The resonant frequency decreases with the volume fraction v. Figure 2.5 shows the dependence of the resonant frequency and the peak ME coupling coefficient on the volume fraction of the piezoelectric phase.

Fig. 2.4 Magnetoelectric effect versus driving frequency. (a) T-L; (b) L-T [18].

Fig. 2.5 Resonant frequency and magnetoelectric effect versus PZT-4 volume fraction. (a) T-L; (b) L-T [18].

As there is interaction between the extension and flexure of the bilayer, the geometric middle plane cannot work as a reference plane. The so-called “neutral plane” is in fact curved. In order to show it, we calculate the locations of the points on the “neutral plane” (0) (0) (0) defined by zero extensional strain from S11 = u1,1 − x3 u3,11 = 0. This determines x3 =

62 (0)

ChunLi Zhang and WeiQiu Chen (0)

u1,1 /u3,11 . For the L-T bilayer (2a = 9.2 mm) in time-harmonic motions, the “neutral plane” is given by 3

∑ B j β j k j cos k j x1

x3 =

j=1 3

∑

(2.75) B j k2j cos k j x1

j=1

and is shown in Fig.2.6. The curves in the figure represent locations of points in the bilayer before deformation. During the harmonic motions from which these curves are (0) determined, the axial strain S11 at these points is zero.

Fig. 2.6 Locations of points in L-T bilayer that have zero axial strains in harmonic motions [18].

For the bilayer considered, the “neutral plane” is very close to the geometric middle plane in the central portion. Near the ends the difference is significant. Clearly, the “neutral plane” is in fact curved and cannot be used as a reference plane.

2.6.2 Magnetoelectric effect of multiferroic spherical shell laminates In this section, a thin bilayer multiferroic spherical shell as shown in Fig.2.7 is considered. The mean radius is R. The outer layer is piezomagnetic material and the inner layer is piezoelectric material with electrodes shown by thick lines. Both materials are all poled in the radial direction. The surfaces of the spherical shell are traction-free and the electrodes are open. For convenience, only the radially symmetric expanding motion (or called breathing mode) is treated. In this case, the equations of zero-order structural theory are sufficient. (0) (0) (0) The sole nonzero displacement is u3 . From Eq.(2.61a), the strain components S11 , S22 are (0) u (0) (0) S11 = S22 = 3 (2.76) R

2 Structural Theories of Multiferroic Plates and Shells

63

Fig. 2.7 A spherical shell model.

And the stresses are (0)

(0) (0)

(0) (0)

(0) (0)

(0)

(0)

(2.77a)

(0)

(0) (0)

(0) (0)

(0) (0)

(0)

(0)

(2.77b)

N11 = T11 = c11 S11 + c12 S22 − e31 E3 − h31 H3 N22 = T22 = c21 S11 + c22 S22 − e32 E3 − h32 H3 where

(0)

c11 = h0 c¯I11 + hc¯II11 , (0)

(0)

c12 = h0 c¯I12 + hc¯II12 , (0) h = hh¯ II ,

c21 = h0 c¯I21 + hc¯II21 , 31 (0) (0) II 0 ¯ h32 = hh32 , e31 = h e¯I31 , cI13 cI13 , cI33 cI cI c¯I21 = cI21 − 23I 31 , c33 II cII c c¯II12 = cII12 − 13II 32 , c33 II h cII h¯ II31 = hII31 − 33II 31 , c33 I e cI e¯I31 = eI31 − 33I 31 , c33 c¯I11 = cI11 −

31 (0) e32

(2.78a)

= h0 e¯I32

cI13 cI32 , cI33 cII cII c¯II11 = cII11 − 13II 13 , c33 II cII c c¯II21 = cII21 − 23II 31 , c33 hII cII h¯ II32 = hII32 − 32II 32 , c33 I e cI e¯I32 = eI32 − 33I 32 c33 c¯I12 = cI12 −

(2.78b)

The electric displacement is (0)

(0) (0)

(0) (0)

(0) (0)

D3 = e31 S11 + e32 S22 + ε33 E3

(2.79)

eI33 eI33 cI33

(2.80)

where (0)

I , ε33 = h0 ε¯33

I I = ε33 + ε¯33

64

ChunLi Zhang and WeiQiu Chen (0)

(0)

As the spherical shell material is spherically isotropic, there are relations of c11 = c22 , (0) (0) (0) (0) (0) (0) c12 = c21 ,e31 = e32 and h31 = h32 . Then, from Eq.(2.76), we have N11 = N22

(2.81)

The equation of motion in the radial direction is 1 (0) − (N11 + N22 ) = (ρ I h0 + ρ II h)u¨3 R

(2.82)

The Gaussian equation is (0)

D3,3 = 0

(2.83)

(0)

When the applied magnetic field is constant H3 , from Eq.(2.82), the static equilibrium equation is N11 + N22 = 0 (2.84) With the free surface conditions and Eq.(2.84), we obtain (0)

(0)

(0)

(0) (0)

(0)

(0)

N11 = (c11 + c12 )S11 − e31 E3 − h31 H3 = 0

(2.85)

From Eq.(2.83) and the open-electrode condition, we have (0)

(0) (0)

(0) (0)

D3 = 2e31 S11 + ε33 E3 = 0

(2.86)

From Eqs.(2.85) and (2.86), the magnetoelectric effect in the spherical shell can be derived as (0) (0) 2e31 h31 α = − (0) (2.87) (0) (0) (0) (0) (c11 + c12 )ε33 + 2e31 e31 (0)

Under the harmonic magnetic field H3 = Heiω t , the equation of motion is shown in Eq.(2.82). Substitution of Eq.(2.77) into Eq.(2.82) yields (0)

(0)

(0)

(0) (0)

(0)

(0)

(0)

2[(c11 + c12 )S11 − e31 E3 − h31 H3 ] = Rω 2 (ρ I h0 + ρ II h)u3

(2.88)

Using the strain-displacement relation in Eq.(2.76), Eq.(2.81) can be rewritten as (0)

(0)

(0)

(0) (0)

(0)

(0)

[2(c11 + c12 ) − R2 ω 2 (ρ I h0 + ρ II h)]S11 − 2e31 E3 − 2h31 H3 = 0

(2.89)

For the open-electrode condition, Eq.(2.86) is still valid. Thus we can obtain the magnetoelectric effect in the spherical shell as (0) (0)

α =−

4e31 h31 (0)

(0)

(0)

(0)

(0) (0)

2(c11 + c12 )ε33 − R2 ω 2 (ρ I h0 + ρ II h)ε33 + 4e31 e31

(2.90)

Equations (2.87) and (2.90) are formulas for computing the magnetoelectric effects in the spherical shell. For numerical calculation, BaTiO3 and PZT-4 are chosen for the

2 Structural Theories of Multiferroic Plates and Shells

65

piezoelectric phase, and CoFe2 O4 is chosen for the piezomagnetic phase. The mean radius R of the spherical shell is 1 cm. We plot the magnetoelectric effects versus the volume fraction of the piezoelectric phase in Figs.2.8 and 2.9 when the spherical shell is subjected to a constant magnetic field. Figure 2.10 is for the harmonic magnetic field.

Fig. 2.8 Magnetoelectric effect (α , α 0 ) versus v (the piezoelectric phase is BaTiO3 ).

Fig. 2.9 Magnetoelectric effect (α 0 ) versus v.

Fig. 2.10 Magnetoelectric effect versus frequency.

66

ChunLi Zhang and WeiQiu Chen

2.7 Summary A complete system of general laminated multiferroic shell equations in orthogonal curvilinear coordinates is derived by expanding the displacement, electric and magnetic potential in a series of powers of the thickness coordinate. The general shell equations can be directly degenerated to equations for plates, and cylindrical and spherical shells when adopting the appropriate values of Lam´e coefficients for the corresponding coordinate systems. The established structural theories can be used in the mechanics analysis of multiferroic structures. They also provide a powerful tool for evaluating the magnetoelectric effects in composite multiferroic structures in a more rigorous sense, when compared to the simplifying equivalent circuit analysis.

Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos. 10832009 and 11090333), and the National Basic Research Program of China (No. 2009CB623204). The authors are very grateful to Professor Jiashi Yang at University of Nebraska-Lincoln for his consistent advices and great helps during our study of the topic discussed in this chapter.

References [1] Spaldin N A, Fiebig M. The renaissance of magnetoelectric multiferroics. Science, 2005, 309, 391-392. [2] Scott J F. Multiferroic memories. Nat. Mater., 2007, 6, 256-257. [3] Fiebig M. Revival of the magnetoelectric effect. J. Phys. D: Appl. Phys., 2005, 38, R123R152. [4] Eerenstein W, Mathur N D, Scott J F. Multiferroic and magnetoelectric materials. Nature, 2006, 442, 759-765. [5] Zhai J Y, Xing Z P, Dong S X, et al. Magnetoelectric laminate composites: an overview. J. Am. Ceram. Soc., 2008, 91, 351-358. [6] Nan C W, Bichurin M I, Dong S X, et al. Multiferroic magnetoelectric composites: historical perspective, status, and future directions. J. Appl. Phys., 2008, 103, 031101. [7] Xing Z P, Li J F, Viehland D. Giant magnetoelectric effect in Pb(Zr,Ti)O3 -Bimorph/ NdFeB laminate device. Appl. Phys. Lett., 2008, 93, 013505. [8] Chen W Q, Lee K Y. Alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates. Int. J. Solids Struct., 2003, 40, 5689-5705. [9] Chen W Q, Lee K Y, Ding H J. General solution for transversely isotropic magneto-electrothermo-elasticity and the potential theory method. Int. J. Eng. Sci., 2004, 42, 1361-1379.

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[10] Chen W Q, Lee K Y, Ding H J. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates. J. Sound Vib., 2005, 279, 237-251. [11] Tzou H S. Piezoelectric Shells: Distributed Sensing and Control of Continua. Kluwer, 1993. [12] Yang J S. The Mechanics of Piezoelectric Structures. Singapore: World Scientific, 2006. [13] Mindlin R D. High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct., 1972, 8, 895-906. [14] Zhang C L, Chen W Q, Yang J S, et al. One-dimensional equations for piezoelectromagnetic beams and magnetoelectric effects in fibers. Smart Mater. Struct., 2009, 18, 095026. [15] Zhang C L, Chen W Q, Li J Y, et al. Two-dimensional analysis of magnetoelectric effects in multiferroic laminated plates. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2009, 56, 1046-1053. [16] Zhang C L, Yang J S, Chen W Q. Theoretical modeling of frequency-dependent magnetoelectric effects in laminated multiferroic plates. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2009, 56, 2750-2759. [17] Zhang C L, Chen W Q, Xie S H, et al. The magnetoelectric effects in multiferroic composite nanofibers. Appl. Phys. Lett., 2009, 94, 102907. [18] Zhang C L, Yang J S, Chen W Q. Magnetoelectric effects in multiferroic bilayers for coupled flexure and extension. J. Intel. Mat. Syst. Str., 2010, 21, 851-855. [19] Zhang C L, Yang J S, Chen W Q. Harvesting magnetic energy using extensional vibration of laminated magnetoelectric plates. Appl. Phys. Lett., 2009, 95, 013511. [20] Zhang C L, Yang J S, Chen W Q. Low-frequency magnetic energy harvest using multiferroic plates. Phys. Lett. A, 2010, 374, 2406-2409. [21] Zhang C L, Chen W Q. A wideband magnetic energy harvester. Appl. Phys. Lett., 2010, 96, 123507. [22] Feng Y C, Tong P. Classical and Computational Solid Mechanics. Singapore: World Scientific, 2001. [23] Tiersten H F. Linear Piezoelectric Plate Vibrations. Plenum Press, 1969. [24] Bichurin M I, Petrov V M, Srinivasan G. Modeling of magnetoelectric effect in ferromagnetic piezoelectric multilayer composites. Ferroelectrics, 2002, 280, 165-175. [25] Huang J H, Chiu Y H, Liu H K. Magneto-electro-elastic Eshelby tensors for a piezoelectricpiezomagnetic composite reinforced by ellipsoidal inclusions. J. Appl. Phys., 1998, 83, 53645370. [26] Ramirez F, Heyliger P R, Pan E N. Free vibration response of two-dimensional magnetoelectro-elastic laminated plates. J. Sound Vib., 2006, 292, 626-644.

Appendix Orthogonal curvilinear coordinate systems In orthogonal curvilinear coordinate systems, the frame vectors (ri ) are mutually orthogonal but are not of unit length. The dimensions of the basis vectors for a point in space may be different. The magnitude of the projection of a vector onto the frame direction is not the corresponding component of the vector. The metric tensors are introduced as ½ (ξi )2 (i = j) (2.A1) gi j = 0 (i 6= j)

68

ChunLi Zhang and WeiQiu Chen

where ξi are the Lam´e coefficients (or scale factors). Note that ξi is the length of the frame vector ri . Let us introduce the frame whose vectors are co-directed with the basis vectors but have unit length: (2.A2) ei = ri /ξi where no summation is over the repeated index i (this also applies to other formulas in this appendix), as in contrast to that in the text. Thus at each point the vectors ei form a Cartesian basis.

A Christoffel symbols of the second kind The Christoffel symbols depend not only on the frame vectors themselves but on their rates of change from point to point. In an orthogonal curvilinear coordinate system, the Christoffel coefficients are Γi kj = 0 (i 6= j 6= k), 1 ∂gjj Γjij = − , ¡2gii√∂ xi¢ ∂ ln g j j (2.A3) Γjij = , ∂ x ¡ √i ¢ ∂ ln g j j j Γj j = ∂xj In a Cartesian frame, all the Christoffel symbols are zero.

B Covariant differentiation The covariant differentiations of a vector Fi and a tensor Ai j are defined as Fi | j = Fi, j − Γjik Fk

(2.A4)

Ai j |k = Ai j,k − Al j Γikl − Ail Γjkl

(2.A5)

C Physical components of a vector and a tensor A vector (or tensor) is invariant when represented with different base vectors. Let us write a vector v and a second-order tensor A as v=

3

3

i=1

i=1

∑ vi ri = ∑ vi

√

3 ri gii √ = ∑ vi ξi ei gii i=1

(2.A6)

The components vi ξi are called the physical components of the vector v. The physical components of the second-order tensor A are ξi ξ j Ai j . We denote the physical components as v for vector components vi and A for tensor components Ai j . Thus, there are the relations

2 Structural Theories of Multiferroic Plates and Shells

69

v = ξi vi

(2.A7)

A = ξi ξ j Ai j

(2.A8)

Chapter 3 Piezoelectric Power/Energy Harvesters

YuanTai Hu, Huan Xue, and HongPing Hu Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China

Abstract A kind of electromechanical conversion devices, piezoelectric power/energy harvesters, to extract energy from ambient vibrations is introduced for wireless energy supply. In general, the former, a piezoelectric power harvester, supplies the scavenged power directly to microelectronic devices without storage, which suits for strong ambient vibrations; The latter harvester possesses the energy-storage function and after a long energy accumulation, such a harvester can supply energy to those microelectronic devices which perform the duties infrequently, for example pacemakers. Therefore, a piezoelectric energy harvester can effectively operate under relatively weak ambient vibrations. The structures of two categories of harvesters are described, respectively. Performances of piezoelectric power/energy harvesters are analyzed and the effective measures to improve the efficiency of harvesters are illustrated in detail. The results are useful in harvester design and applications. Keywords piezoelectric power/energy harvesters, electromechanical coupling, power density, charging efficiency, SSHI, DC-DC converters

72

YuanTai Hu, Huan Xue, and HongPing Hu

3.1 Introduction Since the mid-1900s, the development of computer technology, network technology and wireless sensing technology has been rapid and vigorous. The combination of the former two has changed the production mode and the living habits of people, while that of the latter two has led to the emergence of the wireless sensor network that can be randomly deployed in untraversed adverse environment. By largely and redundantly deploying sensors, it will be possible to monitor, sense, and gather information on various environments or monitored objects real time within the network region and have the information processed to obtain detailed and accurate information and transmit it wirelessly to observers. The technology is currently expeditiously advancing in energy, environment monitoring, and road and bridge building areas. In addition, over the past 10-odd years, great strides have been made and wide applications have been found in miniature and micro-remote-sensing equipments and implanted microelectronic devices no matter whether in the military, medical or civil industries [1-4]. With the accelerated progress of wireless sensors and microelectronic devices, the way to supply energy is arousing everincreasing concern. The conventional battery energy supplying mode is no longer able to meet the development requirement since the battery energy contained is after all limited. Its output power will attenuate with time and the replacement is difficult. This situation will become worse for the sensor network since it is a tremendous workload or even impossible to replace batteries for thousands upon thousands of sensor nodes. Obviously, without the simultaneous development of energy supply technology, deeper development of wireless network sensing technology is seriously hampered, that is, the lagging energy supply technology has become the “bottleneck” constraining further development of modern sensing technology [5]. It is the most pressing matter of the moment to launch researches on advanced new energy supply technology. 1 MiniaThe development of modern microelectronic devices is characterized by: ° 2 Diminishing energy conturized volume, development toward super-scale integration; ° sumption, that is, ever decreasing operating energy consumption of microelectronic de3 Diversified functions, such as, self-calibration, vices, around 10-1000 mW in general; ° wireless control, wireless energy-supply, man-machine compatibility, etc., among which 4 The requirement for wireless energy supply is the foundation of the other functions; ° very long service life. For instance, if powered by a lithium battery, the pacemaker can in general be used for only 6-8 years [5]. Other examples include nuclear reactors, tanks, and armored vehicles with large numbers of microelectronic devices sealed inside. In order to maintain the integrity of structure, once sealed in, it would be very inconvenient to replace or charge the batteries by opening up the sealing. In addition, for electronic instruments working in toxic or corrosive surroundings, it’s also no easy to replace their batteries. For this reason, it will be of extremely great importance to design a device capable of providing microelectronic devices with operation energy directly extracted from environment [6]. Potentially available energy found in environment is in the form of solar energy, temperature difference, vibration or noise, etc. Roundy et al. [4] have made careful investiga-

3

Piezoelectric Power/Energy Harvesters

73

tions on a variety of fixed energy sources (e.g., batteries) and ambient energy sources that are potentially capable of providing microelectronic device with energy, as shown in Table 3.1, which shows that the energy that can be extracted from solar energy and ambient mechanical vibration is equivalent to the energy consumed by microelectronic devices in operation. As solar energy is too apt to be affected by weather and ambient factors, its energy output is obviously decreased when it is cloudy or raining and there may even be no energy output indoors or in sealed surroundings. Therefore, it is a fairly effective method to extract energy from ambient mechanical vibration. At present, direct extraction of mechanical vibration energy from environment has become the focus of research. There are mainly three ways of converting mechanical vibration into electrical energy, namely, the electromagnetic, electrostatic, and piezoelectric way [7]. However, the volume of the electromagnetic structure can hardly be miniaturized while electrostatic energy harvesting is constrained by great damping with low energy harvesting efficiency. Hence piezoelectric conversion is often adopted when extracting energy from ambient vibration, whose operation principle is as follows: the piezoelectric structure is deformed while being driven by the ambient vibration so as to cause the positive and negative charge center inside piezoelectric medium to separate till the generation of polarization voltage, which will drive the free charge on the electrode into directional flow to output electrical energy. Such a device that extracts vibration energy from environment via a piezoelectric structure is called the piezoelectric power/energy harvester. It has been pointed out by Roundy et al. [4], Sodano et al. [8] and Mateu et al. [9] in their comparative analysis of various energy harvesting methods that the piezoelectric conversion mode has very broad prospects for application by virtue of its good electromechanical conversion performance with no need of additional power sources, the ease with which miniaturization is implemented and the Table 3.1 A comparison on energy density of various kinds of energy available. power density/(µW/cm3 ) service term 1 year

power density/(µW/cm3 ) service term 10 years

Solar energy (outdoors)

15000 when fine, 150 when overcast and rainy

15000 when fine, 150 when overcast and rainy

Solar energy (indoors)

6 within office

6 within office

Ambient vibration (piezoelectric conversion)

250

250

Ambient vibration (electrostatic conversion)

50

50

Noise

0.003 at 75 dB, 0.96 at 100 dB

0.003 at 75 dB, 0.96 at 100 dB

Temperature gradient

15 at temperature gradient 10◦ C

15 at temperature gradient 10◦ C

Non-chargeable lithium cell

45

3.5

Chargeable lithium cell

7

0

Hydrocarbon consuming cell

333

33

Methyl consuming cell

280

28

74

YuanTai Hu, Huan Xue, and HongPing Hu

ideal suitability to harvesting the mechanical energy in environment. Usually, the piezoelectric energy harvester can be regarded as half of a piezoelectric transformer. In a piezoelectric transformer, energy should be converted twice, that is, converting the electric energy into mechanical energy, then the mechanical energy is converted back into electric energy. Piezoelectric harvesters are divided into two classes, one operates by direct supply of the output power of piezoelectric structures from ambient vibration excitation without having to store energy and is called piezoelectric power harvesters. The other class has rather weak ambient vibration with the output power of piezoelectric structures lower than the transient energy consumption by devices. By taking into account the fact that certain microelectronic devices are in a dormant state at most time while the energy harvester can harvest energy anytime, it’s clear that after a period of energy accumulation, the energy stored by the energy harvester is still sufficient to meet a short-term operation energy consumption of a device. Obviously, such a harvester should have the energy storing function and is called a piezoelectric energy harvester. This chapter will be devoted to a detailed description of the structures of piezoelectric power/energy harvesters and the two different methods for analyzing the two classes of piezoelectric harvesters. Effective measures to increase the energy harvesting efficiency are presented and the mechanism of improvement is revealed, too.

3.2 Basic structure of a piezoelectric power/energy harvester As a piezoelectric power harvester does not store the energy extracted from ambient vibration but directly provides it to a microelectronic device, its major structure is a piezoelectric transducer with the output end directly connected to a microelectronic device. In an analysis, it is common to have the energy-consuming microelectronic device reduced to an impedance. As a piezoelectric energy harvester is required to possess the energy storing function, its structure is by contrast rather complicated. As shown in Fig.3.1, a 1 The piezoelectric energy harvester is composed of three main components, that is, ° piezoelectric energy harvesting end comprising a piezoelectric transducer and other auxiliary components used to increase the converting efficiency (e.g., a synchronized switch 2 Chargeable storage harvesting inductor with a control switch KI , called SSHI, etc.); ° 3 The energy storing circuit composed of a fullbattery for storing the harvested energy; ° bridge rectifier and a DC-DC converter, of which the latter contains a rectifying capacitor Crect , equivalent to an energy transfer station, and an intermediate circuit component with step-up/step-down function and current stabilizing function (e.g., the Buck-Boost type circuit or Cuk circuit, etc.). Its main function is to step up or down the rectifying capacitor voltage Vrect to match the battery voltage Vb and output to the battery with well-nigh stable charging current. It is clear that the performance of a piezoelectric energy harvester is dependent on the interaction between the piezoelectric structure and the energy storing circuit. This is because, on the one hand, the important parameter of a DC-DC converter,

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the duty cycle D, should be determined by the output power at the piezoelectric end and, on the other hand, the magnitude of the output power of the piezoelectric structure is, in turn, closely related to the physical quantity Vrect in the circuit. There exist three time scales for piezoelectric energy harvesters: corresponding to the natural frequency of the piezoelectric structure, the regulating circuit oscillating frequency and the ambient driving frequency, respectively. For this reason, an analysis of piezoelectric energy harvesters has to be made by proceeding from the interaction between the three time scales so that piezoelectric energy harvesters of excellent performance can be obtained through integral analysis and global optimization.

Fig. 3.1 The structure of a piezoelectric energy harvester.

3.3 Piezoelectric power harvesters 3.3.1 The related researches on piezoelectric power harvesters As a piezoelectric power harvester directly provides power to microelectronic devices without storing energy, its main research is concentrated on performance optimization of the energy-harvesting structure. Earlier research focused on optimizing harvesting structures using equivalent-circuit technique. In the equivalent-circuit technique [10, 11], external force is likened to voltage, velocity to current, and mechanical impedance to electrical impedance, making the mechanically vibrating system equivalent to a circuit system. The 1 Form the coupled electromechanical governing equations and solve basic steps are: ° out the expressions of the quantities, such as displacements and current etc., in terms 2 Set up the impedance matrix using the linear of those to-be-determined coefficients; ° 3 Set up the equivalent-circuit using the circuit electromechanical analogy relationship; ° 4 Perform operation with network theory and by means of the impedance matrix; and ° respect to the equivalent-circuit so as to make it conform with the given boundary conditions. Despite the great ease in using the equivalent-circuit technique, there is always some inappropriateness to simulate the vibration of a continuum structure with a few discrete circuit parameters. Hence, although Umeda et al. [12, 13], Cho et al. [14] and Ha [15] have obtained some conclusions of significance by investigating the bimorph piezoelectric power harvester using this technique, they haven’t given serious consideration to the coupling between the energy harvesting structure and the energy-storage circuit. Keaw-

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boonchuay et al. [16, 17] regarded the voltage and current generated from piezoelectric harvesting end as two independent variables, thereby decoupling the energy-harvesting structure from energy storage device. It was not until recently after Yang et al. [18, 19] and Jiang et al. [20], proceeding from the 3D piezoelectric theory, have established the governing equations for energy harvesting end by simulating the energy consuming device as an impedance and linking the piezoelectric structure output voltage and current with the generalized Ohm’s law that a coupling analysis on piezoelectric power harvester is really accomplished.

3.3.2 Coupling analysis of piezoelectric power harvesters Now we take a piezoelectric bimorph power harvester as an example to show the coupled analysis of piezoelectric power harvesters. As shown schematically in Fig.3.2, the bimorph consists of a pair of identical piezoelectric twin layers, poled along the thickness direction and separated by a metallic layer in the middle [21]. One end of the bimorph is cantilevered into the wall which vibrates along the vertical direction harmonically with a known amplitude A at a given frequency ω , and the other end is connected with a concentrated mass m0 which is used to adjust the vibration characteristics of the structure for optimal performance in converting vibration energy into electric energy. The two electrodes, placed upon the upper and lower surfaces of the bimorph, and the middle metallic layer are connected to the load circuit, whose impedance is denoted by ZL . In this section, we are interested in the performance of this model bimorph and its dependence upon the physical parameters, i.e., the input frequency, the load impedance, and the attached mass.

Fig. 3.2 A piezoelectric bimorph with an elastic layer and an end mass.

We consider the flexural motion of the bimorph in the x3 direction, assuming that the bimorph is slim, i.e., the length l is much larger than the thickness 2(c + h) and the width b. In the flexural motion of such a thin bimorph, the axial strain S1 (x1 ,t) is expressed in terms of the deflection u3 (x1 ,t) as below: S1 = −x3 u3,11

(3.1)

The electric field in the ceramic layers corresponding to the electrode configurations, shown in Fig.3.2, is of the following components:

3

E1 = 0,

E2 = 0,

Piezoelectric Power/Energy Harvesters

E3 = −V /h

77

(3.2)

where V denotes the voltage across each of the two piezoelectric layers. Since the electric potential is constant on each of the electrodes and on the metallic middle layer, V is spatially constant, though it varies with time. Under the usual one-dimensional stress approximation of a beam [22], we have the following stress components: T1 = T1 (x1 ,t), T2 = T3 = T4 = T5 = T6 = 0

(3.3)

The relevant constitutive relations for the piezoelectric ceramic layers can be written as [13] S1 = s11 T1 + d31 E3 , (3.4) D3 = d31 T1 + ε33 E3 In the above, we denote, by s11 , ε33 and d31 , the axial elastic compliance measured with fixed electric field, the transverse dielectric constant measured with fixed stress, and the transverse-axial piezoelectric coefficient. From Eq.(3.4) we solve for the axial stress T1 and the transverse electric displacement D3 as follows: −1 T1 = s−1 11 (−x3 u3,11 ) − s11 d31 E3

(3.5a)

D3 = s−1 11 d31 (−x3 u3,11 ) + ε¯33 E3

(3.5b)

where Eq.(3.1) has been used and 2 ε¯33 = ε33 (1 − k31 ),

2 2 k31 = d31 /(ε33 s11 )

(3.6)

The metallic layer in the middle is assumed to be elastic and its constitutive relation is stated as below: T1 = ES1 = E(−x3 u3,11 ) (3.7) where E is the Young’s modulus. The bending moment is defined by the following integral over a cross section of the beam, which can be integrated explicitly with Eqs.(3.5a) and (3.7): Z V (3.8) M = x3 T1 dx2 dx3 = −Du3,11 + s−1 11 d31 2G h where the bending stiffness D is defined as ½ ¾ 2 3 2 −1 D= Ec + s11 [(c + h)3 − c3 ] b, 3 3 µ ¶ (3.9) h G = c+ hb 2 and G is the first moment of the cross-sectional area of one of the ceramic layers about the x2 axis, shown in Fig.3.2. The shear force in the beam is given by

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Z

N=

T13 dx2 dx3 = M,1 = −Du3,111

(3.10)

The flexural motion of a slim beam is governed by the following equation in terms of the bending moment: M,11 = mu¨3 (3.11) where the mass per unit length of the beam is given as m = ρ 2cb + 2ρ 0 hb

(3.12)

In the above, we denote by ρ and ρ 0 , respectively, the mass densities of the metallic and piezoelectric layers. The electric charge on the top electrode at x3 = c + h is given by [23] Z l

Qe = −b D3 (x3 = c + h)dx1 ½ 0 ¾ V −1 = b s11 d31 (c + h)[u3,1 (l,t) − u3,1 (0,t)] + ε¯33 l h

(3.13)

The current flowing out of this electrode is I = −Q˙ e

(3.14)

When the motion is time-harmonic, the output voltage and current are related by 2I = V /ZL

(3.15)

We note that the above representation of the output circuit effect with a complex impedance ZL is oversimplified for a realistic circuit, but this representation has often been used to evaluate the efficiency in power conversion of such piezoelectric devices. Substitution of Eq.(3.8) into (3.11) yields −Du3,1111 = mu¨3 (3.16) At the cantilevered end of the beam the boundary conditions are u3 (0,t) = A exp(iω t), u3,1 (0,t) = 0

(3.17)

At the right end we have no bending moment but a shear force balancing the inertia force of the attached mass, i.e., M(l,t) = 0, (3.18) −N(l,t) = m0 u¨3 (l,t) For harmonic motions we use the complex notation {u3 (x,t),V, Qe , I} = Re{{U(x), V¯ , Q¯ e , I¯ } exp(iω t)}

(3.19)

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Then Eqs.(3.16)-(3.18) become −DU,1111 = −ω 2 mU (0 < x1 < l), U(0) = A, U,1 (0) = 0, V¯ −DU,11 (l) + s−1 11 d31 2G = 0, h DU,111 (l) = −ω 2 m0U(l)

(3.20)

The general solution to Eq.(3.20)1 can be written as U = B1 sin α x1 + B2 cos α x1 + B3 sinh α x1 + B4 cosh α x1

(3.21)

where B1 -B4 are constants to be determined, and

α=

³m D

ω2

´1/4

(3.22)

Substitution of Eq.(3.21) into Eqs.(3.13) and (3.14) gives the following expression for the current: ½ ¯I = −iω b d31 (c + h)[B1 α cos α l − B2 α sin α l s11 ¾ V¯ +B3 α cosh α l + B4 α sinh α l] + ε¯33 l (3.23) h With the complex notation, the output electrical power is given by 1 P2 = (I¯V¯ ∗ + I¯∗V¯ ) 4

(3.24)

where an asterisk represents complex conjugate. To calculate the mechanical input power, we need the shear force in the beam at the left end. From Eqs.(3.21) and (3.10) we have N(x1 ) = −Dα 3 (−B1 cos α x1 + B2 sin α x1 + B3 cosh α x1 + B4 sinh α x1 )

(3.25)

The velocity of the left end of the beam is given by u˙3 (0,t) = iω A exp(iω t). This leads to the following expression for the input mechanical power: 1 P1 = − [N(0) · (iω A)∗ + N ∗ (0) · (iω A)] 4

(3.26)

The efficiency that describes the device’s ability in energy conversion is defined as

η=

P2 P1

(3.27)

For miniaturized power harvesters, the power density, defined as the output power per unit volume, serves as an important measure for the device performance. For our model

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piezoelectric bimorph, the power density is given as below: p2 =

P2 b(2c + 2h)l

(3.28)

3.3.3 Numerical results To illustrate the performance of this model bimorph and its dependence upon the physical parameters, we present in this section some numerical results generated using PZT-5H [24] as a model piezoelectric material of mass density ρ = 7500 kg/m3 and following elastic, piezoelectric and dielectric constants: s11 = 16.5 × 10−12 m2 /N, s33 = 20.7 × 10−12 m2 /N, s44 = 43.5 × 10−12 m2 /N, s12 = −4.78 × 10−12 m2 /N, s13 = −8.45 × 10−12 m2 /N, d31 = −274 × 10−12 C/N, d15 = 741 × 10−12 C/N, d33 = 593 × 10−12 C/N, ε11 = 3130ε0 , ε33 = 3400ε0 (3.29) For piezoelectric ceramics, damping is often included in modeling by allowing the elastic constants to assume complex values [24], and thus in our numerical calculations, the real elastic constant s11 is replaced by s11 (1 − iQ−1 ), where Q is the quality factor of material. For ceramics Q is on the order of 102 to 103 [25]. We fix Q to be 102 in our calculation. l = 25 mm, b = 8 mm, h = 0.31 mm, the acceleration amplitude ω 2 A = 1.0 m/s2 are fixed in all calculations. The elastic layer is taken to be aluminum alloy with Young’s modulus E = 70 GPa and mass density ρ = 2700 kg/m3 . We plot, in Fig.3.3, the output power density versus the input frequency for load circuits of normalized impedances ZL /(iZ0 ) = 1/5, 1, and 5, respectively, representing three different resistor loads. In the above, Z0 = (iω C0 )−1 stands for a structure parameter with C0 = ε¯33 bl/h. The resonant behavior of the system shown in Fig.3.3 is for the first and the lowest flexure mode, although this system has a series of resonant frequencies corresponding to various flexural modes. It is seen that the system resonates at a frequency near 5 kHz, and both the resonant frequency and the corresponding output power density peak

Fig. 3.3 Power density versus frequency for different impedances ZL (c = h/2, m0 = 0).

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vary considerably with the normalized load impedance. For further illustration, we plot the output power density versus the load impedance, in Fig.3.4, for inputs of frequencies

Fig. 3.4 Power density versus impedance ZL for different frequencies ω (c = h/2, m0 = 0).

4.90 kHz, 5.00 kHz and 5.16 kHz, respectively. At each of these frequencies, the output power density increases essentially linearly for small loads, reaches a maximum, and then decreases monotonically for large loads. Noting that the normalization impedance Z0 is a simple combination of the physical and geometrical parameters of the piezoelectric bimorph and the input frequency, we underscore, with these results, the importance for the load circuit to have impedance desirable by the bimorph, the scavenging structure. We note that the efficiency of the system in converting the input mechanical energy into the output electric energy is insensitive to the input frequency near 5 kHz, as seen in Fig.3.5, in contrast to the output power density which varies considerably with the input frequency, and that the efficiency becomes frequency-sensitive only at very high frequencies, as shown in Fig.3.6. Noting that the output power density is often critically important to a miniaturized power device, we have plotted in Figs.3.7 and 3.8 the power density versus the input frequency for model piezoelectric harvesters of various dimensions and different masses attached. Considering that ambient vibrations are usually of frequencies far lower than 5 kHz, we note, as seen in Figs.3.7 and 3.8, that one can substantially decrease the resonant frequency of the system by reducing the elastic layer thickness or increasing the attached mass, and that both of these adjustments can significantly increase the output power density of the system.

Fig. 3.5 Efficiency versus impedance ZL for different input frequencies ω (c = h/2, m0 = 0).

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Fig. 3.6 Efficiency versus input frequency ω for different impedances ZL (c = h/2, m0 = 0).

Fig. 3.7 Power density versus ω for different thickness ratios c/h (m0 = 0, ZL = iZ0 ).

Fig. 3.8 Power density versus input frequency ω for different end masses m0 (c = h/2, ZL = iZ0 ).

Usually, the ambient vibration frequency is rather low, in general about dozens of Hertz to 100 Hz or even lower. For instance, 3000 r/min of an automobile engine is 50 Hz. In order to make the resonance frequency of the piezoelectric structure close to the ambient driving frequency, the common used cantilever beam type power harvester has to have a very high aspect ratio, or a very large mass should be applied on the free end. Both will make the energy harvester too long in a single direction, which would be unfavorable to

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device miniaturization. Hence Hu et al. [26] proposed a spiral-shaped bimorph structure for the piezoelectric power harvester that can not only effectively reduce the resonance frequency but also help in miniaturizing the structure. Composed of many semicircles of different radii as shown in Fig.3.9, such a structure is characterized by its ability to greatly increase the length-thickness ratio of the structure without worrying about the length of the power harvester getting too large. It is shown by analysis that even if the additional mass m0 is very small, the system resonance can still be reduced to 50 Hz or lower.

Fig. 3.9 A spiral-shaped piezoelectric bimorph.

3.3.4 Investigation on frequency shift of piezoelectric power harvesters In view of the possibility that the ambient vibration frequency may vary and the fact that the piezoelectric structure once formed, its type, dimensions etc., can hardly be changed. It is necessary to design an effective method for regulating the natural frequency of a piezoelectric energy-harvesting structure so as to ensure that, when the ambient vibration frequency varies, the energy harvester can still high efficiently extract energy from the environment. In their discussion of the regulation of resonance frequency for the energy harvester, Roundy and Zhang [27] proposed two frequency regulating methods: active and passive frequency regulations. The former refers to a steady input of energy to the system to enable the power harvester’s natural frequency to match the driving frequency from beginning to end, while the latter refers to the situation where it is only initially necessary to input energy to regulate the natural frequency of the harvester and, once the harvester is operating in the resonant state, input of energy would be stopped. Wu et al. [28] has designed an active frequency-regulating technique using the microcontrol to

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change the natural frequency of the piezoelectric bimorph that consists in mounting the microcontrol on the upper pad of the bimorph with the lower pad responsible for energy conversion. As the microcontrol is capable of changing the capacitance-impedance value of the piezoelectric pad it has glued, it has changed the effective rigidity of the beam and then its natural frequency. Experimental results also show that it is possible to increase the harvester’s average conversion energy by 27.4% with the power harvester under the stochastic vibration condition. Wu et al. [29] also proposed the method of regulating the resonance frequency of a piezoelectric cantilever beam using a center-of-gravity variable mass block, which consists of two parts, one fixed at the free end while the other is a movable bolt mounted on the fixed part. When the bolt moves, the center-of-gravity of the entire mass block undergoes changes till the resonance frequency of the cantilever beam is changed. It is shown by FEM simulation and experimental results that the regulating range of the resonance frequency is 130-180 Hz. Hu et al. [30] and Leland et al. [31] have designed an effective method of biasing field regulation of the power harvester’s resonance frequency by applying the axial force with a bolt. Furthermore, the axial force can either be applied by push or drag, thus achieving the aim of bidirectional frequency regulation. Figure 3.10 shows a schematic illustration of a harvester with axial preload (biasing field).

Fig. 3.10 A schematic illustration of a harvester with an axial preload.

The frequency regulation in Fig.3.10 can be analyzed as follows. The governing equations [30, 32] are Du3,1111 + Fa u3,11 = −mu¨3 ,  −1 −1   T1 = s11 (−x3 u3,11 ) − s11 d31 E3 in c < |x3 | < c + h, D3 = s−1 11 d31 (−x3 u3,11 ) + ε¯33 E3   E1 = 0, E2 = 0, E3 = −Vp /h ½ T1 = E(−x3 u3,11 ) in |x3 | < c φ =0 and the boundary conditions are

(3.30)

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u3 (0,t) = A exp(iω0t),

u3,1 (0,t) = 0, d31Vp M(l,t) = −Du3,11 (l,t) + 2G = 0, s11 h −N(l,t) = Du3,111 (l,t) + Fa u3,1 (l,t) = 0

(3.31)

The charge on the electrode can be obtained from Eq.(3.13) and the output current and voltage have the relationship as shown in Eqs.(3.14) and (3.15). Under a fixed piezoelectric layer width b=10 mm, we choose the piezoelectric layer thickness h=0.50 mm, the metal layer thickness 2c=0.40 mm and the beam length l=74 mm, while the bimorph volume is kept as 2lb(c + h) = 1 cm3 . It is readily obtained that the fundamental natural frequency of this piezoelectric bimorph without any axial preload is 129.3 Hz. In addition, the input acceleration amplitude is kept ω02 A = 1.0 m/s2 in the calculation. Figure 3.11 shows dependence of output power density p2 upon the axial preload for three different external driving frequencies, f =120 Hz, 129.3 Hz, 140 Hz, respectively. Obviously, each external driving frequency is with an optimal axial preload, where the output power density reaches maximum. This maximum is corresponding to the resonance of damping forced vibrations. For example, when the driving frequency f = 120 Hz, 129.3 Hz, 140 Hz, a compressive/tensile axial preload Fa = −11.6 N, 0 N, 9.1 N should be applied at the right edge of the bimorph to induce the bimorph resonance. Moreover, the improvement effect of a proper axial preload becomes more obvious when the natural frequency deviates more from the driving frequency, where the power density becomes much smaller because no resonance happens under that situation.

Fig. 3.11 Output power density versus axial preload.

Figure 3.12 presents the output power density versus the driving frequency for different axial preloads, Fa = −50 N, 0 N, 50 N. It can be observed that a tensile preload Fa = −50 N can induce the resonant frequency of the piezoelectric bimorph to increase from 129.3 Hz to 169.4 Hz, and conversely, a compressive preload Fa = 50 N can make the resonance shift low from 129.3 Hz to 58.1 Hz. This phenomenon indicates that the resonant frequency of a piezoelectric bimorph is able to be adjusted high or low by applying a proper preload according to external driving frequency. In addition, it can be observed

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from Fig.3.12 that a tensile preload arises p2 near resonance better than a compressive preload does, which implies that the natural frequency of a piezoelectric harvesting structure should be designed as low as possible in advance so that it can be adjusted as far as possible through tensile preloads whenever necessary.

Fig. 3.12 Output power density versus driving frequency.

To make sure the mechanism of the above technique, Fig.3.13(a) shows the curvature 1/R0 at the free edge of the bimorph versus driving frequency for three different axial preloads Fa = −50 N, 0 N, 50 N. It follows that 1/|R0 | obtains maximum at resonance, i.e., the piezoelectric bimorph is deformed most seriously at resonance. The more serious mechanical deformation results in more energy conversion from mechanical-to-electric, which is why we always hope to design piezoelectric harvesting structures at their resonant states. It can also be observed from Fig.3.13(a) that a positive/negative preload (compressive/tensile) always lowers/enlarges 1/|R0 |, which results from the smaller/larger de-

Fig. 3.13 (a) Curvature 1/R0 at x1 = l versus driving frequency; (b) Deflection profile along the x1 -axis.

flection amplitude of the bimorph when subjected to a positive/negative axial preload, see

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Fig.3.13(b). This is consistent with the phenomenon observed in Fig.3.12 that a tensile axial preload raise the output power density and a compressive axial preload lowers the power density. Furthermore, Table 3.2 compares the two power densities of a piezoelectric bimorph harvester with and without a preload, which shows very obvious improvement on harvester performance by a proper preload. Table 3.2 Comparison of system output power density for different vibration source. p2 /(µW/cm3 )

ω02 A/(m/s2 )

f /Hz

With axial force

Without axial force

Blender casing

6.4

121

1514.0

167.3

Clothes dryer

3.5

121

452.8

50.0

Door frame just after door closes

3

125

349.3

108.1

Small microwave oven

2.5

121

231.0

25.5

Windows next to a busy road

0.7

100

13.8

0.17

Second story floor of busy office

0.2

100

1.1

0.01

Vibration source

The efficiency versus the driving frequency is shown in Fig.3.14 for different axial preloads. The system efficiency, defined as the quotient of the output power to the input power, keeps high at its operating frequency, although it does not peak there. We note the efficiency is higher than the electromechanical coupling factor of the material. This causes no contradiction because the coupling factor is defined for static processes of energy conversion and does not directly apply to a dynamic problem [33].

Fig. 3.14 Conversion efficiency versus driving frequency.

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3.3.5 Broadband design of piezoelectric power harvesters Shahruz [34], Xue et al. [35] and Ferrari et al. [36] have designed power harvesters that can effectively operate within a broad frequency range without the need of frequency regulation. Its implementation consists in having a number of cantilever beams of different aspect ratios with additional mass made into a power harvester, with the natural frequency of any one of the beams different from the others. If the cantilever beam’s length and additional mass are rationally designed, when the ambient vibration frequency varies in a certain range, there is always one cantilever beam that can operate in the vicinity of resonance, making it look as if the whole system was capable of operation within a broad frequency range. Compared with other schemes for frequency regulation, such a cantilever beam array power harvester has the merit of effective operation within a fairly wide frequency range with no need for additional frequency regulation control devices, avoiding extra energy consumption. The drawback is that its volume can not be too small. The piezoelectric harvester consists of three major components: a main stanchion, a series of piezoelectric bimorphs (PBs) with different thicknesses and a load resistor (Fig.3.15). The main stanchion is fixed on the base which is subjected to multi frequencyspectra environment vibrations. Every PB consists of a pair of identical piezoelectric layers with thickness h, poled along the thickness direction and separated by a metallic layer (thickness 2c) in the middle. The piezoelectric ceramic layers are coated with thin electrodes at their top and bottom surfaces, and the two piezoelectric layers are connected in series in each bimorph. One end of every bimorph is cantilevered into the main stanchion, and the other is free, while the base vibrates in the vertical direction harmonically with amplitude A and frequency f . The energy storage element is not the focus of this section, thus it is represented by simple load impedance.

Fig. 3.15 A schematic illustration of a PBs harvesting system.

Figure 3.16 schematically shows the connection patterns of a PBs system. The whole PBs system consists of n branches in parallel and every branch contains m PBs with

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different thicknesses in series. Whether a PBs branch is on duty or not is determined by the on/off state of the corresponding pilot switch Si (i = 1, 2, · · ·, n).

Fig. 3.16 Connection patterns of PBs system.

In calculations, we use PZT-5H [24] as the piezoelectric material, while the middle metallic layer is taken to be aluminum alloy, their material parameters can be referred to Eq.(3.29) and its following text. Let the piezoelectric layer width b = 10 mm, c/h = 0.4, and the beam length l = 70 mm. The input acceleration amplitude is kept ω02 A = 1.0 m/s2 with ω0 = 2π f . The effective capacitance and " impedance#related to the piezoelectric bin

m

i=1

j=1

morphs can be calculated by C0 = ε¯33 bL ∑ 1/ ∑ h(i, j) and Z0 = 1/(iω0C0 ), respectively. In the calculations, the external load impedance is set to ZL = iZ0 . To make clear the characteristic of a multi bimorph harvesting system step by step, the performance of harvesters consisting of a few PBs in series was examined first. Figure 3.17 shows the dependence of output power on driving frequency for two cases: a single bimorph harvester with h = 0.33 mm; and ten bimorphs harvester in series with h = 0.31 mm, 0.315 mm, 0.32 mm, 0.325 mm, 0.33 mm, 0.335 mm, 0.34 mm, 0.345 mm, 0.35 mm, 0.355 mm, respectively. Comparison of these two cases indicates that not only the output power increases with the use of more bimorphs, but also the frequency-band with output power higher than 10 µW has been widen from (97, 103)Hz to (87, 115)Hz. Clearly the operating frequency band (OFB) can be further extended with more PBs in series. The performance of harvesters consisting of multiple PBs in parallel was also examined. Figure 3.18 shows the dependence of output power on driving frequency for three cases: a single bimorph harvester with h = 0.33 mm and 0.34 mm, respectively; and a harvester with two bimorphs in parallel with h = 0.33 mm, 0.34 mm, respectively. It is found that the resonant frequency of a harvesting system is shifted with increasing a new PB in parallel. The frequency shift is due to the change in the electrical boundary condition when increasing or decreasing PBs in parallel. When a harvester operates in an environment with multi frequency-spectra, it is desirable to design the harvester with a tailorable OFB. Thus, multiple PBs with different

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Fig. 3.17 Dependence of output power on driving frequency for two cases: one is for a single PB and the other is for ten PBs in series with h = 0.31 mm, 0.315 mm, 0.32 mm, 0.325 mm, 0.33 mm, 0.335 mm, 0.34 mm, 0.345 mm, 0.35 mm, 0.355 mm.

Fig. 3.18 Effect of PBs in parallel on harvester performance.

resonant frequency can be connected in series to widen the OFB of a harvesting structure. The branches with PBs connected in series can then be connected in parallel to move the OFB to desirable frequency band. Figure 3.19 shows the output power as a function of frequency for a piezoelectric harvester with three branches (m = 10, n = 3; h = 0.31 mm, 0.315 mm, 0.32 mm, 0.325 mm, 0.33 mm, 0.335 mm, 0.34 mm, 0.345 mm, 0.35 mm, 0.355 mm; 0.33 mm, 0.335 mm, 0.34 mm, 0.345 mm, 0.35 mm, 0.355 mm, 0.36 mm, 0.365 mm, 0.37 mm, 0.375 mm; 0.35 mm, 0.355 mm, 0.36 mm, 0.365 mm, 0.37 mm, 0.375 mm, 0.38 mm, 0.385 mm, 0.39 mm, 0.395 mm). A pilot switch is employed in every PB branch. Figure 3.19 shows the changes in OFB of the harvester by controlling the on/off states of Si (i = 1, 2, 3). Clearly, the results indicate two points: 1 OFB of a harvesting device is able to be obviously widened by connecting multiple °

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PBs with different thicknesses in series so that the harvester can effectively scavenge en2 The location of harvester OFB can ergy from the multi-frequency ambient excitation; ° be moved to different frequency domain by adding or decreasing some PBs branches in parallel through controlling the on/off states of pilot switches. In general, OFB of a traditional energy harvesting device will be fixed, and thus impossible to be adjusted by changing its structure configuration or material according to external forcing, once it has been manufactured. When the frequency of ambient vibration changes, it becomes impossible with such a traditional harvester to scavenge energy from the variable external forcing. Our research on this topic shows that we can effectively adjust OFB through adding or decreasing some PBs branches in parallel.

Fig. 3.19 Movability of the OFB of a PB harvesting system by connecting PBs in mix-pattern.

3.4 Piezoelectric energy harvesters 3.4.1 Component portions of piezoelectric energy harvesters For a piezoelectric energy harvester, there exists not only problems to optimize the energyscavenging structure as before, but also to improve the coupling performance between the energy-scavenging structure and the energy-storage circuit, and to design some auxiliary components to enhance energy conversion efficiency. It is customary to connect an SSHI to the piezoelectric structure in parallel (Lefeuvre et al. [37], Badel et al. [38], and Guyomar et al. [39]): When the piezoelectric structure vibrates to the positive or negative maximums, the rectifying bridge turns from closed to open circuit. The closure of KI at this moment makes the inductor LI in SSHI form an oscillator LI -Cp with the piezoelectric structure equivalent capacitor Cp . If the closing time is set as the half-period of the LI -Cp oscillator, it’s just fit to reverse the output voltage of the piezoelectric structure. Of course, there exists energy loss in the process of reversal. Thus, the LI -Cp os-

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cillator can turn the piezoelectric structure’s output voltage Vp from Vrect · sgn(Vp ) only into [Vrect · sgn(−Vp )]e−π/2QI , QI > 0 being the quality factor. Because the output voltage phases between two adjacent closed-circuit periods are opposite each other, the voltage reversion of LI -Cp is just to make the next closed-circuit period of the energy-harvesting structure arrive earlier, that is, an SSHI has shortened the piezoelectric structure’s opencircuit interval while prolonging its closed-circuit interval. It is well known that the piezoelectric structure’s closed-circuit rigidity is smaller than the open-circuit rigidity. The introduction of an SSHI makes the piezoelectric structure more easily deformable, thereby improving the energy converting efficiency. It is proved by both numerical calculation and test analysis that the energy-storage efficiency of circuit using the SSHI technique is far higher than a circuit that merely uses the rectifying bridge and capacitor, probably 4 times higher [40]. In order to store energy, it is necessary to rectify the AC energy output by a piezoelectric structure into DC electric energy. Ng and Liao [41, 42] input energy to an energystorage capacitor by rectification with a diode. The physical quantity, voltage, was used to monitor the circuit and the capacitor voltage was tested wherever necessary. Once a certain preset value is exceeded by the capacitor voltage (called release voltage), the regulating circuit would begin to release energy to load resistance. When the capacitor voltage is lower than another preset value (called monitoring voltage), the regulating circuit would stop energy release. Such a circuit is the most fundamental energy-storage circuit, but is of low efficiency (about 46%). Sun et al. [43] have designed a circuit that can effectively extract energy from microscale piezoelectric energy-scavenging structures. This kind of circuit is made up of two parts, the synchronous rectifier and the charge-pump type DCDC converter composed of capacitor and MOSFET to facilitate integration into the single chip. The function of the synchronous rectifier is to enhance efficiency. It is shown by results of both theoretical and experimental analysis that, compared with the conventional diode rectifier, the energy extracted from the energy harvesting end using the synchronous rectifier is over 4 times greater. Ottman et al. [44] and Lesieutre et al. [45] matched the rectifying voltage with battery voltage by introducing a step-down DC-DC converter, thus increasing the system energy storing efficiency. Research results show that the optimal duty cycle varies abruptly with the variation of the drive frequency. When the drive frequency is very high, the converter is excited with the duty cycle stabilized somewhere around the optimum value. When the drive frequency is rather low, however, the duty cycle optimum value of the converter will vary frequently, leading to the energy consumed exceeding the energy harvested. Hence the converter will stop work in the circumstances and current is charged directly through the rectifying circuit. Ammar et al. [46] have described an adaptive algorithm for controlling the duty cycle of the DC-DC step-down converter that consists in initially setting a rather small duty cycle while monitoring the battery current. Then, the duty cycle should be increased step by step with the battery current: If the battery current is increased after the increase of duty cycle, the duty cycle should be further increased until the battery current is no longer increasing. Lefeuvre et al. [47] have improved such an adaptive circuit. They had the charge extracting process on the piezoelectric energy harvesting unit synchronized with the system vibration so as

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to enhance the energy transmitting efficiency, referred to as “synchronous charge extraction”, which is a nonlinear process. The circuit for implementing charge synchronization consists of a diode rectifying bridge and a feedback switchover DC-DC converter. The operating process of such a circuit consists in monitoring the voltage at the two ends of the diode rectifying bridge via a control circuit. Once the voltage reaches maximum value, the feedback converter will be activated and the charge on the piezoelectric unit will begin to transmit to the battery. When the charge is completely extracted, the control circuit turns off the converter and stops charge transmission. When the monitoring voltage reaches maximum once again, the system enters the next work cycle, and the charge extraction is in synchronization with the vibration. It is shown by experimental analysis that, compared with the circuit that adopts the linear impedance type converter, adoption of the synchronized converter will make it possible to increase the energy transmitting efficiency over 4 times higher. Chao et al. [48] have designed a new adaptive circuit with rather low energy consumption that is mainly composed of a vibration tracking element, a control element, a step-down converter, and a pulse generator. Analysis shows that there exists an optimal output voltage that makes the energy-harvesting efficiency maximum, while this optimal value is related to the ambient vibration state (amplitude and frequency). As the vibration state in an environment is variable, it is necessary to regulate the output voltage according to the vibration state so as to attain the goal of maximal energy-harvesting efficiency. Chao used a vibration tracking unit to monitor the variation of the state of vibration while regulating the voltage output via a control unit regulating system. In view of the rather slow variation in the vibration state in an everyday environment, the vibration tracking unit is only periodically connected into the system via an MOS switch controlled with a pulse generator. Analysis results show that, with this technique, not only is the energy storing efficiency rather high, but also the energy consumption of the adaptive circuit is very low, merely at the microwatt level. Hu et al. [49] have introduced into the energy-storage circuit with a Buck-Boost type or a Cuk type DC-DC converter aimed to enable the energy harvester to extract energy not only from fairly strong ambient vibrations but also effectively do so from rather weak ambient vibrations. On the basis of an SSHI circuit, Lallart et al. [50] proposed a double synchronous switching circuit (DSHI), which is tantamount to connecting an intermediate capacitor in parallel in the SSHI circuit. When the voltage on the piezoelectric unit reaches maximum, the first control switch turns off and the charge transfers from the piezoelectric unit to the intermediate capacitor. Upon completion of charge transfer, the first control switch turns on. The electrical energy now transmits from the intermediate capacitor to the inductor of the Buck-Boost converter. When the transfer is completed, the second control switch turns on. The inductor of the Buck-Boost converter charges the ultimate energy storing capacitor and then enters the following cycle. As piezoelectric energy harvesters possess the energy-storage function, it is necessary to convert the AC generated by the piezoelectric harvesting end into DC acceptable to the energy-storage circuit. Hence, there exists a rectifying interface between the energy harvesting structure and the energy storing circuit, leading to the coupling between structure and circuit being nonlinear. There is the need to develop a nonlinear analytic model

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for piezoelectric energy harvesters. According to Ref.[51], for the energy-storage circuit the vibration at the piezoelectric end can be considered by introducing a linear mechanical damping term into the governing equation(s) for piezoelectric end vibration. Obviously, such a simulation of the influence of a circuit on structure can not truly reflect the operation behavior of the whole system, since mechanical damping will influence the open/closed state of the rectifying interface while the circuit impedance will not. Although Lu et al. [52] have introduced a rectifying interface between structure and circuit, their further reduction, or reduction of the energy-storage circuit to pure resistance, causes the rectifying interface nonlinear features to vanish completely. Despite their introduction of a DC-DC converter between energy structure and energy-storage battery, Ottman et al. [44, 45, 53] simulated the energy harvesting end merely as a simple current source, which is no real coupling of the structure with the circuit. In recent years, Hu et al. [54] have just adopted the piezoelectric bimorph as their energy-harvesting component and a chargeable battery for storing energy. Relative to a given ambient vibration excitation level there exists a rectifying voltage that gets maximum energy from the piezoelectric energy-harvesting structure. As the battery voltage is not equal to the optimum rectifying voltage, to make the battery voltage and the rectifying voltage (i.e., the voltage of the rectifying capacitor Crect ) match each other, it is necessary to install a DC-DC converter [55].

3.4.2 Integrated analysis of piezoelectric energy harvesters Figure 3.20 shows the three major components of a cantilevered piezoelectric bimorph energy harvester, i.e., a cantilevered piezoelectric bimorph energy harvesting structure connected in parallel to SSHI, the rectifying circuit with a DC-DC converter, and the chargeable battery. Usually, the ambient vibration frequency is very low (about dozens to hundreds of Hertz while the operation frequency of the rectifying circuit with a DCDC converter is in general rather high (about several thousand Hertz and above). As the vibration cycle of the energy-harvesting process is not the same as that of the energy storing process, the intermediate element Crect that links the two parts is equivalent to an energy relay station that should have considerably large capacity to ensure that, within a definite period of time, the charge output from the energy-harvesting end will not greatly change the voltage level of Crect . That is, Crect should be far greater than the piezoelectric structure’s equivalent capacitor Cp . Under this situation, the energy-harvesting process and the energy-storage process can be calculated separately. The former can be calculated from Crect and the left structure with the time scale determined by the external excitation frequency, while the latter can be calculated from Crect and the right circuit with the time scale determined by the circuit oscillation frequency. The two get coupled when the harvested energy is equal to the stored energy. In the energy-harvesting end: The piezoelectric bimorph consists of a pair of identical piezoelectric twin layers, poled along the thickness direction and separated by a metallic

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Fig. 3.20 A schematic of a piezoelectric bimorph energy harvester with a DC-DC converter.

layer in the middle. The ceramic layers are coated with electrodes at their top and bottom surfaces, which are shown by the thick lines in Fig.3.20. The electrodes are connected to a full-bridge rectifier and then the capacitor Crect . One end of the bimorph is cantilevered into the wall, which vibrates along the vertical direction harmonically with a given frequency ω0 . An SSHI is applied to artificially increase the conduct duration of the rectifier and the average output power density of the piezoelectric structure. Thus, an electric condition on the upper electrode in Fig.3.21 is complemented below: ½

ip = 0 (for an open-circuit stage), Vp = ±Vrect (for a closed-circuit stage) 2ip = ±irect

(3.32)

where irect is the output current of the rectifier. Moreover, continuity in displacement and velocity should be imposed at the transition moments between open- and closed-circuit.

Fig. 3.21 A schematic illustration of the energy-scavenging circuit.

Since Cp ¿ Crect , the amount of energy scavenged by the harvesting element in finite time periods produces little influence on the voltage level of Crect such that the scavengingenergy process can be considered as periodic with T0 being the period. Calculation on the scavenging-energy process should be conducted under the time scale T0 . We define the average power density in T0 out from the harvesting structure as 1 γ= 2l(h + c)bT0

Z T0 0

2ipVp dt

and introduce a concept of average rectified current for the below need:

(3.33)

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hIrect i =

1 T0

Z T0 0

|2ip |dt

(3.34)

In the energy-storage end: In actual applications, the external vibrating level may not be constant, so it is not certain for the battery voltage Vb equal to the optimal rectified voltage Vrect obtained above. It is therefore needed for the circuit to have flexibility, i.e., ability to adjust the output voltage of the rectifier matched with the battery voltage effectively. For attaining this purpose, a DC-DC converter is introduced after Crect to step down Vrect in Fig.3.20. In the DC-DC converter with the given input voltage Vrect , a transistor M1 is applied to control the switching on and off durations (ton and toff ), where TS = ton + toff is called the switching period of the DC-DC converter, and fS =1/TS is called the pertinent switching frequency which should be chosen to be in a few kilohertz to a few hundred kilohertz range according to Mohan et al. [56] and Ottman et al. [44, 45]. Obviously, the time scale of the storage circuit, TS , is totally different from T0 . Since Crect is designed much larger, the charge drawn by the DC-DC converter from Crect in finite time period produces little effect on the voltage level of the rectified capacitor, which implies that the storage process can also be regarded to be periodic with TS as the period. The two-time scale approach should be used in analyzing this kind of energy harvesters. In addition, because the time scale corresponding to sc of charging an electrochemical battery is much larger than the above two time scales, it will be certain to take very long time to cause obvious change in the battery voltage by the scavenged energy. We therefore take battery voltage to be constant during the computation. For example, we fix s = sc in this section, i.e., the time that the stored charge in the battery reach 80% of its maxiumum capacity Qm . At this moment: Vb = Vb |s=sc , dVb /ds = (dVb /ds)|s=sc . From Mohan’s book [56], we fix fS = 1000 Hz, and introduce the duty cycle D as follows: ton D= (3.35) TS An inductor L is employed in the DC-DC converter to share the voltage difference Vrect − Vb during the transistor on duration and to store most energy fetched from Crect temporarily. A diode D1 is introduced to conduct the inductor L with C1 after the transistor M1 is off such that the temporarily stored energy in L can be further charged into C1 . The diode will close down automatically after the stored energy in L has been released over, i.e., at the moment of null current out from the inductor. This indicates that nonlinear interaction among Crect , M1 , D1 and L has successfully made Vrect stepped down to match with Vb for effectively charging. In the whole time, the filter capacitor C1 always keeps charging the battery. Thus, C1 is used to take in the scavenged energy from the inductor temporarily and to offer stable energy current to the electrochemical battery. The following is the circuit equations over one circuit period TS : When the transistor M1 is in an on state, the current, iL , flowing through the inductor L satisfies diL Vrect −Vb = dt L

(3.36)

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Once the transistor moves into the off state, the diode D1 will be closed to connect L with C1 until the current iL decreases gradually from its maximum to null, while iL satisfies diL Vb =− dt L

(3.37)

When iL becomes null, corresponding to the state that the temporarily stored energy in L has been released over, D1 is off again. This implies that there exist two sub-stages in an off state of the transistor M1 according to D1 ’s on or off. The inductor will output the whole stored energy to C1 during the on stage of D1 , and then there will not exist current through the inductor during the D1 ’s off state until the next on state of M1 . It should be noted that the capacitor C1 always keeps charging the battery in the whole circuit period TS . We represent the on and off intervals of D1 with ∆1 TS and ∆2 TS , respectively, and obtain from Eqs.(3.36) and (3.37) ·Z t +DT ¸ Z t1 +(D+∆1 )TS 1 S Vrect −V Vb b iL (t) − iL (t1 ) = dζ − dζ (3.38) L L t1 t1 +DTS for any t1 + (D + ∆1 )TS < t 6 t1 + TS with t1 as the beginning of an on duration of the transistor. For simplification in below analysis, we set t1 = 0. Obviously, iL (t) = iL (0) = 0 for (D + ∆1 )TS < t 6 TS , we therefore have Z DTS Vrect −Vb 0

L

dζ −

Z (D+∆1 )TS Vb DTS

L

dζ = 0

(3.39)

which gives a relationship between D and ∆1 : Vb D = Vrect D + ∆1

(3.40)

Considering ∆2 = 1 − D − ∆1 and Eq.(3.40), only one, for example D, among these three parameters requires to be determined according to the energy balance condition. Below we introduce how to determine I0 according to the vibration level of external forcing. For a fixed battery, its allowable maximal charging current is prescribed, written as Im in this section. If we choose Im the same as the maximal charging current offered by the harvester under an external excitation at ω02 A = 5 m/s2 , the battery charging current I0 for ω02 A 6 5 m/s2 can be determined with equality between the average output power from the rectified capacitor to DC-DC converter and the sum of the input power of the battery and the average input power of the filtering capacitor C1 . We assume the power absorbed by the battery and C1 at s = sc equal to the average power harvested by the piezoelectric element as follows: (I0 + iC )Vc = Vrect hIrect i (3.41) where iC = (1/TS )

Z TS 0

i1 dt, i1 denotes the current flowing into C1 . The power in the left

side of Eq.(3.41) is the average over TS from Crect to (Battery, C1 ), and the one in the right

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side of Eq.(3.41) is the average over T0 from the harvesting structure to Crect . Obviously, the two sides of Eq.(3.41) are average powers over two different time scales. As C1 is a filer capacitor, not an energy storage element, most of the energy should be designed to flow into the battery. Under that case, the capacitance C1 is not able to be too large so that the average current of C1 over TS is negligibly small, i.e., iC = 0, compared with I0 . We therefore have I0 = (Vrect /Vc )hIrect i (3.42) It follows that I0 increases monotonously with increasing A when ω02 A 6 5 m/s2 . Once ω02 A > 5 m/s2 happens, I0 will be directly fixed at Im , i.e., ( (Vrect /Vc )hIrect i (ω02 A 6 5 m/s2 ) I0 = (3.43) Im (ω02 A > 5 m/s2 ) The switch of SSHI is almost always open, except when a deflection extremum occurs, i.e., at a transition instant of the rectifier from a closed- to an open-circuit. We take the transition from a closed-circuit, with a positive current flowing out from the upper electrode, to an open-circuit as an example to illustrate the behavior of the SSHI in this chapter. One may discuss the other case with a positive current flowing out from the central electrode similarly. Hereafter, the above-mentioned different transition states will be called as TS1 and TS2, respectively, for simplification. At a TS1, the output voltage Vp between the upper and the central electrodes is positive, which implies that the next closed-circuit of the rectifier will start with Vp = −Vrect . Thus, it becomes surely easier for the earlier arrival of next closed-circuit if Vp can be immediately reversed at a TS1 or a TS2 [39]. After the added switch is closed at a TS1, the equivalent capacitance Cp of the piezoelectric bimorph and the inductance LI constitute an oscillator. Vp will be changed from Vrect to −Vrect e−π/(2QI ) when the LI -Cp oscillator is over [39]. Since QI is always positive, the magnitude of the reversed voltage is always smaller than Vrect , which implies the rectifier will be open initially after the LI -Cp oscillator is over. With decreasing deflection of the bimorph, Vp gradually changes into −Vrect so that next closed-circuit arrives. Then, Vp will block at −Vrect until the bimorph deformation reaches the negative deflection extremum. The above discussion indicates that nonlinear interaction between the rectifier and the SSHI together with Cp can greatly raise the output power of a harvesting structure. It is more convenient to take the first TS1 point during a vibration period T0 of the external forcing as the beginning to compute the piezoelectric structure vibration because the PB is just deformed to its positive deflection extremum and the rectifier is at a transition point TS1. Once the rectifier is open, the switch in the SSHI is on and the oscillator LI -Cp works immediately. After a very short time interval ti , the output voltage of the PB has been changed from Vrect to −Vrect e−π/(2QI ) and the SSHI is switched off again. Following is an open-circuit of the rectifier. With decreasing deflection of the bimorph, Vp reaches −Vrect at last to make the rectifier conduct again and Vp will block at −Vrect until the bimorph deflection reaches the negative deflection extremum.

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The scavenging-energy process over one mechanical period T0 can be solved as follows [49]. We take t = T0 /4 as the beginning and t = 5T0 /4 as the end of the calculation, because it is known that the deflection of the bimorph arrives at its positive deflection extremum at t = T0 /4 or 5T0 /4, i.e., both t = T0 /4 and t = 5T0 /4 are at TS1 transition states. It can be divided into 6 intervals from t = T0 /4 to t =p5T0 /4: Interval 1: T0 /4 6 t 6 T0 /4 + ti (ti = π LICp ), the rectifier is open and the SSHI is switched on such that the SSHI and Cp consists of an oscillator to transport the positive charges from the upper electrode to the central electrode. The output voltage Vp = Vrect will be changed into Vp = −Vrect e−(π/2QI ) after ti . Because the magnitude of the reversed voltage is smaller than Vrect , an open stage of the rectifier follows. Interval 2: T0 /4 + ti 6 t 6 T0 /4 + ti + tu , both the rectifier and the SSHI are in opencircuit. It follows that Vp (T0 /4 + ti ) = −Vrect e−(π/2QI ) , (3.44) Vp (T0 /4 + ti + tu ) = −Vrect which determines the initial charge Qs accumulating on the upper electrode and the time interval tu that Vp changes from −Vrect e−(π/2QI ) into −Vrect . Interval 3: T0 /4 + ti + tu 6 t 6 3T0 /4, the rectifier conducts again because Vp reaches −Vrect . The current, ip , can be obtained: ip = −bs−1 11 d31 (c + h)u˙3,1 (l,t)

(T0 /4 + ti + tu 6 t 6 3T0 /4)

(3.45)

When t = 3T0 /4, the bimorph deflects to its negative extremum, which implies a TS2 transition point. The SSHI is switched on again and consists of an oscillator together with Cp to change the output voltage Vp = −Vrect into Vp = Vrect e−(π/2QI ) for the time from t = 3T0 /4 to t = 3T0 /4 + ti (interval 4). Then an open-circuit stage (interval 5) happens until Vp = Vrect , and a closed-circuit stage (interval 6) with Vp = Vrect follows similarly as before. The current in the closed-circuit interval can be obtained as ip = −bs−1 11 d31 (c + h)u˙3,1 (l,t)

(3T0 /4 + ti + tu 6 t 6 5T0 /4)

(3.46)

Thus, the average current in Eq.(3.34) becomes hIrect i =

2 T0

Z T0 /2 0

2|ip |dt =

2 T0

Z 3T0 /4 T0 /4+tu

2|ip |dt

(3.47)

for ti /T0 ¿ 1. Following, we will solve out the storage process over circuit period TS . According to the on/off states of the transistor M1 and the diode D1 , the charging process in period TS can be divided into the following 3 intervals: Interval 1: 0 6 t 6 DTS , the transistor M1 is switched on and the diode D1 is off. The currents iL , i1 and i2 , respectively, flow through the inductor L, the filter capacitor C1 , and the battery are obtained as follows from the Kirchhoff current law with the initial condition iL = 0 at t = 0:

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1 (Vrect −Vb )t, i2 = I0 , L 1 i1 = iL − i2 = (Vrect −Vb )t − I0 L iL =

(3.48)

In the above, we have assumed the battery voltage is below Vc , and one may discuss the case of the battery voltage larger than Vc with a similar analysis. Interval 2: DTS 6 t 6 (D + ∆1 )TS , the transistor M1 is switched off and the diode D1 is switched on such that the inductor L can release its stored energy to the capacitor C1 . The currents iL , i1 and ¯i2 , respectively, are obtained from the Kirchhoff current law with the initial condition iL ¯t=DTS = (Vrect −Vb )DTS /L, 1 (Vrect DTS −Vbt), i2 = I0 , L 1 i1 = iL − i2 = (Vrect DTS −Vbt) − I0 L iL =

(3.49)

Interval 3: (D + ∆1 )TS 6 t 6 TS , both the transistor M1 and the diode D1 are switched off, and the capacitor C1 is still in charging the battery. The currents iL , i1 and i2 , respectively, are obtained from the Kirchhoff current law: iL = 0, i2 = −i1 = I0 We therefore define the average output current from the inductor L as ·Z DT Z (D+∆ )T ¸ Z S 1 S 1 TS 2 hIout i = iL dt = + iL dt TS 0 TS 0 DTS µ ¶ Vrect Vrect = − 1 D2 TS 2L Vb

(3.50)

(3.51)

The input energy in period TS from Crect to (C1 , battery) through the inductor is Win = hIout iVb TS =

Vrect −Vb Vrect D2 TS2 2L

(3.52)

The absorption energy by (C1 , battery) in period TS is Wb = (I0 + iC )Vb TS

(3.53)

Neglecting the loss in DC-DC converter, we require the equality between the input energy and the storage energy of the battery and the capacitor C1 in a period TS , i.e., Win = Wb

(3.54)

Substituting Eqs.(3.52) and (3.53) into Eq.(3.54) yields Vrect −Vb Vrect D2 TS2 = (I0 + iC )Vb TS 2L

(3.55)

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Thus, the proper duty cycle should be r D=

2L fS Vb · (I0 + iC ) Vrect −Vb Vrect

(3.56)

Obviously, D is dependent of Vrect −Vb and Vb /Vrect . A larger Vrect −Vb requires the inductor L to share a larger electric potential difference, which means a larger current variant rate of the inductor. On the other hand, the smaller is the battery capacity Vb (I0 + iC ), the less electric energy from the rectified capacitor to the battery will be needed in one period TS . Hence, a larger Vrect or a smaller battery capacity corresponds to a smaller duty cycle D for the DC-DC converter, which implies that for a given battery a larger Vrect is with a smaller D; or for a prescribed external forcing a smaller battery capacity is with a smaller D.

3.4.3 Numerical results We present in this section some numerical results for illustration. In the calculations, we use PZT-5H [24] as our piezoelectric material, while the middle metallic layer is taken to be aluminum alloy, their material parameters can be referred to Eq.(3.29) and its following text. We set the vibration frequency of the external forcing is f = 100 Hz = 6000 rpm, typical for vibrating machinery, and we have ω0 = 2π f and T0 = 1/ f = 0.01 s. In all the illustrative calculations, we choose the piezoelectric layer thickness b = 10 mm, the thickness ratio c/h = 0.4, and the bimorph volume 2lb(c + h) = 1 cm3 . The battery parameters Qm = 10 mA · h, V0 = 1.8 V, Vc = 2.3 V and V∞ = 2.31 V [49]. We will set the input acceleration amplitude ω02 A = 3 m/s2 unless stated. In the calculated we have taken l/(c + h) = 125.0, LI = 10 mH, L = 10 mH, Crect = 20 mF and C1 = 2.2 mF. We set the quality factor QI equal to 2.6, which corresponding to a preliminary experimental setup [39]. Figure 3.22 plots the output voltage Vp , and the rectified current irect , versus the time over one period T0 . To avoid plotting the too large oscillating current of LI -Cp oscillator in the current-time figure, we have plotted the rectified current irect , instead of ip , versus the time in Fig.3.22(b). It is shown in Fig.3.22 that a larger Vrect results in shorter time duration of closed-circuit of the rectifier. In addition, it can also be found that the output current, irect , does not depend on the storage circuit after Crect , which means the harvesting structure can be optimized independently. For a smaller Vrect , a longer conduct interval of the rectifier; and conversely, a larger Vrect is corresponding to a shorter conduct interval. Both these two cases cannot reach the optimal performance of the harvesting structure. Figure 3.23 shows dependence of output power density of the harvesting element upon Vrect . The power density initially increases with Vrect , and then decreases after reaching a peak value, which implies that an optimal output voltage Vrect , dependent of the external excitation level, should be chosen properly. Because this optimal voltage is not identical to the battery voltage in the most situations, it is important to add a step down

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DC-DC converter following Crect to match Vrect with the battery voltage when the external forcing level is high.

Fig. 3.22 (a) Output voltage, Vp , and (b) output current, irect , of the harvesting structure versus the time over one period T0 .

Fig. 3.23 Output power density of the harvesting structure versus Vrect for ω02 A = 3 m/s2 .

The DC-DC converter shown in Fig.3.20 can effectively step down the input voltage Vrect to match with the lower battery voltage through adjusting duty cycle D. Figure 3.24(a) plots dependence of D upon Vrect for fixed Vb , or say, dependence of D upon external forcing level for a given battery. As Vrect becomes small, i.e., the external forcing level becomes low, it does not need to step down Vrect too much to match Vb so that the conduct interval of the transistor M1 becomes long gradually; But as Vrect becomes large, that is, the external excitation level arises, Vrect needs to be stepped down much to match Vb . Thus, under the situation with larger Vrect , D should be adjusted to be smaller such

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that higher voltage difference Vrect −Vb is acted on the inductor L, or larger power is temporarily stored in L, during the switching-on duration of M1 . After M1 is open and D1 is on, the inductor will transfer its stored electric energy to the capacitor C1 .

Fig. 3.24 Dependence of duty cycle D upon Vrect (a) and Vb (b), respectively.

Dependence of duty cycle upon the battery voltage is plotted in Fig.3.24(b). Because the electric difference Vrect − Vb decreases with increasing Vb , it will take longer time for the DC-DC converter to absorb enough energy from the rectified capacitor to satisfy the requirement of charging battery in a period TS . As the charging process develops, the battery voltage rises such that the duty cycle should be gradually regulated large following the increasing Vb . Figure 3.25 shows the three currents, iL , i1 , i2 , versus the time in the circuit period TS . During the closed-circuit durations of M1 or D1 , the inductor current iL and the capacitor i1 vary very violently. After both M1 and D1 are open, i1 keeps constant. It shows that the filter capacitor C1 can effectively share the violent current and power during the on in-

Fig. 3.25 Three currents flowing through the inductor L, the filter capacitor C1 and the battery, respectively, versus the time in a period TS .

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tervals of M1 or D1 temporarily, and release the scavenged energy smoothly to the battery during the whole charging period, which indicates the stability effect of the capacitor C1 on the battery current. Figure 3.26 shows the dependence of the charging efficiencies with and without a DCDC converter versus the external excitation levels, while an SSHI is applied in the harvesting structure with ω02 A from 1 m/s2 to 6 m/s2 . For a storage circuit without a DC-DC converter, the charging efficiency is obviously low because the battery voltage is greatly different from the optimal rectified voltage. Furthermore, this phenomenon increases with increasing ω02 A. It follows from Fig.3.27 that the power density of a PB with an SSHI and a DC-DC converter can reach about 2.5 times higher than the one of a PB with an SSHI but without a DC-DC converter when ω02 A = 5 m/s2 .

Fig. 3.26 Comparison between the charging efficiencies of an energy harvester with and without a DC-DC converter in the storage circuit, while an SSHI has been employed in the harvesting structure.

Fig. 3.27 Comparison between the charging efficiencies of an energy harvester with and without an SSHI in the harvesting structure, while a DC-DC converter has been used in the storage circuit.

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We show the dependence of the charging efficiencies with and without an SSHI at the harvesting element versus the external excitation levels in Fig.3.27, while ω02 A is from 1 m/s2 to 6 m/s2 . For a piezoelectric structure without SSHI, the charging efficiency is obviously lower, and this phenomenon increases with increasing ω02 A. It is found from Fig.3.27 that the power density of a harvester with an SSHI and a DC-DC converter is about 5 times higher than the one only with a DC-DC converter but without an SSHI when ω02 A = 5 m/s2 . We also note in Fig.3.27 that the harvesting structure with SSHI is always able to scavenge energy for ω02 A from 1 m/s2 to 6 m/s2 , even when there does not exist any rectifier conduction under ω02 A < 3.2 m/s2 for the case without SSHI. The reason that the latter cannot extract energy from the ambient vibrations under ω02 A < 3.2 m/s2 is that its output voltage magnitude |Vp | < Vrect always, see Fig.3.28. The mechanism of an SSHI in improving the efficiency of a PB in scavenging-energy is illustrated as follows: an SSHI can reverse the output voltage Vp through charge transfer between the output electrodes at the transition moments from closed- to open-circuit. Voltage reversal induces earlier arrival of next rectifier conduction because the output voltage phases of any two adjacent closed-circuit shapes are just opposite each other, see Fig.3.28. We know that a PB is with a smaller flexural stiffness under closed-circuit condition than under open-circuit condition. Thus, the PB subjected to longer closed-circuit condition will be easier to be accelerated. A larger flexural velocity, see Fig.3.29(a), makes the PB to deform toward its reverse deflection with larger amplitude (Fig.3.29(b)), which implies that more mechanical energy will be converted into electric one. Specially, a PB with SSHI is still able to effectively scavenge energy even when the external forcing level is very low (ω02 A < 3.2 m/s2 ).

Fig. 3.28 Output voltages of the PB with and without SSHI versus the time over one mechanical period T0 for ω02 A = 3 m/s2 and 4 m/s2 , respectively.

Usually, the vibrating intensity of an ambient vibration source is not immutable. Therefore, when extracting vibration energy using an energy harvester, it is very likely to face

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Fig. 3.29 Deflection velocities (a) and displacements (b) of the PB with and without an SSHI versus the time over one mechanical period T0 for ω02 A = 3 m/s2 and 4 m/s2 , respectively.

a situation: when the projected strong vibration source changes from strong to weak or the projected weak vibration source changes from weak to strong, neither the operating efficiency of the energy harvester simply designed for the strong vibration source nor that for the weak vibration source will be very high. Sometimes the charging efficiency is so low that no operation can be performed. To remedy such a situation, Xue et al. [57] have proposed a novel piezoelectric energy harvester with a Cuk type DC-DC converter in its analogue circuit. This device is capable of effectively extracting vibration energy from various strong or weak ambient vibration sources and storing the energy harvested in chargeable chemical batteries. In order to enable the energy harvester to operate normally in an environment in which the vibrating amplitude is varying, it is required that its charging circuit possess the function of rectifier step up and step down at once. Only in this way will it be possible to ensure sustained highly efficient operation of the energy harvester,

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while the conventional Buck and Boost type DC-DC converter is capable of performing either the rectifying step-up or step-down function separately. That is to say, when ambient vibration is very weak, it is simply impossible for the energy harvester using the Buck type converter to operate normally. On the other hand, when the amplitude of the external excitation is very violent, the energy harvester using the Boost type converter will lose its efficiency. And it is only the Buck-Boost type DC-DC converter and Cuk type DC-DC converter that both have such a function to step up/down the rectified voltage simultaneously. By contrast, the performance of the Cuk type converter is even better since the two inductors L1 and L2 are placed with the same iron core as shown in Fig.3.30 so that there is a rather strong effect of mutual inductance generated between them. As rational use of its unique mutual inductance effect can appreciably reduce the magnitude of the transient impulse current, Xue et al. have chosen the Cuk type DC-DC converter, and it is precisely the said converter that has enabled the entire system to possess the capability of adapting to ambient amplitude.

Fig. 3.30 Schematic diagram of Cuk type energy storing circuit.

A schematic diagram of the Cuk type energy storing circuit is shown in Fig.3.30, in which C1 is the energy relay unit inside the Cuk type DC-DC converter that is responsible for temporarily storing the energy transmitted from the rectifying capacitor. The inductor L1 is responsible for extracting energy from Crect when M1 is in the closed stage. When M1 is in the off stage, it releases the extracted energy to C1 . Similarly, the inductor L2 extracts energy from C1 when M1 is closed and releases the energy to C2 and the battery when M1 is off. The introduction of the filter capacitor C2 can effectively smooth the charging current and prolong the useful life of the charging battery. When M1 is cut off, D1 is spontaneously conducted under the action of the positive voltage until the energy stored in L1 and L2 is completely released. It is shown by numerical calculation that a piezoelectric energy harvester with the Cuk type energy storing circuit is capable of more efficient energy extraction from a variable-amplitude vibrating environment. It is also shown from the calculation results that, by means of mutual inductance design, it is possible to eliminate current pulse by about 20%. The effect of such a gentle current can prolong the lifetime of the energy harvester. The dynamic calculations of piezoelectric structures are very complicated in general, because its electric boundary condition is intermittently exchanged between open- and closed-circuit. Obviously, finite element method is suitable for analyzing dynamic characteristics of piezoelectric structures [55, 58-60]. FEM is based on the variational principle

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and subdivided interpolation. What is adopted for solving the vibration of a linear piezoelectric elastomer is the Hamilton variational principle. Subdivided interpolation consists in subdividing the entire continuum structure into finite units that are connected to one another at their boundary surfaces. An interpolation function should be constructed that can give the relationship between the unknowns’ node values and an arbitrary point value within the unit. Finally, a set of equations for approximate satisfaction of the governing equations and the boundary conditions is established and the unknowns can be obtained through an FEM program.

Acknowledgments This work was supported by National Natural Science Foundation of China (Nos. 10932004, 51121002, and 11272127) and Major State Basic Research Development Program of China (973 Program) (No.2009CB724205).

References [1] Joseph A D. Energy harvesting projects. IEEE Pervas. Comput., 2005, 4(1), 69-71. [2] Lal A, Duggirala R, Li H. Pervasive power: a radioisotope-powered piezoelectric generator. IEEE Pervas. Comput., 2005, 4(1), 53-61. [3] Roundy S, Leland E S, Baker J. Improving power output for vibration-based energy scavengers. IEEE Pervas. Comput., 2005, 4(1), 28-36. [4] Roundy S, Wright P K, Rabaey J. A study of low lever vibrations as a power source for wireless sensor nodes. Computer Com., 2003, 26(11), 1131-1144. [5] Davis W R, Zhang N, Camera K. A design environment for high-throughput low-power dedicated signal processing systems. IEEE J. Solid-St. Circ., 2002, 37(3), 420-431. [6] Ching N N, Chan G M, Li W J, et al. PCB integrated micro-generator for wireless systems. International Symposium on Smart Structures and Microsystems, 2000, 19-21. [7] Shu Y C, Lien I C. Analysis of power output for piezoelectric energy harvesting systems. Smart Mater. Struct., 2006, 15(6), 1499-1512. [8] Sodano H A, Inman D J, Park G. A review of power harvesting from vibration using piezoelectric materials. Shock Vib. Digest, 2004, 36(3), 197-205. [9] Mateu L, Moll F. Review of energy harvesting techniques and applications for microelectronics. Proceeding of SPIE: Conference on VLSI Circuits and Systems II, 2005, 5837, 359-373. [10] Van Dyke K S. The electric network equivalent of a piezo-electric resonator. Proceeding of IRE, 1928, 16, 742. [11] Mason W P. Piezoelectric Crystals and Their Application to Ultrasonics. New York: Van Nostrand, 1950. [12] Umeda M, Nakamura K, Ueha S. Analysis of the transformation of mechanical impact energy to electric energy using piezoelectric vibrator. Jpn. J. Appl. Phys., Part 1: Regular Papers & Short Notes & Review Papers, 1996, 35(5), 3267-3273.

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[13] Umeda M, Nakamura K, Ueha S. Energy storage characteristics of a piezo-generator using impact induced vibration. Jpn. J. Appl. Phys., Part 1: Regular Papers & Short Notes & Review Papers, 1997, 36(5B), 3147-3151. [14] Cho J, Anderson M, Richards R, et al. Optimization of electromechanical coupling for a thinfilm PZT membrane. I: modeling. J. Micromech. Microeng., 2005, 15(10), 1797-1803. [15] Ha S K. Analysis of the asymmetric triple-layered piezoelectric bimorph using equivalent circuit models. J. Acoust. Soc. Am., 2001, 110, 856-864. [16] Keawboonchuay C, Engel T G. Electrical power generation characteristics of piezoelectric generator under quasi-static and dynamic stress conditions. IEEE Trans. Plasma Sci., 2003, 50(10), 1377-1382. [17] Keawboonchuay C, Engel T G. Maximum power generation in a piezoelectric pulse generator. IEEE Trans. Plasma Sci., 2003, 31(1), 123-128. [18] Yang J, Zhou H, Hu Y, et al. Performance of a piezoelectric harvester in thickness-stretch mode of a plate. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2005, 52(10), 1872-1876. [19] Yang J S, Chen Z G, Hu Y T. An exact analysis of a rectangular plate piezoelectric generator. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2007, 54(1), 190-195. [20] Jiang S N, Hu Y T. Analysis of a piezoelectric bimorph plate with a central-attached mass as an energy harvester. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2007, 54(7), 14631469. [21] Jiang S N, Li X F, Guo S H, et al. Performance of a piezoelectric bimorph for scavenging vibration energy. Smart Mater. Struct., 2005, 14(4), 769-774. [22] IEEE Standard on Piezoelectricity. [23] Yang J S, Fang H Y. Analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2002, 49, 798-804. [24] Auld B A. Acoustic Fields and Waves in Solids. Vol. 1. New York: Wiley, 1973, 357-382. [25] Holland R, Eernisse E P. Design of Resonant Piezoelectric Devices. Cambridge: MIT Press, 1969, 12-16. [26] Hu Y T, Hu H P, Yang J S. A low frequency piezoelectric power harvester using a spiralshaped bimorph. Sci. China-Phys. Mech. Astron., 2006, 49(6), 649-659. [27] Roundy S, Zhang Y. Toward self-tuning adaptive vibration based micro-generators. Proceeding of SPIE: Conference on Smart Structures, Devices, and Systems, 2005, 5649, 373-384. [28] Wu W J, Chen Y Y, Lee B S, et al. Tunable resonant frequency power harvesting devices. Proceedings of SPIE: Conference on Smart Structures and Materials, 2006, 6169, 61690A. [29] Wu X, Lin J, Kato S, et al. A frequency adjustable vibration energy harvester. Proc. PowerMEMS, 2008 (Sendai, Japan, Nov.), 245-248. [30] Hu Y T, Xue H, Hu H P. A piezoelectric power harvester with adjustable frequency through axial preloads. Smart Mater. Struct., 2007, 16(5), 1961-1966. [31] Leland E S, Wright P K. Resonance tuning of piezoelectric vibration energy scavenging generators using compressive axial preload. Smart Mater. Struct., 2006, 15(5), 1413-1420. [32] Bokaian A. Natural frequencies of beams under compressive axial loads. J. Sound Vib., 1998, 126, 49-65. [33] Sung K H. Admittance matrix of asymmetric piezoelectric bimorph with two separate electrical ports under general distributed loads. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr, 2001, 48, 976-984. [34] Shahruz S M. Design of mechanical bandpass filters for energy scavenging. J. Sound Vib., 2006, 292(3-5), 987-998.

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[35] Xue H, Hu Y T, Wang Q M. Broadband piezoelectric energy harvesting devices using multiple bimorphs with different operating frequencies. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2008, 55(9), 2104-2108. [36] Ferrari M, Ferrari V, Guizzetti M, et al. Piezoelectric multifrequency energy converter for power harvesting in autonomous microsystems. Sensors Actuat. A-Phys., 2008, 142(1), 329335. [37] Lefeuvre E, Badel A, Richard C, et al. High performance piezoelectric vibration energy reclamation. Proceedings of SPIE: Conference on Smart Structures and Materials, 2004, 5390, 379-387. [38] Badel A, Guyomar D, Lefeuvre E, et al. Efficiency enhancement of a piezoelectric energy harvesting device in pulsed operation by synchronous charge inversion. J. Intel. Mat. Syst. Str., 2005, 16(10), 889-901. [39] Guyomar D, Badel A, Lefeuvre E, et al. Toward energy harvesting using active materials and conversion improvement by nonlinear processing. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2005, 52(4), 584-594. [40] Guan M, Liao W H. On the energy storage devices in piezoelectric energy harvesting. Proceedings of SPIE: Conference on Smart Structures and Materials, 2006, 6169, 61690C. [41] Ng T H, Liao W H. Sensitivity analysis and energy harvesting for a self-powered piezoelectric sensor. J. Intel. Mat. Syst. Str., 2005, 16(10), 785-797. [42] Ng T H, Liao W H. Feasibility study of a self-powered piezoelectric sensor. Proceedings of SPIE: Conference on Smart Structures and Materials, 2004, 5389, 377-388. [43] Sun J, Mitchell D M, Greuel R M. Averaged modeling of PWM converters in discontinuous conduction mode. IEEE Trans. Power Electr., 2001, 16(4): 482-492. [44] Ottman G K, Hofmann H F, Lesieutre G A. Optimized piezoelectric energy harvesting circuit using step-down converter in discontinuous conduction mode. IEEE Trans. Power Electr., 2003, 18(2), 696-703. [45] Lesieutre G A, Ottman G K, Hofmann H F. Damping as a result of piezoelectric energy harvesting. J. Sound Vib., 2004, 269(3-5), 991-1001. [46] Ammar Y, Buhrig A, Marzencki M, et al. Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator. Proceedings of the 2005 Joint Conference on Smart Objects and Ambient Intelligence: Innovative Context-aware Services: Usages and Technologies, 2005, 121, 287-292. [47] Lefeuvre E, Badel A, Richard C, et al. Piezoelectric energy harvesting device optimization by synchronous electric charge extraction. J. Intel. Mat. Syst. Str., 2005, 16(10), 865-876. [48] Chao L, Tsui C Y, Ki W H. A batteryless vibration-based energy harvesting system for ultra low power ubiquitous applications. IEEE International Symposium on Circuits and Systems, 2007, 1349-1352. [49] Hu Y T, Xue H, Hu T, et al. Nonlinear interface between the piezoelectric harvesting structure and the modulating circuit of an energy harvester with a real storage battery. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2008, 55(1), 148-160. [50] Lallart M, Garbuio L, Petit L, et al. Double synchronized switch harvesting (DSSH): a new energy harvesting scheme for efficient energy extraction. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2008, 55(10), 2119-2130. [51] Roundy S, Wright P K. A piezoelectric vibration based generator for wireless electronics. Smart Mater. Struct., 2004, 13, 1131-1142. [52] Lu F, Lee H P, Lim S P. Modeling and analysis of micro piezoelectric power generators for micro-electromechanical-systems applications. Smart Mater. Struct., 2004, 13(1), 56-63.

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[53] Ottman G K, Hofmann H F, Bhatt A C, et al. Adaptive piezoelectric energy harvesting circuit for wireless remote power supply. IEEE Trans. Power Electr., 2002, 17, 669-676. [54] Hu Y T, Hu T, Xue H, et al. An energy harvester with an improved piezoelectric harvesting structure and an adaptive storage circuit. Smart Mater. Struct., 2006, 54(6), 1177-1187. [55] Hwang W S, Park H C. Finite element modeling of piezoelectric sensors and actuators. AIAA J., 1993, 31(5), 930-937. [56] Mohan N, Undeland T, Robbins W. Power Electronics: Converters, Applications and Design. New York: Wiley, 1995. [57] Xue H, Hu H P, Hu Y T. An improved piezoelectric harvester available in scavenging-energy from the operating environment with either weaker or stronger vibration levels. Sci. ChinaPhys. Mech. Astron., 52(2), 218-225. [58] Chee C Y K, Tong L Y, Steven G P. Review on the modeling of piezoelectric sensors and actuators incorporated in intelligent structures. J. Intel. Mat. Syst. Str., 1998, 9(1), 3-19. [59] Auricchio F, Bisegna P, Lovadina C. Finite element approximation of piezoelectric plates. Int. J. Numer. Meth. Eng., 2001, 50(6), 1469-1499. [60] Wang S Y, Quek S T, Ang K K. Dynamic stability analysis of finite element modeling of piezoelectric composite plates. Int. J. Solids Struct., 2004, 41 (3-4), 745-764.

Chapter 4 A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates

Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, 315211, China

Abstract The analysis of surface acoustic waves in finite elastic solids is of fundamental and practical importance, giving the fact that such typical problems concerning solutions of the well-known wave propagation equations are hard to solve. In searching of an accurate analytical method, we compare this problem with the known bulk acoustic wave problems in finite elastic solids and the corresponding Mindlin and Lee plate theories which have been instrumental in solving these problems with accuracy and simplicity. Analogous to power and trigonometric series expansions of displacements, we use exponential functions obtained from solutions of surface acoustic waves in semi-infinite solids to represent the decaying displacements along thickness direction. For the finite isotropic elastic plate with small thickness, both the exponentially decaying and growing modes are needed in general for establishing the two-dimensional theory for surface acoustic waves in finite plates. But for anisotropic and piezoelectric plates in which thicknesses are more than five wavelengths, the exponentially decaying modes alone will be able to predict vibrations of surface acoustic wave modes accurately. Eventually we present the twodimensional theory specifically for surface acoustic waves in finite isotropic, anisotropic and piezoelectric solids with goals to use it for surface acoustic wave resonator analysis and design. Keywords Rayleigh, surface, wave, vibration, frequency, plate, resonator

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4.1 Introduction The surface acoustic wave is one of the most commonly utilized wave modes in acoustic wave resonators in various electronic applications, while the essential analysis is done with known Rayleigh waves in semi-infinite solids [1-9]. The wave propagation phenomenon has wide applications in many engineering fields, but without exception all the analyses are based on classical solutions from a semi-infinite solid, which is considered as a typical physical model of many devices and applications [10-20]. Of course, as we know, surface acoustic waves, or Rayleigh waves, essentially vanish in the depth below three wavelengths or more, or this can justify the approximation for many elastic solids that carry surface acoustic waves as semi-infinite [1-5]. This approximation is acceptable in most cases, with surface acoustic wave resonator design as one of the most typical applications that need wave propagation parameters such as the wave velocity and decaying indices for the determination of device properties [21,22]. Clearly, the accurate determination of such parameters is important in the prediction or the selection of these parameters with design parameters of the surface acoustic wave resonator structure, which includes essential parameters like the thickness of the piezoelectric substrate. On the other hand, it is also known that in an infinite plate, surface acoustic waves take a different form in comparison with the familiar solutions from semi-infinite solids [21,22]. In case the plate thickness increases, the displacements still exhibit the surface acoustic wave characteristics, especially along the thickness of the plate, showing a clear variation. The early researches are mainly carried out with isotropic material which has given some important properties of surface acoustic wave propagation in different structures [1-6]. For example, as the surface acoustic wave velocity in an isotropic plate approaches that of a semi-infinite solid, it is also natural that the combined displacements of both antisymmetric and symmetric modes about the thickness will gradually resemble a Rayleigh wave mode [21,22]. As the thickness approaches infinity, the surface acoustic wave velocities associated with both anti-symmetric and symmetric modes approach Rayleigh wave velocity also. Since the modes are symmetric and anti-symmetric regarding the mid-plane or thickness of the isotropic plate, the displacements in the lower surface will disappear. The combined displacements will be very close to that of Rayleigh waves for a substrate of larger thickness. Since there are symmetric and anti-symmetric modes, or exponentially decaying and growing modes in an isotropic plate, the dominant modes we need to consider in an isotropic plate should include both [21,22]. Using known dominant modes in an isotropic plate as basis functions, the two-dimensional theory for surface acoustic waves in finite isotropic elastic solids can be extended to include more vibration modes [21-26]. A two-dimensional theory with the inclusion of the four vibration modes will be able to predict the surface wave velocity and mode coupling in a finite isotropic elastic plate more accurately. The derivation of the two-dimensional theory follows the same procedure, which is standard in the plate theories we are familiar with, such as Mindlin [27,28], Lee [29], Peach [30], and others [31]. There will be three dominant modes in case of anisotropic materials [24]. For piezoelectric materials, there will be four dominant modes from a semi-infinite solid and the

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115

expansion of displacements and electrical potential with these eigensolutions will result in much larger number equations [25]. Solutions of these equations, even in the simplest straight-crested wave modes, are difficult to obtain due to the coupled wave modes. All these will be reflected in the two-dimensional theory, which will be material-dependent in the order of the expansion and equations. For practical solutions, it is always important to reduce the number of wave modes through techniques such as identifying the strong couplings and dominant modes from the higher-order variables. In addition, another frequent technique in dealing with piezoelectric materials is the use of piezoelectrically stiffened elastic constants after eliminating the electrical variables from the coupled equations. Since it has been proven that the piezoelectrically stiffened constants can generally give good approximation in some weakly coupled materials like the quartz crystal, we also employed this technique in the two-dimensional analysis of surface acoustic waves in finite piezoelectric materials. The solutions, which show clear difference in comparison with the negligence of piezoelectric effect, can be considered as an approximation and used as a reference for the complete solutions from the two-dimensional equations without eliminating electrical variables. These solutions are considered as the gradual progress we are making in establishing three types of the two-dimensional theory for surface acoustic waves in finite isotropic, anisotropic, and piezoelectric plates, which are required for the design of typical surface acoustic wave resonators of different materials. The two-dimensional equations can be further applied to complicated structures with the presence of electrodes for direct applications in surface acoustic wave resonator design by using the finite element method [32-41].

4.2 A two-dimensional analysis of surface acoustic waves in finite isotropic elastic plates 4.2.1 Surface acoustic waves in an infinite isotropic elastic plate For surface acoustic waves in a plate with finite thickness, as shown in Fig.4.1, we assume that the displacements take the same form as in a half-space with exponential variation along the thickness coordinate [1-4, 21-24]: u1 = Aekβ x2 eik(x1 −ct) , u2 = Bekβ x2 eik(x1 −ct) , u3 = 0

(4.1)

where u j ( j = 1, 2, 3), A, B, k, β , xi (i = 1, 2), c and t are displacements, amplitudes, wavenumber, decaying index, coordinates, wave velocity and time, respectively. These are typical Rayleigh waves in solids with the distinct thickness dependence while traveling along x1 direction.

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Fig. 4.1 An infinite isotropic elastic plate.

With abbreviated notations [42-44], the strains from Eq.(4.1) are S1 = ikAekβ x2 eik(x1 −ct) , S2 = kβ Bekβ x2 eik(x1 −ct) , S6 = k(β A + iB)ekβ x2 eik(x1 −ct) , S3 = S4 = S5 = 0

(4.2)

For isotropic materials, with strains in Eq.(4.2), the corresponding stresses, again in abbreviated notations, are T1 = k[(λ + 2µ )iA + λ β B]ekβ x2 eik(x1 −ct) , T2 = k[λ iA + (λ + 2µ )β B]ekβ x2 eik(x1 −ct) , T3 = kλ (iA + β B)ekβ x2 eik(x1 −ct) ,

(4.3)

T6 = k µ (β A + iB)ekβ x2 eik(x1 −ct) , T4 = T5 = 0 where λ and µ are Lam´e constants. Using the stress equations of motion for an isotropic elastic solid with displacements given in Eq.(4.1) and stresses in Eq.(4.3), after simplification, we have [β 2 c2T + (c2 − c2L )]A + iβ (c2L − c2T )B = 0, (4.4) iβ (c2L − c2T )A + [β 2 c2L + (c2 − c2T )]B = 0 with c2L =

λ + 2µ , ρ

c2T =

µ ρ

(4.5)

defined as the longitudinal and transverse wave velocities of material and ρ as the density of material, respectively. By solving Eq.(4.4) for the decaying index β , we have two neat and elegant solutions:

β12 = 1 −

c2 , c2L

β22 = 1 −

c2 c2T

(4.6)

We further define the decaying index vector = {β11 , β12 , β21 , β22 } = {β1 , β2 , −β1 , −β2 } which gives the corresponding amplitude ratios as

(4.7)

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A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates

µ ¶ B = −iβ1 , A 1

µ ¶ B 1 = , A 2 iβ2

µ ¶ B = iβ1 , A 3

µ ¶ B 1 =− A 4 iβ2

117

(4.8)

As we already know, they represent the longitudinal and transverse modes of surface acoustic waves propagating in solids, respectively [5]. The phase velocity c is to be determined by boundary conditions of the free surfaces. Then solutions of non-trivial displacement and stress based on Eqs.(4.1) and (4.3) are [21,22] u1 = (A1 ekβ1 x2 + A2 ekβ2 x2 + A3 e−kβ1 x2 + A4 e−kβ2 x2 )eik(x1 −ct) , ¶ µ 1 1 kβ2 x2 −kβ2 x2 kβ1 x2 −kβ1 x2 + A2 e − β1 A3 e − A4 e eik(x1 −ct) , u2 = −i β1 A1 e β2 β2 T1 = ik[(λ + 2µ − λ β12 )A1 ekβ1 x2 + 2µ A2 ekβ2 x2 + (λ + 2µ − λ β12 )A3 e−kβ1 x2 +2µ A4 e−kβ2 x2 ]eik(x1 −ct) , T2 = ik[(λ − 2µβ12 − λ β12 )A1 ekβ1 x2 − 2µ A2 ekβ2 x2 + (λ − 2µβ12 − λ β12 )A3 e−kβ1 x2 −2µ A4 e−kβ2 x2 ]eik(x1 −ct) , µ ¶ β22 + 1 β22 + 1 kβ1 x2 kβ2 x2 −kβ1 x2 −kβ2 x2 T6 = k µ 2β1 A1 e + − 2β1 A3 e − eik(x1 −ct) A2 e A4 e β2 β2 (4.9) where the amplitudes Ai (i = 1, 2, 3, 4) are to be determined. Now applying the traction-free boundary conditions on the free surfaces: T2 (x2 = 0) = T6 (x2 = 0) = 0, T2 (x2 = −h) = T6 (x2 = −h) = 0

(4.10)

with the stresses in Eq.(4.9), we have (λ − λ β12 − 2µβ12 )A1 − 2µ A2 + (λ − λ β12 − 2µβ12 )A3 − 2µ A4 = 0, 2β1 A1 +

1 + β22 1 + β22 A2 − 2β1 A3 − A4 = 0, β2 β2

(λ − λ β12 − 2µβ12 )A1 e−kβ1 h − 2µ A2 e−kβ2 h + (λ − λ β12 − 2µβ12 )A3 ekβ1 h (4.11) −2µ A4 ekβ2 h = 0, 1 + β22 1 + β22 2β1 A1 e−kβ1 h + A2 e−kβ2 h − 2β1 A3 ekβ1 h − A4 ekβ2 h = 0 β2 β2 as the equations for amplitudes. To obtain the velocity of surface acoustic waves in the plate, we must have the determinant of boundary value equations of Eq.(4.11)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

¯ ¯ ¯ ¯ −2µ −2µ λ − λ β12 − 2µβ12 λ − λ β12 − 2µβ12 ¯ ¯ 2 2 ¯ 1 + β2 1 + β2 ¯¯ ¯ 2β1 −2β1 − ¯ ¯ β2 β2 ¯ ¯ ¯ ¯ ¯ (λ − λ β12 − 2µβ12 )e−kβ1 h −2µ e−kβ2 h (λ − λ β12 − 2µβ12 )ekβ1 h −2µ ekβ2 h ¯ ¯ ¯ ¯ ¯ 2 2 1 + β 1 + β ¯ 2 −kβ2 h 2 kβ2 h ¯ 2β1 e−kβ1 h −2β1 ekβ1 h − e e ¯ ¯ β2 β2 (4.12) to vanish. By evaluating Eq.(4.12) with known material constants, we should be able to obtain the velocity and mode shapes of surface acoustic waves in plates of different thickness. Using isotropic material with Poisson’s ratio ν = 0.2, we also define dimensionless plate thickness c h (4.13) H = , kς = 2π, V = ς cT where ς is the wavelength. With this plate material, we have the velocity in Fig.4.2 and displacements in Figs. 4.3-4.8. The displacements are calculated with u1 = A1 (ekβ1 x2 + α1 ekβ2 x2 + α2 e−kβ1 x2 + α3 e−kβ2 x2 )eik(x1 −ct) , µ ¶ 1 1 u2 = −iA1 β1 ekβ1 x2 + α1 ekβ2 x2 − β1 α2 e−kβ1 x2 − α3 e−kβ2 x2 eik(x1 −ct) β2 β2

(4.14)

where the amplitude ratios are also defined as

αi =

Ai+1 A1

(i = 1, 2, 3)

(4.15)

from Eq.(4.12).

Fig. 4.2 Normalized surface acoustic wave phase velocity c/cT versus thickness to wavelength ratio H for a strip with Poisson’s ratio ν = 0.2.

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A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates

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Figures 4.3-4.6 show the displacements of symmetric and anti-symmetric surface acoustic wave modes at corresponding phase velocities with thickness H = 2. It is clear that the largest displacements appear on the plate faces. As plate thickness H is larger than 3, the velocity will be that of Rayleigh waves and the combined displacements also resemble those of Rayleigh waves, as shown in Figs.4.7 and 4.8. The dependence of surface waves on plate thickness once again shows that the phase velocity and mode shapes should be analyzed with the consideration of the plate thickness, particularly when the thickness is less than five wavelengths.

Fig. 4.3 Displacement u1 (symmetric) of an infinite plate of normalized thickness H = 2 at normalized phase velocity V = 0.915410657.

Fig. 4.4 Displacement u2 (anti-symmetric) of an infinite plate of normalized thickness H = 2 at normalized phase velocity V = 0.915410657.

Fig. 4.5 Displacement u1 (anti-symmetric) of an infinite plate of normalized thickness H = 2 at normalized phase velocity V = 0.9071792147.

Fig. 4.6 Displacement u2 (symmetric) of an infinite plate of normalized thickness H = 2 at normalized phase velocity V = 0.9071792147.

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

Fig. 4.7 Combined displacement u1 of an infinite plate with normalized thickness H = 6 at surface wave mode.

Fig. 4.8 Combined displacement u2 of an infinite plate with normalized thickness H = 6 at surface wave mode.

4.2.2 Two-dimensional equations for finite isotropic elastic plate Deriving two-dimensional theories with eigenmodes of elastic plates has been widely practiced in a few familiar two-dimensional theories and methods in the vibration analysis of plates, such as Mindlin [27,28], Lee [29], and Peach [30]. Based on the similar principle and procedure, Wang and Hashimoto et al. [23-26] derived a two-dimensional theory for the analysis of surface acoustic waves in finite solids, aiming at the accurate prediction of surface acoustic wave velocity and modes in resonators. This method has been applied to anisotropic and piezoelectric materials, but the thickness of substrates has been assumed to be large enough, at least five wavelengths or more to ensure the vanishing of exponentially decaying waves in depth. In case of a solid with finite thickness, we need to include more eigenmodes in the displacement expansion scheme. Naturally, these eigenmodes should include the two pairs of surface wave family in Eq.(4.8), making the two-dimensional expansion adequate in comparison with previous work. Following the two-dimensional expansion procedure [23-30], we have the displacement expansion in the thickness coordinate as 4

u j (x1 , x2 , x3 ,t) =

(n)

∑ uj

(x1 , x3 ,t)φn (x2 )

( j = 1, 2),

n=1

φn (x2 ) = exp(kβn x2 ), (n)

(4.16)

βn = {β1 , β2 , −β1 , −β2 }

where u j ( j = 1, 2; n = 1, 2, 3, 4) are the higher-order displacements. This definition takes the same form as the two-dimensional theories by Mindlin [27,28], Lee et al. [29], and Peach [30]. The difference of the expansion scheme in Eq.(4.16) and the earlier paper by Wang and Hashimoto [23,24] is that the exponentially growing modes are now included, in addition to the dominant exponentially decaying ones.

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Following the standard procedure in the derivation of the two-dimensional higher-order theory, we have the strain components from Eq.(4.16) as (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

S1 = u1,1 , S2 = kβn u2 , S3 = u3,3 , S4 = u2,3 + kβn u3 , S5 = u3,1 + u1,3 , (n)

S6 = u2,1 + kβn u1

(n = 1, 2, 3, 4)

(4.17)

The constitutive relation of higher-order stresses and strains takes the form 4

(n)

∑ Tˆp

Tp =

φn

(p = 1, 2, 3, 4, 5, 6)

(4.18)

n=1

with stress terms defined as (n) (n) (n) (n) (n) (n) Tˆp = c p1 u1,1 + c p2 kβn u2 + c p3 u3,3 + c p4 (u2,3 + kβn u3 ) (n)

(n)

(n)

(n)

+c p5 (u3,1 + u1,3 ) + c p6 (u2,1 + kβn u1 )

(4.19)

where c pq (p, q = 1, 2, 3, 4, 5, 6) are elastic constants of material. For a finite elastic solid with identical face A = A0 and thickness h as shown in Fig.4.9, we have the variational equation of motion as Z 0

Z

−h

dx2

A

(Ti j,i − ρ u¨ j )δu j dA = 0

(4.20)

or "

Z

∑

n=1 A

(n) (n) (n) Ti j,i − kβn T 2 j + F2 j − ρ

#

4

∑

m=1

(m) Amn u¨ j

(n)

δu j dA = 0

(i, j = 1, 2, 3) (4.21)

Fig. 4.9 A finite elastic plate with coordinate system.

after integration over the thickness. The two-dimensional equations of motion from Eq.(4.21) is (n)

(n)

(n)

Ti j,i − T 2 j + F2 j = ρ

4

(m)

∑ Amn u¨ j

m=1

(i, j = 1, 2, 3; n = 1, 2, 3, 4)

(4.22)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

with the higher-order stresses defined as (n)

Ti j = (n) T2j (n) F2 j

Z 0 −h

Ti j φn dx2 , (n)

= kβn T2 j ,

(4.23)

= T2 j (0)φn (0) − T2 j (−h)φn (−h),

Amn =

Z 0

−h

exp(kβm x2 ) exp(kβn x2 )dx2

(i, j = 1, 2, 3; m, n = 1, 2, 3, 4)

With definition in Eqs.(4.18) and (4.19), we have the higher-order stress as (n)

Tp

4

=

∑ Amn

m=1

£ (n) (n) (n) (n) (n) c p1 u1,1 + c p2 kβn u2 + c p3 u3,3 + c p4 (u2,3 + kβn u3 )

(n) (n) (n) (n) ¤ +c p5 (u3,1 + u1,3 ) + c p6 (u2,1 + kβn u1 )

(4.24)

The two-dimensional equations of motion in Eq.(4.22) can be further expanded as (n)

(n)

(n)

(n)

(n)

T1,1 + T6,2 + T5,3 − kβn T6 + F6 (n)

(n)

(n)

(n)

(n)

T6,1 + T2,2 + T4,3 − kβn T2 + F2 (n)

(n)

(n)

(n)

(n)

T5,1 + T4,2 + T3,3 − kβn T4 + F4

4

=ρ

m=1 4

=ρ

,

(m)

,

∑ Amn u¨2

m=1 4

=ρ

(m)

∑ Amn u¨1

(m)

∑ Amn u¨3

(4.25) (n = 1, 2, 3, 4)

m=1

where the integration constants are Z 0

Z 0

φm φn dx2 = exp k(βm + βn )x2 dx2 −h · ¸ 1 1 = 1− k(βm + βn ) exp 2π(βm + βn )H

Amn =

−h

(4.26)

It is important to note that since the decaying indices may have opposite values, we have A13 = A31 = A24 = A42 = h (4.27) These integration constants denote the weak coupling of the paired exponentially decaying and growing modes for plates of larger thickness. The boundary conditions corresponding to the displacements in Eq.(4.16) and stresses in Eq.(4.24) can be obtained through Z 0 −h

Z

dx2

C

ni Ti j δu j ds = 0

or after integration over the thickness,

(i, j = 1, 2, 3)

(4.28)

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A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates 4

I

∑

n=1 C

(n)

(n)

ni Ti j δu j ds = 0

(i, j = 1, 2, 3; n = 1, 2, 3, 4)

(n)

123

(4.29)

(n)

which implies that we need to specify Ti j or δu j on the cylindrical face. These equations complete the two-dimensional theory for surface acoustic waves in finite isotropic elastic plates. The comparison of the known plate theories, like Mindlin [27,28], Lee et al. [29], Peach [30], shows that the major difference is that only the dominant surface wave modes are considered in the analysis along with their couplings. If the thickness is large enough, we can neglect the exponentially growing modes to reduce the equations to pure surface acoustic waves, as shown in an earlier study by Wang and Hashimoto [23,24]. Other typical modes in finite plates, like the flexural ones, should be carefully examined for their coupling with surface waves before they can be included in the equations. We leave this part to future investigations using the same principle and procedure outlined in this chapter. To demonstrate the applications of these equations, we consider the surface acoustic wave propagating in a rectangular strip, as shown in Fig.4.9, which is the typical substrate of surface acoustic wave resonators. By neglecting the effect of width of the substrate, we assume the displacements take the form of (m)

¯ 1 )eikct , = A2m−1 sin(kkx (m) ¯ 1 )eikct , u = A2m cos(kkx u1 2

(4.30)

m = 1, 2, 3, 4 ¯ c, and t are the amplitudes, wavenumber, associated where Ai (i = 1, 2, 3, 4, 5, 6, 7, 8), k, k, wavenumber, velocity, and time, respectively. By substituting Eq.(4.30) into Eq.(4.25), we have the equations in expansion for amplitudes. As a result, we have amplitude ratios as

αi j =

Ai+1 (k¯ j ) , A1 (k¯ j )

Ri = A1 (k¯ j )

(i = 1, 2, 3, · · · , 7; j = 1, 2, 3, · · · , 8)

(4.31)

Then, the displacements will be 8

(1) u1 = ∑ Ri sin(kk¯ i x1 ), i=1 8

(2) u1 = ∑ Ri α2i sin(kk¯ i x1 ), (3)

i=1 8

u1 = ∑ Ri α4i sin(kk¯ i x1 ), (4)

i=1 8

u1 = ∑ Ri α6i sin(kk¯ i x1 ), i=1

8

(1) u2 = ∑ Ri α1i cos(kk¯ i x1 ), i=1

8

(2) u2 = ∑ Ri α3i cos(kk¯ i x1 ), (3)

i=1 8

u2 = ∑ Ri α5i cos(kk¯ i x1 ), (4)

i=1 8

u2 = ∑ Ri α7i cos(kk¯ i x1 ) i=1

(4.32)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

and stresses will be (1)

T1

8

=

∑ Ri k

£ (λ + 2µ )(A11 + A21 α2i + A31 α4i + A41 α6i )k¯ i

n=1

¤ +λ (A11 β1 α1i + A21 β2 α3i − A31 β1 α5i − A41 β2 α7i ) cos(kk¯ i x1 ), (1)

T6

8

=

∑ Ri µ k

£ −(A11 α1i + A21 α3i + A31 α5i + A41 α7i )k¯ i

n=1

¤ +(A11 β1 + A21 β2 α2i − A31 β1 α4i − A41 β2 α6i ) cos(kk¯ i x1 ), (2)

T1

8

=

∑ Ri k

£ (λ + 2µ )(A12 + A22 α2i + A32 α4i + A42 α6i )k¯ i

n=1

¤ +λ (A12 β1 α1i + A22 β2 α3i − A32 β1 α5i − A42 β2 α7i ) cos(kk¯ i x1 ), (2)

T6

8

=

∑ Ri µ k

£ −(A12 α1i + A22 α3i + A32 α5i + A42 α7i )k¯ i

n=1

¤ +(A12 β1 + A22 β2 α2i − A32 β1 α4i − A42 β2 α6i ) cos(kk¯ i x1 ), (3)

T1

8

=

∑ Ri k

£ (λ + 2µ )(A13 + A23 α2i + A33 α4i + A43 α6i )k¯ i

n=1

¤ +λ (A13 β1 α1i + A23 β2 α3i − A33 β1 α5i − A43 β2 α7i ) cos(kk¯ i x1 ), (3)

T6

8

=

∑ Ri µ k

£ −(A13 α1i + A23 α3i + A33 α5i + A43 α7i )k¯ i

n=1

¤ +(A13 β1 + A23 β2 α2i − A33 β1 α4i − A43 β2 α6i ) cos(kk¯ i x1 ), (4)

T1

8

=

∑ Ri k

£ (λ + 2µ )(A14 + A24 α2i + A34 α4i + A44 α6i )k¯ i

n=1

¤ +λ (A14 β1 α1i + A24 β2 α3i − A34 β1 α5i − A44 β2 α7i ) cos(kk¯ i x1 ), (4)

T6

8

=

∑ Ri µ k

£ −(A14 α1i + A24 α3i + A34 α5i + A44 α7i )k¯ i

n=1

¤ +(A14 β1 + A24 β2 α2i − A34 β1 α4i − A44 β2 α6i ) cos(kk¯ i x1 ) (4.33) For a rectangular strip with coordinate shown in Fig.4.9, the free ends require (1)

T1

(1)

= T6

(2)

= T1

(2)

= T6

(3)

= T1

(3)

= T6

(4)

= T1

(4)

= T6

=0

(x1 = ±a)

(4.34)

Again, the following dimensionless variables are used in the computation: X=

x1 , ς

kx1 = 2πX,

kk¯ i x1 = 2πk¯ i X

(4.35)

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125

The velocity c versus length a for thickness H = 2 and 5 are shown in Figs.4.10 and 4.11, respectively. It shows that the coupling of surface wave modes will be much stronger at smaller thickness, as H = 2 in Fig.4.10. As the thickness increases, the velocity versus length in Fig.4.11 should approach the earlier results [16]. It should also be noted that the exponential functions in the boundary determinant would be difficult to evaluate for large thickness and length.

Fig. 4.10 Normalized phase velocity of surface waves in a finite strip with normalized thickness H = 2 and Poisson’s ratio ν = 0.2.

Fig. 4.11 Normalized phase velocity of surface waves in a finite strip with normalized thickness H = 5 and Poisson’s ratio ν = 0.2.

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

4.2.3 Conclusions To analyze surface acoustic waves in plates, we start with the general assumption on displacements associated with the exponentially decaying feature along the thickness direction. It is found that in a plate, both exponentially decaying and growing modes exist, representing the symmetric and anti-symmetric surface waves. These paired modes have large deformation on the two parallel plate faces, while the deformation in the midplane of the plate is changed significantly by the plate thickness. As the plate thickness increases, the combined deformation, or displacements, of the plate approaches that of a semi-infinite solid with the surface wave velocity. This implies that surface acoustic waves in finite solids, or plates to be exact, have different characteristics from known semi-infinite solids with a clear dependence on the size of the solids. In other words, the effect of plate thickness cannot be neglected, especially when the thickness is relatively small, say less than three wavelengths. Based on our analysis of surface acoustic waves in an infinite plate, we recognize that the two-dimensional theory for surface acoustic waves in finite plates should include both symmetric and anti-symmetric modes, or the exponentially decaying and growing modes, to have a full consideration of all participating modes. The consideration of the plate thickness offers much improved analytical capability of our two-dimensional theory. As the results show, the increase of the plate thickness will change the phase velocity spectrum, or the velocity versus plate thickness, significantly. As the plate thickness increases, the phase velocity will have a much simpler spectrum, implying less coupling and interaction between modes. This simply validates what we have known for waves in elastic solids. Using this two-dimensional theory, the analysis and design of surface acoustic wave resonators can be improved through the accurate prediction of essential wave propagating parameters like velocity, mode shapes, and couplings. These analyses are what have been lacking in surface acoustic wave analysis in comparison with relatively sophisticated techniques for bulk acoustic wave resonators, which can be analyzed with Mindlin and Lee plate theories.

4.3 A two-dimensional analysis of surface acoustic waves in finite anisotropic elastic plates 4.3.1 Surface acoustic waves in semi-infinite anisotropic solids As we know, due to fact that most actual acoustic device are made of anisotropic materials such as quartz crystals and ceramics, it is practically important to investigate surface acoustic wave propagation in finite anisotropic elastic plates [43]. First, we have to investigate surface acoustic wave propagation in semi-infinite anisotropic elastic solids which will provide us some important parameters for following derivation of two-dimensional analysis. For surface acoustic waves propagating along x1 direction, as shown in Fig.4.12,

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A Two-dimensional Analysis of Surface Acoustic Waves in Finite Piezoelectric Plates

127

we assume the displacements in the anisotropic solids are [23,24] u1 = Aekβ x2 eik(x1 −ct) , u2 = Bekβ x2 eik(x1 −ct) , u3 = Cekβ x2 eik(x1 −ct)

(4.36)

where u j ( j = 1, 2, 3), A(B,C), k, β , xi (i = 1, 2, 3), c and t are displacements, amplitude, wavenumber, decaying index, coordinates, wave velocity and time, respectively. It should pointed out in this section we will consider displacements and stresses of the x3 direction due to material properties.

Fig. 4.12 A semi-infinite anisotropic solid.

With displacements in Eq.(4.36), we have strains as S1 = iku1 ,

S2 = kβ u2 ,

S5 = iku3 ,

S6 = kβ u1 + iku2

S3 = 0,

S4 = kβ u3 , (4.37)

With displacements in Eq.(4.36), we can define stresses in the anisotropic elastic solid in abbreviated notations as Tp = k[(c p6 β + ic p1 )u1 + (c p2 β + ic p6 )u2 + (c p4 β + ic p5 )u3 ]

(p = 1, 2, 3, 4, 5, 6) (4.38)

where c pq (p, q = 1, 2, 3, 4, 5, 6) is the elastic constant tensor. With displacements in Eq.(4.36) and stresses in Eq.(4.38), and use the stress equations of motion of elastic solids including the x3 direction, T1,1 + T6,2 = −ρ k2 c2 u1 , T6,1 + T2,2 = −ρ k2 c2 u2 , T5,1 + T4,2 = −ρ k2 c2 u3 where ρ is the density of quartz crystal solid. Substituting Eqs.(4.36) and (4.38) into Eq.(4.39), we have

(4.39)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

(ρ c2 + c66 β 2 + 2c16 iβ − c11 )A + [c26 β 2 + (c12 + c66 )iβ − c16 ]B +[c46 β 2 + (c14 + c56 )iβ − c15 ]C = 0, [c26 β 2 + (c12 + c66 )iβ − c16 ]A + (ρ c2 + c22 β 2 + 2c26 iβ − c66 )B +[c24 β 2 + (c25 + c46 )iβ − c56 ]C = 0,

(4.40)

2

[c46 β + (c14 + c56 )iβ − c15 ]A + [c24 β 2 + (c25 + c46 )iβ − c56 ]B +[ρ c2 + c44 β 2 + 2c45 iβ − c55 ]C = 0 and the displacements in the semi-infinite anisotropic elastic solid as 3

u1 =

∑ α1r Ar ekβr x2 eik(x1 −ct) ,

3

u2 =

r=1 3

u3 =

∑ α2r Ar ekβr x2 eik(x1 −ct) ,

r=1

∑ Ar ekβr x2 eik(x1 −ct)

(4.41)

r=1

where amplitudes Ar (r = 1, 2, 3) are to be determined and the amplitude ratios are

α1r =

A(βr ) , C(βr )

α2r =

B(βr ) , C(βr )

α3r = 1

(r = 1, 2, 3)

(4.42)

The decaying index βr in the anisotropic solid can be obtained from Eq.(4.40) with given phase velocity c [21,22]. Consequently, the stresses in Eq.(4.38) can be expressed in amplitudes given in Eqs.(4.41) and (4.42): 3

T2 = k ∑ [(c26 βr + ic21 )α1r + (c22 βr + ic26 )α2r + (c24 βr + ic25 )α1r ]Ar ekβr x2 eik(x1 −ct) , r=1 3

T4 = k ∑ [(c46 βr + ic41 )α1r + (c42 βr + ic46 )α2r + (c44 βr + ic45 )α2r ]Ar ekβr x2 eik(x1 −ct) , r=1 3

T6 = k ∑ [(c66 βr + ic61 )α1r + (c62 βr + ic66 )α2r + (c64 βr + ic65 )α3r ]Ar ekβr x2 eik(x1 −ct) r=1

(4.43) Referring to Fig.4.12, traction-free boundary conditions on the top surface of solid are T2 (x2 = 0) = T4 (x2 = 0) = 0,

T6 (x2 = 0) = 0

(4.44)

For surface acoustic waves in a semi-infinite anisotropic solid, solutions include the wave velocity and three parameters related to the decaying of displacements along the depth of solids for the calculation of vibration modes [21,22].

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4.3.2 Two-dimensional equations for finite anisotropic elastic plates Since we assume the surface acoustic waves in finite anisotropic solids are dominated by modes appearing in the semi-infinite solid of the same material in the Section 4.3.1, following Mindlin [27,28] and Lee et al. [29], we write displacements with three components as 3

u j (x1 , x2 , x3 ,t) =

(n)

∑ uj

(x1 , x3 ,t)φn (x2 )

( j = 1, 2, 3; n = 1, 2, 3)

(4.45)

n=1 (n)

with x j ,t and u j as the coordinates, time, and the nth-order displacements and expansion functions defined as

φn (x2 ) = {exp(kβ1 x2 ), exp(kβ2 x2 ), exp(kβ3 x2 )}

(n = 1, 2, 3)

(4.46)

where k is wavenumber and the decaying parameters βn (n = 1, 2, 3) are obtained from Eq.(4.40) with known phase velocity c from Eq.(4.44). As shown in Section 4.2, there exist two decaying parameters for isotropic materials. For anisotropic materials, there will be three decaying parameters in general with complicated displacement expansion. At the meantime, we only consider exponentially decaying modes in this section because the thickness of plate is more than 5 wavelengths which are large enough to neglect the exponentially growing modes, as shown in an earlier study by Wang and Hashimoto [23,24]. Therefore, all the equations in this section are different with that in Section 4.2. With displacements in Eq.(4.45) and basis functions for expansion in Eq.(4.46), corresponding strains in abbreviated notation [27-29, 42] are defined as 3

Sp = (n) S5

(n)

∑ Sp

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

φn , S1 = u1,1 , S2 = kβn u2 , S3 = u3,3 , S4 = u2,3 + kβn u3 ,

n=1 (n) (n) = u3,1 + u1,3 ,

(n)

(n)

(n)

S6 = u2,1 + kβn u1

(p = 1, 2, 3, 4, 5, 6; n = 1, 2, 3) (4.47)

Consequently, for anisotropic materials the stresses are 3

Tp =

(n)

∑ Tˆp

(n) (n) φn , Tˆp = c pq Sq

(p, q = 1, 2, 3, 4, 5, 6; n = 1, 2, 3)

(4.48)

n=1

For an elastic plate with thickness h shown in Fig.4.9, stress equations of motion in variational form are Z Z 0 A −h

3

(Ti j,i − ρ u¨ j )

(m)

∑ δu j

φm dx2 dA = 0

(4.49)

m=1

and through integration over the thickness coordinate we have the corresponding twodimensional equations as

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du (n)

(n)

3

(n)

Ti j,i − T 2 j + F2 j = ρ

(m)

∑ Amn u¨ j

(i, j = 1, 2, 3; n = 1, 2, 3)

(4.50)

m=1

where the two-dimensional quantities are defined as (n)

Ti j = (n)

Z 0 −h

Ti j φn dx2 ,

(n) (n) T¯2 j = kβn T2 j ,

(4.51)

F2 j = T2 j (0)φn (0) − T2 j (−h)φn (−h), Amn =

Z 0

−h

φm φn dx2

(i, j = 1, 2, 3; n = 1, 2, 3)

With two-dimensional variables, we have the nth-order stresses in Eq.(4.48) as (n)

Tp

3

=

∑ Amn

m=1

£ (n) (n) (n) (n) (n) c p1 u1,1 + c p2 kβn u2 + c p3 u3,3 + c p4 (u2,3 + kβn u3 )

(n) ¤ (n) (n) (n) +c p5 (u3,1 + u1,3 ) + c p6 (u2,1 + kβn u1 )

(4.52)

The integral Eq.(4.51) with known expansion functions from Eq.(4.48) are · ¸ Z 0 1 1 Amn = φm φn dx2 = 1− , k(βm + βn ) exp 2π(βm + βn )H −h h H= (m, n = 1, 2, 3) ς

(4.53)

where ς is wavelength. For a finite plate-like solid, we can always use the wavelength as a measure of plate thickness. Since the value of βm (m = 1, 2, 3) is close to unity (1.0), larger H will make the integral in Eq.(4.53) to a simple expression Amn =

1 k(βm + βn )

(m, n = 1, 2, 3)

(4.54)

Referring to Fig.4.9, boundary conditions are I Z 0 C −h

ni Ti j δu j dsdx2 =

I Z 0 C −h

3

ni Ti j

(n)

∑ δu j φn dsdx2 =

n=1

3

∑

I

n=1 C

(n)

(n)

ni Ti j δu j ds = 0 (4.55)

where C is the cylindrical surface. We now examine surface acoustic wave travelling along the x1 direction. For simplicity, the solutions are assumed as (m)

(m)

¯ 1 )eikct , = A1 sin(kkx (m) (m) ¯ 1 )eikct , u = A cos(kkx u1

2 (m) u3

2 (m) ¯ 1 )eikct = A3 cos(kkx

(4.56)

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(m) ¯ c, and t are amplitudes, wavenumber, associated wavenumber, where Ai (i = 1, 2, 3), k, k, velocity, and time, respectively. It should be emphasized that the associated wavenumbers k¯ only appear in the trigonometric function arguments. By substituting Eq.(4.56) into Eq.(4.50), we have the equations in expansion for amplitudes [24]. As a result, we have amplitude ratios as (n)

αmi =

Ak (k¯ i ) (3) A (k¯ i )

(n, k = 1, 2, 3; i = 1, 2, 3, · · · , 9; m = 3(n − 1) + k)

(4.57)

3

and we can rewrite stresses as (n)

T1

9

∑

=

© A1n [c11 α1i + c12 β1 α2i + c14 β1 α3i ] + A2n [c11 α4i k¯ i + c12 β2 α5i + c14 β2 α6i ]

i=1

ª (3) + A3n [c11 α7i k¯ i + c12 β3 α8i + c14 β3 α9i ] A3 (k¯ i )k cos k¯ i kx1 eikct , (n)

T6

9

∑

=

© A1n [c66 (β1 α1i − α2i k¯ i ) − c65 k¯ i α2i ] + A2n [c66 (β2 α4i − α5i k¯ i ) − c65 k¯ i α6i ]

i=1

ª (3) + A3n [c66 (β3 α7i − α8i k¯ i ) − c65 k¯ i α9i ] A3 (k¯ i )k sin k¯ i kx1 eikct , (n)

T5

9

∑

=

© A1n [c56 (β1 α1i − α2i k¯ i ) − c55 k¯ i α2i ] + A2n [c56 (β2 α4i − α5i k¯ i ) − c55 k¯ i α6i ]

i=1

ª (3) + A3n [c56 (β3 α7i − α8i k¯ i ) − c55 k¯ i α9i ] A3 (k¯ i )k sin k¯ i kx1 eikct (4.58) The traction-free boundary conditions are (n)

(n)

(n)

T1 (x1 = ±a) = T6 (x1 = ±a) = T5 (x1 = ±a) = 0

(n = 1, 2, 3)

(4.59)

By substituting Eq.(4.58) into Eq.(4.59), we have frequency spectra which is the most important solutions for surface acoustic waves propagating in finite solids. As a numerical example, we study an AT-cut quartz crystal strip of H p= 10 with straight-crested waves. The surface acoustic wave velocity normalized by c66 /ρ versus. length is given in Fig.4.13. A similar study for different materials is done by Wang and Hashimoto [23]. The results show that velocities of surface acoustic wave mode and its overtone modes are generally sensitive to the plate length, while mode conversion occurs at certain lengths which should be avoided in the selection of substrates for surface acoustic wave resonators. Although there are complicated factors such as the periodic electrodes (interdigital transducers, IDTs) affect the surface acoustic wave propagation in finite anisotropic solids, proper selection of substrates is essential in resonator performance improvement. To this purpose, the velocity spectra based on our two-dimensional analysis is of practical importance, and further refinement with the consideration of electrodes will be more useful.

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

Fig. 4.13 Normalized surface wave velocity versus length in an AT-cut quartz plate with thickness H = 10.

4.3.3 Conclusions A two-dimensional theory for the analysis of surface acoustic waves in finite anisotropic solids is established based on the exponential expansion of displacements in the thickness direction with representative parameters from semi-infinite substrates. The twodimensional theory is analogous to the well-known plate theories by Mindlin and Lee for the propagation of bulk acoustic waves in finite elastic solids. These two-dimensional equations can be used for analytical solutions in a manner similar to plate equations for bulk acoustic wave problems, or they can be implemented in finite element method to improve the efficiency of numerical techniques. The limited order of variables and the absence of correction procedure make the two-dimensional theory simple and effective in practical applications involving the analysis of surface acoustic wave in finite anisotropic solids. With the calculated velocity (frequency) spectra and identified wave modes, the selection of substrate of surface acoustic wave resonators can be more optimal, thus easing the burden of trial and error iteration in the design process. With the inclusion of consideration of periodic electrodes and other complications, the accuracy of substrate selection will eliminate many uncertainties in the design process and improve the resonator performance based on a refined precise analytical model.

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4.4 A two-dimensional analysis of surface acoustic waves in finite piezoelectric plates 4.4.1 Surface acoustic waves in semi-infinite piezoelectric solids In order to obtain precise characteristics of surface acoustic wave propagation in the piezoelectric materials, we have to consider piezoelectric effect on wave velocity or frequency which will result in much larger number equations. In this section, we will establish the two-dimensional equations for surface acoustic waves in finite piezoelectric plate, and employ piezoelectrically stiffened elastic constants to eliminate the electrical variables from these complex equations. First, we study surface acoustic wave propagation in semi-infinite piezoelectric elastic solids, as shown in Fig.4.12. For surface acoustic waves propagating along x1 direction, we assume the displacements in the anisotropic solid are [24-26] u1 = A1 ekβ x2 eik(x1 −ct) , u2 = A2 ekβ x2 eik(x1 −ct) , (4.60) u3 = A3 ekβ x2 eik(x1 −ct) , φ = A4 ekβ x2 eik(x1 −ct) where A j ( j = 1, 2, 3, 4), φ , k, β , c, xi (i = 1, 2), and t are amplitudes, electrical potential, wavenumber, decaying index, phase velocity, coordinates, and time, respectively. With known displacements in Eq.(4.60), we can obtain strain components and electrical field in displacements and potential in abbreviated notation as S1 = iku1 , S2 = kβ u2 , S3 = 0, S4 = kβ u3 , S5 = iku3 , S6 = kβ u1 + iku2 , E1 = −ikφ , E2 = −kβ φ , E3 = 0

(4.61)

Consequently, the stresses components and the electrical displacements in abbreviated notation are [42,43] Tp = k[(c p6 β + ic p1 )u1 + (c p2 β + ic p6 )u2 + (c p4 β + ic p5 )u3 + (e2p β + ie1p )φ ], Di = k[(ei6 β + iei1 )u1 + (ei2 β + iei6 )u2 + (ei4 β + iei5 )u3 + (εi2 β + iεi1 )φ ] (p = 1, 2, 3, 4, 5, 6; i = 1, 2, 3)

(4.62)

where c pq , eip , and εi j are elastic, piezoelectric, and dielectric constants of material, respectively. The equations of motion of piezoelectric solids are T1,1 + T6,2 = −ρ k2 c2 u1 , T6,1 + T2,2 = −ρ k2 c2 u2 , T5,1 + T4,2 = −ρ k2 c2 u3 , D1,1 + D2,2 = 0 where ρ is the density of quartz crystal. Substituting Eqs.(4.60) and (4.62) into Eq.(4.63), we have (ρ c2 + c66 β 2 + 2c16 iβ − c11 )A1 + [c26 β 2 + (c12 + c66 )iβ − c16 ]A2 +[c46 β 2 + (c14 + c56 )iβ − c15 ]A3 + [e26 β 2 + (e16 + e21 )iβ − e11 ]A4 = 0,

(4.63)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

[c26 β 2 + (c12 + c66 )iβ − c16 ]A1 + (ρ c2 + c22 β 2 + 2c26 iβ − c66 )A2 +[c24 β 2 + (c25 + c46 )iβ − c56 ]A3 + [e22 β 2 + (e12 + e26 )iβ − e16 ]A4 = 0, [c46 β 2 + (c14 + c56 )iβ − c15 ]A1 + [c24 β 2 + (c25 + c46 )iβ − c56 ]A2 +(ρ c2 + c44 β 2 + 2c45 iβ − c55 )A3 + [e24 β 2 + (e25 + e14 )iβ − e15 ]A4 = 0, [e26 β 2 + (e16 + e21 )iβ − e11 ]A1 + [e22 β 2 + (e12 + e26 )iβ − e16 ]A2 +[e24 β 2 + (e25 + e14 )iβ − e15 ]A3 + [ε11 − (ε12 + ε21 )iβ − ε22 β 2 ]A4 = 0 (4.64) To obtain nontrivial solutions of unknown amplitudes in Eq.(4.64), we must let the determinant to vanish. With known elastic, piezoelectric, and dielectric constants, and velocity, we will obtain four decaying index β = (β1 , β2 , β3 , β4 ,). Then, displacements and potential in Eq.(4.60) can be written as [24-26] 4

uj =

∑ A jn exp(kβn ) exp ik(x1 − ct)

( j = 1, 2, 3),

n=1 4

φ=

(4.65)

∑ A4n exp(kβn ) exp ik(x1 − ct)

n=1

and amplitude ratios are defined as

αir =

Ai (βr ) A4 (βr )

(i, r = 1, 2, 3, 4)

(4.66)

where amplitudes A jn ( j = 1, 2, 3, 4; n = 1, 2, 3, 4) are to be determined. For analysis, the next step is to substitute the displacements and electrical potential expressions in Eq.(4.66) into the stress and electrical displacements expressions in Eq.(4.62), we obtain 4

Tp = k ∑ [(c p6 βr + ic p1 )α1r + (c p2 βr + ic p6 )α2r + (c p4 βr + ic p5 )α3r r=1

+(e p2 βr + ie p1 )]A4r ekβr x2 eik(x1 −ct) , 4

D j = k ∑ [(e j6 βr + ie j1 )α1r + (e j2 βr + ie j6 )α2r + (e j4 βr + ie j5 )α3r r=1

−(ε j2 βr + iε j1 )]A4r ekβr x2 eik(x1 −ct) (p = 1, 2, 3, 4, 5, 6; j = 1, 2, 3)

(4.67)

We now need to apply the traction-free and charge-free boundary conditions on plate surfaces, which are T2 (x2 = 0) = T4 (x2 = 0) = T6 (x2 = 0) = D2 (x2 = 0) = 0

(4.68)

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to calculate the wave velocity.

4.4.2 Two-dimensional equations for finite piezoelectric elastic plates We expand the displacements and electrical potential in known basis functions of the thickness coordinate as 4

u j (x1 , x2 , x3 ,t) =

(n)

∑ uj

n=1 4

∑φ

φ (x1 , x2 , x3 ,t) =

(x1 , x3 ,t)φn (x2 )

( j = 1, 2, 3), (4.69)

(n)

(x1 , x3 ,t)φn (x2 )

n=1 (n)

where u j ( j = 1, 2, 3), φ (n) and φn (n = 1, 2, 3, 4) are higher-order displacements, electrical potential components, and basis functions, respectively. The basis function will be from surface acoustic wave solutions of semi-infinite piezoelectric solids in the form

φn (x2 ) = {exp(kβ1 x2 ), exp(kβ2 x2 ), exp(kβ3 x2 ), exp(kβ4 x2 )}

(n = 1, 2, 3, 4) (4.70)

where decaying parameters βn (n = 1, 2, 3, 4) are obtained from Section 4.4.1. Based on the theory of piezoelectricity and displacements and electrical potential in Eqs.(4.69) and (4.70), we have the corresponding strain and electrical field as 4

Sp =

(n)

∑ Sp

4

φn , Ei =

n=1 (n)

(n)

∑ Ei

φn

(p = 1, 2, 3, 4, 5, 6; i = 1, 2, 3)

(4.71)

n=1

(n)

where S p and Ei (n = 1, 2, 3, 4) are higher-order two-dimensional strain and electrical filed components, respectively. With displacements and electrical potential in Eq.(4.69), it is straight-forwarded to have the following components: (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

S1 = u1,1 , S2 = kβn u2 , S3 = u3,3 , S4 = u2,3 + kβn u3 , (n)

(n)

(n)

(4.72)

S5 = u3,1 + u1,3 , S6 = u2,1 + kβn u1 ; (n)

(n)

(n)

(n)

(n)

E1 = −φ,1 , E2 = −kβn φ (n) , E3 = −φ,3

(n = 1, 2, 3, 4)

Consequently, we have the stresses and electrical displacements for an anisotropic piezoelectric material as 4

Tp =

(n) ∑ Tˆp φn , Di =

n=1

with

4

(n)

∑ Dˆ i

φn

(p = 1, 2, 3, 4, 5, 6; i = 1, 2, 3)

(4.73)

n=1

(n) (n) (n) (n) (n) (n) Tˆp = c pq Sq − e pk Ek , Dˆ i = eip S p + εik Ek (p, q = 1, 2, 3, 4, 5, 6; i, k = 1, 2, 3; n = 1, 2, 3, 4)

(4.74)

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

With displacements and electrical potential in Eq.(4.69) and strains and electrical fields in Eq.(4.74), we follow Mindlin [27,28] and Lee et al. [29] with a piezoelectric solid illustrated in Fig.4.9, we have Z Z 0 A −h

Z Z 0

A −h

4

(Ti j,i − ρ u¨ j )

(m)

∑ δu j

φm dx2 dA = 0,

m=1 4

Di,i

∑

m=1

(4.75)

(m) δφ j φm dx2 dA

=0

where A is the faces, or the domain of the solids. Through integration of Eq.(4.75), we have (n) (n) (n) Ti j,i − T¯2 j + F2 j = ρ (n) (n) Di,i − D¯ 2 + D(n)

4

(m)

∑ Amn u¨ j

,

(4.76)

m=1

=0

(i, j = 1, 2, 3; n = 1, 2, 3, 4; H = 10)

with two-dimensional quantities defined as (n)

Ti j = (n)

Di

=

Amn =

Z 0 −h Z 0 −h

Z 0

−h

(n) (n) (n) Ti j φn dx2 , T¯2 j = kβn T2 j , F2 j = T2 j (0)φn (0) − T2 j (−h)φn (−h), (n) (n) Di φn dx2 , D¯ 2 = kβn D2 , D(n) = D2 (0)φn (0) − D2 (−h)φn (−h),

φm φn dx2

(4.77)

(i, j = 1, 2, 3; n = 1, 2, 3, 4; H = 10)

Now we have the stress and electrical displacement components as (n)

Tp

4

=

∑ Amn

£ (m) (m) (m) (m) (m) (m) c p1 u1,1 + c p2 kβm u2 + c p3 u3,3 + c p4 (u2,3 + kβm u3 ) + c p5 (u3,1

m=1 (m) (m) (m) (m) (m) ¤ +u1,3 ) + c p6 (u2,1 + kβm u1 ) + e1p φ,1 + e2p kβm φ (m) + e3p φ,3 , (n)

Di

4

=

∑ Amn

m=1

£ (m) (m) (m) (m) (m) (m) (m) ei1 u1,1 + ei2 kβm u2,2 + ei3 u3,3 + ei4 (u2,3 + kβm u3 ) + ei5 (u3,1 + u1,3 ) (m)

(m)

(m)

(m) ¤

+ei6 (u2,1 + kβm u1 ) − εi1 φ,1 − εi2 kβm φ (m) − εi3 φ,3

(H = 10) (4.78)

We now examine surface acoustic wave travelling along the x1 direction in a straightcrested manner. For simplicity, solutions are assumed as (m)

(m) ¯ 1 eikct , u(m) = A(m) cos kkx ¯ 1 eikct , = A1 sin kkx 2 2 (m) (m) ¯ 1 eikct , φ (m) = A(m) sin kkx ¯ 1 eikct u3 = A3 cos kkx 4

u1

(4.79)

The usual solution procedure, which is to substitute Eq.(4.79) back into the 16 equations in Eq.(4.77), does not yield the results we are expected. Namely, associated wavenumbers are not conjugated complex numbers which can be utilized.

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The approximation can be made with the surface acoustic wave velocity of piezoelectric solids using the piezoelectrically stiffened elastic constants [9] c¯ pq = c pq +

e1p e1q ε11

(p, q = 1, 2, 3, 4, 5, 6)

(4.80)

where e1p , e1q , and ε11 are specific piezoelectric and dielectric constants, respectively. Accordingly, the two-dimensional equations can be simplified also. This approximation can be traced to the three-dimensional equation of charge of piezoelectricity with the truncation

ε11 φ,11 = e11 u1,11 + e15 u3,11 + e16 u2,11

(4.81)

Through integration of this equation, we have

φ=

e11 u1 + e15 u3 + e16 u2 + f1 (x2 )x1 + f2 (x2 ) ε11

(4.82)

where two arbitrary functions fi (x2 )(i = 1, 2) are to be determined. Naturally, we can approximate the two-dimensional electrical potential with Eq.(4.82) as

φ=

eikct ε11

9

(3)

∑ A3i (e11 sin kk¯ i x1 + e16 cos kk¯ i x1 + e15 cos kk¯ i x1 ) + f1 (x2 )x1 + f2 (x2 )

i=1

(4.83) For typical piezoelectric devices with electrodes on the upper and lower faces shown in Fig.4.9, we can treat the two arbitrary functions as the static electrical potentials and let φ (0) φ (h) f1 (x2 ) = 0, f2 (x2 ) = (h − x2 ) + x2 (4.84) h h where φ (0) and φ (h) are the potentials on the lower and upper faces. The two-dimensional equations are given as a complete set for solutions of surface acoustic wave propagating in a finite piezoelectric plate, but the accuracy has to be verified. As a numerical example of applications, we studied an ST-cut quartz crystal strip of H = 10 with straight-crested surface acoustic waves. The surface acoustic wave velocity p normalized by c66 /ρ versus length is given in Fig.4.14. It is shown that the piezoelectricity has noticeable effect on the surface acoustic wave velocity in a finite structure based on the analysis with the two-dimensional theory and piezoelectrically stiffened elastic constants. Our approximation is based on the fact that the piezoelectric effect through three-dimensional equations is very small [9]. The simplified consideration of piezoelectric effect will make the analysis of surface acoustic wave resonators with larger number of pairs of periodic electrodes more convenient [26-39].

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Ji Wang, RongXing Wu, JinBo Lin, QiaoQiao Pan, and JianKe Du

Fig. 4.14 Normalized surface acoustic wave velocity versus length in an ST-cut quartz plate with thickness H = 10. The computed frequency spectra from the mechanical vibrations (circle ◦) is compared with the piezoelectrically stiffened constants (dot •).

4.4.3 Conclusions In this section, we first studied surface acoustic wave propagation in semi-infinite piezoelectric solid and obtained four decaying parameters for the following deduction. Similar with anisotropic material, we established two-dimensional equations of surface acoustic wave propagation in finite piezoelectric plate, and found that these equations are too complicated to solve with available methods. By an introduction of stiffened elastic constants, we eliminated the electrical variables form these coupling equations. Compared with mechanical vibration of surface acoustic wave propagation in finite ST-cut quartz crystal plate, our approximation still revealed that the piezoelectricity has noticeable effect on the surface acoustic wave velocity. These equations without simplification can also used for the future simulation of surface acoustic wave propagation in finite piezoelectric plates by finite element method or other numerical methods.

4.5 Summary Based on our analysis of surface acoustic waves in an infinite isotropic elastic plate, the two-dimensional theory for surface acoustic waves in finite isotropic plates with small thickness has been established by including both exponentially decaying and growing modes which can be found from semi-infinite substrates. We found that the consideration of plate thickness offers much improved analytical capability of our two-dimensional theory. As the results show, the increase of the plate thickness will change the phase velocity

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139

spectrum, or the velocity versus plate thickness, significantly. As the plate thickness increases, the phase velocity will have a much simpler spectrum, implying less coupling and interaction between modes. Following this direction, we extend our research to anisotropic and piezoelectric material. Though investigating surface acoustic wave propagation in semi-infinite anisotropic and piezoelectric elastic solids, we have obtained three and four decaying parameters in these two different substrates. For the thickness of plate larger than five wavelengths, we don’t need to consider the exponentially growing modes for surface acoustic wave propagation in anisotropic and piezoelectric plate. The two-dimensional equations for surface acoustic wave propagation in finite anisotropic plate can provide us reliable results, while the equations of piezoelectric plate are too complicated to give any accurate results. We employed stiffened elastic constants to consider piezoelectric effects. All these two-dimensional theories are analogous to the well-known plate theories by Mindlin and Lee for the propagation of bulk acoustic waves in finite elastic solids. These two-dimensional theories can improve analysis and design of surface acoustic wave resonator through the accurate prediction of essential wave propagating parameters like velocity, mode shapes, and couplings. These analyses are what have been lacking in surface acoustic wave analysis in comparison with relatively sophisticated techniques for bulk acoustic wave resonators, which can be analyzed with Mindlin and Lee plate theories effectively.

Acknowledgment This research is supported in part by grants from the National Natural Science Foundation of China (Nos. 10572065, 10932004, and 10772087) and the Doctoral Program Fund of Ministry of Education of China (No. 20093305110003). Additional supports are from the K. C. Wang Magana Fund and the Doctoral Dissertation Enhancement Fund administered by Ningbo University (No. PY2009002).

References [1] Auld B A. Acoustic Fields and Waves in Solids (I & II). Malabar, Florida: Krieger, 1990. [2] Royer D, Dieulesaint E. Elastic Waves in Solids (I & II). Berlin: Springer, 2000. [3] Hashimoto K Y. Surface Acoustic Wave Devices in Telecommunications: Modelling and Simulation. Berlin: Springer, 2000. [4] Love A E H. A Treatise on the Mathematical Theory of Elasticity II. Cambridge: Cambridge University Press, 1893, 16-17. [5] Victorov I A. Rayleigh and Lamb Waves: Physical Theory and Applications. New York: Plenum Press, 1967. [6] Liu G R, Xi Z C. Elastic Waves in Anisotropic Laminates. Boca Raton, Florida: CRC Press, 2001.

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[7] Achenbach J D. Wave Propagation in Elastic Solids. North-Holland, 1987. [8] Graff K F. Wave Motion in Elastic Solids. Dover, 1991. [9] Farnell G W. Properties of elastic surface waves. In: Mason W P, Thurston R N. ed. Physical Acoustics. Academic Press, 1970, 6, 109-166. [10] Wu Y. Waves in elastic solids. In: Zhang F, Wang L. ed. Modern Piezoelectricity I. Beijing: Science Press, 2001, 100-140. (in Chinese) [11] Tiersten H F. Elastic surface waves guided by thin films. J. Appl. Phys., 1969, 40(2), 770-789. [12] Liu G R, Tani J. Surface waves in functionally gradient piezoelectric plates. ASME J. Vib. Acoust., 1994, 116, 440-448. [13] B¨ovik P. A comparison between the Tiersten model and O(h) boundary conditions for elastic surface acoustic waves guided by thin layers. ASME J. Appl. Mech., 1996, 63, 162-167. [14] Liang W, Shen Y P. Gradient surface ply model of SH wave propagation in SAW sensors. Acta Mech. Sin., 1999, 15(2), 155-163. [15] Fang H Y, Yang J S, Jiang Q. Surface acoustic waves propagating over a rotating piezoelectric half-space. IEEE Trans. Ultrason., Ferroelectr. Freq. Contr., 2001, 48(4), 998-1044. [16] Alshits V I, Maugin G A. Dynamics of multilayers: elastic waves in an anisotropic graded or stratifield plate. Wave Motion, 2005, 41(4), 357-394. [17] Wang Q, Quek S T, Varadan V K. Analytical solution for shear horizontal wave propagation in piezoelectric coupled media by interdigital transducer. ASME J. Appl. Mech., 2005, 72(3), 341-350. [18] Song B, Hong W. A new algorithm for the extraction of the surface waves for the Green’s function in layered dielectrics. Sci. China-Inf. Sci., 2002, 45(2), 143-151. [19] Wang X M, Zhang H L. Modeling of elastic wave propagation on a curved free surface using an improved finite-difference algorithm. Sci. China-Phys. Mech. Astron., 2004, 47(5), 633648. [20] Chen Z J, Han T, Ji X J, et al. Lamb wave sensors array for nonviscous liquid sensing. Sci. China-Phys. Mech. Astron., 2006, 49(4), 461-472. [21] Wang J, Du J K, Pan Q Q. A two-dimensional analysis of surface acoustic waves in finite plates with eigensolutions. Sci. China-Phys. Mech. Astron., 2007, 50(5), 631-649. [22] Wang J, Du J K, Lu W Q, et al. Exact and approximate analysis of surface acoustic waves in an infinite elastic plate with a thin metal layer. Ultrasonics, 2006, 44(S1), 941-945. [23] Wang J, Hashimoto K Y. A two-dimensional theory for the analysis of surface acoustic waves in anisotropic elastic solids. Proceedings of 2003 IEEE International Ultrasonics Symposium, 2003, 637-640. [24] Wang J, Hashimoto K Y. A two-dimensional theory for the analysis of surface acoustic waves in finite elastic solids. J. Sound Vib., 2006, 295, 838-855. [25] Wang J, Lin J B. A two-dimensional theory for surface acoustic wave analysis in finite piezoelectric solids. J. Intel. Mat. Syst. Str., 2005, 16(7-8), 623-629. [26] Wang J, Lin J B, Wan Y P, et al. A two-dimensional analysis of surface acoustic waves in finite solids with considerations of electrodes. Int. J. Appl. Electrom. Mech., 2005, 22(1-2), 530-568. [27] Mindlin R D. In: Yang J S. ed. An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. Hackensack, New Jersey: World Scientific, 2006. [28] Mindlin R D. High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct., 1972, 8, 891-906. [29] Lee P C Y, Yu J D, Lin W S. A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces. J. Appl. Phys., 1998, 83(3), 1213-1223.

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[30] Peach R C. A normal mode expansion for the piezoelectric plates and certain of its applications. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 1988, 35, 593-611. [31] Wang J, Yang J S. Higher-order theories of piezoelectric plates and applications. Appl. Mech. Rev., 2000, 53(4), 87-99. [32] Wang J, Yong Y K, Imai T. Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory. Int. J. Solids Struct., 1999, 36(15), 2303-2319. [33] Wang J, Yu J D, Yong Y K, et al. A new theory for electroded piezoelectric plates and its finite element application for the forced vibrations of quartz crystal resonators. Int. J. Solids Struct., 2000, 37, 5653-5673. [34] Yong Y K. Analysis of periodic structures for BAW and SAW resonators. Proceedings of 2001 IEEE Ultrasonics Symposium, 2001, 781-789. [35] Yoon S, Yu J D, Kanna S, et al. Finite element analysis of the substrate thickness on traveling leaky surface acoustic waves. Proceedings of 2003 IEEE International Ultrasonics Symposium, 2004, 1696-1699. [36] Lee P C Y, Syngellakis S, Hou J P. A two-dimensional theory for high frequency vibrations of piezoelectric crystal plates with or without electrodes. J. Appl. Phys., 1987, 61(4), 1249-1262. [37] Wang J, Du J K, Lin J B, et al. Two-dimensional analysis of the effect of an electrode layer on surface acoustic waves in a finite anisotropic plate. Ultrasonics, 2006, 44(S1), 935-939. [38] Wang J, Li Z, Du J K. An analysis of the effect of periodic electrodes on surface acoustic wave resonators. Proceedings of IEEE 2006 International Frequency Control Symposium, 2006, 161-164. [39] Rahman S, Langtangen H P, Barnes C H W. A finite element method for modelling electromechanical wave propagation in anisotropic piezoelectric media. Commun. Comput. Phys., 2006, 2(2), 271-292. [40] Mackerle J. Crystals and polycrystals: FEM and BEM material modeling: a bibliography (1998-2000). Finite Elem. Anal. Des., 2002, 38, 461-475. [41] Zhang W, Tang J C. Constitutive computational modeling foundation of piezoelectric microstructures and application to high-frequency microchip DSAW resonators. Acta Mech. Sin., 2002, 18(2), 170-180. [42] Tiersten H F. Linear Piezoelectric Plate Vibrations. New York: Plenum Press, 1969. [43] Yang J S. An Introduction to the Theory of Piezoelectricity. Berlin: Springer, 2005. [44] ANSI/IEEE std 176-1987. IEEE Standard on Piezoelectricity. IEEE Inc., Piscataway, New Jersey, 1988.

Chapter 5 Wave Characteristics in the Functionally Graded Piezoelectric Waveguides: Legendre Polynomial Approach

JianGong Yu School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, 454000, China

Abstract Functionally graded piezoelectric material (FGPM) has more advantages than the traditional piezoelectric material (PEM). Various FGPM structures can be used for fabricating the different acoustic sensors. The behavior of the selected wave mode directly affects the performance of the device. By the Legendre orthogonal polynomial approach, this chapter presents an overview of the guided wave characteristics in three typical FGPM structures: plate, hollow cylinder and hollow sphere. Dispersion curves for FGPM structures and the corresponding non-piezoelectric structures are calculated to show the piezoelectric effect. Mechanical displacement distribution and electric potential distribution are also illustrated. The influence of the ratio of radius to thickness is discussed. The influence of the polarizing direction on the piezoelectric effect is studied. The differences of the wave characteristics between PEM and FGPM structures are also concerned. In this chapter, the open circuit surface is assumed. Keywords functionally graded piezoelectric material, waveguide structures, Legendre polynomial, dispersion curves, polarizing direction

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5.1 Introduction To enhance the effect of piezoelectric materials (PEM), the concept of functionally graded piezoelectric material has been proposed and termed FGPM. Various techniques for fabricating FGPM have been developed [1-8]. Zhu and Meng [9] and Wu et al. [10] have reported the development of FGPM actuators with gradation of compositions from high to low piezoelectric property and from low to high dielectric property. Mahapatra et al. [11] discussed the advantages of FGPM interdigital transducers (IDT) over the traditional IDT. The behavior of the selected wave mode directly affects the performance of the device. Thus, in the applications of FGPM to SAW devices, the study of wave propagation behavior in FGPM structures is of considerable importance. Different FGPM structures can be used for fabricating the different acoustic sensors, such as cylindrical curved plate for line-focus sensor and spherical curved plate for pointfocus sensor. The investigations of wave propagation in the FGPM can be classified as two structures. One is the half infinite medium that is composed of various combinations FGPM layer and elastic or piezoelectric substrates. The researching objectives are Love waves and Rayleigh waves. In this area, the research team of Xi’an Jiaotong University (Liu J, Wang Z K, Jin F, et al.) obtained many results [12-20] by the WKB method, power series method and analytical technique. Furthermore, Du et al. achieved some significant work. They obtained the analytical solutions of Love wave dispersion relations [21] and considered the effect of viscous dissipation [22]. Eskandari and Shodja [23] also investigated the Love wave propagation in quadratic variation FGPM medium. The other is the waveguide structures of finite thickness. The researching objectives are guided elastic waves. As early as 1991, Liu and Tani [24, 25] proposed a hybrid numerical method for characteristics and response of wave propagation in FGPM plates, where the FGPM layer is divided into a number of inhomogeneous thin layers and the Fourier transformation method was employed. Han et al. [26] used linearly inhomogeneous elements to analyze the wave characteristics in FGPM cylinders. Afterwards, Liu et al. [27] proposed quadratic inhomogeneous elements for the dispersion of waves in FGPM plates. Chakraborty et al. [28] formulated a set of finite elements to analyze wave propagation in FGPM plates when subjected to mechanical, thermal loading or piezoelectric actuation, where the material properties are allowed to vary both in length and in thickness directions. Using the spectral element method, Chakraborty et al. [29] characterized the wave propagation in FGPM plates by the thin-layer model, and modeled an ultrasonic transducer composed of FGPM. Based on higher-order shear deformation theory, Mahapatra et al. [11] used spectral element method investigating the Lamb wave characteristics in FGPM plates, and pointed that there was the reduced dispersion of the Lamb wave modes in FGPM plates. The above research work have a common point, they divided the FGPM structure into many homogeneous or inhomogeneous layers. Lefebvre et al. treated the FGPM structures as continuous gradient medium. They developed the Legendre orthogonal polynomial series method to investigate the wave propagation in FGPM plates [30]. Then using the polynomial method Yu et al. investigated the wave

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145

characteristics in FGPM cylindrical curved plates [31], spherical curved plates [32] and hollow cylinders [33]. A specificity of the polynomial approach lies in how the mechanical and electrical boundary conditions are incorporated into the equations of motion and are automatically accounted for by the device of assuming position-dependent material physical constants. The equations of motion are then converted into a matrix eigenvalue problem thanks to an expansion of the independent mechanical and electrical variables in an appropriate series of orthonormal functions: this makes possible the semi-variational determination of the frequencies of modes and associated profiles. The orthogonal polynomial method was first applied to line acoustic waves guided in homogeneous semi-infinite wedges and ridges [34-39], then to surface acoustic waves in layered [40, 41] and graded [42] semiinfinite structures. It has subsequently been applied to optimization of SAW-based signal processing and communications devices [43, 44]. These investigating objects were half infinite medium. So the used orthogonal polynomial is Laguerre polynomial. Its orthogonal integral interval is (0, ∞). Later on, this method was extended to analyze guided wave propagation in finitethickness structures, such as multilayered PEM [45] and FGPM [30] plates. Here, the orthogonal integral interval of the used orthogonal polynomial can not be infinite. This mapped orthogonal functions technique with automatically satisfied boundary conditions is not limited to only flat surfaces but is capable to calculate the vibration modes of topographic waveguides of various geometries. It has been used to calculate axial waves in anisotropic homogeneous [46] and FGM hollow cylinders [47] and then to axial [33] and circumferential [31] waves in FGPM hollow cylinders. It has also been applied to calculate toroidal waves in the FGM [48] and FGPM [32] spherical curved plates. This mapped orthogonal functions technique is not limited to only either anisotropic elastic media or piezoelectric elastic media. Very recently, it has also been extended to the application of the wave propagation in piezoelectric-piezomagnetic FGM to study the magneto-electric effect both in plates [49] and hollow cylinders [50, 51]. In this chapter, resorted to the Legendre orthogonal polynomial method and based on the references [30-33], we present an overview of the guided wave characteristics in three typical FGPM structures: plate, hollow cylinder and hollow sphere. Dispersion curves for FGPM structures and the corresponding non-piezoelectric structures are calculated to show the piezoelectric effect. Mechanical displacement distribution and electric potential distribution are also illustrated. The influence of the ratio of radius to thickness in curved waveguides is discussed. The influence of the polarizing direction on the piezoelectric effect is studied. The differences of the wave characteristics between PEM and FGPM structures are also concerned. In this chapter, the open circuit surface is assumed.

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5.2 Wave propagation in the FGPM plate In this section, through solving the wave equation of the FGPM plate we introduce the application of the orthogonal functions technique (developed by Lefebvre et al. [30]) to the FGPM structures. The obtained solution is compared with the published data to confirm the availability of method.

5.2.1 Mathematics and formulation Consider an anisotropic FGPM plate which is infinite horizontally with a thickness h. We place the horizontal (x,y)-plane of a Cartesian coordinate system on the bottom surface and let the plate be in the positive z-region. For the wave propagation considered in this paper, the body forces, electric charge are assumed to be zero. Thus, the dynamic equation for the piezoelectric plate is governed by

∂ 2 ux ∂ Txx ∂ Txy ∂ Txz + + =ρ 2 , ∂x ∂y ∂z ∂t ∂ Txz ∂ Tyz ∂ Tzz ∂ 2 uz + + =ρ 2 , ∂x ∂y ∂z ∂t

∂ Txy ∂ Tyy ∂ Tyz ∂ 2 uy + + =ρ 2 , ∂x ∂y ∂z ∂t ∂ Dx ∂ Dy ∂ Dz + + =0 ∂x ∂y ∂z

(5.1)

where ui , Ti j , and Di are the elastic displacement, stress and electric displacement, and ρ is the density of the material. By introducing the rectangular window function π (z) ½ 1 (0 6 z 6 h) π (z) = 0 (elsewhere) the stress-free and open circuit surface boundary conditions (Tzz = Txz = Tyz = 0, and Dz = 0 at z = 0, z = h) are automatically incorporated in the constitutive relations of the linear, anisotropic, piezoelectric solid:      C11 C12 C13 C14 C15 C16     Txx    εxx       εyy   Tyy  C22 C23 C24 C25 C26                  Tzz C C C C ε zz 33 34 35 36  = π (z)  Tyz  C44 C45 C46  2εyz              Txz  symmetry C55 C56   2εxz               Txy C66 2εxy   e11 e21 e31  e12 e22 e32       e13 e23 e33   Ex   − (5.2a)  e14 e24 e34   Ey  π (z)   Ez  e15 e25 e35  e16 e26 e36

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

  εxx           εyy      e11 e12 e13 e14 e15 e16   Dx   εzz   Dy = e21 e22 e23 e24 e25 e26 π (z)    2εyz   Dz e31 e32 e33 e34 e35 e36     2εxz        2εxy    11 12 13  Ex  Ey π (z) + 22 23    symmetry 33 Ez

147

(5.2b)

where εi j and Ei are the strain and electric field; Ci j , ei j and i j are the elastic, piezoelectric and dielectric coefficients, respectively. In fact, the rectangular window function π (z) that added in Txx , Txy ,Tyy , Dx and Dy is not necessary, but it dose not influence the results because the latter integration of Eq.(5.7) over z is from 0 to h. The formulation of Eq.(5.2) is to facilitate. The relationship between the general strain and general displacement can be expressed as

∂ uy ∂ ux ∂ uz , εyy = , εzz = , ∂x ∂y ∂z µ ¶ µ ¶ 1 ∂ uy ∂ uz 1 ∂ ux ∂ uz εyz = + , εxz = + , 2 ∂z ∂y 2 ∂z ∂x µ ¶ 1 ∂ ux ∂ uy ∂Φ ∂Φ εxy = + , Ex = − , Ey = − , 2 ∂y ∂x ∂x ∂y εxx =

Ez = −

∂Φ ∂z

(5.3)

Because the material properties vary in the thickness direction, the elastic constants of the medium are the function of z: ³ z ´1 ³ z ´2 ³ z ´L C(z) = C(0) +C(1) +C(2) + · · · +C(L) h h h With implicit summation over repeated indices, C(z) can be written compactly as C(z) = C(l) zl

(l = 0, 1, 2, · · · , L)

(5.4)

And other material constants can be treated in the same way:

ρ (z) = ρ (l)

³ z ´l

(l)

ei j (z) = ei j

³ z ´l h

,

(l) i j (z) = i j

³ z ´l

(l = 0, 1, 2, · · · , L) (5.5) For a free harmonic wave being propagated in the x direction in a plate, we assume the displacement components, electric potential to be of the form h

,

h

ux (x, y, z,t) = exp i(kx − ω t)U(z)

(5.6a)

uy (x, y, z,t) = exp i(kx − ω t)V (z)

(5.6b)

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JianGong Yu

uz (x, y, z,t) = exp i(kx − ω t)W (z)

(5.6c)

Φ (x, y, z,t) = exp i(kx − ω t)X(z)

(5.6d)

where U(z), V (z), W (z) represent the amplitude of vibration in the x, y, z directions respectively and X(z) represents the amplitudes of electric potential. k is the magnitude of the wave vector in the propagation direction, and ω is the angular frequency. Substituting Eqs.(5.2)-(5.6) into Eq.(5.1), the governing differential equations in terms of displacement components and electric potential can be obtained: ³ z ´l

(l)

(l)

(l)

(l)

(l)

(l)

(l)

[C55 U 00 +C45 V 00 +C35 W 00 + e35 X 00 + 2ikC15 U 0 + ik(C14 +C56 )V 0 h (l) (l) (l) (l) (l) (l) (l) (l) +lz−1 (C55 +C45 V 0 +C35 W 0 + e35 X 0 ) + ik(C13 +C55 )W 0 + ik(e15 + e31 )X 0 (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

−k2 (C11 U +C16 V +C15 W + e11 X) +likz−1 (C15 U +C56 V +C55 W + e15 X)]π (z) ³ z ´l (l) (l) (l) (l) (l) (l) +(δ (z − 0) − δ (z − h)) (C55 U 0 +C45 V 0 +C35 W 0 + e35 X 0 + ikC15 U + ikC56 V h ρ (l) zl ω 2 (l) (l) +ikC55 W + ike15 X) = − U π (z) (5.7a) hl ³ z ´l (l) (l) (l) (l) (l) (l) (l) (l) [C45 U 00 +C44 V 00 +C34 W 00 + e34 X 00 + ik(C14 +C56 )U 0 + ik(C36 +C45 )W 0 h (l) (l) (l) (l) (l) (l) (l) +2ikC46 V 0 + lz−1 (C45 U 0 +C44 V 0 +C34 W 0 + e34 X 0 ) + ik(e14 + e36 )X 0 (l)

(l)

(l)

(l)

−k2 (C16 U +C66 V +C56 W + e16 X)+likz−1(C14 U +C46 V +C45 W + e14 X)]π (z) ³ z ´l (l) (l) (l) (l) (l) +(δ (z − 0) − δ (z − h)) (C45 U 0 +C44 V 0 +C34 W 0 + e34 X 0 + ikC14 U h ρ (l) zl ω 2 (l) (l) (l) +ikC46 V + ikC45 W + ike14 X) = − V π (z) (5.7b) hl ³ z ´l (l) (l) (l) (l) (l) (l) (l) [C35 U 00 +C34 V 00 +C33 W 00 + e33 X 00 + ik(C13 +C55 )U 0 + 2ikC35 W 0 h (l) (l) (l) (l) (l) (l) (l) (l) +lz−1 (C35 U 0 +C34 V 0 +C33 W 0 + e33 X 0 ) + ik(e13 + e35 )X 0 + ik(C36 +C45 )V 0 (l)

(l)

(l)

(l)

−k2 (C15 U +C56 V +C55 W + e15 X) +likz−1 (C13 U +C36 V +C35 W + e13 X)]π (z) ³ z ´l (l) (l) (l) (l) (l) (l) +(δ (z − 0) − δ (z − h)) (C35 U 0 +C34 V 0 +C33 W 0 + e33 X 0 + ikC13 U + ikC36 V h ρ (l) zl ω 2 (l) (l) +ikC35 W + ike13 X) = − W π (z) (5.7c) hl ³ z ´l (l) (l) (l) (l) (l) (l) (l) (l) [e35 U 00 + e34 V 00 + e33 W 00 − 33 X 00 + ik(e15 + e31 )U 0 + ik(e14 + e36 )V 0 h (l) (l) (l) (l) (l) (l) (l) +lz−1 (e35 U 0 + e34 V 0 + e33 W 0 − 33 X 0 ) + ik(e13 + e35 )W 0 − 2ik 13 X 0 (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

−k2 (e11 U + e16 V + e15 W − 11 X) +likz−1 (e31 U + e36 V + e35 W − 13 X)]π (z) ³ z ´l (l) (l) (l) (l) (l) +(δ (z − 0) − δ (z − h)) (e35 U 0 + e34 V 0 + e33 W 0 − 33 X 0 + ike31 U h

5 (l)

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides (l)

+ike36 V + ike35 W − ik

(l) 13 X) = 0

149

(5.7d)

U(z), V (z), W (z), and X(z) can be expanded to Legendre orthogonal polynomial series as ∞

U(z) =

∑ p1m Qm (z),

m=0 ∞

X(z) =

∞

V (z) =

∑ p2m Qm (z),

∞

W (z) =

m=0

∑ p3m Qm (z),

m=0

∑ rm Qm (z)

(5.8)

m=0

where pim (i = 1, 2, 3) and rm are the expansion coefficients and r µ ¶ 2m + 1 2z − h Pm Qm (z) = h h with Pm being the mth Legendre polynomial. Theoretically, m runs from 0 to ∞. In practice, the summation over the polynomials in Eq.(5.8) can be halted at some finite value m = M, when higher order terms become essentially negligible. Multiplying by Q∗j (z) with j running from 0 to M, integrating over z from 0 to h, and taking advantage of the orthonormality of the functions Qm (z), gives the following systems: l

j,m 1 j,m 2 j,m 3 j,m A11 pm + l A12 pm + l A13 pm + l A14 rm = −ω 2 · l Mmj p1m

j,m 1 j,m 2 j,m 3 j,m A21 pm + l A22 pm + l A23 pm + l A24 rm j,m j,m j,m j,m l A31 p1m + l A32 p2m + l A33 p3m + l A34 rm j,m 2 j,m 3 j,m l j,m 1 A41 pm + l A42 pm + l A43 pm + l A44 rm l

2 l

(5.9a)

= −ω · Mmj p2m = −ω 2 · l Mmj p3m

(5.9b)

=0

(5.9d)

(5.9c)

j,m where l Aαβ (α , β = 1, 2, 3, 4) and l Mmj are the elements of a non-symmetric matrix. They can be obtained according to Eq.(5.7). Removing those terms whose subscript include 4 from Eq.(5.9), the equation represents the wave propagation equation in pure elastic FGM plate. Equation (5.9d) can be written as j,m −1 l j,m 1 j,m 2 j,m 3 rm = −(l A44 ) ( A41 pm + l A42 pm + l A43 pm )

(5.10)

j,m −1 j,m where (l A44 ) denotes the inverse of the matrix (l A44 ). Substituting Eq.(5.10) into Eqs.(5.9a), (5.9b), (5.9c) has j,m j,m j,m −1 j,m 1 j,m j,m j,m −1 j,m 2 [A11 − A14 (A44 ) A41 ]pm + [A12 − A14 (A44 ) A42 ]pm j,m j,m j,m −1 j,m 3 +[A13 − A14 (A44 ) A43 ]pm = −ω 2 Mmj p1m

(5.11a)

j,m j,m j,m −1 j,m 1 j,m j,m j,m −1 j,m 2 [A21 − A24 (A44 ) A41 ]pm + [A22 − A24 (A44 ) A42 ]pm j,m j,m j,m −1 j,m 3 2 j 2 +[A23 − A24 (A44 ) A43 ]pm = −ω Mm pm

(5.11b)

150

JianGong Yu j,m j,m j,m −1 j,m 1 j,m j,m j,m −1 j,m 2 [A31 − A34 (A44 ) A41 ]pm + [A32 − A34 (A44 ) A42 ]pm j,m j,m j,m −1 j,m 3 +[A33 − A34 (A44 ) A42 ]pm = −ω 2 Mmj p3m

(5.11c)

Equation (5.11) can be written as  j,m j,m j,m     j  1  A¯ 11 A¯ 12 A¯ 13  p1m  Mm 0 0   pm      j,m j,m j,m   j 2  A¯ 21 A¯ 22 A¯ 23  p2m = −ω 2  p 0 M 0   m m      3  j,m ¯ j,m ¯ j,m  3  pm pm A¯ 31 A32 A33 0 0 Mmj

(5.12)

So, Eq.(5.12) yields a form of the eigenvalue problem. The eigenvalue ω 2 gives the angular frequency of the guided wave; eigenvectors pim (i = 1, 2, 3) allow the components of the particle displacement to be calculated; rm , which can be gotten according to Eq.(5.10), determines the electric potential distribution. According to Vph = ω /k, the phase velocity can be obtained. The matrix Eq.(5.12) can be solved numerically making use of standard computer programs for the diagonalization of non-symmetric square matrices. 3(M + 1) eigenmodes are generated from the order M of the expansion. Acceptable solutions are those eigenmodes for which convergence is obtained as M is increased. We determine that the eigenvalues obtained are converged solutions when a further increase in the matrix dimension does not result in a significant change in the eigenvalues. The computer program was written using Mathematica. A special case is concerned. When the material is orthotropic or has fewer independent material constants in the wave propagating direction, and is polarized in the thickness direction, the governing differential equations are reduced to ³ z ´l

(l)

(l)

(l)

(l)

(l)

(l)

(l)

[C55 U 00 + lz−1C55 U 0 + ik(C13 +C55 )W 0 + ik(e15 + e31 )X 0 − k2C11 U ³ z ´l (l) (l) (l) +likz−1 (C55 W + e15 X)]π (z) + (δ (z − 0) − δ (z − h)) (C55 U 0 h ρ (l) zl ω 2 (l) (l) +ikC55 W + ike15 X) = − U π (z) (5.13a) hl ³ z ´l (l) (l) (l) [C44 V 00 + lz−1C44 V 0 −k2C66 V ]π (z) + (δ (z − 0) − δ (z − h)) h ³ z ´l ρ (l) zl ω 2 (l) × C44 V 0 = − V π (z) (5.13b) h hl ³ z ´l (l) (l) (l) (l) (l) (l) (l) [C33 W 00 + e33 X 00 + ik(C13 +C55 )U 0 + lz−1 (C33 W 0 + e33 X 0 ) + likz−1C13 U h ³ z ´l (l) (l) (l) (l) −k2 (C55 W + e15 X)]π (z) + (δ (z − 0) − δ (z − h)) (C33 W 0 + e33 X 0 h ρ (l) zl ω 2 (l) (l) + ikC13 U + ike13 X) = − W π(z) (5.13c) hl ³ z ´l (l) (l) (l) (l) (l) (l) (l) [e33 W 00 − 33 X 00 + ik(e15 + e31 )U 0 + lz−1 (e33 W 0 − 33 X 0 ) + likz−1 e31 U h h

5 (l)

−k2 e15 W + k2

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

³ z ´l

(l) 11 X]π (z) + (δ (z − 0) − δ (z − h))

h

(l)

(e33 W 0 −

151

(l) 0 33 X

(l)

+ike31 U) = 0

(5.13d)

Obviously, Eq.(5.13b) is independent of the other three equations. In fact, Eq.(5.13b) represents the propagating SH wave. It is not influenced by the electric field. Equations (5.13a) and (5.13c) control the propagating Lamb-like wave and are coupled with the piezoelectric equation.

5.2.2 Numerical results Because the effective moduli of the FGM is not yet the last word, the Voigt-type model is used here to calculate the effective modulli of two combined FGM. It is expressed as P(z) = P1V1 (z) + P2V2 (z)

(5.14)

where Vi (z) and Pi respectively denote the volume fraction of the ith material and the corresponding property of the ith material. Here, ∑ Vi (z) = 1. So, the properties of the graded material can be expressed as P(z) = P2 + (P1 − P2 )V1 (z)

(5.15)

According to Eq.(5.4), the gradient field of the material volume fraction can be expressed as a power series expansion. The coefficients of the power series can be determined using the Mathematica function ‘Fit’. For example, if the gradient field is µ V1 (z) = then P(z) = P(0) + P(1)

h−z h

³ z ´1 h

where P(l) = P2 + (P1 − P2 )

¶L

+ P(2)

(0 6 z 6 h) ³ z ´2 h

³ z ´l L! l!(L − l)! h

+ · · · + P(L)

³ z ´L h

(0 6 l 6 L)

In order to validate the computer program, guided waves dispersion curves in an FGM plate and an FGPM plate are calculated to compare the results with the published data. The published data are from two different methods, reverberation matrix method for the FGM plate [52] and inhomogeneous layer element method for the FGPM plate [27]. As shown in Figs.5.1 and 5.2, our solutions are coincident to the published data. The used material properties can be seen in the references.

152

JianGong Yu

Fig. 5.1 Dispersion curves for the pure elastic FGM plate. (a) From Chen et al. [52]; (b) Author’s results.

Fig. 5.2 Dispersion curves for the FGPM plate. (a) From Liu et al. [27] (dashed line: FGPM plate, solid line: FGM plate); (b) Author’s results for the FGPM plate.

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

153

This section does not discuss the concrete wave characteristics of the FGPM plate. In the next section, we will introduce the wave propagating characteristics in curved waveguides. In fact, the plate can be thought as a curved plate with an infinite ratio of radius to thickness.

5.3 Circumferential wave in the FGPM cylindrical curved plate In this section, we investigate the wave characteristics in the FGPM cylindrical curved plate by the orthogonal functions technique. Circumferential wave dispersion curves for FGPM and the corresponding FGM cylindrical curved plates are obtained and the piezoelectric effect is shown. The influence of the ratio of radius to thickness on the piezoelectric effect and electric potential distribution are discussed. Finally, the influence of polarizing direction is illustrated. Some work of this section is from Refs. [31] and [53]. The author expresses his sincere thanks to the publishing houses.

5.3.1 Mathematics and formulation Based on linear three-dimensional piezoelasticity, consider an anisotropic FGPM hollow cylinder of infinite length. As shown in Fig. 5.3, in the cylindrical coordinate system (θ , z, r), a, b, h are the inner and outer radii and the thickness respectively.

Fig. 5.3 The scheme of the circumferential wave in a hollow cylinder.

The stress-free and open circuit surface boundary conditions (Trr = Trθ = Trz = 0 and Dr = 0 at z = a, z = b) are automatically incorporated in the constitutive relations of the linear, anisotropic, piezoelectric cylindrical medium:

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JianGong Yu

     C11 C12 C13 C14 C15 C16  Tθ θ  εθ θ              Tzz  C22 C23 C24 C25 C26  εzz                 Trr C33 C34 C35 C36  εrr   = π (r) C44 C45 C46  2εrz   Trz             symmetry C55 C56   T  2ε          rθ    rθ   C66 Tθ z 2εθ z   e11 e21 e31  e12 e22 e32       e13 e23 e33   Eθ    Ez π (r) −   e14 e24 e34   Er   e15 e25 e35  e16 e26 e36   εθ θ            εzz    e11 e12 e13 e14 e15 e16   Dθ    ε rr Dz =  e21 e22 e23 e24 e25 e26  π (r)   2εrz    Dr e31 e32 e33 e34 e35 e36     2ε      rθ   2εθ z     11 12 13  Eθ  Ez π (r) + 22 23    symmetry 33 Er

(5.16a)

(5.16b)

The rectangular window function π (r) is ½ 1 (a 6 r 6 b) π (r) = 0 (elsewhere) Generalized geometric equations under the cylindrical coordinate are 1 ∂ uθ ur ∂ uz ∂ ur + , εzz = , εrr = , r ∂θ r ∂z ∂r µ ¶ µ ¶ 1 1 ∂ ur ∂ uθ uθ 1 ∂ ur ∂ uz εrθ = + − , εrz = + , 2 r ∂θ ∂r r 2 ∂z ∂r µ ¶ 1 ∂ uθ ∂ uz εθ z = , + 2 ∂z r∂ θ 1 ∂Φ ∂Φ ∂Φ Eθ = − , Ez = − , Er = − r ∂θ ∂z ∂r

εθ θ =

And the field equations governing wave propagation are

(5.17)

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

∂ 2 ur ∂ Trr 1 ∂ Trθ ∂ Trz Trr − Tθ θ + + + =ρ 2 , ∂r r ∂θ ∂z r ∂t ∂ Tθ r 1 ∂ Tθ θ ∂ Tθ z 2Trθ ∂ 2 uθ + + + =ρ 2 , ∂r r ∂θ ∂z r ∂t ∂ Tzr 1 ∂ Tzθ ∂ Tzz Trz ∂ 2 uz + + + =ρ 2 , ∂r r ∂θ ∂z r ∂t ∂ Dr 1 ∂ Dθ ∂ Dz Dr + + + =0 ∂r r ∂θ ∂z r

155

(5.18)

Similar to previous section, for the FGPM material properties varying in the radial direction, its elastic, dielectric, piezoelectric constants and mass density of the medium are the function of radius: ³ r ´l ³ r ´l C(r) = C(l) , e(r) = e(l) , h h ³ r ´l ³ r ´l (r) = (l) , ρ (r) = ρ (l) (l = 0, 1, 2, · · · , L) (5.19) h h where l, C(l) et al. are the order number and coefficients to be determined in order to fit the polynomials (4) to the original material constants of the elastic medium inside the material. In the homogeneous case, C(r) = C(0) ,C(l) = 0(l > 0). For a free harmonic wave being propagated in the circumferential direction in a circular cylinder of infinite length, we assume the displacement components and electric potential to be of the form ur (r, θ , z,t) = exp i(kbθ − ω t)U(r)

(5.20a)

uθ (r, θ , z,t) = exp i(kbθ − ω t)V (r)

(5.20b)

uz (r, θ , z,t) = exp i(kbθ − ω t)W (r)

(5.20c)

Φ (r, θ , z,t) = exp i(kbθ − ω t)X(r)

(5.20d)

U(r), V (r),W (r) represent the amplitude of vibration in the radial, circumferential and axial directions respectively and X(r) represents the amplitude of electric potential. Substituting Eqs.(5.16), (5.17), (5.19), (5.20) into Eq.(5.18), the governing differential equations in terms of displacement components and electric potential can be obtained: 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (C33 U +C35 V 00 +C34 W 00 + e33 X 00 ) + rl+1 [((l + 1)C33 + 2ikbC35 )U 0 hl (l) (l) (l) (l) (l) (l) +(lC35 −C15 )V 0 + ikb(C13 +C55 )V 0 + ((l + 1)C34 −C14 )W 0 (l)

(l)

(l)

(l)

(l)

(l)

+ikb(C36 +C45 )W 0 + (l + 1)e33 X 0 − e31 X 0 + ikb(e13 + e35 )X 0 ] (l)

(l)

(l)

(l)

(l)

(l)

+rl [l(C13 + ikbC35 )U − (C11 + k2 b2C55 )U + l(ikbC13 −C35 )V (l)

(l)

(l)

(l)

(l)

(l)

(l)

+(C15 − k2 b2C15 − ikb(C11 +C55 ))V + (ikb(lC36 −C16 ) − k2 b2C56 )W (l)

(l)

(l)

+(l + 1)ikbe13 X−(ikbe11 + k2 b2 e15 )X]}π (r) + (δ (r − a) − δ (r − b))

156

JianGong Yu

1 l+2 (l) 0 (l) (l) (l) (l) (l) {r (C33 U +C35 V 0 +C34 W 0 + e33 X 0 ) + rl+1 [(C13 + ikbC35 )U hl ρ (l) rl+2 ω 2 (l) (l) (l) (l) +(ikbC13 −C35 )V + ikbC36 W + ikbe13 X]} = − U π (r) hl ×

(5.21a)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (C35 U +C55 V 00 +C45 W 00 + e35 X 00 ) + rl+1 [((l + 2)C35 +C15 + ikb(C13 hl (l) (l) (l) (l) (l) (l) +C55 ))U 0 + ((l + 1)C55 + 2ikbC15 )V 0 + (l + 2)C45 W 0 + ikb(C14 +C56 )W 0 (l)

(l)

(l)

(l)

(l)

+ikb(e31 + e15 )X 0 + (l + 2)e35 X 0 ] + rl [(l + 1)(C15 + ikbC55 )U (l)

(l)

(l)

(l)

(l)

+(ikbC11 − k2 b2C15 )U + (likbC15 − (l + 1)C55 − k2 b2C11 )V (l)

(l)

(l)

(l)

+((l + 1)ikbC56 − k2 b2C16 )W + (l + 2)ikbe15 X − k2 b2 e11 X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l {rl+2 (C35 U 0 +C55 V 0 +C45 W 0 + e35 X 0 ) h (l) (l) (l) (l) (l) (l) +rl+1 [(C15 + ikbC55 )U + (ikbC15 −C55 )V + ikbC56 W + ikbe15 X]} =−

ρ (l) rl+2 ω 2 V π (r) hl

(5.21b)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (C34 U +C45 V 00 +C44 W 00 + e34 X 00 ) + rl+1 [((l + 1)C34 +C14 + ikb(C36 hl (l) (l) (l) (l) (l) (l) (l) +C45 ))U 0 + (lC45 + ikb(C14 +C56 ))V 0 + ((l + 1)C44 + 2ikbC46 )W 0 + (ikb(e36 (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

+e14 ) + (l + 1)e34 )X 0 ] + rl (l(C14 + ikbC45 ) + ikbC16 − k2 b2C56 )U (l)

(l)

(l)

(l)

+(l(ikbC14 −C45 ) − ikbC56 − k2 b2C16 )V + (likbC46 − k2 b2C66 )W (l)

(l)

+((l + 1)ikbe14 −k2 b2 e16 )X]}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) (l) (l) (l) (l) × l {rl+2 (C34 U 0 +C45 V 0 +C44 W 0 + e34 X 0 ) + rl+1 [(C14 + ikbC45 )U h ρ (l) rl+2 ω 2 (l) (l) (l) (l) +(ikbC14 −C45 )V + ikbC46 W + ikbe14 X]} = − W π (r) hl

(5.21c)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (e33 U + e35 V 00 + e34 W 00 − 33 X 00 ) + rl+1 [(e31 + (l + 1)e33 + ikb(e13 hl (l) (l) (l) (l) (l) (l) (l) +e35 ))U 0 + (ikb(e31 + e15 ) + le35 )V 0 + (ikb(e36 + e14 ) + (l + 1)e34 )W 0 (l) (l) (l) (l) (l) 0 l 2 2 (l) 33 + 2ikb 13 )X ] + r [(l(e31 + ikbe35 ) − k b e15 + ikbe11 )U (l) (l) (l) (l) (l) +(l(ikbe31 − e35 ) − ikbe15 − k2 b2 e11 )V + (likbe36 − k2 b2 e16 )W (l) (l) −((l + 1)ikb 13 − k2 b2 11 )X]}π (r) + (δ (r − a) − δ (r − b))

−((l + 1)

1 l+2 (l) 0 (l) (l) (l) (l) (l) {r (e33 U + e35 V 0 + e34 W 0 − 33 X 0 ) + rl+1 [(e31 + ikbe35 )U hl (l) (l) (l) (l) +(ikbe31 − e35 )V + ikbe36 W − ikb 13 X]} = 0 ×

(5.21d)

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

157

where, U(r), V (r), W (r) and X(r) can be expanded to Legendre orthogonal polynomial series ∞

U(r) =

∑

V (r) =

∑

p3m Qm (r),

X(r) =

m=0 ∞

W (r) =

∞

p1m Qm (r),

m=0

∑ p2m Qm (r),

m=0 ∞

∑

(5.22) rm Qm (r)

m=0

where pim (i = 1, 2, 3) and rm is the expansion coefficients and s µ ¶ 2m + 1 2r − (b + a) Qm (r) = Pm (b − a) (b − a) The latter solving process of Eq.(5.21) is similar to the previous section. They are not repeated here. Next, we observe three special cases of Eq.(5.21). When the material is orthotropic or has fewer independent constants in the wave propagation direction, and is polarized in the radial direction, the governing differential equations are reduced to 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (C33 U + e33 X 00 ) + rl+1 [(l + 1)C33 U 0 + ikb(C13 +C55 )V 0 + ((l + 1)e33 hl (l) (l) (l) (l) (l) (l) (l) −e31 )X 0 ] + rl [lC13 U − (C11 + k2 b2C55 )U + likbC13 V − ikb(C11 +C55 )V 1 (l) (l) (l) −k2 b2 e15 X]}π (r) + (δ (r − a) − δ (r − b)) l [rl+2 (C33 U 0 + e33 X 0 ) h ρ (l) rl+2 ω 2 (l) (l) +rl+1 (C13 U + ikbC13 V )] = − U π (r) (5.23a) hl 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r C55 V + rl+1 [ikb(C13 +C55 )U 0 + (l + 1)C55 V 0 + ikb(e31 + e15 )X 0 ] hl (l) (l) (l) (l) (l) +rl [(ikb(l + 1)C55 +C11 )U − ((l + 1)C55 + k2 b2C11 )V +(l + 2)ikbe15 X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l [rl+2C55 V 0 + rl+1 (ikbC55 U −C55 V + ikbe15 X)] h ρ (l) rl+2 ω 2 =− V π (r) (5.23b) hl 1 l+2 (l) 00 (l) (l) [r C44 W + rl+1 (l + 1)C44 W 0 − rl k2 b2C66 W ]π (r) + (δ (r − a) − δ (r − b)) hl 1 ρ (l) rl+2 ω 2 (l) × l rl+2C44 W 0 = − W π (r) (5.23c) h hl 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (e33 U − 33 X 00 ) + rl+1 [(e31 + (l + 1)e33 )U 0 + ikb(e31 + e15 )V 0 hl (l) (l) (l) (l) (l) (l) −(l + 1) 33 X 0 ] + rl [(le31 − k2 b2 e15 )U + ikb(le31 − e15 )V + k2 b2 11 X]}π (r) 1 (l) (l) (l) +(δ (r − a) − δ (r − b)) l [rl+2 (e33 U 0 − 33 X 0 ) + rl+1 (U + ikbV )e31 ] = 0 h

(5.23d)

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JianGong Yu

Here, Eq.(5.23c) is independent of the other three equations, and Eqs.(5.23a), (5.23b) and (5.23d) are coupled with each other. Equation (5.23c) represents the propagating circumferential SH wave. It is not influenced by the piezoelectricity. Equation (5.23a) and (5.23b) controlled the propagating Lamb-like wave. When the material is orthotropic and is polarized in the circumferential direction, the governing differential equations are 1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r C33 U + rl+1 [(l + 1)C33 U 0 + ikb(C13 +C55 )V 0 + ikbe35 X 0 ] + rl [lC13 U − (C11 hl (l) (l) (l) (l) (l) (l) +k2 b2C55 )U + likbC13 V − ikb(C11 +C55 )V +ikb((l + 1)e13 − e11 )X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l [rl+2C33 U 0 + rl+1 (C13 U + ikbC13 V + ikbe13 X)] h ρ (l) rl+2 ω 2 =− U π (r) (5.24a) hl 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (C55 V + e35 X 00 ) + rl+1 [ikb(C13 +C55 )U 0 + (l + 1)C55 V 0 + (l + 2)e35 X 0 ] hl (l) (l) (l) (l) (l) +rl [(ikb(l + 1)C55 +C11 )U − ((l + 1)C55 + k2 b2C11 )V − k2 b2 e11 X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l [rl+2 (C55 V 0 + e35 X 0 ) + rl+1 (ikbC55 U −C55 V )] h ρ (l) rl+2 ω 2 =− V π (r) (5.24b) hl 1 l+2 (l) 00 (l) (l) [r C44 W + rl+1 (l + 1)C44 W 0 − rl k2 b2C66 W ]π (r) + (δ (r − a) − δ (r − b)) hl 1 ρ (l) rl+2 ω 2 (l) × l rl+2C44 W 0 = − W π (r) (5.24c) h hl 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (e35 V − 33 X 00 ) + rl+1 [ikb(e13 + e35 )U 0 + le35 V 0 − (l + 1) 33 X 0 ] hl (l) (l) (l) (l) +rl [ikb(le35 + e11 )U − (le35 + k2 b2 e11 )V + k2 b2 11 )X]}π (r) + (δ (r − a) 1 (l) (l) (l) (l) (5.24d) −δ (r − b)) l [rl+2 (e35 V 0 − 33 X 0 ) + rl+1 (ikbe35 U − e35 V )] = 0 h Similarly, the independent SH wave is not affected by the piezoelectricity. When the material is orthotropic and is polarized in the axial direction, the governing differential equations are 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r C33 U + rl+1 [(l + 1)C33 U 0 + ikb(C13 +C55 )V 0 ] + rl [lC13 U − (C11 hl (l) (l) (l) (l) +k2 b2C55 )U + likbC13 V −ikb(C11 +C55 )V ]}π (r) + (δ (r − a) − δ (r − b)) ×

ρ (l) rl+2 ω 2 1 l+2 (l) 0 (l) l+1 (l) [r C U + r (C U + ikbC V )] = − U π (r) 33 13 13 hl hl

1 l+2 (l) 00 (l) (l) (l) (l) {r C55 V + rl+1 [ikb(C13 +C55 )U 0 + (l + 1)C55 V 0 ] + rl [(ikb(l + 1)C55 hl

(5.25a)

5 (l)

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides (l)

159

(l)

+C11 )U − ((l + 1)C55 + k2 b2C11 )V ]}π (r) + (δ (r − a) − δ (r − b)) ×

ρ (l) rl+2 ω 2 1 l+2 (l) 0 (l) (l) l+1 [r C V + r (ikbC U −C V )] = − V π (r) 55 55 55 hl hl

(5.25b)

1 l+2 (l) 00 (l) (l) (l) (l) {r (C44 W + e34 X 00 ) + rl+1 [(l + 1)C44 W 0 + (l + 1)e34 X 0 ] − rl k2 b2 (C66 W hl 1 (l) (l) (l) (l) +e16 X)}π (r) + (δ (r − a) − δ (r − b)) l [rl+2 (C44 W 0 + e34 X 0 ) + rl+1 ikbC46 W ] h ρ (l) rl+2 ω 2 =− W π (r) (5.25c) hl 1 l+2 (l) 00 (l) (l) (l) {r (e34 W − 33 X 00 ) + rl+1 [(l + 1)e34 W 0 − (l + 1) 33 X 0 ] hl (l) (l) −rl k2 b2 (e16 W − 11 X)}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) × l rl+2 (e34 W 0 − 33 X 0 ) = 0 (5.25d) h Here, only the independent circumferential SH wave is influenced by the piezoelectricity.

5.3.2 Numerical results and discussion 5.3.2.1 Comparison with flat plate As is well known, the wave characteristics for a flat plate are almost the same to those for a hollow cylinder with very large ratio of radius to thickness. In this example, a comparison is made of the results of an FGPM flat plate obtained by Lefebvre et al. [30] and the results of three FGPM cylindrical curved plates with three different ratios of radius to thickness by the Mathematica program. The inner/bottom surface of the cylindrical plate is LiTaO3 and three ratios of outer radius to thickness are η = 100, η = 10, η = 2. The material constants of the inner and outer surface and the gradient variation can be found in the paper of Lefebvre et al. [30]. Corresponding phase velocity dispersion curves are presented in Fig.5.4. For the results of cylindrical curved plate, the series expansion Eq.(5.22) is truncated at M = 11. As is apparent, when η = 100, the dispersion curves for FGPM cylindrical plate are almost the same as the FGPM flat plate except that there is a very low cut-off frequency in the third mode of the cylindrical curved plate. In addition, from Fig.5.4 one can also see that the influences of η on the dispersion curves are obvious. The first cut-off frequency becomes higher as η decreases. With the decrease of η , the dispersion becomes more serious and the phase velocity becomes higher.

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JianGong Yu

Fig. 5.4 Phase velocity dispersion curves for the FGPM waveguides. (a) Flat plate from Lefebvre et al. [30]; (b) Cylindrical plate with η = 100; (c) Cylindrical plate with η = 10; (d) Cylindrical plate with η = 2.

5.3.2.2 FGPM cylindrical curved plates with different gradient variations The circumferential wave behavior of the FGPM cylindrical curved plates composed of PZT-4 (inner surface) and Ba2 NaNb5 O15 (outer surface) with a = 9 mm and b = 10 mm is calculated. In the direction of wave propagation, material constants of the two materials with radial polarization are listed in Table 5.1. The gradient field used here is µ ¶ r−a n V1 (r) = (5.26) h where n is selected as 0.5, 1, 2 and 3, respectively, i.e., four different gradient fields are considered.

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

161

Table 5.1 Material constants of two homogeneous piezoelectric materials (units: Ci j , 1010 N/m2 ; −11 F/m2 ; e , C/m; ρ , kg/m3 ). i j , 10 ij Property

C11

C12

Ba2 NaNb5 O15

23.9

10.4

PZT-4

13.9

7.8

Property

e15

e24

C13

C22

C23

C33

C44

5

24.7

5.2

13.5

6.5

6.6

7.6

7.4

13.9

7.4

11.5

2.56

2.56

3.05

e31

e32

e33

11

22

C55

C66

33

ρ

Ba2 NaNb5 O15

2.8

3.4

–0.4

–0.3

4.3

196

201

28

5.3

PZT-4

12.7

12.7

–5.2

–5.2

15.1

650

650

560

7.5

Figures 5.5 and 5.6 are the frequency spectra and phase velocity spectra for the four FGPM cylindrical plates and their corresponding FGM cylindrical plates to show the piezoelectric effect. It can be seen that the effect of piezoelectricity is so strong as not to be ignored and the phase velocity in FGPM cylindrical plate is higher than in the FGM cylindrical plate. For a definite mode, the piezoelectric effect becomes stronger as wavenumber increases. Furthermore, an important point should be concerned that for the four FGPM cylindrical plates, the piezoelectric effect becomes stronger as the power n increases.

Fig. 5.5 Frequency spectra for cylindrical plate (dotted line, FGM; solid line, FGPM).

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Fig. 5.6 Phase velocity dispersion curves for cylindrical plate (dotted line, FGM; solid line, FGPM).

In order to explain this phenomenon, Figs.5.7 and 5.8 give the frequency spectra and phase velocity spectra for the pure PZT-4 and Ba2 NaNb5 O15 cylindrical plates and their corresponding non-piezoelectric cases. Obviously, the piezoelectric effect of PZT-4 plate is stronger than that of Ba2 NaNb5 O15 plate. It is known from Eq.(5.26) that the volume fraction of PZT-4 in the FGPM increases as the power n increases.

Fig. 5.7 Frequency spectra for PEM cylindrical plates: solid line, piezoelectric; dotted line, non- piezoelectric. (a) PZT-4; (b) Ba2 NaNb5 O15 .

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163

Fig. 5.8 Phase velocity dispersion curves for PEM cylindrical plates: solid line, piezoelectric; dotted line, non- piezoelectric. (a) PZT-4; (b) Ba2 NaNb5 O15 .

The electric potential distribution of the FGPM cylindrical plate is investigated. Figure 5.9 shows electric potential distributions of the lowest 9 modes for the FGPM cylindrical plate of the Fig.5.6(b) when k = 300000 rad/m. It can be seen that the electric potential in the FGPM cylindrical plate distribute mostly near the outer or inner surface. For the lower modes, electric potential distribute mostly near the inner surface. As the mode order increases, the electric potential distribution turns to outer surface. Up to the 8th mode, the electric potential has distributed mostly near the outer surface.

Fig. 5.9 Electric potential distribution in the thickness direction of the FGPM cylindrical plate that inner surface is PZT-4 and outer surface is Ba2 NaNb5 O15 (k = 300000 rad/m).

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Next, we consider a similar structure to the Fig.5.6(b), but exchange the materials of outer and inner surfaces. The inner surface is Ba2 NaNb5 O15 and the outer surface is PZT-4. The other conditions are consistent with those of Fig.5.6(b). Figure 5.10 is the dispersion curves for this structure. No much difference between the dispersion curves for the two structures can be found. Their piezoelectric effects are also similar. In fact, the volume fractions of the two materials in the two structures are almost equal. The electric potential distribution of this FGPM cylindrical plate is shown in Fig.5.11. Similarly to the Fig.5.9, the electric potential distribute mostly near the outer or inner surface. However, in contrast to the Fig.5.9, for the lower modes, electric potential distribute mostly near the outer surface. As the mode order increases, the electric potential distribution turns to inner surface. Up to the 8th mode, the electric potential has distributed mostly near the inner surface.

Fig. 5.10 Dispersion curves for cylindrical curved plate that inner surface is Ba2 NaNb5 O15 and outer surface is PZT-4: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

5.3.2.3 Influence of the ratio of radius to thickness In order to investigate the effect of η , the ratio of the outer radius to thickness, the frequency spectra and group velocity spectra for the FGPM cylindrical curved plate are shown in Figs.5.12 and 5.13, in which b is 10 mm and 2 mm respectively but keeping

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165

Fig. 5.11 Electric potential distribution in the thickness direction of FGPM cylindrical plate that inner surface is Ba2 NaNb5 O15 and outer surface is PZT-4 (k = 300000 rad/m).

Fig. 5.12 Frequency spectra for FGPM cylindrical plate. (a) η = 10; (b) η = 2.

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h = 1 mm unchanged, i.e.,η = 10 and 2. The gradient field and material are the same to the structure of Fig.5.6(b). From the two figures, it can be found that the ratio of radius to thickness has considerable influence on the dispersion curves for all the modes, and that the larger the wavenumber, the stronger this effect. In addition, the effect is far stronger on lower order modes than on higher order modes at smaller wavenumber. For a smaller η , the group velocity changes more sharply, and the maximum of the group velocity is larger except for the second mode.

Fig. 5.13 Group velocity spectra for FGPM cylindrical plate. (a) η = 10; (b) η = 2.

Figure 5.14 is the electric potential distribution of the FGPM cylindrical plate of η = 2. It is different from that of the large ratio, η = 10, in Fig.5.10. For the first three order modes of little ratio, the electric potentials distribute mostly near the outer surface. But they distribute mostly near the inner surface for the large ratio. Figure 5.15 is the electric potential distribution of the FGPM cylindrical plate that the inner surface is Ba2 NaNb5 O15 and the outer surface is PZT-4 when η = 2. As is known before, when the ratio is large, if the material of inner and outer surface exchanges, their electric potential distributions were reversed. But as shown in Figs.5.14 and 5.15, for the little ratio, whatever the inner surface is, the electric potential always distributes mostly near the outer surface for the first three modes. It is well known that the circumferential wave characteristics of large ratio cylindrical plates are very close to the wave characteristics of flat

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167

Fig. 5.14 Electric potential distribution in the thickness direction of FGPM cylindrical curved plate when η = 2 that inner surface is PZT-4 and outer surface is Ba2 NaNb5 O15 (k = 300000 rad/m).

Fig. 5.15 Electric potential distribution in the thickness direction of FGPM cylindrical curved plate when η = 2 that inner surface is Ba2 NaNb5 O15 and outer surface is PZT-4 (k = 300000 rad/m).

plate. So, for the large ratio FGPM cylindrical plate, the circumferential wave electric potential distributions are similar to the FGPM plate. But for the little ratio FGPM cylindrical plate, the circumferential wave electric potential distributions are different from the FGPM plate and are similar to those of the PEM cylindrical plate. For the PEM cylindrical

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curved plate, whatever the ratio is, the electric potential distributes mostly near the outer surface for the first three modes at large wavenumber, as shown in Fig.5.16, the electric potential distribution of the PZT-4 cylindrical plate which wall thickness is 1 mm when η = 10 and η = 2.

Fig. 5.16 Electric potential distribution in the thickness direction of PZT-4 cylindrical curved plate (k = 300000 rad/m). (a) η = 10; (b) η = 2.

5.3.2.4 Influence of the polarizing direction In this section, the FGPM cylindrical curved plates that are polarized in circumferential and axial direction are studied. The geometry, material and the gradient field are all the same to the structure of Fig.5.6(b). Under the circumferential and axial polarization, the dielectric and piezoelectric constants of material are listed in the Table 5.2 and the elastic constants are kept unchanged for more easy comparison. Table 5.2 Piezoelectric and dielectric constants of PZT-4 with axial and circumferential polarization (units: i j , 10−11 F/m2 ; ei j ,C/m). Polarizing direction Axial

e16

e21

e22

e23

e34

11

22

33

PZT-4

–12.7

–15.1

5.2

5.2

–12.7

650

560

650

Ba2 NaNb5 O15

–3.4

0.3

–4.3

0.4

–2.8

201

28

196

Polarizing direction Circumferential

e11

e12

e13

e26

e35

11

22

33

PZT-4

–15.1

5.2

5.2

–12.7

–12.7

560

650

650

Ba2 NaNb5 O15

–4.3

0.4

0.3

–2.8

–3.4

28

196

201

Figure 5.17 gives the dispersion curves for the FGPM cylindrical curved plates that are polarized in circumferential direction. It can be seen that the piezoelectric effect is stronger on the higher order modes than on lower order modes; the piezoelectric effect becomes weaker with frequency increasing and does not change considerably with wavenumber varying. It is different from the radial polarization. For the radial polariza-

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169

tion, the piezoelectric effect becomes stronger with wavenumber increasing and does not change obviously with the frequency varying.

Fig. 5.17 Dispersion curves for cylindrical plate under circumferential polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

For all the above discuss except for the Fig.5.4, only the circumferential Lamb-like waves are considered because the circumferential SH wave is not influenced by the piezoelectricity when the material is orthotropic and polarized in the radial and circumferential direction. Figure 5.18 is the circumferential SH wave dispersion curves for the FGPM cylindrical plate that is polarized in axial direction. This time, the circumferential Lamblike wave is not affected by the piezoelectricity. As can be seen from Fig.5.18, for a definite mode, the piezoelectric effect becomes stronger with wavenumber increasing and does not change obviously with the frequency varying, which is similar to the radial polarization. Moreover, the piezoelectric effect is stronger on the higher order modes than on lower order modes, which is similar to the circumferential polarization. Another point should be paid attention to. For the first mode of FGM cylindrical plate, the dispersion

Fig. 5.18 Circumferential SH wave dispersion curves for cylindrical plate under axial polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

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is serious, but for the FGPM cylindrical plate, the dispersion is weaker. In fact, this phenomenon that piezoelectricity weakens the guided wave dispersion in the FGPM cylindrical plate can also be observed in Figs.5.6 and 5.10(b). However, for the PEM cylindrical plate, as shown in Fig.5.8, the phenomenon can not be seen. In order to elaborate the weakened dispersion phenomenon, Fig.5.19 shows the SH wave group velocity dispersion curves for axial polarized FGPM cylindrical plates with six different ratios of outer radius to thickness. Their thicknesses are all 1 mm and their ratios are η = 10, 5, 3.5, 3, 2.5 and 2, respectively. Obviously, the ratio has an influence on the phenomenon of piezoelectricity weakening dispersion. As the η decreases, the dispersive weakening extent becomes weaker. When η = 2.5, the dispersive weakening phenomenon almost disappear. When the ratio becomes smaller, η = 2, piezoelectricity even strengthen the dispersion. It can also be seen that as η decreases, the group velocity of both FGM and FGPM cylindrical plates increases.

Fig. 5.19 SH wave group velocity dispersion curves under different ratios of radius to thickness: dotted line, FGM; solid line, FGPM. (a) η = 10; (b) η = 5; (c) η = 3.5; (d) η = 3; (e) η = 2.5; (f) η = 2.

Figure 5.20 shows the electric potential mode shapes in the thickness direction for the first three modes when η = 5, 3.5, 3 and 2.5 at k = 300000 rad/m. It is can be seen once more that when the ratio η is larger, the electric potential distributions of the FGPM cylindrical plate are similar to those of the FGPM flat plate, i.e., the electric potential distribute near the surface with stronger piezoelectric effect; with η decreases, the electric potential distributions of the FGPM cylindrical plate deviate from the FGPM flat plate, and increasingly present the characteristic of hollow cylinders, i.e., the electric potential always distribute near the outer surface. The ratio has also obvious influence on the piezoelectric effect of displacement distributions, as shown in Figs.5.21 and 5.22, the displacement distributions of the first order

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171

Fig. 5.20 SH wave electric potential mode shapes of different ratio of radius to thickness. (a) η = 5; (b) η = 3.5; (c) η = 3; (d) η = 2.5.

Fig. 5.21 SH wave displacement mode shapes of the first mode when η = 10: dotted line, FGM; solid line, FGPM. (a) k = 1000 rad/m; (b) k = 3000 rad/m; (c) k = 5000 rad/m.

SH mode when η = 10 and η = 2. It can be seen that the piezoelectric effect becomes more considerable as the wavenumber increases. Importantly, the displacement distributions are asymmetric about the middle point of thickness. For the large ratio FGPM

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cylindrical plate, piezoelectricity weakens this asymmetry. But for the little ratio case, piezoelectricity strengthens this asymmetry.

Fig. 5.22 SH wave displacement mode shapes of the first mode when η = 2: dotted line, FGM; solid line, FGPM. (a) k = 1000 rad/m; (b) k = 3000 rad/m; (c) k = 5000 rad/m.

5.3.3 Brief summary This section discusses the circumferential wave characteristics in the FGPM cylindrical curved plate. Dispersion curves, electric potential distribution are illustrated. Based on the calculated results, the following conclusions can be drawn: (1) For the circumferential wave in FGPM cylindrical curved plates, the piezoelectric effect is considerable especially when wavenumber is large. (2) The variety of the gradient shape of the material can easily changes the piezoelectric effect on the wave characteristics. (3) Electric potential distribute mostly near the outer or inner surface for large wavenumbers. (4) Ratio of radius to thickness has a significant influence both on dispersion curves and on electric potential distribution. (5) Polarizing direction can obviously changes the piezoelectric effect on the dispersion curves for the FGPM cylindrical curved plates. (6) In the FGPM cylindrical plate, piezoelectricity can change the guided wave dispersive extent and the displacement shape.

5.4 Axial wave in the FGPM hollow cylinders The investigating object of this section is the same to the previous section, but the wave propagating direction is different. This section considers the axial wave propagation, which include the axial symmetric waves (L modes), flexural waves (F modes) and torsional waves (T modes). The piezoelectric effect and the influence of the polarizing direction on the three waves are discussed respectively. The work of this section has been published in Ref.[33]. The author expresses his sincere thanks to the publishing house.

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

173

5.4.1 Mathematics and formulation Because only the wave propagating direction is different from previous section, the fundamental equations are the same to the previous section except for the assumed harmonic wave solutions of displacement components and electric potential. They are of the form ur (r, θ , z,t) = exp i(kz − ω t) exp(iN θ )U(r)

(5.27a)

uθ (r, θ , z,t) = exp i(kz − ω t) exp(iN θ )V (r)

(5.27b)

uz (r, θ , z,t) = exp i(kz − ω t) exp(iN θ )W (r)

(5.27c)

Φ (r, θ , z,t) = exp i(kz − ω t) exp(iN θ )X(r)

(5.27d)

where U(r), V (r),W (r) represent the amplitudes of vibration in the radial, tangential and axial directions respectively and, X(r) represents the amplitude of electric potential. N is the circumferential wavenumber. The obtained governing differential equations in terms of displacement components and electric potential are obtained as 1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (C33 U +C35 V 00 +C34 W 00 + e33 X 00 ) + rl+1 [((l + 1)C33 + 2iNC35 − 2irkC34 )U 0 hl (l) (l) (l) (l) (l) (l) (l) (l) +(lC35 −C15 )V 0 + iN(C13 +C55 )V 0 − irk(C36 +C45 )V 0 + ((l + 1)C34 −C14 )W 0 (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

+iN(C36 +C45 )W 0 − irk(C23 +C44 )W 0 + ((l + 1)e33 − e31 + iN(e13 + e35 ) (l)

(l)

(l)

(l)

(l)

(l)

−irk(e23 + e34 ))X 0 ] + rl [l(C13 + iNC35 − irkC34 )U − (C11 + N 2C55 + irkC34 (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

+k2 r2C44 − 2NrkC45 )U + l(iNC13 −C35 − irkC36 )V + (irk(C45 +C16 −C36 ) (l)

(l)

(l)

(l)

(l)

(l)

+Nrk(C56 +C14 ) + (1 − N 2 )C15 − k2 r2C46 − iN(C11 +C55 ))V (l)

(l)

(l)

(l)

(l)

(l)

(l)

+(l(iNC36 − irkC23 ) + Nrk(C25 +C46 ) − iNC16 − irk(C23 −C12 ) (l)

(l)

(l)

(l)

(l)

(l)

−N 2C56 − k2 r2C24 )W + (l(iNe13 − irke23 ) + Nrk(e25 + e14 ) (l)

(l)

(l)

(l)

(l)

(l)

+iN(e13 − e11 ) − irk(e23 − e21 ) − N 2 e15 − k2 r2 e24 )X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l {rl+2 (C33 U 0 +C35 V 0 +C34 W 0 + e33 X 0 ) h (l) (l) (l) (l) (l) (l) +rl+1 [(C13 + iNC35 − irkC34 )U + (iNC13 −C35 − irkC36 )V (l)

(l)

(l)

(l)

+(iNC36 − irkC23 )W + (iNe13 − irke23 )X]} = −

ρ (l) rl+2 ω 2 U π (r) hl

(5.28a)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (C35 U +C55 V 00 +C45 W 00 + e35 X 00 ) + rl+1 [((l + 2)C35 +C15 + iN(C13 hl (l) (l) (l) (l) (l) (l) +C55 ))U 0 − irk(C36 +C45 )U 0 + ((l + 1)C55 + 2iNC15 − 2irkC56 )V 0 (l)

(l)

(l)

(l)

(l)

(l)

(l)

+(l + 2)C45 W 0 + iN(C14 +C56 )W 0 − irk(C25 +C46 )W 0 + iN(e31 + e15 )X 0 (l)

(l)

(l)

(l)

+(l + 2)e35 X 0 − irk(e25 + e36 )X 0 ] + rl (l + 1 − N 2 )C15 U

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JianGong Yu (l)

(l)

(l)

(l)

+rl [(iN((l + 1)C55 +C11 ) − irk((l + 2)C45 +C16 ) (l)

(l)

(l)

(l)

(l)

(l)

−k2 r2C46 + Nrk(C56 +C14 ))U − k2 r2 e26 X + (liNC15 − (l + 1)C55 (l)

(l)

(l)

(l)

(l)

−k2 r2C66 − N 2C11 + 2NrkC16 − (l + 1)irkC56 )V + (l + 2)iNe15 (l)

(l)

(l)

(l)

(l)

(l)

×((l + 1)iNC56 − k2 r2C26 − N 2C16 + Nrk(C12 +C66 ) − (l + 2)irkC25 )W (l)

(l)

(l)

(l)

+Nrk(e21 + e16 )X − N 2 e11 X − (l + 2)irke25 X]}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) (l) (l) (l) (l) (l) × l {rl+2 (C35 U 0 +C55 V 0 +C45 W 0 + e35 X 0 ) + rl+1 [(C15 + iNC55 − irkC45 )U h (l) (l) (l) (l) (l) (l) (l) +(iNC15 −C55 − irkC56 )V + (iNC56 − irkC25 )W +(iNe15 − irke25 )X]} =−

ρ (l) rl+2 ω 2 V π (r) hl

(5.28b)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) (l) {r (C34 U +C45 V 00 +C44 W 00 + e34 X 00 ) + rl+1 [((l + 1)C34 +C14 + iN(C36 +C45 ) hl (l) (l) (l) (l) (l) (l) (l) (l) −irk(C23 +C44 ))U 0 + (lC45 + iN(C14 +C56 ) − irk(C25 +C46 ))V 0 + ((l + 1)C44 (l)

(l)

(l)

(l)

(l)

(l)

(l)

+2iNC46 − 2irkC24 )W 0 + iN(e36 + e14 )X 0 + (l + 1)e34 X 0 − irk(e24 + e32 )X 0 ] (l)

(l)

(l)

(l)

(l)

(l)

(l)

+rl [l(C14 + iNC45 − irkC44 )U + iNC16 U + Nrk(C25 +C46 )U − N 2C56 U (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

−k2 r2C24 U − irk(C12 +C44 )U + l(iNC14 −C45 − irkC46 )V + Nrk(C12 +C66 )V (l)

(l)

(l)

(l)

(l)

(l)

(l)

−iNC56 V − N 2C16 V − k2 r2C26 V + irk(C25 −C46 )V + l(iNC46 − irkC24 )W (l)

(l)

(l)

(l)

(l)

(l)

−irkC24 W + 2NrkC26 W − N 2C66 W − k2 r2C22 W + ((l + 1)(iNe14 − irke24 ) (l)

(l)

(l)

(l)

+Nrk(e12 + e26 ) − N 2 e16 − k2 r2 e22 )X]}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) (l) (l) (l) (l) (l) × l {rl+2 (C34 U 0 +C45 V 0 +C44 W 0 + e34 X 0 ) + rl+1 [(C14 + iNC45 − irkC44 )U h (l) (l) (l) (l) (l) (l) (l) +(iNC14 −C45 − irkC46 )V + i(NC46 − rkC24 )W + i(Ne14 − rke24 )X]} =−

ρ (l) rl+2 ω 2 W π (r) hl

(5.28c)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) (l) {r (e33 U + e35 V 00 + e34 W 00 − 33 X 00 ) + rl+1 [(e31 + (l + 1)e33 + iN(e13 + e35 ) hl (l) (l) (l) (l) (l) (l) (l) (l) (l) −irk(e23 + e34 ))U 0 + (iN(e31 + e15 ) + le35 − irk(e25 + e36 ))V 0 + (iN(e36 + e14 ) (l)

(l)

(l)

(l) (l) (l) 0 33 + 2iN 13 − 2irk 23 )X ] (l) (l) (l) (l) (l) (l) (l) +rl [l(e31 + iNe35 − irke34 )U + Nrk(e14 + e25 )U − N 2 e15 U − k2 r2 e24 U (l) (l) (l) (l) (l) (l) +iNe11 U − irk(e21 + e34 )U + l(iNe31 − e35 − irke36 )V (l) (l) (l) (l) (l) +Nrk(e16 + e21 )V − iNe15 V − N 2 e11V − k2 r2 e26V + irk(e25 − e36 )V (l) (l) (l) (l) (l) (l) +(liNe36 − (l + 1)irke32 + Nrk(e12 + e26 ) − N 2 e16 − k2 r2 e22 )W (l) (l) (l) (l) (l) −((l + 1)(iN 13 − irk 23 ) + 2Nrk 12 − N 2 11 − k2 r2 22 )X]}π (r)

+(l + 1)e34 − irk(e24 + e32 ))W 0 − ((l + 1)

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

1 l+2 (l) 0 (l) (l) {r (e33 U + e35 V 0 + e34 W 0 − hl (l) (l) (l) (l) (l) (l) +rl+1 [(e31 + iNe35 − irke34 )U + (iNe31 − e35 − irke36 )V +(δ (r − a) − δ (r − b))

(l)

(l)

+i(Ne36 − rke32 )W − i(N

(l) (l) 13 − rk 23 )X]}

175

(l) 0 33 X )

=0

(5.28d)

The latter solving process of Eq.(5.28) is similar to previous sections.

5.4.2 Numerical results and discussion 5.4.2.1 FGPM hollow cylinders with different gradient variations Firstly, we calculated three FGPM hollow cylinders composed of PZT-4 (inner surface) and Ba2 NaNb5 O15 (outer surface) with a = 9 mm and b = 10 mm. In the direction of wave propagation, material constants are listed in Table 5.1. The gradient field used here is also Eq.(5.26), in which n is selected as 1, 2 and 3, respectively, i.e., three different gradient fields are considered. Figures 5.23 and 5.24 show the axial symmetric mode frequency spectra and phase

Fig. 5.23 Axial symmetric mode frequency spectra for hollow cylinders: solid line, FGPM; dotted line, FGM. (a) n =1; (b) n =2; (c) n =3.

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velocity spectra. Obviously, some modes are not influenced by the piezoelectricity. They are torsional modes. For the longitudinal modes, the effect of piezoelectricity is similar to that of circumferential waves: the phase velocity of the FGPM cylinder is higher than that of the FGM cylinder; for a definite longitudinal mode, the piezoelectric effect becomes stronger as wavenumber increases; for the three FGPM cylinders, the piezoelectric effect becomes stronger as the power n increases.

Fig. 5.24 Axial symmetric mode phase velocity spectra for hollow cylinders: solid line, FGPM; dotted line, FGM. (a) n =1; (b) n =2; (c) n =3.

Figures 5.25-5.27 are the corresponding first, second and third order flexural mode dispersion curves when the power n =1. For the flexural longitudinal modes, the piezoelectric effect is similar to the axial symmetric modes. For the torsional modes, although the piezoelectric effect is not zero, it is too weak to be concerned from these dispersion curves. Figure 5.28 is the radial and axial displacement distributions of the first order flexural longitudinal modes (circumferential displacement is too small relative to the other two directions to be considered here). Figure 5.29 is circumferential displacement distributions of the first order flexural torsional modes. Obviously, the piezoelectric effect is strong on the flexural longitudinal modes but is very weak on the flexural torsional modes.

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

177

Fig. 5.25 The first order flexural mode dispersion curves for hollow cylinders: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.26 The second order flexural mode dispersion curves for FGM hollow cylinders: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.27 The third order flexural mode dispersion curves for hollow cylinders: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

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Fig. 5.28 Displacement distributions of the first order flexural longitudinal modes when k = 3000 rad/m: solid line, FGPM; dashed line, FGM. (a) ur ; (b) uz .

Fig. 5.29 Displacement distributions of the first order flexural torsional modes for hollow cylinder when k = 3000 rad/m: solid line, FGPM; dotted line, FGM.

Figures 5.30 and 5.31 are the corresponding electric potential distributions of flexural longitudinal and torsional modes. For the lowest three modes of the first three order flexural longitudinal modes, their electric potential distributions are almost one and the same. Therefore, Fig.5.9 only gives the electric potentials of the first order flexural longitudinal modes. Obviously, electric potentials of flexural torsional modes are not zero, but they are far below those of flexural longitudinal modes. Furthermore, the electric potential tendencies of the three order flexural torsional modes are identical. But their amplitudes are different. The amplitudes of the second order modes are almost twice as those of the first order modes, and the third almost thrice. So, the piezoelectric effect on flexural torsional modes becomes stronger as the order increases. 5.4.2.2 Influence of the ratio of radius to thickness In order to investigate the influence of η , Figs.5.32-5.34 show the dispersion curves of the axial symmetric modes and the first two order flexural modes for the FGPM hollow

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

179

Fig. 5.30 Electric potential distributions of the first order flexural longitudinal modes for the FGPM hollow cylinder when k = 3000 rad/m.

Fig. 5.31 Electric potential distributions of the flexural torsional modes for the FGPM hollow cylinder when k = 3000 rad/m. (a) The first order; (b) The second order; (c) The third order.

cylinder with little ratio of radius to thickness when the power n = 1. Keeping the thickness unchanged, the outer radius is 2 mm, i.e., η = 2. The influence of the ratio on the dispersion curves can be read in many literatures. Here, only its influence on the piezoelectric effect is discussed. For the longitudinal modes, the piezoelectric effect on dispersion curves is similar to that of the large ratio. But for the flexural torsional modes, the piezoelectric effect is obviously stronger than that of the large ratio. This can be also seen from

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JianGong Yu

Fig. 5.32 The axial symmetric mode dispersion curves for hollow cylinders with η =2: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.33 The first order flexural mode dispersion curves for hollow cylinders with η = 2: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.34 The second order flexural mode dispersion curves for hollow cylinders with η = 2: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

181

the circumferential displacement distributions of the first order flexural torsional modes, as shown in Fig.5.35.

Fig. 5.35 Displacement distribution of the first order flexural torsional mode for hollow cylinders with η = 2 when k = 3000 rad/m: solid line, FGPM; dotted line, FGM.

Figures 5.36 and 5.37 show the corresponding electric potential distributions of the

Fig. 5.36 Electric potential distribution of the flexural longitudinal mode for the FGPM hollow cylinder with η = 2 when k = 3000 rad/m. (a) The first order; (b) The second order.

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JianGong Yu

flexural longitudinal and torsional modes. It can be seen that the electric potentials of flexural torsional modes at η = 2 outclass those at η = 10, and are even close to the corresponding flexural longitudinal modes. Furthermore, the electric potentials of the first order flexural torsional modes are slight below those of the second order, but the electric potentials of the first order flexural longitudinal modes are slight above those of the second order, which is also different from the situation at large ratio. In sum, for all the flexural torsional modes, the electric potential amplitudes of the higher modes are larger than those of the lower modes whatever the ratio is.

Fig. 5.37 Electric potential distribution of the flexural torsional mode for the FGPM hollow cylinder with η = 2 when k = 3000 rad/m. (a) The first order; (b) The second order.

5.4.2.3 Influence of the polarizing direction In this section, the FGPM hollow cylinders that are polarized in axial and circumferential direction are considered. Similar to the previous section, the elastic constants and mass

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

183

density are assumed unchanged, and the piezoelectric and dielectric constants are listed in Table 5.2. Figures 5.38 and 5.39 are the dispersion curves for the FGPM hollow cylinder with axial polarization when h = 1 mm, η = 10 and the power n =1. Similar to the radial polarization, piezoelectricity does not affect the axial symmetric torsional modes and has weak effect on the flexural torsional modes. However, for the longitudinal mode, the piezoelectric effect is different from the radial polarization. For the radial polarization, the piezoelectric effect becomes stronger with the wavenumber increasing and does not change considerably with frequency increasing for all modes. For the axial polarization, the piezoelectric effect becomes stronger with the wavenumber increasing for lower modes but becomes weaker with the frequency and wavenumber increasing for higher modes.

Fig. 5.38 The axial symmetric mode dispersion curves for hollow cylinders with axial polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.39 The first order flexural mode dispersion curves for hollow cylinders with axial polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

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JianGong Yu

Figures 5.40 and 5.41 are the dispersion curves for the FGPM hollow cylinder with circumferential polarization when h = 1 mm, η = 10 and n = 1. It can be seen that the piezoelectric effect is contrary to the other two polarizing directions. Piezoelectricity does not affect the axial symmetric longitudinal modes and has weak effect on the flexural longitudinal modes. But for the torsional mode, the piezoelectric effect is considerable. For a definite torsional mode, the piezoelectric effect becomes stronger with wavenumber increasing and does not change considerably with the frequency varying, which is similar to the situation with radial polarization. Moreover, the piezoelectric effect is stronger on the higher order modes than on lower order modes. Furthermore, the phenomenon that piezoelectricity weakens the guided wave dispersion can be easily observed for the first torsional mode. In fact, this phenomenon can also be discerned in other dispersion curves for the FGPM hollow cylinders.

Fig. 5.40 The axial symmetric mode dispersion curves for hollow cylinders with circumferential polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

Fig. 5.41 The first order flexural mode dispersion curves for hollow cylinders with circumferential polarization: solid line, FGPM; dotted line, FGM. (a) Frequency spectra; (b) Phase velocity spectra.

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

185

5.4.3 Brief summary This section characterizes the axial guided waves in FGPM hollow cylinders. Dispersion curves, electric potential distribution are illustrated. The differences of the piezoelectric effect on axial symmetric modes and flexural modes are discussed. Electric potential distributions are used to show the piezoelectric effect on flexural torsional modes. The influences of the ratio of radius to thickness and the polarizing direction on the piezoelectric effect are illustrated. Based on the calculated results, the following conclusions can be drawn: (1) When the cylinder is polarized in the radial and axial directions, whatever for axial symmetric waves or for flexural waves, piezoelectric effect occurs mainly on the longitudinal modes. The weak piezoelectric effect of flexural torsional modes becomes stronger as the mode order increases. When the cylinder is polarized in the circumferential direction, piezoelectric effect occurs mainly on the torsional modes. (2) Ratio of radius to thickness has a significant influence on piezoelectric effect, especially for torsional waves. Piezoelectric effect on torsional waves becomes stronger as the ratio decreases. (3) Similar conclusions to previous section are also be drawn: polarizing direction obviously changes the piezoelectric effect; piezoelectricity can weaken the guided wave dispersion; gradient variation can change the piezoelectric effect easily.

5.5 Wave propagation in FGPM spherical curved plates As is well known, the wave characteristics in spherical curved plate are very similar to those of circumferential wave in cylindrical curved plate. This section firstly obtained guided wave differential equations of FGPM spherical curved plates in terms of displacement components and electric potential, and then mainly discussed the differences of wave characteristics between the spherical plate and cylindrical plate. Some work of this section is from Ref. [32]. The author expresses his sincere thanks to the publishing house.

5.5.1 Mathematics and formulation Based on linear three-dimensional piezoelasticity, consider an anisotropic FGPM hollow sphere. In the spherical coordinate system (θ , φ , r), as shown in Fig.5.42, let a, b, h be the inner and outer radii and the thickness. The stress-free and open circuit surface boundary conditions (Trr = Trφ = Trθ = 0 and Dr = 0 at z = a, z = b) are automatically incorporated in the constitutive relations of the linear, anisotropic, piezoelectric cylindrical medium:

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JianGong Yu

Fig. 5.42 The scheme of the guided wave propagation in a hollow sphere.

     Tθ θ  C11 C12 C13 C14 C15 C16  εθ θ             εφ φ   Tφ φ  C22 C23 C24 C25 C26                 Trr C C C C ε rr 33 34 35 36  = π (r)  C44 C45 C46  2εrφ    Trφ              symmetry C55 C56   2ε      Trθ       rθ   C66 Tθ φ 2εθ φ   e11 e21 e31  e12 e22 e32       e13 e23 e33   Eθ   −  e14 e24 e34   Eφ  π (r)    e15 e25 e35  Er e16 e26 e36   εθ θ            εφ φ    e11 e12 e13 e14 e15 e16   Dθ    ε rr Dφ =  e21 e22 e23 e24 e25 e26  π (r)    2εrφ   Dr e31 e32 e33 e34 e35 e36     2ε      rθ   2εθ φ     11 12 13  Eθ  Eφ π (r) + 22 23    symmetry 33 Er

(5.29a)

(5.29b)

Generalized geometric equations under the spherical coordinate are 1 ∂ uθ ur 1 ∂ uφ ur cot θ ∂ ur + , εφ φ = + + uθ , εrr = , r ∂θ r r sin θ ∂ φ r r ∂r µ ¶ µ ¶ 1 1 ∂ ur ∂ uθ uθ 1 1 ∂ ur 1 ∂ uφ εrθ = + − uφ + − , εrφ = , (5.30) 2 r ∂θ ∂r r 2r sin θ ∂ φ 2 ∂r µ ¶ 1 1 ∂Φ 1 ∂ uθ ∂ uφ εθ φ = + − uφ cot θ , Eθ = − , 2r sin θ ∂ φ ∂θ r ∂θ

εθ θ =

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

Eφ = −

1 ∂Φ , r sin θ ∂ φ

Er = −

187

∂Φ ∂r

And the field equations governing wave propagation are

∂ Trr 1 ∂ Trθ 1 ∂ Trφ 2Trr + Trθ cot θ − Tθ θ − Tφ φ ∂ 2 ur + + + =ρ 2 , ∂r r ∂θ r sin θ ∂ φ r ∂t ¡ ¢ 2 ∂ Trθ 1 ∂ Tθ θ 1 ∂ Tθ φ 3Trθ + cot θ Tθ θ − Tφ φ ∂ uθ + + + =ρ 2 , ∂r r ∂θ r sin θ ∂ φ r ∂t 2 ∂ Trφ 1 ∂ Tθ φ 1 ∂ Tφ φ 3Trφ + 2Tθ φ cot θ ∂ uz + + + =ρ 2 , ∂r r ∂θ r sin θ ∂ φ r ∂t ∂ D 1 2Dr + Dθ cot θ ∂ Dr 1 ∂ Dθ φ + + + =0 ∂r r ∂θ r sin θ ∂ φ r

(5.31)

Material properties of the FGPM spherical plate are treated in the same way to Section 5.3. According to Kargl et al. [54] and Towfighi et al. [55], the wave front on the surface of a spherical shell is assumed to be toroidal. In addition, to study wave propagation from point A to B in a spherical plate segment, the two points A and B can always be aligned along the equator of a sphere by adjusting the positions of the north and south poles. Therefore, to study the wave propagation between two points in a spherical plate segment, it is sufficient to solve the governing equations for θ = π/2 only. Thus the propagating wave is independent of θ . Then the electric potential and the displacement components of this toroidal wave can be written as ur (r, θ , φ ,t) = exp i(kbφ − ω t)U(r)

(5.32a)

uθ (r, θ , φ ,t) = exp i(kbφ − ω t)V (r)

(5.32b)

uφ (r, θ , φ ,t) = exp i(kbφ − ω t)W (r)

(5.32c)

Φ (r, θ , φ ,t) = exp i(kbφ − ω t)X(r)

(5.32d)

where U(r), V (r),W (r) represent the amplitudes of vibration in the radial, and two tangential directions respectively and X(r) represents the amplitude of electric potential. Substituting Eqs.(5.29), (5.30), (5.32) and θ = π/2 into Eq.(5.31), the governing differential equations in terms of displacement components and electric potential can be obtained: 1 l+2 (l) 00 (l) (l) (l) (l) (l) {r (C33 U +C35 V 00 +C34 W 00 + e33 X 00 ) + rl+1 [((l + 2)C33 + 2ikbC34 )U 0 l h (l) (l) (l) (l) (l) (l) (l) (l) +((l + 1)C35 −C15 −C25 )V 0 + ikb(C36 +C45 )V 0 + ((l + 1)C34 −C14 −C24 )W 0 (l)

(l)

(l)

(l)

(l)

(l)

(l)

+ikb(C23 +C44 )W 0 + ((l + 2)e33 − e31 − e32 + ikb(e23 + e34 ))X 0 ] (l)

(l)

(l)

(l)

(l)

(l)

(l)

+rl [(l + 1)(C13 +C23 + ikbC34 )U − (C11 +C22 + 2C12 + k2 b2C44 )U (l)

(l)

(l)

(l)

(l)

(l)

+(C15 − (l + 1)C35 − k2 b2C46 )V + ikb((l + 1)C36 −C16 −C26 −C45 )V

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JianGong Yu (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

+ikb((l + 1)C23 −C12 −C22 −C44 )W + (C14 +C24 +C56 − (l + 1)C34 (l)

(l)

(l)

(l)

(l)

−k2 b2C24 )W + ikb((l + 1)e23 − e21 − e22 )X − k2 b2 e24 X]}π (r) 1 (l) (l) (l) (l) +(δ (r − a) − δ (r − b)) l {rl+2 (C33 U 0 +C35 V 0 +C34 W 0 + e33 X 0 ) h (l) (l) (l) (l) (l) (l) (l) +rl+1 [(C13 +C23 + ikbC34 )U + (ikbC36 −C35 )V + (ikbC23 −C34 )W (l)

+ikbe23 X]} = −

ρ (l) r2 ω 2 U π (r) hl

(5.33a)

1 l+2 (l) 00 (l) (l) (l) (l) (l) (l) {r (C35 U +C55 V 00 +C45 W 00 + e35 X 00 ) + rl+1 [((l + 3)C35 +C15 +C25 hl (l) (l) (l) (l) (l) +ikbC36 + ikbC45 )U 0 + ((l + 2)C55 + ikbC56 )V 0 + (l + 2)C45 W 0 (l)

(l)

(l)

(l)

(l)

+ikb(C25 +C46 )W 0 + ikb(e36 + e25 )X 0 + (l + 3)e35 X 0 ] (l)

(l)

(l)

(l)

(l)

+rl [(l + 2)(C15 +C25 + ikbC45 )U + (ikbC26 − k2 b2C46 )U (l)

(l)

(l)

(l)

−((l + 2)C55 +C12 − (l + 1)ikbC56 + k2 b2C66 )V (l)

(l)

(l)

(l)

(l)

(l)

+(C16 − (l + 2)C45 − k2 b2C26 )W + ikb((l + 2)C25 −C46 +C16 )W (l)

(l)

+(l + 2)ikbe25 X − k2 b2 e26 X]}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) (l) (l) × l {rl+2 (C35 U 0 +C55 V 0 +C45 W 0 + e35 X 0 ) h (l) (l) (l) (l) (l) +rl+1 [(C15 +C25 + ikbC45 )U + (ikbC56 −C55 )V (l)

(l)

(l)

+(ikbC25 −C45 )W + ikbe25 X]} = −

ρ (l) r2 ω 2 V π (r) hl

(5.33b)

1 l+2 (l) 00 (l) (l) (l) {r (C34 U +C45 V 00 +C44 W 00 + e34 X 00 ) hl (l) (l) (l) (l) (l) +rl+1 [((l + 3)C34 +C14 +C24 + ikbC23 + ikbC44 )U 0 (l)

(l)

(l)

(l)

(l)

+((l + 2)C45 + ikbC25 + ikbC46 )V 0 + ((l + 2)C44 + ikbC24 )W 0 (l)

(l)

(l)

+(ikb(e32 + e24 ) + (l + 3)e34 )X 0 ] (l)

(l)

(l)

(l)

(l)

(l)

+rl [((l + 2)C14 + (l + 2)C24 − k2 b2C24 )U + ikb((l + 2)C44 +C12 +C22 )U (l)

(l)

(l)

(l)

(l)

+ikb((l + 2)C46 −C25 )V − ((l + 2)C45 +C26 + k2 b2C26 )V (l)

(l)

(l)

(l)

+(C66 − (l + 2)C44 + (l + 1)ikbC24 − k2 b2C22 )W (l)

(l)

+(ikb(l + 2)e24 −k2 b2 e22 )X]}π (r) + (δ (r − a) − δ (r − b)) 1 (l) (l) (l) (l) (l) (l) (l) × l {rl+2 (C34 U 0 +C45 V 0 +C44 W 0 + e34 X 0 ) + rl+1 [(C14 +C24 + ikbC44 )U h (l) (l) (l) (l) (l) +(ikbC46 −C45 )V + (ikbC24 −C44 )W + ikbe24 X]} =−

ρ (l) rl+2 ω 2 W π (r) hl

(5.33c)

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

189

1 l+2 (l) 00 (l) (l) (l) {r (e33 U + e35 V 00 + e34 W 00 − 33 X 00 ) hl (l) (l) (l) (l) (l) +rl+1 [(e31 + e32 + (l + 2)e33 + ikb(e23 + e34 ))U 0 (l)

(l)

(l)

(l)

(l)

(l)

+(ikb(e36 + e25 ) + (l + 1)e35 )V 0 + (ikb(e32 + e24 ) + (l + 1)e34 )W 0 (l) (l) 0 33 + 2ikb 23 )X ] (l) (l) (l) (l) (l) (l) +rl [((l + 1)(e31 + e32 + ikbe34 ) − k2 b2 e24 + ikb(e21 + e22 ))U (l) (l) (l) +(ikb((l + 1)e36 − e25 ) − (l + 1)e35 (l) (l) (l) (l) (l) (l) −e12 − k2 b2 e26 )V + (ikb((l + 1)e32 − e24 ) + e16 − (l + 1)e34 − k2 b2 e22 )W (l) (l) +(ikb(l + 1) 23 − k2 b2 22 )X]}π (r) + (δ (r − a) − δ (r − b))

−((l + 2)

1 l+2 (l) 0 (l) (l) (l) (l) (l) (l) {r (e33 U + e35 V 0 + e34 W 0 − 33 X 0 ) + rl+1 [(e31 + e32 + ikbe34 )U hl (l) (l) (l) (l) (l) +(ikbe36 − e35 )V + (ikbe32 − e34 )W − ikb 23 X]} = 0 ×

(5.33d)

The latter solving process of Eq.(5.33) is similar to previous sections. Next, we concern a special case of Eq.(5.33). When the material is orthotropic in the wave propagation direction, and is polarized in the radial direction, the governing differential equations are reduced to (l)

(l)

(l)

(l)

(l)

(l)

h−l {rl+2 (C33 U 00 + e33 X 00 + q33 Y 00 ) + rl+1 [(l + 2)C33 U 0 + ikb(C23 +C44 )W 0 (l)

(l)

(l)

(l)

(l)

+((l + 2)e33 − e31 − e32 )X 0 ] + rl [(l + 1)(C13 +C23 )U (l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

−(C11 +C22 + 2C12 + k2 b2C44 )U + ikb((l + 1)C23 −C12 −C22 −C44 )W (l)

(l)

(l)

(l)

−k2 b2 e24 X]}π (r) + (δ (r − a) − δ (r − b))h−l {rl+2 (C33 U 0 + e33 X 0 + q33 Y 0 ) (l)

(l)

(l)

+rl+1 [(C13 +C23 )U+ikbC23 W ]} = −h−l ρ (l) rl+2 ω 2U π (r) (l)

(l)

(l)

(l)

(5.34a) (l)

h−l [rl+2C55 V 00 + rl+1 (l + 2)C55 V 0 + rl ((l + 2)C55 +C12 + k2 b2C66 )V ]π (r) (l)

(l)

+(δ (r − a) − δ (r − b))h−l rl+1 (rC55 V 0 −C55 V ) = −h−l ρ (l) rl+2 ω 2V π (r) (l)

(l)

(l)

(l)

(l)

(5.34b)

(l)

h−l {rl+2C44 W 00 + rl+1 [ikb(C23 +C44 )U 0 + (l + 2)C44 W 0 + ikb(e32 + e24 )X 0 ] (l)

(l)

(l)

(l)

(l)

(l)

+rl [ikb((l + 2)C44 +C12 +C22 )U − rl k2 b2C22 W + (C66 − (l + 2)C44 )W (l)

(l)

+ikb(l + 2)e24 X]}π (r) + (δ (r − a) − δ (r − b))h−l {rl+2C44 W 0 (l)

(l)

(l)

+rl+1 [ikbC44 U−C44 W + ikbe24 X]} = −h−l ρ (l) rl+2 ω 2W π (r) (l)

(l) 00 (l) 00 (l) (l) (l) l+1 [(e31 + e32 + (l + 2)e33 )U 0 33 X − g33 Y ) + r (l) (l) (l) (l) (l) (l) +ikb(e32 + e24 )W 0 −(l + 2) 33 X 0 ] + rl [((l + 1)(e31 + e32 ) − k2 b2 e24 )U (l) (l) (l) +ikb((l + 1)e32 − e24 )W −k2 b2 22 X]}π (r)

h−l {rl+2 (e33 U 00 −

(5.34c)

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JianGong Yu (l)

+(δ (r − a) − δ (r − b))h−l {rl+2 (e33 U 0 − (l)

(l)

(l) 0 (l) 0 33 X − g33 Y )

(l)

+rl+1 [(e31 + e32 )U + ikbe32 W ]} = 0

(5.34d)

Here, Eq.(5.34b) represents the propagating SH wave, which is independent of the other three coupled equations. Only the Lamb-like wave controlled by Eqs.(5.34a) and (5.34c) is coupled with the piezoelectric equation (5.34d).

5.5.2 Numerical results and discussion As related before, the circumferential wave characteristics in cylindrical curved plate with large ratio of radius to thickness (or large radius of curvature) are very close to those in flat plate. From the view of geometry, the difference between the circumferential waves in cylindrical plate and waves in flat plate lies in that in the wave propagating direction, the cylindrical plate is curved but the flat plate is plane. When the radius of curvature is close to infinite, the difference does not exist. The difference between the spherical plate and cylindrical plate lies in that the shear horizontal direction of wave propagation in spherical plate is also curved. Similarly, when the radius of curvature is close to infinite, the difference does not exist. So, we firstly observed their differences in dispersion curves, as shown in Fig.5.43, the phase velocity dispersion curves for FGPM curved plates of 1 mm thickness with two ratios of radius to thickness (η = 10 and η = 2), in which the dotted lines represent spherical plate and solid lines represent cylindrical plate. It is obvious that the difference in the little ratio curved plate is larger than that in the large ratio curved plate. For the large ratio curved plate, the difference exists mainly at low frequencies; as the frequency increases, the difference disappears quickly and does not exist in higher order modes. For the little ratio curved plate, the difference exists at low frequencies of each mode; as the frequency increases, the difference disappears slowly.

Fig. 5.43 Phase velocity dispersion curves for FGPM curved plate: dotted line, spherical plate; solid line, cylindrical plate. (a) η = 10; (b) η = 2.

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Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

191

Next, we observe the difference from the displacement and electrical potential distributions. The curved plate with large ratio (η = 10) is firstly concerned when the wavenumber k = 500 rad/m, 1000 rad/m and 1500 rad/m, as shown in Figs.5.44-5.49. When k = 500 rad/m, the difference of displacement on the first two modes are obvious but that on the third mode is very weak; the difference of electrical potential on the first three modes are all obvious. When k = 1000 rad/m, the difference of displacement on the first two modes still exists, but it becomes weaker; similarly, the difference of electrical potential also becomes weaker, and the difference can not be discerned on the third mode. When wavenumber is increased to k = 1500 rad/m, the difference of displacement and electrical potential distributions only can be seen on the first mode, and it is very weak.

Fig. 5.44 Displacement distribution in the thickness direction of FGPM curved plate with η = 10 at k = 500 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

Fig. 5.45 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 10 at k = 500 rad/m. (a) Cylindrical plate; (b) Spherical plate.

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JianGong Yu

Fig. 5.46 Displacement distribution in the thickness direction of FGPM curved plate with η = 10 at k = 1000 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

Fig. 5.47 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 10 at k = 1000 rad/m. (a) Cylindrical plate; (b) Spherical plate.

Fig. 5.48 Displacement distribution in the thickness direction of FGPM curved plate with η = 10 at k = 1500 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

5

Wave Characteristics in the Functionally Graded Piezoelectric Waveguides

193

Fig. 5.49 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 10 at k = 1500 rad/m. (a) Cylindrical plate; (b) Spherical plate.

Then, we concern the case of curved plate of little ratio (η = 2). As the difference of little ratio case is larger than the large ratio case, the wavenumber is selected at k = 1000 rad/m, 3000 rad/m and 5000 rad/m, as shown in Figs.5.50-5.55. When k = 1000 rad/m, the differences of displacement and electrical potential are all obvious on the first three modes. As the wavenumber is increased to k = 3000 rad/m, even to k = 5000 rad/m, the differences of displacement and electrical potential just become weaker and weaker, but they still exist.

Fig. 5.50 Displacement distribution in the thickness direction of FGPM curved plate with η = 2 at k = 1000 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

Finally, through the case of the FGPM spherical curved plate (η = 10), we illustrate the validity of the automatic boundary condition approach. As is shown in Fig.5.56, the stress and electrical displacement distributions of at k = 3000 rad/m. It can be seen that

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Fig. 5.51 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 2 at k = 1000 rad/m. (a) Cylindrical plate; (b) Spherical plate.

Fig. 5.52 Displacement distribution in the thickness direction of FGPM curved plate with η = 2 at k = 3000 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

Fig. 5.53 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 2 at k = 3000 rad/m. (a) Cylindrical plate; (b) Spherical plate.

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Fig. 5.54 Displacement distribution in the thickness direction of FGPM curved plate with η = 2 at k = 5000 rad/m. (a) Cylindrical plate: dashed line ur , solid line uθ ; (b) Spherical plate: dashed line ur , solid line uφ .

Fig. 5.55 Electrical potential distribution in the thickness direction of FGPM curved plate with η = 2 at k = 5000 rad/m. (a) Cylindrical plate; (b) Spherical plate.

Fig. 5.56 Stress and electric displacement distributions of FGPM spherical curved plate with η = 10 at k = 3000 rad/m. (a) Stress: dashed line Trr , solid line Tφ φ , long short line Trφ ; (b) Electrical displacement: dashed line Dr , solid line Dφ .

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the normal components of the stress and electric displacement are all zeros at the inner and outer surfaces (r = a and r = b).

5.6 Summary By the Legendre orthogonal polynomial method, this chapter presents an overview of the guided wave characteristics in three typical FGPM structures: plate, hollow cylinder and hollow sphere. As concluded above, wave characteristics in FGPM structures are different from those in conventional PEM structures, such as the weakened dispersion and the electrical potential mode shapes in curved plates. For different geometrical FGPM structures, wave characteristics have similarities and differences. The ratio of radius to thickness and the polarizing direction have significant influences on the wave characteristics and piezoelectric effect. In this chapter, the open circuit surface is assumed. However, the orthogonal polynomial method can also solve the short circuit boundary, as shown in Ref.[45]. For the case of circumferential wave in FGPM cylindrical curved plate, eliminating the term π (r) from Eq.(5.16b) and changing the term X(r) in Eq.(5.22) to ∞

X(r) = (r − a)(r − b)

∑ rm Qm (r)

(5.35)

m=0

Equation (5.35) automatically satisfies the short circuit boundary. Furthermore, the orthogonal polynomial method has been used to solve the wave propagation in piezoelectric-piezomagnetic FGM structures [49-51], and the thermoelastic wave characteristics [56,57]. It is also promising to solve the visco-elastic structures, and to take into account the influence of the electrical sources, force and pressure. If extending the displacement and electrical potential into the product of two orthogonal polynomials in two vertical directions, we can also solve the problems of the three dimensional modal analysis and harmonic response in various FGPM resonators. The shortcoming of the orthogonal polynomial method lies in that it is hard to obtain the solutions of very high order modes, which would result in a too large dimensional matrix to solve its eigenvalues.

Acknowledgments Some of the work on functionally graded piezoelectric structures is supported by the National Natural Science Foundation of China (No. 10802027).

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References [1] Kawai T, Miyazaki S, Araragi M. A new method for forming a piezoelectric FGM using a dual dispenser system. In: Yamanouchi, et al. Ed. Proc. First Int. Symp. Func. Grad. Mate., Sendai, Japan, 1990, 191-196. [2] Takagi K, Li J F, Yokoyama S, et al. Fabrication and evaluation of PZT/Pt piezoelectric composites and functionally graded actuators. J. Eur. Ceram. Soc., 2003, 23, 1577-1583. [3] Zhang H L, Li J F, Zhang B P. Fabrication and evaluation of PZT/Ag composites and functionally graded piezoelectric actuators. J. Electroceram., 2006, 16, 413-417. [4] Zhu X, Wang Q, Meng Z. A functionally gradient piezoelectric actuator prepared by powder metallurgical process in PNN-PZ-PT system. J. Mater. Sci. Lett., 1995, 14, 516-518. [5] Chen Y H, Ma J, Li T. A functional gradient ceramic monomorph actuator fabricated using electrophoretic deposition. Ceram. Int., 2004, 30, 683-687. [6] Qui J, Tani, J, Ueno T, et al. Fabrication and high durability of functionally graded piezoelectric bending actuators. Smart Mater. Struct., 2003, 12, 115-121. [7] Alexander P, Brei D. The design tradeoffs of linear functionally graded piezoceramic actuators. ASME Int. Mech. Eng. Cong. & Expo., 2003, 68, 171-180. [8] Jin D, Meng Z. Functionally graded PZT/ZnO piezoelectric composites. J. Mater. Sci. Lett., 2003, 22, 971-974. [9] Zhu X, Meng Z. Operational principle, fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator. Sensor. Actuat. A-Phys., 1995, 48, 169-176. [10] Wu C C M, Khan M, Moy W. Piezoelectric ceramics with functional gradient: a new application of material design: a new application in material design. J. Am. Ceram. Soc., 1996, 79, 809-812. [11] Mahapatra D R, Singhal A, Gopalakrishnan S. Lamb wave characteristics of thickness-graded piezoelectric IDT. Ultrasonics, 2005, 43, 736-746. [12] Liu J, Wang Z K. The propagation behavior of Love waves in a functionally graded layered piezoelectric structure. Smart Mater. Struct., 2005, 14, 137-146. [13] Li X Y, Wang Z K, Huang S H. Love waves in functionally graded piezoelectric materials. Int. J. Solids Struct., 2004, 41, 7309-7328. [14] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a layered structure with a functionally graded piezoelectric substrate and a hard dielectric layer. Ultrasonics, 2009, 49, 293-297. [15] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in functionally graded piezoelectric materials with exponential variation. Smart Mater. Struct., 2008, 17. [16] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a functionally graded piezoelectric substrate coated with a finite-thickness metal waveguide layer. Appl. Phys. Lett., 2009, 94, 023501. [17] Qian Z H, Jin F, Lu T J, et al. Propagation behavior of Love waves in a functionally graded half-space with initial stress. Int. J. Solids Struct., 2009, 46, 1354-1361. [18] Cao X S, Jin F, Jeon I, et al. Propagation of Love waves in a functionally graded piezoelectric material (FGPM) layered composite system. Int. J. Solids Struct., 2009, 46, 4123-4132. [19] Liu J, Cao X S, Wang Z K. Love waves in a smart functionally graded piezoelectric composite structure. Acta Mech., 2009, 208, 63-80. [20] Cao X S, Jin F, Wang Z K, et al. Bleustein-Gulyaev waves in a functionally graded piezoelectric material layered structure. Sci. China-Phys. Mech. Astron, 2009, 52, 613-625. [21] Du J, Jin X, Wang J, et al. Love wave propagation in functionally graded piezoelectric material layer. Ultrasonics, 2007, 46, 13-22.

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[22] Du J, Xian K, Yong Y K. Effect of dissipation on a shear horizontal surface acoustic wave in a functionally graded piezoelectric material structure. Smart Mater. Struct., 2009, 18, 095012. [23] Eskandari M, Shodja H M. Love waves propagation in functionally graded piezoelectric materials with quadratic variation. J. Sound Vib., 2008, 313, 195-204. [24] Liu G R, Tani J. Characteristics of wave propagation in functionally gradient piezoelectric material plates and its response analysis: Part 1, Theory; Part 2, Calculation results. Trans. Jpn. Soc. Mech. Eng., 1991, 57, 2122-2133. [25] Liu G R, Tani J. Surface waves in functionally gradient piezoelectric plates. J. Vib. Acoust., 1994, 116, 440-448. [26] Han X, Liu G R. Elastic waves in a functionally graded piezoelectric cylinder. Smart Mater. Struct., 2003, 12, 962-971. [27] Liu G R, Dai K Y, Han X, et al. Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates. J. Sound Vib., 2003, 268, 131-147. [28] Chakraborty A, Mahapatra R D, Gopalakrishnan S. Finite element simulation of BAW propagation in inhomogeneous plate due to piezoelectric actuation. Lect. Notes Comp. Sci., 2003, 2668, 715-724. [29] Chakraborty A, Gopalakrishnan S, Kausel E. Wave propagation analysis in inhomogeneous piezocomposite layer by the thin-layer method. Int. J. Numer. Meth. Eng., 2005, 64, 567-569. [30] Lefebvre J E, Zhang V, Gazalet J. Acoustic wave propagation in continuous functionally graded plates, an extension of the Legendre polynomial approach. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2001, 48, 1332-1340. [31] Yu J G, Ma Q J. Circumferential wave in functionally graded piezoelectric cylindrical curved plates. Acta Mech., 2008, 198, 171-190. [32] Yu J G, Wu B, Huo H L, et al. Wave propagation in functionally graded piezoelectric spherical curved plates. Phys. Status Solidi B, 2007, 244, 3377-3389. [33] Yu J G, Wu B, Chen G Q. Wave characteristics in functionally graded piezoelectric hollow cylinders. Arch. Appl. Mech., 2009, 79, 807-824. [34] Maradudin A A, Wallis R F, Mills D L, et al. Vibrational edge modes in finite crystals. Phys. Rev. B,1972, 6, 1106-1111. [35] Moss S L, Maradudin A A, Cunningham S L. Vibrational edge modes for wedges with arbitrary interior angles. Phys. Rev. B, 1973, 8, 2999-3008. [36] Sharon T M, Maradudin A A, Cunningham S L. Vibrational modes on a rectangular ridge. Lett. Appl. Eng. Sci., 1974, 2, 161-174. [37] Maradudin A A. Edge modes. Jpn. J. Appl. Phys. Suppl., 1974, 2, 871-878. [38] Maradudin A A, Subbaswamy K R. Edge localized vibration modes on a rectangular ridge. J. Appl. Phys., 1977, 48, 3410-3414. [39] Datta S, Hunsinger B J. Analysis of line acoustical waves in general piezoelectric crystals. Phys. Rev. B, 1977, 16, 4224-4229. [40] Datta S, Hunsinger B J. Analysis of surface waves using orthogonal functions. J. Appl. Phys., 1978, 49, 475-479. [41] Kim Y, Hunt W D. Acoustic fields and velocities for surface-acoustic-wave propagation in multilayered structures: an extension of the Laguerre polynomial approach. J. Appl. Phys., 1990, 68, 4993-4997. [42] Gubernatis J E, Maradudin A A. A Laguerre series approach to the calculation of wave properties for surfaces of inhomogeneous elastic materials. Wave Motion, 1987, 9, 111-121. [43] Kim Y, Hunt W D. An analysis of surface acoustic wave propagation in a piezoelectric film over a GaAs/AlGaAs heterostructure. J. Appl. Phys., 1992, 71, 2136-2142.

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[44] Thompson C, Weiss B L. Characteristics of surface acoustic wave propagation in III-V semiconductor quantum well structures. J. Appl. Phys., 1995, 78, 5002-5007. [45] Lefebvre J E, Zhang V, Gazalet J, et al. Legendre polynomial approach for modeling free ultrasonic waves in multilayered plates. J. Appl. Phys., 1999, 85, 3419-3427. [46] Elmaimouni L, Lefebvre J E, Zhang V, et al. A polynomial approach to the analysis of guided waves in anisotropic cylinders of infinite length. Wave Motion, 2005, 42, 177-189. [47] Elmaimouni L, Lefebvre J E, Zhang V, et al. Guided waves in radially graded cylinders: a polynomial approach. NDT & E Int., 2005, 38, 344-353. [48] Yu J G, Wu B, He C F. Characteristics of guided waves in graded spherical curved plates. Int. J. Solids Struct., 2007, 44, 3627-3637. [49] Wu B, Yu J G, He C F. Wave propagation in non-homogeneous magneto-electro-elasitc plates. J. Sound Vib., 2008, 317, 250-264. [50] Yu J G, Ma Q J, Su S. Wave propagation in non-homogeneous magneto-electro-elasitc hollow cylinders. Ultrasonics, 2008, 48, 664-677. [51] Yu J G, Wu B. Circumferential wave in magneto-electro-elastic functionally graded cylindrical curved plates. Eur. J. Mech. A-Solid., 2009, 28, 560-568. [52] Chen W Q, Wang H M, Bao R H. On calculating dispersion curves of waves in a functionally graded elastic plate. Compos. Struct., 2007, 81, 233-242. [53] Yu J G, Xue T L, Zhang X M. Circumferential SH waves in functionally graded piezoelectric hollow cylinders. SPIE Proc., 2009, 7493, 74936G-1. [54] Kargl S G, Marston P L. Ray synthesis of lamb wave contributions to the total scattering cross section for an elastic spherical shell. J. Acoust. Soc. Am., 1990, 88, 1103-1113. [55] Towfighi S, Kundu T. Elastic wave propagation in anisotropic spherical curved plates. Int. J. Solids Struct., 2003, 40, 5495-5510. [56] Yu J G, Wu B, He C F. Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation. Ultrasonics, 2010, 50, 416-423. [57] Yu J G, Xue T L. Generalized thermoelastic waves in spherical curved plates without energy dissipation. Acta Mech., 2010, 212, 39-50.

Chapter 6 Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures as Sensors and Actuators

HuiMing Wang Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

Abstract Radial vibration analysis is presented for layered piezoelectric cylindrical and spherical structures acting as sensors and actuators. The piezoelectric layered composites are subjected to dynamic mechanical and electrical excitations at the internal and external surfaces. To solve this problem, the normal mode expansion technique is employed for determining the mechanical field. The governing equation for the electrical field is transferred into a Volterra integral equation of the second kind with respect to a function of time. By solving the integral equation, the mechanical and electrical fields are completely determined. To illustrate the applications of the present solution, several examples are performed and the basic understanding of the dynamic behaviors of layered piezoelectric cylindrical and spherical structures under different dynamic mechanical and electrical excitations is addressed. The solution can be used in design, control and optimization of the piezoelectric integrated sensors and actuators when serving in the radial vibration environment. Keywords cylindrical structure, layered composite, piezoelectric, radial vibration, spherical structure

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6.1 Introduction Piezoelectric materials can transform electrical energy to mechanical energy and vice versa. Due to this special capability, the piezoelectric materials have been widely used as sensors and actuators in the area of smart structures and systems [1-6]. In engineering practices, the structures always need to undertake some kinds of loadings, such as concentrated forces, bending moments, torques, pressures, electrical loadings, temperature loadings, etc. If the loadings are time dependent, we call them as dynamic loadings. When the structures are subjected to dynamic loads, the maximum dynamic stresses are always much higher than the static stresses. Therefore, to make sure the safety and reliability of the engineering structures when subjected to the dynamic loadings, it is an important stage to carry out the dynamic analysis in the design process. The dynamic analysis of piezoelectric medium has attracted many attentions in the last decades. In most smart devices and systems, the active element is a piezoelectric ceramic. For different purposes, the active element can be designed in various configurations (beams, plates, tubes, rods, spheres, rings, discs or shells etc.). Due to the diversity of the structural configurations and the variety of the solving methods, great achievements have been made in the dynamic analysis of piezoelectric structures. Here, we just mentioned some literatures in the dynamic analysis of piezoelectric cylindrical and spherical structures. The dynamic analysis for homogeneous cylindrical structures has been widely investigated. Haskins and Walsh [7] studied the vibrations of radially polarized, thin-wall ferroelectric cylindrical shells and Martin [8] investigated those of longitudinally polarized ones. Paul [9] obtained the frequency equation of a piezoelectric cylindrical shell. The free vibrations of the radially and axially polarized piezoelectric ceramic cylinders have been studied by Adelman et al. [10,11]. Ding et al. [12,13] investigated the three-dimensional free vibrations of empty and compressible fluid-filled piezoelectric cylindrical shells. Kim et al. [14] studied the radial vibration characteristics of piezoelectric cylindrical transducers. Ebenezer et al. [15,16] investigated the frequency dependent behaviors of the radially and axially polarized piezoelectric cylindrical shells. Huang et al. [17] studied the steady state electromechanical responses of a long piezoelectric tube subjected to harmonic loadings. As for the dynamic responses, Ding et al. [18,19] obtained the dynamic solution of a piezoelectric hollow cylinder for axisymmetric plane strain problems under the mechanical and thermal excitations. Dai and Wang [20] studied the dynamic responses of piezoelectric hollow cylinders placed in an axial magnetic field. The topic on the electroelastic waves in solid and hollow cylinders have also been widely carried out. The torsional wave propagation in an infinite piezoelectric cylinder (622) crystal class has been investigated by Srinivasamoorthy and Anandam [21]. The analysis of asymmetric modes of wave propagation in a piezoelectric solid cylinder was studied by Paul and Raju [22]. The wave propagation problems in hollow piezoelectric cylinders have also been considered by many researchers [23-25]. For homogeneous spherical structures, the free vibrations of piezoceramic hollow sphere for different vibration modes have been widely investigated [26-30]. The three-

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dimensional free vibrations of submerged and compressible fluid-filled piezoelectric hollow sphere are further studied in Ref.[31,32]. The steady-state radial vibrations of a piezoelectric hollow sphere in compressible fluid have been investigated in Ref.[33]. Ding et al. [34] investigated the transient responses in a piezoelectric hollow sphere subjected to dynamic mechanical and electrical loads. The studies for the structures made of functionally graded materials and multilayered materials have also attracted considerable attentions. Siao et al. [35] investigated the frequency spectra of laminated piezoelectric cylinders. Kharouf and Heyliger [36] dealt with the free vibration problems of laminated piezoelectric cylinders. Li et al. [37] studied the free vibration of a piezoelectric laminated cylindrical shell under the hydrostatic pressure. The three-dimensional vibrations of layered piezoelectric cylinders have been considered by Hussein and Heyliger [38]. Chen et al. [39,40] investigated the free vibrations of functionally graded hollow cylinders and spheres. Sun et al. [41] analyzed the axial vibration characteristics of a cylindrical, radially polarized piezoelectric transducer with different electrode patterns. Li et al. [42] investigated the spherical-symmetric steady-state response of fluid-filled laminate piezoelectric spherical shell under external excitation. Recently, a theoretical analysis method is developed to study the transient responses in functionally graded and multilayered piezoelectric hollow spheres and cylinders [4347]. Dai et al. [48] obtained the dynamic solutions of functionally graded piezoelectric solid cylinder and sphere placed in a uniform magnetic field. The wave characteristics in functionally graded and layered piezoelectric cylinders have also been studied [49,50]. In this investigation, a theoretical method is presented systematically for solving the elastodynamic problem of layered piezoelectric cylindrical and spherical structures under radial vibration. The solving method is as follows: the dynamic solution is firstly divided into two parts by the method of superposition: One is quasi-static and the other is dynamic. The static part is obtained by the state space method and the dynamic part is derived by the separation of variables method. Then by using the electric boundary conditions and electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully [51]. After solving the integral equation, the mechanical and electrical fields are then completely determined.

6.2 Basic equations for piezoelectric media 6.2.1 Basic equations for three-dimensional problems The three-dimensional equations of piezoelectric media are summarized here as [52,53]

σi j, j = ρ u¨i

(6.1)

Di,i = 0

(6.2)

σi j = ci jkl εkl − eki j Ek

(6.3)

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HuiMing Wang

Di = ei jk ε jk + εik Ek

(6.4)

εi j = (ui, j + u j,i )/2

(6.5)

Ei = −Φ,i

(6.6)

In the above equations, ui is the mechanical displacement, σi j and εi j are the stress and strain, respectively, ρ is the mass density, Di is the electric displacement, Ei is the electric field and Φ is the electric potential, ci jkl , eki j and εi j are the elastic, piezoelectric and dielectric constants, respectively.

6.2.2 Basic equations for radial motion In the following analysis, the cylindrical coordinate system (r, θ , z) is employed to describe the problem for cylindrical structure and the spherical coordinate system (r, θ , ϕ ) for spherical structure. For radial motion (ur = ur (r,t) and Φ = Φ (r,t). Especially, we have εzz = 0 for cylindrical structures and εθ θ = εϕϕ for spherical structures), the basic equations for cylindrical and spherical structures can be written in a systemic form as

∂ σrr α (σrr − σθ θ ) ∂ 2 ur + =ρ 2 ∂r r ∂t 1 ∂ α (r Drr ) = 0 rα ∂ r

(6.7) (6.8)

where α is structural configuration parameter (α = 1 is for cylindrical structure and α = 2 for spherical structure). For radially polarized piezoelectric media, the constitutive relations (6.3) and (6.4) can be simplified as [53,54]

σθ θ = [c11 + (α − 1)c12 ]εθ θ + c13 εrr − e31 Er , σrr = α c13 εθ θ + c33 εrr − e33 Er Drr = α e31 εθ θ + e33 εrr + ε33 Er

(6.9) (6.10)

In Eqs.(6.9) and (6.10), the compressed subscript notations for the material constants are employed. Equations (6.5) and (6.6) for radial motion case can be simplified as ur ∂ ur , εθ θ = ∂r r ∂Φ Er = − ∂r

εrr =

(6.11) (6.12)

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6.3 Layered model and solution 6.3.1 Layered model and governing equations Consider a n-layer hollow structure. The inner and outer radii as well as the radii of the interfaces are denoted as r0 = a, rn = b and ri (i = 1, 2, · · · , n − 1), see Fig.6.1. In the following equations, the symbols with a superscript (i) denote the quantities in the ith layer.

Fig. 6.1 Geometry of layered hollow structures.

In the ith layer (ri−1 6 r 6 ri ), with the aid of Eqs.(6.11) and (6.12), the constitutive relations (6.9) and (6.10) can be rewritten as (i)

(i)

(i)

σθ θ = [c11 + (α − 1)c12 ] (i)

(i) (i) ur

σrr = α c13

(i)

(i) (i) ∂ ur

+ c33

(i)

(i) ur (i) ∂ ur (i) ∂ Φ + c13 + e31 , r ∂r ∂r (i) ∂ Φ

+ e33

(i)

∂r ∂r (i) (i) (i) (i) (i) ur (i) ∂ ur (i) ∂ Φ Drr = α e31 + e33 − ε33 r ∂r ∂r r

,

(6.13)

The equation of motion of the ith layer is (i)

(i)

(i) (i) σrr − σθ θ ∂ σrr ∂ 2 ur +α = ρ (i) ∂r r ∂ t2

(6.14)

and the charge equation of electrostatics is 1 ∂ α (i) [r Drr ] = 0 rα ∂ r The boundary conditions are

(6.15)

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HuiMing Wang (1)

(n)

σrr (a,t) = q0 (t), Φ

(1)

σrr (b,t) = qn (t)

(a,t) = Φ0 (t),

Φ

(n)

(b,t) = Φn (t)

(6.16) (6.17)

where q0 (t) and qn (t) are prescribed dynamic pressures acting on the inner and outer surfaces, respectively. And Φ0 (t) and Φn (t) are known electric potential acting on the inner and outer surfaces, respectively. The continuity conditions at the interfaces can be expressed as (i+1)

σrr Φ

(i)

(i+1)

(ri ,t) = σrr (ri ,t),

(i+1)

ur

(i)

(ri ,t) = ur (ri ,t)

(i+1) (i) Drr (ri ,t) = Drr (ri ,t)

(i)

(ri ,t) = Φ (ri ,t),

(6.18) (6.19)

where i = 1, 2, · · · , n − 1. The initial conditions (t = 0) are expressed as (i)

(i)

(i)

ur (r, 0) = U0 (r), (i)

(i)

u˙r (r, 0) = V0 (r)

(6.20)

(i)

where i = 1, 2, · · · , n. U0 (r) and V0 (r) are known functions of the radial coordinate r and a dot over a quantity denotes its partial derivative with respect to time t. For convenience, the following non-dimensional quantities are introduced: (i)

c11P =

(i)

c11

, (1) c33

(i)

(i)

c12P =

c12

(i)

(i)

c33 ρ (i) (i) , c33P = (1) , ρ¯ (i) = (1) , (1) ρ c33 c33 c13

(i)

, c13P = (1)

c33

(i) e31

(i)

(i) ε ei ur r (i) (i) (i) (i) e1 = q , e3 = q 33 , ε3 = 33 , u = , ξ= , (1) b b (1) (1) (1) (1) ε33 c33 ε33 c33 ε33 v u (1) (i) (i) u ε Φ (i) σjj Drr (i) (i) (i) σ j = (1) ( j = r, θ ), Dr = q , φ = t 33 , (6.21) (1) (1) (1) c33 c33 b c33 ε33 v v u (1) u (1) u u ε Φn ε Φ0 cv ri t 33 ξi = (i = 0, 1, · · · , n), φ0 = t 33 , φ , τ = t, = n (1) b (1) b b b c33 c33 s (i) (i) (1) U V c33 q0 qn (i) (i) u0 = 0 , v0 = 0 , p0 = (1) , pn = (1) , cv = b cv ρ (1) c c 33

33

By virtue of Eq.(6.21), Eqs.(6.13)-(6.15) can be rewritten as (i)

(i)

(i)

σθ = [c11P + (α − 1)c12P ]

(i) (i) u(i) (i) ∂ u (i) ∂ φ + c33P + e3 , ξ ∂ξ ∂ξ

(i)

(i)

(i)

(i) u

σr = α c13P Dr = α e1

(i) (i) u(i) (i) ∂ u (i) ∂ φ + c13p + e1 , ξ ∂ξ ∂ξ

(i)

ξ

(i) (i) ∂ u

+ e3

∂ξ

(i) ∂ φ

− ε3

(i)

∂ξ

(6.22)

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Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures (i)

207

(i)

(i) σr − σθ ∂ σr ∂ 2 u(i) +α = ρ¯ (i) ∂ξ ξ ∂ τ2 1 ∂ α (i) [ξ Dr ] = 0 ξα ∂ξ

(6.23) (6.24)

and the boundary conditions (6.16) and (6.17), the continuity conditions (6.18) and (6.19) as well as the initial condition (6.20) can be rewritten as (1)

(n)

(6.25)

(1)

(n)

(6.26)

σr (ξ0 , τ ) = p0 (τ ), σr (ξn , τ ) = pn (τ ) φ

(ξ0 , τ ) = φ0 (τ ), φ (ξn , τ ) = φn (τ ) (i+1) (i) σr (ξi , τ ) = σr (ξi , τ ), u(i+1) (ξi , τ ) = u(i) (ξi , τ ) (i+1) (i) φ (i+1) (ξi , τ ) = φ (i) (ξi , τ ), Dr (ξi , τ ) = Dr (ξi , τ ) (i) (i) u(i) (ξ , 0) = u0 (ξ ), u˙(i) (ξ , 0) = v0 (ξ )

(6.27) (6.28) (6.29)

In Eq.(6.29) and hereafter, a dot over a quantity denotes its partial derivative with respect to non-dimensional time τ . The solution of Eq.(6.24) can take the form as (i)

Dr (ξ , τ ) = ξ −α η (i) (τ )

(6.30)

where η (i) (τ ) is an unknown function with respect to non-dimensional time τ . With the aid of the second equation in Eq.(6.28), we then obtain

η (1) (τ ) = η (2) (τ ) = · · · = η (n) (τ ) = η (τ )

(6.31)

Utilizing Eqs.(6.30) and (6.31), the third equation in Eq.(6.22) can be rewritten as (i)

(i)

∂ φ (i) α e1 u(i) e3 ∂ u(i) 1 η (τ ) = (i) + (i) − (i) α ∂ξ ε3 ξ ε3 ∂ ξ ε3 ξ

(6.32)

The substitution of Eq.(6.32) into the first two equations in Eq.(6.22) comes out (i)

(i)

(i)

σθ = [c11D + (α − 1)c12D ] (i) σr

(i) (i) u = α c13D

ξ

(i) u(i) (i) ∂ u (i) η (τ ) + c13D − e1D α , ξ ∂ξ ξ

(i) (i) ∂ u + c33D

∂ξ

(6.33)

(i) η (τ ) − e3D α

ξ

where (i)

(i)

c11D = c11P + (i) c33D

(i) (i)

e1 e1 (i)

ε3

(i)

(i)

(i) (i)

, c12D = c12P +

(i) (i) e e (i) = c33P + 3 (i)3 , ε3

(i) e1D

(i)

=

e1

(i)

ε3

,

e1 e1 (i)

ε3

(i) e3D

(i)

(i)

=

(i)

, c13D = c13P + e3

(i)

ε3

(i) (i)

e1 e3 (i)

ε3

, (6.34)

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HuiMing Wang

Equations (6.33) and (6.23) can be rewritten as (i)

(i)

(i)

(i)

(i)

Σθ = [c11D + (α − 1)c12D ]u(i) + c13D ∇u(i) − e1D (i) Σr

η (τ ) ξ α −1

(i) (i) (i) = α c13D u(i) + c33D ∇u(i) − e3D

(i) (i) (i) ∇Σr + (α − 1)Σr − αΣθ = ρ¯ (i) ξ 2

∂ 2 u(i) ∂ τ2

where (i)

(i)

(i)

η (τ ) , ξ α −1

(i)

Σ r = ξ σr , Σ θ = ξ σθ , ∇ = ξ

(6.35)

(6.36)

∂ ∂ξ

(6.37)

Then Eqs.(6.25) and (6.27) can be rewritten as (1)

(n)

Σr (ξ0 , τ ) = ξ0 p0 (τ ), Σr (ξn , τ ) = ξn pn (τ )

(6.38)

(i+1) (i) Σr (ξi , τ ) = Σr (ξi , τ ),

(6.39)

u(i+1) (ξi , τ ) = u(i) (ξi , τ )

6.3.2 The method of superposition By means of the method of superposition [55], the solution is divided into two parts. One is the quasi-static part and the other is the dynamic part. Therefore, the displacement and stresses can be assume as (i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

u(i) = us + ud , Σr = Σrs + Σrd , Σθ = Σθ s + Σθ d (i)

(i)

(6.40)

(i)

where us , Σrs and Σθ s are the quasi-static part satisfying the following equations: (i)

(i)

(i)

(i)

(i)

(i)

(i)

Σθ s = [c11D + (α − 1)c12D ]us + c13D ∇us − e1D ξ −(α −1) η (τ ), (i)

(i)

(i)

(i)

(i)

(i)

Σrs = α c13D us + c33D ∇us − e3D ξ −(α −1) η (τ ) (i)

(i)

(i)

∇Σrs + (α − 1)Σrs − αΣθ s = 0 (1)

(6.41) (6.42)

(n)

Σrs (ξ0 , τ ) = ξ0 p0 (τ ), Σrs (ξn , τ ) = ξn pn (τ )

(6.43)

(i+1) (i) Σrs (ξi , τ ) = Σrs (ξi , τ ),

(6.44)

(i+1) (i) us (ξi , τ ) = us (ξi , τ )

Substituting Eq.(6.40) into Eqs.(6.35), (6.36), (6.38), (6.39) and considering Eqs.(6.41)(i) (i) (i) (6.44), the governing equations for the dynamic part ud , Σrd and Σθ d are then derived as (i)

(i)

(i)

(i)

(i)

(i)

Σθ d = [c11D + (α − 1)c12D ]ud + c13D ∇ud , (i)

(i)

(i)

(i)

(i)

Σrd = α c13D ud + c33D ∇ud

(6.45)

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures (i)

(i)

(i)

(i)

(i)

209

∇Σrd + (α − 1)Σrd − αΣθ d = ρ¯ (i) ξ 2 [u¨d + u¨s ]

(6.46)

(1) (n) Σrd (ξ0 , τ ) = 0, Σrd (ξn , τ ) = 0 (i+1) (i) (i+1) (i) Σrd (ξi , τ ) = Σrd (ξi , τ ), ud (ξi , τ ) = ud (ξi , τ )

(6.47) (6.48)

Furthermore, Eq.(6.29) can be rewritten as (i)

(i)

(i)

(i)

(i)

(i)

ud (ξ , 0) = u0 (ξ ) − us (ξ , 0), u˙d (ξ , 0) = v0 (ξ ) − u˙s (ξ , 0)

(6.49)

6.3.3 The quasi-static part (i)

(i)

(i)

The quasi-static part us , Σrs and Σθ s can be determined by the state space method. The second equation in Eq.(6.41) can be rewritten as (i)

(i) (i)

(i)

(i)

(i)

∇us = a11 us + a12 Σrs + a13 ξ −(α −1) η (τ ) where

(i)

(i)

a11 = −α

c13D (i)

c33D

1

(i)

, a12 =

(i)

(i)

c33D

(6.50)

(i)

, a13 =

e3D

(6.51)

(i)

c33D

Substituting the first equation in Eq.(6.41) into Eq.(6.42) and utilizing Eq.(6.50), we obtain (i) (i) (i) (i) (i) (i) ∇Σrs = a21 us + a22 Σrs + a23 ξ −(α −1) η (τ ) (6.52) where

(i)

(i)

(i)

(i)

(i)

a21 = α [c11D + (α − 1)c12D + c13D a11 ], (i)

(i)

(i)

(i)

a22 = α c13D a12 − (α − 1),

(i)

(i)

(6.53)

(i)

a23 = α (c13D a13 − e1D )

Equations (6.50) and (6.52) can be rewritten in a matrix form as ∇X(i) (ξ , τ ) = N(i) X(i) (ξ , τ ) + L(i) η (τ )ξ −(α −1)

(6.54)

where ( X(i) (ξ , τ ) =



)

(i)

us (ξ , τ ) (i) Σrs (ξ , τ )

N(i) = 

,

(i)

(i)

(i) a21

(i) a22

a11 a12

 ,

L(i) =

   a(i)  13

 a(i) 

(6.55)

23

The solution of Eq.(6.54) is X(i) (ξ , τ ) = T(i) (ξ )(X(i) (ξi−1 , τ ) + G(i) (ξ )η (τ )) where

µ T(i) (ξ ) =

ξ ξi−1

¶N (i) ,

G(i) (ξ ) =

Z ξ ξi−1

[T(i) (ζ )]−1 L(i) ζ −α dζ

(6.56)

(6.57)

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HuiMing Wang

Here T(i) (ξ ) is a 2 × 2 matrix, and G(i) (ξ ) is a column vector with two elements. The continuity conditions in Eq.(6.44) can be rewritten as X(i+1) (ξi , τ ) = X(i) (ξi , τ ) (i = 1, 2, · · · , n − 1)

(6.58)

Setting ξ = ξi in Eq.(6.56) and repeatedly using Eq.(6.58), we obtain X(i) (ξi , τ ) = H(i) X(1) (ξ0 , τ ) + M(i) η (τ ) (i = 1, 2, · · · , n) where (i) H(i) = Tˆ 1 ,

M(i) =

i

(6.59)

(i)

∑ Tˆ m G(m) (ξm ),

m=1

m

(6.60)

(i) Tˆ m = ∏ T( j) (ξ j ) (m = 1, 2, · · · , i) j=i

in which H(i) is a 2 × 2 matrix and M(i) is a column vector with two elements. Setting i = n in Eq.(6.59) and utilizing Eq.(6.43), gives   (n) (n) ½ ½ (n) ¾ ¾ ( (n) ) (1) H H M1 12 us (ξn , τ ) =  11 u ( ξ , τ ) s 0  + (6.61) (n) η (τ ) (n) (n) ξn pn (τ ) ξ p ( τ ) M2 0 0 H21 H22 (1)

Then us (ξ0 , τ ) can be determined from the second equation in Eq.(6.61): (1)

(n)

(n)

(n)

us (ξ0 , τ ) = [ξn pn (τ ) − H22 ξ0 p0 (τ ) − M2 η (τ )]/H21

(6.62)

Rewrite Eq.(6.56) as (



)

(i)

us (ξ , τ ) (i) Σrs (ξ , τ )

=

(i)

 

(i)

T11 (ξ ) T12 (ξ ) (i)

(i)

 

(i−1)

H11

(i−1)

(i−1)

H12

(i−1)



½



T21 (ξ ) T22 (ξ ) H21 H22       M (i−1)   G(i) (ξ )  1 1 + η (τ ) + η (τ )  M (i−1)   G(i) (ξ )  2 2

(1)

us (ξ0 , τ ) ξ0 p0 (τ )

¾

(6.63)

Substituting Eq.(6.62) into the first equation in Eq.(6.63), we obtain (i)

(i)

(i)

(i)

us (ξ , τ ) = f1 (ξ )p0 (τ ) + f2 (ξ )pn (τ ) + f3 (ξ )η (τ )

(6.64)

where (" (i) f1 (ξ ) = ξ0 (i)

f2 (ξ ) =

ξn (n) H21

(i−1) H12 −

(n)

H22

(i−1) H (n) 11 H21

#

" (i) T11 (ξ ) +

(i−1) (i) (i−1) (i) T11 (ξ ) + H21 T12 (ξ )],

[H11

(i−1) H22 −

(n)

H22

(i−1) H (n) 21 H21

#

) (i) T12 (ξ )

,

(6.65)

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

" (i) (i) f3 (ξ ) = T11 (ξ )

(i−1) M1 −

211

#

(n)

M2

(i−1) (i) H + G1 (ξ ) (n) 11 H21 " # (n) M2 (i) (i−1) (i−1) (i) +T12 (ξ ) M2 − (n) H21 + G2 (ξ ) H21

6.3.4 The dynamic solution Substituting Eq.(6.45) into Eq.(6.46), we obtain " # (i) (i) (i) (i) ∂ 2 ud α ∂ ud βi2 (i) 1 ∂ 2 ud ∂ 2 us + − 2 ud = 2 + ∂ξ2 ξ ∂ξ ξ ∂ τ2 ∂ τ2 ci where

v u (i) u c + (α − 1)(c(i) − c(i) ) 12D 13D βi = tα 11D , (i) c33D

s ci =

(6.66)

(i)

c33D ρ¯ (i)

(6.67)

A new dependent variable is introduced as (i)

1

ud (ξ , τ ) = ξ − 2 (α −1) w(i) (ξ , τ )

(6.68)

Substituting Eq.(6.68) into Eq.(6.66) and utilizing Eq.(6.64), we obtain " ∂ 2 w(i) 1 ∂ w(i) µi2 (i) 1 ∂ 2 w(i) ∂ 2 p0 (τ ) (i) + − w = + g ( ξ ) 1 ∂ξ2 ξ ∂ξ ξ2 ∂ τ2 ∂ τ2 c2i

∂ (i) +g2 (ξ ) where

2 p (τ ) n ∂ τ2

∂ (i) + g3 (ξ )

2 η (τ )

¸

∂ τ2

(6.69)

r

1 1 (i) (i) βi2 + (α − 1)2 , g1 (ξ ) = ξ 2 (α −1) f1 (ξ ), 4 1 1 (i) (i) (i) (i) g2 (ξ ) = ξ 2 (α −1) f2 (ξ ), g3 (ξ ) = ξ 2 (α −1) f3 (ξ )

µi =

(6.70)

By means of the separation of variables method, w(i) (ξ , τ ) can be assumed as w(i) (ξ , τ ) =

∞

(i)

∑ Rm (ξ )Ωm (τ )

(6.71)

m=1

(i)

where Ωm (τ ) is an undetermined function. Rm (ξ ) is a known function and the detailed (i) procedure for determining Rm (ξ ) is presented in Appendix A. Substituting Eq.(6.71) into Eq.(6.69), we have

212

HuiMing Wang ∞

∑

m=1

· (i)

Rm (ξ )

¸ d2 Ωm (τ ) d2 p0 (τ ) d2 pn (τ ) (i) (i) 2 ω Ω τ ) = −g1 (ξ ) + ( − g2 (ξ ) m m 2 2 dτ dτ dτ 2 (i)

−g3 (ξ )

d2 η (τ ) dτ 2

(6.72)

(i)

By virtue of the orthogonal property of Rm (ξ ) [46, 47, 56], the following equation can be derived from Eq.(6.72): d2 Ωm (τ ) + ωm2 Ωm (τ ) = qm (τ ) dτ 2

(m = 1, 2, · · · , ∞)

(6.73)

where qm (τ ) = I1m p¨0 (τ ) + I2m p¨n (τ ) + I3m η¨ (τ ), n

I jm = − ∑ ρ¯ (i) i=1

n

Jm = ∑ ρ¯ (i) i=1

Z ξi

Z ξi

ξi−1

(i)

(i)

ξ g j (ξ )Rm (ξ )dξ /Jm

( j = 1, 2, 3),

(6.74)

(i)

ξi−1

ξ [Rm (ξ )]2 dξ

The solution of Eq.(6.73) is

Ωm (τ ) = Ωm (0) cos ωm τ +

Z τ Ω˙ m (0) 1 sin ωm τ + qm (p) sin ωm (τ − p)dp ωm ωm 0

(6.75)

Then u(i) (ξ , τ ) can be written as 1

u(i) (ξ , τ ) = ξ − 2 (α −1)

∞

(i)

(i)

∑ Rm (ξ )Ωm (τ ) + f1

(ξ )p0 (τ )

m=1 (i)

(i)

+ f2 (ξ )pn (τ ) + f3 (ξ )η (τ )

(6.76)

It is noted here that the time function η (τ ) in Eq.(6.76) is still unknown. By following a similar procedure as shown in Refs. [46] and [47], a Volterra integral equation of the second kind [57] can be obtained (Also see Appendix B), from which η (τ ) can be determined efficiently by the recursion formula [51]. After η (τ ) is obtained, the mechanical and electrical fields can then be completely determined.

6.4 Numerical results and analysis In this section, some examples about the dynamic responses of layered piezoelectric spherical and cylindrical structures subjected to dynamic electrical and mechanical loads will be presented. The material constants are listed in Table 6.1 [10, 36]. In the following analysis, we suppose that the piezoelectric hollow structures are initially at the rest. That (i) (i) is, u0 (ξ , 0) = 0 and v0 (ξ , 0) = 0 (i = 1, 2, · · · , n).

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

213

Table 6.1 Elastic, piezoelectric and dielectric constants of piezoelectric materials. Material constants

PZT-4

BaTiO3

PZT-5H

c11 /GPa

139.0

150.0

126.0

c12 /GPa

77.8

66.0

79.5

c13 /GPa

74.3

66.0

84.1

c23 /GPa

74.3

66.0

84.1

c33 /GPa

115.0

146.0

117.0

e31 /(C/m2 )

−5.2

−4.35

−6.5

e32 /(C/m2 )

−5.2

−4.35

−6.5

/(C/m2 )

15.1

17.5

23.3

ε33 /(×10−9 F/m)

e33

5.62

15.04

13.0

ρ /(×103 kg/m3 )

7.5

5.7

7.5

Example 1

Two-layer cylindrical structure subjected to shock pressure

In this example, the dynamic responses of a two-layer (n = 2) piezoelectric composite hollow cylinder subjected to dynamic mechanical excitation are considered. The inner part of the piezoelectric composite hollow cylinder is made of PZT-4 and the outer part is BaTiO3 . The geometric parameters adopted here are: ξ0 = 0.6, ξ1 = 0.8 and ξ2 = 1.0, respectively. Here we consider the case that the composite hollow cylinder is subjected to mechanical excitation. For this case, the piezoelectric hollow cylinder acts as a sensor. Suppose the piezoelectric cylinder is electrically shorted at internal and external surfaces and is subjected to a sudden constant pressure (shock pressure) at the inner surface. The boundary conditions are p0 (τ ) = −H(τ ),

φ

(1)

(ξ0 , τ ) = 0, φ

pn (τ ) = 0

(6.77)

(n)

(6.78)

(ξn , τ ) = 0

where H( · ) denotes the Heaviside function. The time histories of the non-dimensional radial and hoop stresses at the inner surface (ξ = 0.6), the interface (ξ = 0.8), as well as the outer surface (ξ = 1.0) are shown in Figs.6.2 and 6.3. It is observed that the dominant stress is hoop stress and appears at the internal surface. Figures 6.4 and 6.5 show the distributions of radial and hoop stresses at the times τ = 0.05, 0.15 and 0.25 in the two-layer piezoelectric hollow cylinder subjected to shock pressure at the internal surface. In Figs.6.4 and 6.5, the phenomenon that the wavefront moves from the inner to out is clearly presented. Also, the discontinuity of the stress at the wavefront is observed. The distributions of the non-dimensional electric potential φ at different times (τ = 0.05, 0.15 and 0.25) are shown in Fig.6.6. We find that at the different times, the maximum induced electric potential appears at the different positions.

214

HuiMing Wang

Fig. 6.2 Time histories of radial stress at different locations in two-layer piezoelectric cylindrical structure subjected to dynamic mechanical load at the internal surface.

Fig. 6.3 Time histories of hoop stress at different locations in two-layer piezoelectric cylindrical structure subjected to shock pressure at the internal surface.

Example 2

Fig. 6.4 Distributions of radial stresses at different times in two-layer piezoelectric cylindrical structure subjected to shock pressure at the internal surface.

Two-layer cylindrical structure subjected to dynamic electrical excitation

In this example, a different external excitation form will be considered. We simulate the dynamic responses of a two-layer (n = 2) piezoelectric composite hollow cylinder subjected to dynamic electrical excitation at the external surface. For this case, the piezoelectric hollow cylinder acts as an actuator. The geometric configuration and the material properties of the composite hollow cylinder are the same as those in Example 1. The boundary conditions are prescribed as

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

Fig. 6.5 Distributions of hoop stresses at different times in two-layer piezoelectric cylindrical structure subjected to shock pressure at the internal surface.

p0 (τ ) = 0,

φ

(1)

Fig. 6.6 Distributions of electric potential at different times in two-layer piezoelectric cylindrical structure subjected to shock pressure at the internal surface.

pn (τ ) = 0

(ξ0 , τ ) = 0, φ

215

(n)

(6.79)

(ξn , τ ) = 1 − exp(−0.5τ )

(6.80)

where exp( · ) denotes the exponential function. Figure 6.7 shows the time histories of the non-dimensional radial stress at the interface (ξ = 0.8). The oscillation behavior is observed and the radial stress with maximum amplitude is compressive. Figure 6.8 depicts the time histories of the non-dimensional hoop stress at the inner surface (ξ = 0.6), the interface (ξ = 0.8), as well as the outer surface (ξ = 1.0), respectively. One can find that the maximum tensile hoop stress appears at the internal surface of the BaTiO3 layer. The distributions of radial and hoop stresses as well as the electric potential at the times τ = 0.05, 0.10, 0.15 and 0.20 in the two-layer piezoelectric hollow cylinder subjected to dynamic electrical excitation at the external surface are shown in Figs.6.9-6.11. From Figs.6.9 and 6.10, we find that under this kind of external electrical excitation, no distinct wavefront with strong stress discontinuity can be observed. Example 3

Three-layer spherical structure subjected to dynamic mechanical load

Consider a three-layer (n = 3) piezoelectric spherical structure subjected to dynamic mechanical load at the internal surface. The composite mode is taken as PZT-4[inner] / BaTiO3[middle] /PZT-4[outer] . The geometric parameters adopted here are ξ0 = 0.5, ξ1 = 0.6, ξ2 = 0.9 and ξ3 = 1.0, respectively. The boundary conditions are assumed as p0 (τ ) = 0,

φ

(1)

pn (τ ) = exp(−2τ ) − 1

(ξ0 , τ ) = 0,

φ

(n)

(ξn , τ ) = 0

(6.81) (6.82)

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HuiMing Wang

Fig. 6.7 Time histories of radial stress at the interface in two-layer piezoelectric cylindrical structure subjected to dynamic electrical excitation at the external surface.

Fig. 6.8 Time histories of hoop stress at different locations in two-layer piezoelectric cylindrical structure subjected to dynamic electrical excitation at the external surface.

Fig. 6.9 Distributions of radial stresses at different times in two-layer piezoelectric cylindrical structure subjected to dynamic electrical excitation at the external surface.

Fig. 6.10 Distributions of hoop stresses at different times in two-layer piezoelectric cylindrical structure subjected to dynamic electrical excitation at the external surface.

The time histories of the radial stress at ξ = 0.5, 0.6, 0.9 and 1.0 are shown in Fig.6.12. Figure 6.13 depicts the time histories of the hoop stress at the internal and external surfaces of the PZT-4 layers (ξ = 0.5 and 0.6 are for inner PZT-4 layer and ξ = 0.9 and 1.0 for outer PZT-4 layer). The time histories of the hoop stress and the electric potential at the internal and external surfaces of the BaTiO3 layer (ξ = 0.6 and 0.9) are presented in Figs.6.14 and 6.15. The oscillation behaviors of the mechanical and elec-

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

217

Fig. 6.11 Distributions of electric potential at different times in two-layer piezoelectric cylindrical structure subjected to dynamic electrical excitation at the external surface.

trical fields are illustrated. With the time processing, although the external mechanical excitation approaches a constant value, the stress and electric potential still keep their oscillation characteristics. The maximum tensile hoop stress appears at the inner surface.

Fig. 6.12 Time histories of radial stress at different locations in three-layer piezoelectric spherical structure subjected to dynamic mechanical load at the internal surface.

Fig. 6.13 Time histories of hoop stress at the internal and external surfaces of the PZT-4 layers in three-layer piezoelectric spherical structure subjected to dynamic mechanical load at the internal surface.

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HuiMing Wang

Fig. 6.14 Time histories of hoop stress σθ at the internal and external surfaces of the BaTiO3 layer in three-layer piezoelectric spherical structure subjected to dynamic mechanical load at the internal surface.

Example 4

Fig. 6.15 Time histories of electric potential at the internal and external surfaces of the BaTiO3 layer in three-layer piezoelectric spherical structure subjected to dynamic mechanical load at the internal surface.

Three-layer spherical structure subjected to dynamic electrical excitation

The dynamic behaviors of a three-layer (n = 3) piezoelectric spherical structure subjected to dynamic electrical excitation at the external surface will be illustrated in this example. Here we consider that the electrical excitation varies as a sinusoidal form. Except the boundary conditions, all the other computational parameters are the same as those in Example 3. The boundary conditions are taken as p0 (τ ) = 0,

φ

(1)

pn (τ ) = 0

(ξ0 , τ ) = 0, φ

(n)

(ξn , τ ) = sin 5τ

(6.83) (6.84)

For a three-layer piezoelectric spherical structure subjected to dynamic electrical excitation at the external surface, Fig. 6.16 shows the time histories of radial stress σr at two interfaces and Fig.6.17 shows the time histories of hoop stress σθ at the internal and external surfaces. It can be seen that the maximum amplitude of the tensile radial stress at ξ = 0.6 and that of the tensile hoop stress at ξ = 0.5 are in same order. Figure 6.18 illustrates that at the beginning times τ = 0.05, 0.10, 0.15, 0.20, 0.25 and 0.30, the distributions of electric potential φ in the three-layer piezoelectric spherical structure increase monotonously from the inner to the outer.

6

Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

Fig. 6.16 Time histories of radial stress at two interfaces in three-layer piezoelectric spherical structure subjected to dynamic electrical excitation at the external surface.

219

Fig. 6.17 Time histories of hoop stress at the internal and external surfaces in three-layer piezoelectric spherical structure subjected to dynamic electrical excitation at the external surface.

Fig. 6.18 Distributions of electric potential at different times in three-layer piezoelectric spherical structure subjected to dynamic electrical excitation at the external surface.

6.5 Summary Based on the linear piezoelectricity theory, the analytical solutions are obtained for layered piezoelectric cylindrical and spherical structures subjected to dynamic mechanical and electrical excitations at the internal and external surfaces. By employing the state

220

HuiMing Wang

space method, the derivation procedure is completed via operating the 2 × 2 matrices in spite of the number of layers. To show the applications of the present solutions, several examples are performed. The characteristics of the transient responses and the stress wave propagation phenomena are discussed. The presented solutions provide an effective way in performing the parametric analysis of layered piezoelectric cylindrical and spherical sensors and actuators under radial vibration.

Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 10872179, 10725210, and 10832009), the Zhejiang Provincial Natural Science Foundation of China (No. Y7080298), and the Key Team of Technological Innovation of Zhejiang Province (Grant 2011R09025-05).

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[13] Ding H J, Guo Y M, Yang Q D, et al. Free vibration of piezoelectric cylindrical shells. Acta Mech. Solida Sin., 1997, 10, 48-55. [14] Kim J O, Hwang K K, Jeong H G. Radial vibration characteristics of piezoelectric cylindrical transducers. J. Sound Vib., 2004, 276, 1135-1144. [15] Ebenezer D D, Abraham P. Closed-form analysis of thin radially polarized piezoelectric ceramic cylindrical shells with loss. Curr. Sci., 2002, 83, 981-988. [16] Ebenezer D D, Ramesh R. Exact analysis of axially polarized piezoelectric ceramic cylinders with certain uniform boundary conditions. Curr. Sci., 2003, 85, 1173-1179. [17] Huang J H, Shiah Y C, Lee B J. Electromechanical responses of a long piezoelectric tube subjected to dynamic loading. J. Phys. D: Appl. Phys., 2008, 41, 025404(8pp). [18] Ding H J, Wang H M, Hou P F. The transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems. Int. J. Solids Struct., 2003, 40, 105-123. [19] Ding H J, Wang H M, Ling D S. Analytical solution of a pyroelectric hollow cylinder for piezothermoelastic axisymmetric dynamic problems. J. Thermal Stresses, 2003, 26, 261-276. [20] Dai H L, Wang X. Dynamic responses of piezoelectric hollow cylinders in an axial magnetic field. Int. J. Solids Struct., 2004, 41, 5231-5246. [21] Srinivasamoorthy V R, Anandam C. Torsional wave propagation in an infinite piezoelectric cylinder (622) crystal class. J. Acoust. Soc. Am., 1980, 67, 2034-2035. [22] Paul H S, Raju D P. Asymmetric analysis of the modes of wave propagation in a piezoelectric solid cylinder. J. Acoust. Soc. Am., 1982, 71, 255-263. [23] Shul’ga N A, Grigorenko A Y, Loza I A. Axisymmetric electroelastic waves in a hollow piezoelectric ceramic cylinder. Sov. Appl. Mech., 1984, 20, 23-28. [24] Paul H S, Venkatesan M. Vibration of a hollow circular cylinder of piezoelectric ceramics. J. Acoust. Soc. Am., 1987, 82, 952-956. [25] Shul’ga N A. Propagation of harmonic waves in anisotropic piezoelectric cylinders, homogeneous piezoelectric waveguides. Int. Appl. Mech., 2002, 38, 933-953. [26] Loza I A, Shul’ga N A. Axisymmetric vibrations of a hollow piezoceramic sphere with radial polarization. Sov. Appl. Mech., 1984, 20, 113-117. [27] Loza I A, Shul’ga N A. Forced axisymmetric vibrations of a hollow piezoceramic sphere with an electrical method of excitation. Sov. Appl. Mech., 1990, 26, 818-822. [28] Shul’ga N A. Electroelastic oscillation of a piezoceramic sphere with radial polarization. Sov. Appl. Mech., 1986, 22, 497-500. [29] Shul’ga N A. Radial electroelastic vibrations of a hollow piezoceramic sphere. Sov. Appl. Mech., 1990, 22, 731-734. [30] Shul’ga N A. Harmonic electroelastic oscillations of spherical bodies. Sov. Appl. Mech., 1993, 29, 812-817. [31] Cai J B, Chen W Q, Ye G R, et al. Nature frequencies of submerged piezoceramic hollow spheres. Acta Mech. Sin., 2000, 16, 55-62. [32] Chen W Q, Ding H J, Xu R Q. Three dimensional free vibration analysis of a fluid-filled piezoelectric hollow sphere. Comput. Struct., 2001, 79, 653-663. [33] Borisyuk A I, Kirichok I F. Steady-state radial vibrations of piezoceramic spheres in compressible fluid. Sov. Appl. Mech., 1979, 15, 936-940. [34] Ding H J, Wang H M, Chen W Q. Transient responses in a piezoelectric spherically isotropic hollow sphere for symmetric problems. ASME J. Appl. Mech., 2003, 70, 436-445. [35] Siao J C T, Dong S B, Song J. Frequency spectra of laminated piezoelectric cylinders. J. Vib. Acoust., 1994, 116, 364-370. [36] Kharouf N, Heyliger P R. Axisymmetric free vibrations of homogeneous and laminated piezoelectric cylinders. J. Sound Vib., 1994, 174, 539-561.

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[37] Li H Y, Lin Q R, Liu Z X, et al. Free vibration of piezoelectric laminated cylindrical shells under hydrostatic pressure. Int. J. Solids Struct., 2001, 38, 7571-7585. [38] Hussein M, Heyliger P. Three-dimensional vibrations of layered piezoelectric cylinders. J. Eng. Mech., 1998, 124, 1294-1298. [39] Chen W Q, Bian Z G, Lv C F, et al. 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Int. J. Solids Struct., 2004, 41, 947-964. [40] Chen W Q, Wang L Z, Lu Y. Free vibrations of functionally graded piezoelectric hollow spheres with radial polarization. J. Sound Vib., 2002, 251, 103-114. [41] Sun D, Wang S, Hata S, et al. Axial vibration characteristics of a cylindrical, radially polarized piezoelectric transducer with different electrode patterns. Ultrasonics, 2010, 50, 403-410. [42] Li H, Liu Z, Lin Q. Spherical-symmetric steady-state response of fluid-filled laminate piezoelectric spherical shell under external excitation. Acta Mech., 2001, 150, 53-66. [43] Hou P F, Wang H M, Ding H J. Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder. Int. J. Eng. Sci., 2003, 41, 1849-1868. [44] Ding H J, Wang H M, Chen W Q. Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere. Arch. Appl. Mech., 2003, 73, 49-62. [45] Wang H M, Chen Y M, Ding H J. Dynamic responses of a multilayered piezoelectric hollow cylinder under electric potential excitation. J. Zhejiang Univ. Sci., 2005, 6A, 933-937. [46] Wang H M, Ding H J, Chen Y M. Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere. Arch. Appl. Mech., 2005, 74, 581-599. [47] Wang H M, Ding H J, Chen Y M. Dynamic solution of a multilayered orthotropic piezoelectric hollow cylinder for axisymmetric plane strain problems. Int. J. Solids Struct., 2005, 42, 85-102. [48] Dai H L, Fu Y M, Yang J H. Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere. Acta Mech. Sin., 2007, 23, 55-63. [49] Yu J G, Wu B, Chen G Q. Wave characteristics in functionally graded piezoelectric hollow cylinders. Arch. Appl. Mech., 2009, 79, 807-824. [50] Du J, Jin X, Wang J, et al. SH wave propagation in a cylindrically layered piezoelectric structure with initial stress. Acta Mech., 2007, 191, 59-74. [51] Ding H J, Wang H M, Chen W Q. New numerical method for Volterra integral equation of the second kind in piezoelectric dynamic problems. Appl. Math. Mech., 2004, 25, 16-23. [52] Tiersten H F. Linear Piezoelectric Plate Vibrations. New York: Plenum Press, 1969. [53] Ding H J, Chen W Q. Three Dimensional Problems of Piezoelasticity. New York: Nova Science Publishers, 2001. [54] Nye J F. Physical Properties of Crystals. Oxford: Oxford University Press, 1985. [55] Berry J G, Naghdi P M. On the vibration of elastic bodies having time-dependent boundary conditions. Quart. Appl. Math., 1956, 14, 43-50. [56] Yin X C, Yue Z Q. Transient plane-strain response of multilayered elastic cylinders to axisymmetric impulse. ASME J. Appl. Mech., 2002, 69, 825-835. [57] Kress R. Linear Integral Equations. Berlin: Springer-Verlag, 1989.

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Appendix A (i)

The detailed procedure for determining the function Rm (ξ ) is shown as below. (i) According to the differential form at the left-hand side of Eq.(6.69), we know that Rm (ξ ) must i i be a linear combination of Jµi (km ξ ) and Yµi (km ξ ). That is, (i)

(i)

(i)

i i Rm (ξ ) = D1 Jµi (km ξ ) + D2 Yµi (km ξ) (i)

(6.A1)

(i)

where D1 and D2 are undetermined constants, Jµi ( · ) and Yµi ( · ) are Bessel functions of the first and second kinds of order µi , and ωm i km = (6.A2) ci where ωm is a series of positive real numbers. Substituting Eq.(6.52) into the second equation in Eq.(6.45) and utilizing Eq.(6.68), we obtain (i)

1

Σrd (ξ , τ ) = ξ − 2 (α −1) where

∞

(6.A3)

m=1

(i)

and

(i)

∑ σm (ξ )Ωm (τ ) (i)

σm (ξ ) = P(i) (∇)Rm (ξ )

(6.A4)

· ¸ 1 (i) (i) (i) P(i) (∇) = c33D ∇ + α c13D − (α − 1)c33D 2

(6.A5)

Substituting Eq.(6.A1) into Eq.(6.A4), we obtain (i)

(i)

(i)

i i σm (ξ ) = D1 P(i) (∇)Jµi (km ξ ) + D2 P(i) (∇)Yµi (km ξ)

(6.A6)

Setting ξ = ξi−1 in Eqs.(6.A1) and (6.A6), we then obtain two equations: (i)

(i)

(i)

(i)

(i)

i ξ i Rm (ξi−1 ) = D1 Jµi (km i−1 ) + D2 Yµi (km ξi−1 ),

(6.A7)

(i)

i ξ (i) i σm (ξi−1 ) = D1 P(i) (∇)Jµi (km i−1 ) + D2 P (∇)Yµi (km ξi−1 ) (i)

(i)

(i)

(i)

From Eq.(6.A7), we can obtain D1 and D2 . Then substituting the obtained D1 and D2 into Eqs. (6.A1) and (6.A6), we have (i) (i) (i) Zm (ξ ) = Sm (ξ )Zm (ξi−1 ) (6.A8) where

( (i)

Zm (ξ ) = and

(i)

Rm (ξ ) (i)

σm (ξ )



) ,

(i)

Sm (ξ ) = 

(i)

(i)

(i)

(i)

Sm,11 (ξ ) Sm,12 (ξ ) Sm,21 (ξ ) Sm,22 (ξ )

 

(6.A9)

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HuiMing Wang (i)

i ,ξ i i i Sm,11 (ξ ) = [PY (i, km i−1 )Jµi (km ξ ) − PJ (i, km , ξi−1 )Yµi (km ξ )]/Λi , (i)

i ξ i i i Sm,12 (ξ ) = [Jµi (km i−1 )Yµi (km ξ ) −Yµi (km ξi−1 )Jµi (km ξ )]/Λi , (i)

i ,ξ i i i Sm,21 (ξ ) = [PY (i, km i−1 )PJ (i, km , ξ ) − PJ (i, km , ξi−1 )PY (i, km , ξ )]/Λi , (i)

(6.A10)

i ξ i i i Sm,22 (ξ ) = [Jµi (km i−1 )PY (i, km , ξ ) −Yµi (km ξi−1 )PJ (i, km , ξ )]/Λi , i i i i ,ξ Λi = PY (i, km i−1 )Jµi (km ξi−1 ) − PJ (i, km , ξi−1 )Yµi (km ξi−1 ), i , ξ ) = P(i) (∇)J (ki ξ ), PJ (i, km µi m

i , ξ ) = P(i) (∇)Y (ki ξ ) PY (i, km µi m (i)

Equation (6.A8) indicates that an arbitrary quantity Zm (ξ ) in ith layer (ξi−1 6 ξ 6 ξi ) can be expressed by its value at ξ = ξi−1 . For one-dimensional problem, this method is called as initial (i) parameter method and the initial value Zm (ξi−1 ) is just called as initial parameter. With the aid of Eqs.(6.A3), (6.68) and (6.71), we can derive the following conditions from Eqs.(6.47) and (6.48) as (1)

σm (ξ0 ) = 0, (i+1)

σm

(i)

(ξi ) = σm (ξi ),

(i+1)

Rm

(n)

(6.A11)

(i)

(6.A12)

σm (ξn ) = 0 (m = 1, 2, · · · , ∞)

(ξi ) = Rm (ξi ) (m = 1, 2, · · · , ∞; i = 1, 2, · · · , n − 1)

By the newly introducing notation in the first equation in Eq.(6.A9), Eq.(6.A12) can be rewritten in a matrix form as (i+1) (i) Zm (ξi ) = Zm (ξi ) (i = 1, 2, · · · , n − 1) (6.A13) Setting ξ = ξi in Eq.(6.A8) and repeatedly applying Eq.(6.A13), we obtain (i)

(1)

Zm (ξi ) = Q(i) Zm (ξ0 ) (i = 1, 2, · · · , n) where

1

( j)

Q(i) = ∏ [Sm (ξ j )]

(6.A14)

(6.A15)

j=i

Setting i = n in Eq.(6.A14) and utilizing Eq.(6.A11), then we have (

(n)

Rm (ξn ) 0

)

" =

) (n) # ( (1) Rm (ξ0 ) (n) (n) 0 Q21 Q22 (n)

Q11 Q12

(6.A16)

From the second equation in Eq.(6.A16), we obtain (n)

Q21 = 0

(6.A17)

Equation (6.A17), a transcendental equation, is the eigenequation from which a series of positive real roots ωm (m = 1, 2, · · · , ∞) can be obtained. After ωm (m = 1, 2, · · · , ∞), arranged in an ascending order, have been obtained, Eq.(6.A8) can then be rewritten in the following form with the aid of Eq.(6.A14). " ( (i) )  (i) ) (i) (i−1) (i−1) # ( (1) Sm,11 (ξ ) Sm,12 (ξ ) Rm (ξ ) Q11 Q12 Rm (ξ0 )   = (6.A18) (i) (i) (i) (i−1) (i−1) 0 Sm,21 (ξ ) Sm,22 (ξ ) σm (ξ ) Q21 Q22

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Radial Vibration Analysis of Layered Piezoelectric Cylindrical and Spherical Structures

225

Then from the first equation in Eq.(6.A18), we derive (i)

(i−1) (i) (i−1) (i) (1) Sm,11 (ξ ) + Q21 Sm,12 (ξ )]Rm (ξ0 )

Rm (ξ ) = [Q11 (1)

(6.A19) (1)

In Eq.(6.A19), Rm (ξ0 ) is a common constant for each layer, which can be taken as Rm (ξ0 ) = 1 in (i) the calculation. Thus Rm (ξ ) is determined completely.

B In this appendix, the derivation of a Volterra integral equation of the second kind is presented. By applying the integration-by-parts formula, Eq.(6.75) can be rewritten as

Ωm (τ ) = Ω1m (τ ) + I3m η (τ ) − I3m ωm

Z τ 0

η (p) sin ωm (τ − p)dp

(6.A20)

where

Ω1m (τ ) = I1m p0 (τ ) + I2m pn (τ ) + {Ωm (0) − [I1m p0 (0) + I2m pn (0) +I3m η (0)]} cos ωm τ +

1 ˙ {Ωm (0) − [I1m p˙0 (0) + I2m p˙n (0) ωm

+I3m η˙ (0)]} sin ωm τ − I1m ωm −I2m ωm

Z τ 0

Z τ 0

p0 (p) sin ωm (τ − p)dp

pn (p) sin ωm (τ − p)dp

(6.A21)

Utilizing Eqs.(6.64) and (6.68), the initial conditions (6.49) can be rewritten as ∞

1

ξ − 2 (α −1)

(i)

∑ Rm (ξ )Ωm (0)

m=1 (i)

(i)

(i)

(i)

= u0 (ξ ) − f1 (ξ )p0 (0) − f2 (ξ )pn (0) − f3 (ξ )η (0), ∞

1

ξ − 2 (α −1)

∑

(i)

(6.A22)

Rm (ξ )Ω˙ m (0)

m=1 (i)

(i)

(i)

(i)

= v0 (ξ ) − f1 (ξ ) p˙0 (0) − f2 (ξ ) p˙n (0) − f3 (ξ )η˙ (0) (i)

By virtue of the orthogonal property of Rm (ξ ), Eq.(6.A22) can be reformed as

Ωm (0) = I1m p0 (0) + I2m pn (0) + I3m η (0) + I4m , Ω˙ m (0) = I1m p˙0 (0) + I2m p˙n (0) + I3m η˙ (0) + I5m where

(6.A23)

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HuiMing Wang

I4m = I5m =

n

Z ξi

i=1 n

ξi−1 Z ξi

i=1

ξi−1

∑ ρ¯ (i)

∑ ρ¯ (i)

(i)

(i)

(i)

(i)

1

ξ 2 (α +1) u0 (ξ )Rm (ξ )dξ /Jm , 1

(6.A24)

ξ 2 (α +1) v0 (ξ )Rm (ξ )dξ /Jm

Utilizing Eq.(6.A23), Eq.(6.A22) can be rewritten as

Ω1m (τ ) = I1m p0 (τ ) + I2m pn (τ ) + I4m cos ωm τ + −I1m ωm

Z τ 0

p0 (p) sin ωm (τ − p)dp − I2m ωm

Z τ 0

I5m sin ωm τ ωm

pn (p) sin ωm (τ − p)dp

(6.A25)

Now the function Ω1m (τ ) becomes known. Substituting Eq.(6.76) into Eq.(6.32) and integrating it at the intervals [ξi−1 , ξi ] (i = 1, 2 · · · , n), then summarizing the n equations and utilizing Eq.(6.26) and the first equation in Eq.(6.28), we obtain ∞

ψ1 (τ ) = K1 η (τ ) +

∑ K2m Ωm (τ )

(6.A26)

m=1

where

ψ1 (τ ) = φn (τ ) − φ0 (τ ) − K3 p0 (τ ) − K4 pn (τ ), ( (i) Z ) (i) i n ξi f (i) (ξ ) e3 h (i) α e1 1 (i) 3 K1 = ∑ d ξ + f ( ξ ) − f ( ξ ) − [Y ( ξ ) −Y ( ξ )] , i i i−1 i−1 3 3 (i) ξ (i) (i) ξ i−1 ε3 ε3 ε3 i=1 ( (i) Z ¸) (i) · n ξi R(i) (ξ ) e3 α e1 − 12 (α −1) (i) − 12 (α −1) (i) m K2m = ∑ Rm (ξi ) − ξi−1 Rm (ξi−1 ) , dξ + (i) ξi 1 (i) ξ (α +1) i−1 ξ 2 ε3 ε3 i=1 ( (i) Z ) (i) i n ξi f (i) (ξ ) e3 h (i) α e1 (i) 1 f ( ) − f ( K3 = ∑ d ξ + ξ ξ ) , i i−1 1 1 (i) ξ (i) ξ i−1 ε3 ε3 i=1   ln ξ (α = 1) Y (ξ ) = , 1 − (α = 2) ξ ( (i) Z ) (i) i n ξi f (i) (ξ ) e3 h (i) α e1 (i) 2 K4 = ∑ dξ + (i) f2 (ξi ) − f2 (ξi−1 ) (6.A27) (i) ξ ξ i−1 ε ε i=1 3

3

Substituting Eq.(6.A20) into Eq.(6.A26), we obtain

ψ (τ ) = κ1 η (τ ) +

∞

Z τ

m=1

0

∑ κ2m

where

η (p) sin ωm (τ − p)dp

(6.A28)

∞

ψ (τ ) = ψ1 (τ ) − ∞

κ1 = K1 +

∑ K2m Ω1m (τ ),

m=1

∑ K2m I3m ,

(6.A29)

κ2m = −ωm K2m I3m

m=1

Equation (6.A28) is the Volterra integral equation of the second kind.

Chapter 7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids for Electro-acoustic Devices

Zheng-Hua Qian1 and Feng Jin2 1 State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China 2 State Key Laboratory of Mechanical Structural Strength and Vibration, Xi’an Jiaotong University, Xi’an, 710049, China

Abstract Layered solids are extensively-used structures for the design of electro-acoustic wave devices in which surface waves are excited and propagate. Transverse surface waves existing in such structures are attractive to the electro-acoustic device designers due to their unique features like high performance and simple particle motion and so on. One kind of the transverse surface wave is the so-called Love wave which exists only when the bulk-shear-wave velocity in the layer material is less than that in the substrate. Besides the well-known Love wave, another type of the transverse surface wave can exist in such layered structures under some specific condition. This type of transverse surface wave could be regarded as a modified version to the conventional Bleustein-Gulyaev (BG) wave which exists and propagates only at the free surface of a piezoelectric half-space due to the interconnection between elastic and electric fields. It is closely related to a bulk shear wave, and though little reported and used at present it may have certain advantages over the traditional surface wave utilized for electro-acoustic devices. This chapter presents a brief overview on this type of transverse surface wave based on the recent work in our group. Furthermore, the propagation of such modified B-G wave in layered structures with initial stresses and/or material gradient is also introduced. Keywords transverse surface waves, layered structures, piezoelectricity, dispersion characteristics, electro-acoustic devices

228

Zheng-Hua Qian and Feng Jin

7.1 Introduction Electro-acoustic devices, such as oscillators, amplifiers, filters and delayers and so on, are extensively applied in various modern industrial fields, like electronic technology, mechanical engineering and medical appliance, ranging from almost all commodities of our daily life like mobile phones or personal computers to high-tech products like atomic force microscopes. Therefore, it is of practical importance and interest to design electroacoustic devices with high performance and low cost. Usually, these devices consist of a substrate with a thin layer deposited on it. The properties of the covering layer have a great influence on the performance of the electro-acoustic devices. The simplest one is a layered isotropic elastic structure. The more efficient one is a layered piezoelectric structure, i.e., an elastic substrate coated with a piezoelectric layer or a piezoelectric substrate with an elastic layer. Transverse surface waves existing in such layered structures are attractive to the electro-acoustic device designers due to their unique features like high performance and simple particle motion. One kind of the transverse surface wave is the so-called Love wave which exists only when the bulk-shear-wave velocity in the layer material is less than that in the substrate. In the past three decades, a lot of work was done on the Love wave propagation in the above two types of layered structure, which will not be overviewed in this chapter. For those who are interested in the basic properties of Love waves and their utilization in sensor devices, please refer to the review work in Ref. [1] done by Jakoby and Vellekoop. Some recent work on the propagation and dispersion behavior of Love waves in such piezoelectric layered structures can be found in Refs. [2-7]. Besides the well-known Love wave, another type of the transverse surface wave can also exist in such layered structures under some specific condition. This type of transverse surface wave could be regarded as a modified version to the conventional Bleustein-Gulyaev (B-G) wave which exists and propagates only at the free surface of a piezoelectric half-space due to the interconnection between elastic and electric fields [8,9]. It is closely related to a bulk shear wave, and though little reported and used at present may have certain advantages over the traditional surface wave utilized for electro-acoustic devices. On this aspect, Curtis et al. [10] first analyzed the transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness. It was discovered in their work that the piezoelectricity of the substrate allows a non-leaky but dispersive wave to exist under conditions in which no Love wave is possible, namely, when the bulk-shear-wave velocity in the layer is greater than that in the substrate. After many-year silence, Wang et al. [2] studied the well-known transverse surface wave, i.e., Love wave, on a piezoelectric coupled solid which consists of a metal substrate carrying a piezoelectric layer. Recently, Qian et al. [11] made a revisit to the work in Ref. [2] and found out that not only could the Love wave exist in such piezoelectric coupled solid media but also another type of transverse surface wave (hereafter called modified B-G wave) could exist under the similar physical condition to that in Ref. [1]. However, there is another dispersive mode of the modified B-G wave in the case of electrically short circuit, as shown in the Ref. [11]. In order to remove the undesired dispersive mode, a hard metal interlayer was introduced in our recent work [12],

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

229

which illustrated that the presence of the hard metal interlayer could not only get rid of the undesired mode appearing in the case without interlayer but also shorten the existence range of phase velocity within which a non-leaky but dispersive mode exists. A detailed introduction on the modified B-G wave will be given in Section 7.2 of this chapter. For the layered structures, however, due to the non-uniform material properties, coefficients of thermal expansion and chemical/nucleation shrinkage/growth during the processing and cooling down to operating or room temperature, there exist initial stresses in the layered structures. Besides, to prevent the piezoelectric material from brittle fracture, the layered piezoelectric structures are usually pre-stressed during the manufacture process, where the initial stress is negative with the magnitude of 100 MPa, even up to 1000 MPa [13]. Two major causes of thin film stress are intrinsic stress and thermal stress. Excessive initial stress in the layered structures can lead to delamination, microcracking, debonding and degradation of the layer, it can also lead to dramatically change of dispersion relation corresponding to the wave propagation in the above-mentioned structures. During the past one decade, many researchers studied the effects of initial stresses on the propagation of Love wave [14-18] and B-G wave [19,20], which will not be stated detailedly in this chapter. In Section 7.3, we present our recent work [21] on the modified B-G wave propagation in a piezoelectric material carrying a prestressed metal layer of finite-thickness. For simplicity, only constant initial stresses are considered in the current work. Those who are interested in conducting the work on inhomogeneous initial stresses can follow the similar procedure on Love wave propagation introduced in our previous work [16,17]. Functionally graded materials have been known to have good performances in many applications. Much work has been done on the research like thermomechanical response and fracture behavior of functionally graded materials since its appearance [22]. With the development of the material technology, the functionally graded piezoelectric materials (FGPMs) can be manufactured and used to be substrates in electro-acoustic devices to improve their efficiency and natural life, which could solve not only the residual stress problem in layered structures but also the penetration depth problem in pure piezoelectric substrate. Hence, the research of transverse surface wave propagation behaviors and characteristics in FGPMs has become a topic of practical interest and importance. Recently, many researchers investigated the effects of material gradient on the Love wave propagation in the above-mentioned layered structures [23-33], which will not be overviewed in this chapter. In Section 7.4, we will present the modified B-G wave propagation in functionally graded piezoelectric layered structures and the influence of material gradient on its dispersion characteristics [34-37]. In general, this chapter is organized as follows: a modified B-G wave in a layered structure with a piezoelectric layer and a metal/dielectric substrate is first presented in the first subsection of Section 7.2, followed by the introduce of a hard metal interlayer to the layered structure in order to remove the undesired mode in the case of electrically short circuit in the second subsection of Section 7.2. Then the effects of initial stresses, inevitably coming with such layered structures, on the modified B-G wave propagation are introduced in Section 7.3. After that the material gradient in the substrate or covering layer

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of such layered structures is introduced to show its effects on the dispersion characteristics of the modified B-G wave in Section 7.4. Finally some conclusions are summarized for future reference in Section 7.5.

7.2 Transverse surface waves in piezoelectric layered solids 7.2.1 Piezoelectric layer and metal/dielectric substrate Consider a half-space occupying x1 > 0 with a piezoelectric ceramic layer of uniform thickness h (see Fig.7.1). Here the piezoelectric material, taken to be of class 6mm (or ∞ m), is poled along the x3 direction of Cartesian coordinates (x1 , x2 , x3 ). There is an ideal electrode at the interface that is grounded. On the electrode, the electric potential has to

Fig. 7.1 A metal/dielectric half-space covered by a piezoelectric layer of uniform thickness.

vanish. The half-space is of either a metal or an isotropic (nonpiezoelectric) dielectric. It is assumed that the waves propagate in the positive direction of the x2 -axis, such that the movement denoted by mechanical displacement u1 , u2 , u3 and electric potential ϕ can be written as u1 ≡ u2 ≡ 0, u3 = u3 (x1 , x2 ,t), (7.1) ϕ = ϕ (x1 , x2 ,t) Let u and ϕ denote separately the mechanical displacement and electric potential function of the piezoelectric layer. Following Bleustein [8], the governing equations are ∇2 u − (1/c2p )u¨ = 0, ∇2 [ϕ − (e15 /ε11 )u] = 0

(7.2)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 , and cp = [(c44 + e215 /ε11 )/ρ ]1/2 is the bulk-shear-wave velocity in the piezoelectric material, with c44 , e15 , ε11 and ρ representing the elastic, piezoelectric, dielectric constants and mass density, respectively. For the metal substrate, let w0 denote its mechanical displacement in the z direction. The governing field equation is [10] ∇2 w0 − (1/c2m )w¨ 0 = 0

(7.3)

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where cm = (c044 /ρ 0 )1/2 is the bulk-shear-wave velocity in the metal material, with c044 and ρ 0 representing separately the shear modulus and mass density. The wave propagation problem specified by Eqs.(7.1) and (7.2) should satisfy the fol2 w = w0 , zx = 0zx , 1 zx = 0 at x = −h. ° lowing boundary and continuity conditions: ° 3 w0 → 0 as x → +∞. The electrical conditions at the free surface can ϕ = 0 at x = 0. ° 4 electrically open circuit: Dx = 0 at x = −h and be classified into two categories, i.e., ° 5 electrically short circuit (or metalized surface): ϕ = 0 at x = −h, based on that the ° space above the piezoelectric layer is vacuum or air and its permittivity is much less than that of the piezoelectric material. Built upon an earlier work [2, 10], we consider the following transverse waves satisfy3 ing radiation condition °:  u(x1 , x2 ,t) = (A1 e−bkx1 + A2 ebkx1 ) exp[ik(x2 − ct)]       −kx kx A3 e 1 + A4 e 1 + (−h 6 x1 6 0) (7.4)  exp[ik(x2 − ct)]  ϕ (x1 , x2 ,t) =  e15  −bkx bkx  1 1 (A1 e + A2 e )  ε11 0

u0 = A5 e−b kx1 exp[ik(x2 − ct)] (x1 > 0)

(7.5)

where A1 , A2 , A3 , A4√and A5 are arbitrary constants, k(= 2π/λ ) is the wavenumber, λ is the wavelength, i = −1, and c is the phase velocity. Equations (7.4) and (7.5) satisfy separately Eqs. (7.2) and (7.3) if q b2 = 1 − c2 /c2p , b0 = 1 − c2 /c2m (7.6) The stress component and electric displacement needed for the boundary and continuity conditions are # "  c¯44 bk(−A1 e−bkx1+A2 ebkx1 )  exp[ik(x2 − ct)]  31 (x1 , x2 ,t)= (−h 6 x1 6 0) (7.7) +e15 k(−A3 e−kx1 + A4 ekx1 )   −kx kx D1 (x1 , x2 ,t) = −ε11 k(−A3 e 1 + A4 e 1 ) exp[ik(x2 − ct)] 0 (x , x ,t) = −c0 kb0 A e−kb0 x1 exp[ik(x − ct)] 2 5 31 1 2 44

(x1 > 0)

(7.8)

Substitution of Eqs.(7.4), (7.5) and the corresponding mechanical and electrical com1 ° 2 and ° 4 or ° 5 yields ponents into the remaining boundary and continuity conditions °, the following homogeneous linear algebraic equations for coefficients A1 , A2 , A3 , A4 and A5 : A1 + A2 = A5 , A3 + A4 + (A1 + A2 )e15 /ε11 = 0, c¯44 b(A2 − A1 ) + e15 (A4 − A3 ) = −c044 b0 A5 , (7.9) c¯44 b(A2 e−khb − A1 ekhb ) + e15 (A4 e−kh − A3 ekh ) = 0, −A3 ekh + A4 e−kh = 0 The existence condition of nontrivial solutions of these coefficients leads to the following dispersion relations of the transverse surface waves described by Eqs.(7.4) and (7.5)

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for the electrically open case: kp2 tanh(kh) − b tanh(bkh) − c044 b0 /c¯44 = 0

(7.10)

where kp2 = e215 /ε11 c¯44 is the piezoelectric coupling factor in the piezoelectric layer with c¯44 = c44 + e215 /ε11 being the piezoelectrically stiffened elastic constant [10]. For the electrically shorted case (the free surface of the piezoelectric layer is plated with an infinite thin metal strip), the last equation in Eq.(7.9) should be replaced by the 5 following expression which corresponds to the condition °: A3 ekh + A4 e−kh + (A1 ekhb + A2 e−khb )e15 /ε11 = 0

(7.11)

Then, a set of homogeneous linear algebraic equations with respect to A1 , A2 , A3 , A4 , and A5 can be obtained. By the similar procedure as that in the electrically open case, we can obtain the corresponding phase velocity equation for the electrically shorted case: c0 (kp4 + b2 ) tanh(kh) tanh(bkh) − 44 kp2 b0 tanh(bkh) c¯44 · ¸ 0 c 1 + 44 bb0 tanh(kh) + 2kp2 b −1 = 0 c¯44 cosh(kh) cosh(bkh)

(7.12)

It is readily seen from Eq.(7.10) and Eq.(7.12) that the phase velocity c is related to the wavenumber, layer thickness, elastic, dielectric and piezoelectric constants. With Eq.(7.6), Eqs.(7.10) and (7.12) determine c versus k or ω versus k. For convenience, we introduce the dimensionless wavenumber H = h/λ . Using Eq.(7.6), we can then rewrite Eqs.(7.10) and (7.12) separately as kp2 tanh(2πH) − b tanh(b2πH) − c044 b0 /c¯44 = 0

(7.13a)

c044 2 0 k b tanh(b2πH) c¯44 p

(kp4 + b2 ) tanh(2πH) tanh(b2πH) − · ¸ c044 0 1 2 + bb tanh(2πH) + 2kp b −1 = 0 c¯44 cosh(2πH) cosh(b2πH)

(7.13b)

We make the following observations from Eqs.(7.13): It can be seen from Eq.(7.6) that the parameter b in Eqs.(7.13) can not only take real values but also imaginary values, depending on whether the surface wave velocity c is smaller or greater than the bulkshear-wave velocity in the piezoelectric layer, cp . 7.2.1.1 Love-type wave When the bulk-shear-wave velocity in the piezoelectric layer is less than that in the substrate, i.e., cp < cm , multivalued roots for the phase velocity c versus the dimensionless wavenumber H are expected from Eqs.(7.13) if only if the parameter b is imaginary, deq 2 2 noted by b = ib1 = i c /cp − 1. The dispersion equations (7.13) can thus be rewritten

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as kp2 tanh(2πH) + b1 tan(b1 2πH) − c044 b0 /c¯44 = 0 c0 (kp4 − b21 ) tanh(2πH) tan(b1 2πH) − 44 kp2 b0 tan(b1 2πH) c¯44 · ¸ c044 1 0 2 + b1 b tanh(2πH) + 2kp b1 −1 = 0 c¯44 cosh(2πH) cos(b1 2πH)

(7.14a)

(7.14b)

in which Eq.(7.14b) corresponds to the result in Ref. [2]. The dispersion relation given in Ref. [2] is in determinantal form which is inconvenient to use in practice, so the above dispersion relation (7.14b) still appears new, from which continuous dispersion curves rather than some discrete points can be easily calculated. A comparison between our result and the result in Ref. [2] can be found in our recent work [11], where very nice agreement is observed. If we further set kp = 0, i.e., no piezoelectricity in the layer, the two equations in Eqs.(7.14) reduce to the same frequency equation for Love waves in an elastic half-space carrying an elastic layer [38]. 7.2.1.2 B-G-type wave q When the parameter b is real, denoted by b = b2 = 1 − c2 /c2p , roots for the phase velocity c versus the dimensionless wavenumber H are only expected when the bulk-shearwave velocity in the piezoelectric layer cp and that in the substrate cm satisfy cp > cm > cBG . Here, cBG = cp (1 − kp4 )1/2 is the phase velocity of the B-G wave in the same piezoelectric material as the layer coated with an infinitely thin layer of conducting material. The dispersion equations (7.13) can thus be rewritten as kp2 tanh(2πH) − b2 tanh(b2 2πH) − c044 b0 /c¯44 = 0

(7.15a)

c044 2 0 k b tanh(b2 2πH) c¯44 p

(kp4 + b22 ) tanh(2πH) tanh(b2 2πH) − · ¸ c0 1 + 44 b2 b0 tanh(2πH) + 2kp2 b2 −1 = 0 c¯44 cosh(2πH) cosh(b2 2πH)

(7.15b)

from which the roots obtained are neither single-valued nor multivalued. Under electrically open circuit condition for which Eq. (7.15a) is responsible, only one mode of wave propagation is expected. While under electrically short circuit condition for which Eq.(7.15b) is responsible, one additional mode appears besides the mode corresponding to that in the case of electrically open circuit. This “new created” mode seems due to the appearance of electrode on the free surface of the piezoelectric layer. If we further set kp = 0, i.e., no piezoelectricity in the layer, no roots exist from Eq.(7.15), which means that the modes of wave propagation under this situation do not exist in the absence of piezoelectricity. The phenomenon is illustrated by the numerical example shown in Fig.7.2 which refers to a PZT-4 deposited on a zinc substrate. The material parameters for both the met-

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als and the piezoelectric material are taken from Refs. [2, 10]. In order to make a deep insight into the dispersion characteristics of these modes, the plots of group velocity are included correspondingly in Fig.7.2, denoted by thinner lines. In the case of electrically open circuit, the former part of the mode when H < 0.1901 corresponds to normal dispersion [38], while the latter part when H > 0.1901 anomalous dispersion [38]. On the other hand, in the case of electrically short circuit, it can be seen from the group velocity plots in Fig.7.2 that the first mode is a combination of partly normal dispersion [38] and partly anomalous dispersion [38] in which the phase velocity first decreases from the substrate velocity cm (2440 m/s) to a minimum value at H = 0.3901, then increases monotonically to the B-G wave velocity (2412 m/s), whilst the second mode is totally normal dispersion in which the phase velocity decreases monotonically from the substrate velocity cm (2440 m/s) to the same final value of 2412 m/s as that in the case of electrically open circuit.

Fig. 7.2 Phase velocity c and group velocity cg of the B-G-type wave in a zinc substrate carrying a PZT-4 layer plotted as a function of H = h/λ for electrically open and short cases. cm = 2440 m/s, cp = 2597 m/s, cBG = 2258 m/s.

This B-G-type wave is the transverse surface wave that we focus on in this chapter. However, there is another dispersive mode in the case of electrically short circuit which may not be good to practical design of electro-acoustic devices. In the next subsection, we talk about how to remove this undesired mode by introducing a hard metal interlayer between the covering layer and the substrate.

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7.2.2 Piezoelectric layered solids with a hard metal interlayer Consider a piezoelectric layer/metal substrate system with a hard metal interlayer of finitethickness h1 , occupying the half-space x > −h0 , as shown in Fig.7.3. Suppose the piezoelectric layer of uniform thickness h0 is deposited perfectly on the hard metal interlayer, which results in a surface at x = −h0 free of external forces. Here the piezoelectric material is taken to be of class 6 mm (or ∞ m), with its polar axis oriented along the z direction of Cartesian coordinates (x, y, z). It is assumed that the waves propagate in the positive direction of the y-axis, such that the nonzero field quantities representing the motion are only functions of the coordinates (x, y) and time t.

Fig. 7.3 A piezoelectric layered structure with one hard metal interlayer.

Let w and ϕ denote separately the mechanical displacement and electrical potential function in the piezoelectric layer. Similarly, the coupled field equations are given by ∇2 w − (1/c2p )w¨ = 0, ∇2 [ϕ − (e15 /ε11 )w] = 0

(7.16)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 , and cp = [(c44 + e215 /ε11 )/ρ ]1/2 is the bulk-shear-wave velocity in the piezoelectric material, with c44 , e15 , ε11 and ρ representing the elastic, piezoelectric, dielectric constants and mass density, respectively. For the hard metal interlayer, let w1 denotes its mechanical displacement in the z direction. The governing field equation is [10] ∇2 w1 − (1/c2m1 )w¨ 1 = 0

(7.17)

where cm1 = (µ1 /ρ1 )1/2 is the bulk-shear-wave velocity in the interlayer, with µ1 and ρ1 representing separately the shear modulus and mass density. Similarly, for the metal substrate, let w2 denotes its mechanical displacement in the z direction. The governing field equation is [10] ∇2 w2 − (1/c2m2 )w¨ 2 = 0

(7.18)

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where cm2 = (µ2 /ρ2 )1/2 is the bulk-shear-wave velocity in the metal substrate, with µ2 and ρ2 representing separately the shear modulus and mass density. The wave propagation problem specified by Eqs.(7.16)-(7.18) should satisfy the fol1 zx = 0 at x = −h0 . ° 2 w = w1 , zx = 1zx , lowing boundary and continuity conditions: ° 4 w2 → 0 as x → +∞. The electrical 3 w1 = w2 , 1zx = 2zx , at x = h1 . ° ϕ = 0 at x = 0. ° 5 electrically conditions at the free surface can be classified into two categories, i.e., ° 6 electrically short circuit (or metalized surface): open circuit: Dx = 0 at x = −h0 and ° ϕ = 0 at x = −h0 , based on the fact that the space above the piezoelectric layer is vacuum or air and its permittivity is much less than that of the piezoelectric material. Built upon a previous work [11], we consider the following transverse waves satisfying 4 attenuation condition °:  w(x, y,t) = (A1 e−bkx + A2 ebkx ) exp[ik(y − ct)]       −kx kx A3 e + A4 e + (−h0 6 x 6 0) (7.19)  exp[ik(y − ct)]   ϕ (x, y,t) =  e15   (A1 e−bkx + A2 ebkx ) ε11 w1 = (A5 e−b1 kx + A6 eb1 kx ) exp[ik(y − ct)] (0 6 x 6 h1 ) (7.20) w2 = A7 e−b2 kx exp[ik(y − ct)] (x > h1 )

(7.21)

where A1 , A2 , A3 , A4 , A5 , A6 and √ A7 are arbitrary constants, k(= 2π/λ ) is the wavenumber, λ is the wavelength, i = −1, and c is the phase velocity. Equations (7.19)-(7.21) satisfy separately Eqs.(7.16)-(7.18) if b2 = 1 − c2 /c2p ,

b21 = 1 − c2 /c2m1 ,

b22 = 1 − c2 /c2m2

(7.22)

Substitution of Eqs.(7.19)-(7.21) and the corresponding stress components into the 1 °, 2 ° 3 and ° 5 or ° 6 yields seven linear, remaining boundary and continuity conditions °, homogeneous algebraic equations for coefficients A1 through A7 . The existence condition of nontrivial solutions of these coefficients leads to the following dispersion relations of the transverse surface waves described by Eqs.(7.19)-(7.21): kp2 tanh(2πH) − b tanh(2πHb) −

µ1 b1 µ1 b1 tanh(2πHb1 hr ) + µ2 b2 =0 c¯44 µ1 b1 + µ2 b2 tanh(2πHb1 hr )

(7.23)

for the case of electrically open circuit, and · (kp4 + b2 ) tanh(2πH) tanh(2πHb) + 2kp2 b +

1 −1 cosh(2πH) cosh(2πHb)

¸

µ1 b1 µ1 b1 tanh(2πHb1 hr ) + µ2 b2 [b tanh(2πH) − kp2 tanh(2πHb)] = 0 (7.24) c¯44 µ1 b1 + µ2 b2 tanh(2πHb1 hr )

for the case of electrically short circuit, respectively.

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237

In Eqs.(7.23) and (7.24), H = h0 /λ , hr = h1 /h0 , and kp2 = e215 /ε11 c¯44 is the piezoelectric coupling factor in the piezoelectric material with c¯44 = c44 + e215 /ε11 being the piezoelectrically stiffened elastic constant [10, 11]. One observation can be made that Eqs.(7.23) and (7.24) degenerate exactly to the results in the previous subsection Eqs.(7.13a) and (7.13b) when the interlayer does not exist, i.e., hr = 0. This serves to demonstrate the validity of the present mathematic formulation to some extent. And for the case that there are two interlayers between the piezoelectric layer and the substrate, as shown in Fig.7.4, the dispersion relations can be obtained readily by replacing µ2 b2 in Eqs.(7.23) and (7.24) with the expression defined as

µ2 b 2

µ2 b2 tanh(b2 kh2 ) + µ3 b3 µ2 b2 + µ3 b3 tanh(b2 kh2 )

(7.25)

where µ2 , b2 and h2 denote the corresponding quantities of the second interlayer, and µ3 , b3 denote those of the substrate. The detailed process to obtain Eq.(7.25) is omitted here for brevity. It is thus concluded that the dispersion relations for multiple- interlayer case can be deduced by analog. For simplicity, we only discuss the case when one interlayer exists in the following numerical examples.

Fig. 7.4 A piezoelectric layered structure with two hard metal interlayers.

Based on a previous work [11], two physical situations are possible for transverse surface waves in a piezoelectric layer/metal substrate system without the interlayer, i.e., 1 Type 1: cm2 > cp > cBG , b real or imaginary; ° 2 Type 2: cp > cm2 > cBG , b hr = 0: ° always real. Here, cBG = cp (1−kp4 )1/2 is the phase velocity of the B-G wave on the surface of a piezoelectric substrate coated with an infinitely thin layer of conducting material. The Type 1 wave is a kind of Love wave, which has been well known. In this subsection, we thus focus on the effect of the appearance of a hard metal interlayer on the Type 2 wave, i.e., the wave introduced in previous subsection. The dispersion curves plotted for selected values of thickness ratio hr are illustrated in Figs.7.5-7.7, corresponding to a PZT-4/steel/zinc system for the cases of both electrically open circuit and electrically short circuit, respectively. In order to clearly show the effect

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of the hard metal interlayer on the second mode in the case of electrically short circuit, we individually plot in Fig.7.7 the second mode for selected values of thickness ratio hr different from those in Figs.7.5 and 7.6. The material parameters for both the metal and the piezoelectric material are taken from Refs. [2,8]. For comparison, in each plot, the dispersion curves in the case without interlayer (i.e., hr = 0) are included. It is readily seen from these three plots that the appearance of the hard metal interlayer has a significant effect on the dispersion curves, regardless of the electrically boundary conditions.

Fig. 7.5 Phase velocity c of the transverse surface waves in a PZT-4/steel/zinc system as a function of H = h/λ for selected values of thickness ratio hr in electrically open case. cm1 = 3281 m/s, cm2 = 2440 m/s, cp = 2597 m/s, cBG = 2258 m/s.

Fig. 7.6 Phase velocity c of the transverse surface waves in a PZT-4/steel/zinc system as a function of H = h/λ for selected values of thickness ratio hr for the first mode in electrically short case. cm1 = 3281 m/s, cm2 = 2440 m/s, cp =2597 m/s, cBG = 2258 m/s.

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Fig. 7.7 Phase velocity c of the transverse surface waves in a PZT-4/steel/zinc system as a function of H = h/λ for selected values of thickness ratio hr for the second mode in electrically short case. cm1 = 3281 m/s, cm2 = 2440 m/s, cp = 2597 m/s, cBG = 2258 m/s.

In the case of electrically open circuit, it can be seen from Fig.7.5 that the main changes induced by the hard metal interlayer consist in the minimum velocity value and the wavenumber range for the existence of the mode. For the case without the hard metal interlayer (i.e., hr = 0), the phase velocity tends to the B-G wave velocity under electrically open circuit as the dimensionless wavenumber approaches infinity, which means that the mode exists for all the wavenumber values. With the increase in the thickness ratio hr , the wavenumber range for the existence of the mode is gradually shortened to a limited range with both its starting and end values corresponding to the bulk-shear-wave velocity in the metal substrate. However, the wavenumber corresponding to the minimum velocity almost does not change at all with the increase in the thickness ratio, which is an interesting phenomenon and appears new. That feature has the potential application to increase the operating frequency of acoustic devices at some circumstance. It will be noted that the total change in phase velocity produced in the case of electrically open circuit is relatively small, compared with that in the case of electrically short circuit. And the appearance of the hard metal interlayer makes the difference more obvious. In the case of electrically short circuit, it can be seen from Figs.7.6 and 7.7 that the thickness ratio affects the two modes in different ways. For the first mode, the main influence induced by the appearance of the hard metal interlayer is the dispersive type, i.e., the mode is changed gradually from partly normal dispersion [38] plus partly anomalous dispersion [38] to totally normal dispersion, as shown in Fig.7.6. When the thickness ratio hr > 0.2, the local minimum phase velocity in the low frequency range of the first mode disappears, i.e., the mode starts from the bulk-shear-wave velocity cm (2440 m/s) of the metal substrate and decreases monotonically to the B-G wave velocity (2260 m/s) under electrically short circuit. For the second mode which does not have a counterpart in the case of electrically open circuit, however, it is clearly seen from Fig.7.7 that even

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very small thickness ratio affects the mode significantly. The wavenumber range for the existence of the second mode is quickly shortened by the appearance of the hard metal interlayer. This feature can be used to remove the second mode (undesired in practical applications) in the case of electrically short circuit by using a very thin hard metal interlayer, for example, one can set the thickness ratio hr = 0.03.

7.3 Transverse surface waves in prestressed piezoelectric layered solids Consider a piezoelectric material, here taken to be of class 6 mm (or ∞ m), occupying halfspace x1 > 0, with its polar axis oriented along the x3 direction of Cartesian coordinates (x1 , x2 , x3 ), as shown in Fig.7.8. Let a metal layer of uniform thickness h be deposited perfectly on the piezoelectric substrate, with its surface at x1 = −h free of external forces. It is assumed that initial stresses due to processing are present in the layered structure. The substrate is in general much thicker than the coating layer, resulting in a layered piezoelectric half-space problem with negligible initial stresses in the substrate. For wave motion of small amplitude, the elastic field equation with initial stresses reads as [15, 19] 0 i j, j + (ui,k k j ), j

= ρ u¨ i

(7.26)

where ρ is the mass density, ui is the displacement vector, i j is the stress tensor, 0k j is the initial stress tensor, and no presence of body forces is assumed. A comma followed by subscript i denotes differentiation with respect to xi , whereas a dot atop of a symbol represents differentiation with respect to time. Throughout this section, Einstein’s summation convention is adopted for all repeated Latin indices (1, 2, 3).

Fig. 7.8 A piezoelectric half-space covered by a prestressed metal/dielectric layer of uniform thickness.

It is assumed that the waves propagate in the positive direction of the y-axis, such that the nonzero field quantities representing the motion are only functions of the coordinates (x1 , x2 ) and time t. Only one initial stress component (either positive or negative), denoted here by 0 , exists in the metal layer along the propagating direction [15,19]. Let w0 denote the mechanical displacement of the thin metal layer. The governing field equation is thence

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

∇2 w0 + (

0

/c044 )w0,yy − (1/c21 )w¨ 0 = 0

241

(7.27)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 , and c1 = (c044 /ρ 0 )1/2 is the bulk-shear-wave velocity in the metal layer, with c044 and ρ 0 representing separately the shear modulus and mass density of the metal material. Let w and ϕ denote the mechanical displacement and electrical potential function of the piezoelectric substrate. Following Bleustein [8], the coupled field equations are ∇2 w − (1/c22 )w¨ = 0,

(7.28)

∇2 [ϕ − (e15 /ε11 )w] = 0

where c2 = [(c44 + e215 /ε11 )/ρ ]1/2 is the bulk-shear-wave velocity in the substrate, with c44 , e15 , ε11 and ρ representing the elastic, piezoelectric, dielectric constants and mass density of the piezoelectric material, respectively. The wave propagation problem specified by Eqs.(7.27) and (7.28) should satisfy the 1 0zx = 0 at x = −h. ° 2 w0 = w, 0zx = zx , following boundary and continuity conditions: ° 3 w and ϕ → 0 as x → +∞. ϕ = 0 at x = 0. ° Built upon an earlier work [10], we consider the following transverse waves satisfying 3 attenuation condition °: ) w(x, y,t) = A1 e−bkx exp[ik(y − ct)] (x > 0) (7.29) ϕ (x, y,t) = [A2 e−kx + (e15 /ε11 )A1 e−bkx ] exp[ik(y − ct)] 0

0

w0 = [A3 e−b kx + A4 eb kx ] exp[ik(y − ct)] (−h 6 x 6 0)

(7.30)

where A1 , A2 , A3 and A4 are arbitrary √ constants yet to be determined, k(= 2π/λ ) is the wavenumber, λ is the wavelength, i = −1, and c is the phase velocity. Equations (7.29) and (7.30) satisfy separately Eqs.(7.27) and (7.28) if b = (1 − c2 /c22 )1/2 ,

b0 = (1 − c2 /c21 +

0

/c044 )1/2

(7.31)

Substitution of Eqs.(7.29), (7.30) and the corresponding stress components into the re1 and ° 2 yields four linear, homogeneous maining boundary and continuity conditions ° algebraic equations for coefficients A1 , A2 , A3 and A4 . The existence condition of nontrivial solutions of these coefficients leads to the following dispersion relation for transverse surface waves described by Eqs.(7.27) and (7.28): 0

0

c¯44 b + c044 b0 (e4πHb − 1)/(e4πHb + 1) − e215 /ε11 = 0

(7.32)

where H = h/λ is the ratio of layer thickness to wavelength, and c¯44 = c44 + e215 /ε11 is the piezoelectrically stiffened elastic constant of the substrate material [10]. When 0 = 0, i.e., no initial stress exists in the metal layer, the problem reduces to that solved by Curtis and Redwood [10]. This limiting case is used to check the validity of the present predictions on the influence of initial stress on the dispersion behavior of transverse surface

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waves. The present problem can be classified into three physical situations for which dispersion curves of different types may be found [10]: Type 1: c1 > c2 > cBG , b0 always real Type 2: c2 > cBG > c1 , b0 always imaginary Type 3: c2 > c1 > cBG , b0 real or imaginary where cBG is the velocity of the B-G wave in a piezoelectric material with shorted electrode on its surface. In this section, we just discuss the effects of initial stress on the dispersion behavior of the Type 1 wave. For those who are interested in the effects of initial stress on the other two types of wave, please refer to our recent work [21]. Using Eq.(7.31), we can rewrite Eq.(7.32) as s s s à ! 0 0 e2 c2 c2 c2 0 c¯44 1 − 2 + c44 1 − 2 + 0 tanh 2πH 1 − 2 + 0 − 15 = 0 (7.33) ε11 c2 c1 c44 c1 c44 for real b0 , and s c¯44

c2 1 − 2 − c044 c2

s

s à ! 0 0 e215 c2 c2 − 1 − tan 2πH − 1 − − =0 0 0 c44 c44 ε11 c21 c21

(7.34)

for imaginary b0 , respectively. Type 1 dispersion curves shown in Fig.7.9 for different values of 0 are obtained by solving Eq.(7.33) for a PZT-4/aluminum layered structure. For comparison, the dispersion curve in the limit when no initial stress is present (i.e., 0 = 0) is included too, and excellent agreement with the predictions of Curtis and Redwood [10] is observed. The Type 1 situation permits only one mode of wave propagation, corresponding to a B-G wave modified by a thin aluminum layer that does not exist in the absence of piezoelectricity, the initial stress 0 being present or not. The presence of 0 could increase or reduce the phase velocity significantly when its magnitude is greater than 1 GPa, although the starting value of the mode at h/λ = 0 is not affected (Fig.7.9). Type 1 wave does not always exist as a nonleaky wave for all values of h/λ , as shown in Fig.7.9 for the case of 0 = 13 GPa. For instance, its velocity reaches the bulk-shear-wave velocity c2 (2597 m/s) of the substrate when h/λ ≈ 0.48, at which the mode breaks off, becoming leaky. The general criterion for the existence of phase velocity c for all values of h/λ is [10] µ ± ¶ µ 2 ± ¶2 0 e15 ε11 c22 e215 ε11 2 c22 6 1− 2 + 0 6 (7.35) − 2 c¯44 c044 c1 c1 c44 Here, we only present the results in the structure which consists of a piezoelectric substrate and a metal/dielectric covering layer. As for the structure composed of a metal/dielectric substrate and a piezoelectric covering layer, readers can readily get the similar discussion by following the above-mentioned procedure.

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

243

Fig. 7.9 Phase velocity c of Type 1 transverse surface waves in a PZT-4 substrate carrying an aluminum layer plotted as a function of H = h/λ for selected values of initial stress. c1 = 3251 m/s, c2 = 2597 m/s, cBG = 2258 m/s, and c(H = ∞) = 2568 m/s.

7.4 Transverse surface waves in graded piezoelectric layered solids 7.4.1 Functionally graded covering layer Consider a layered structure, as shown in Fig.7.10, involving a graded metal/dielectric layer with uniform thickness of h bonded perfectly to a transversely isotropic piezoelectric substrate (here taken to be of class 6 mm or ∞ m) occupying the half-space x > 0, with its polar axis in the z direction of a set of rectangular Cartesian coordinates. The properties in the metal layer change gradually along the x-axis direction. The upper surface of the metal layer is traction free. Without loss of generality, the structure can be treated as a layered half-space problem since the thickness of the substrate is usually much greater than that of the layer. It is assumed without loss of generality that the waves propagate in the positive direction of y-axis, such that the nonzero mechanical displacement component

Fig. 7.10 A piezoelectric half-space covered by a graded metal/dielectric layer of uniform thickness.

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and the electrical potential function representing the motion are only functions of x, y and t. We can thus treat wave propagation in this structure as a two-dimensional plain strain problem. The main objective is to obtain an analytical solution for the waves in this structure. For the gradient metal/dielectric layer occupying the region −h < x < 0, let w1 denote the mechanical displacement and assume that the shear modulus µ and the mass density ρ depend on the x-coordinate, i.e., µ (x) = µ0 f (x) and ρ (x) = ρ0 g(x), where f (x) and g(x) are arbitrary functions of x, and µ0 = µ (0) and ρ0 = ρ (0). Then from the basic equilibrium equation and the constitutive equation in elasticity, we derive [32] f ∇2 w1 + f 0 ∂ w1 /∂ x = gw¨ 1 /c20

(7.36)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 , f 0 denotes the first order differentiation with respect to x, and c0 = (µ0 /ρ0 )1/2 is the bulk-shear-wave velocity in the layer at x = 0. For the piezoelectric substrate occupying x > 0, let w2 and ϕ2 denote the mechanical displacement and electrical potential function, then we have the following coupled electro-mechanical field equations [28]: c44 ∇2 w2 + e15 ∇2 ϕ2 = ρ2 w¨ 2 , e15 ∇2 w2 − ε11 ∇2 ϕ2 = 0

(7.37)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 and c44 , e15 and ε11 are the elastic, piezoelectric and dielectric constants in the piezoelectric substrate, and ρ2 the mass density, respectively. The actual motion must satisfy the following boundary conditions, omitting those which are trivially satisfied. (1) The mechanical traction-free condition at x = −h, (1) zx (−h, y) = 0

(7.38)

(2) The continuity conditions at x = 0, (1) zx (0, y) =

(2) zx (0, y),

w1 (0, y) = w2 (0, y), ϕ2 (0, y) = 0

(7.39)

(3) The attenuation conditions at x → +∞, w2 → 0,

ϕ2 → 0

(7.40)

Then the propagation problem of the transverse surface waves in the half-space studied here becomes the solution of Eqs.(7.36) and (7.37) under conditions (7.38)-(7.40). For the gradient metal layer occupying the region −h < x < 0, it is very difficulty to obtain the analytical solution of Eq.(7.36). We thus assume the following formal solution and get the analytically asymptotic solution by WKB (Wentzel-Kramers- Brillouin) technique [32]:

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

w1 (x, y,t) = f −1/2 ξ (x) exp[ik(y − ct)]

245

(7.41)

where f is the arbitrary function f (x) describing the √ gradient shear modulus, k(k = 2π/λ ) is the wavenumber with λ being the wavelength, i = −1, c is the phase velocity and ξ (x) is the unknown function. Substitution of Eq.(7.41) into Eq.(7.36) yields

ξ 00 + k2 q2 ξ = 0

(7.42)

where 00 denotes the second order differentiation with respect to the coordinate x and q q = c2 g/(c20 f ) − 1 − [2 f 00 − f −1 ( f 0 )2 ]/(4k2 f ) Usually, for high-frequency short waves, i.e., k À 1, the WKB solution of Eq.(7.42) can be obtained [39]. The detailed solution process of such problem can be found in our previous work [32]. Hence, the solutions of the problem as a plane harmonic wave satisfying the attenuation condition (7.40) are listed directly here: w1 (x, y,t) = [A1 P(x) + A2 P∗ (x)] exp[ik(y − ct)] (7.43) µ ¶ Z √ where A1 and A2 are unknown constants, P(x) = exp −ik qdx / f q and P∗ (x) is the conjugate complex of P(x). For the piezoelectric substrate occupying x > 0, the mechanical displacement and electrical potential function can be obtained directly from Eq.(7.37) [28] w2 (x, y,t) = A3 e−kbx exp[ik(y − ct)], µ ¶ e15 ϕ2 (x, y,t) = A4 e−kx + A3 e−kbx exp[ik(y − ct)] ε11

(7.44)

where A3 and A4 are the unknown constants, and b = (1 − c2 /c2s )1/2 with cs = (c¯44 /ρ2 )1/2 and c¯44 = c44 + e215 /ε11 being the bulk-shear-wave velocity and the effective shear modulus of the piezoelectric substrate, respectively. Substitution of Eqs.(7.43) and (7.44) and their corresponding stress components into the boundary conditions (7.38) and (7.39) yields four linear, homogeneous algebraic equations for coefficients A1 , A2 , A3 and A4 . For nontrivial solutions of these coefficients, the determinant of the coefficient matrix of the linear has to vanish. This leads to the dispersion relation determining phase velocity versus wavenumber of the transverse surface waves: e215 /ε11 Im[Q(0)Q∗ (−h)] c¯44 + b − =0 (7.45) kIm[P(0)Q∗ (−h)] µ0 µ0 µ ¶ µ ¶ Z p 1 0 where Im denotes the imaginary part, Q(x) = f /q − f / f − ikq exp −ik qdx 2 and Q∗ (x) is the conjugate complex of Q(x).

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We examine some special cases below. From Eq.(7.45), it follows that, the transverse surface waves are dispersive. When f (x) = g(x) = 1, i.e., the material of the layer is not gradient, we note that P(0) = (c2 /c20 − 1)−1/4 ,

Q(0) = −ik(c2 /c20 − 1)1/4 ,

Q∗ (−h) = ik(c2 /c20 − 1)1/4 exp[−ikh(c2 /c20 − 1)1/2 ] Substitution of the above expressions into Eq.(7.45) yields s s " s # e2 /ε11 c¯44 c2 c2 c2 1− 2 − − 1 tan kh − 1 − 15 =0 2 2 µ0 cs µ0 c0 c0

(7.46)

which is exactly the same as that of the Type 2 transverse surface wave in Ref. [10]. When e15 = 0, i.e., the material of the layer is not piezoelectric, we have Im[Q(0)Q∗ (−h)] c44 + b=0 kIm[P(0)Q∗ (−h)] µ0

(7.47)

which is exactly the same as the dispersion relation obtained in Ref. [32]. The above discussion on the dispersion equation (7.45) confirms to some extent the validity of the mathematic formulation obtained here. The dispersion relation of Eq.(7.45) is a transcendental equation where the gradient variation of the material properties in the metal layer is arbitrary. To study the propagation of transverse surface waves in this structure and to discuss the effects of the gradient coefficient on the dispersion curves and the phase velocity, as a special case, we assume f (x) and g(x) to be of the same exponential form [40]: f (x) = g(x) = exp α x

(7.48)

so that q(x) in Eq.(7.42) can be expressed as q(x) = q = [(c/c0 )2 −C2 ]1/2

(7.49)

in which C = (1+m2 /16π2 H 2 )1/2 with the following dimensionless quantities introduced m = α h and H = h/λ

(7.50)

where the quantity m denotes the degree of material gradient inside the metal layer. And the functions P(x) and Q(x) can thus be obtained easily by integration of q(x), the details of which can be found in our previous work [32]. Substitution of the these expressions into the Eq.(7.45) yields c¯44 b − e215 /ε11 (c2 /c20 − 1) tan(−2πHq) + =0 q − (m/4πH) tan(−2πHq) µ0

(7.51)

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247

Actually, q in Eq.(7.49) can not only take real values but also imaginary values, and whether it does depends on whether the surface wave velocity c is less than or greater than the velocity of the bulk shear waves in the metal layer c0 . Therefore, it is convenient to classify the problem into three physical situations in which dispersion curves of different types are found [10]: Type 1, c0 > cs > cBG , q always imaginary Type 2, cs > cBG > c0 , q always real Type 3, cs > c0 > cBG , q real or imaginary where cBG is the velocity of a B-G wave on the surface of a piezoelectric material coated with an infinitely thin layer of conducting material. In this subsection, we discuss the effects of material gradient on the dispersion behavior of the Type 1 wave only. For those who are interested in the effects of material gradient on the other two types of wave, please refer to our recent work [36]. Dispersion curves in the Type 1 situation are calculated from Eq.(7.49) when q is always imaginary, which is shown in Fig.7.11. In the plot, dispersion curves for nongraded metal layer case are attached for comparison to show a good agreement with the demonstrations by Curtis and Redwood [10]. The Type 1 situation permits only one mode of propagation, which does not exist in the absence of piezoelectricity, no matter whether the aluminum layer is gradient or not. The phase velocity c in the case of nongraded aluminum layer increases monotonically from the B-G wave velocity 2258 m/s to a final asymptotic value 2568 m/s equal to the velocity as H = ∞, which corresponds to a B-G wave modified into the totally anomalous dispersion mode by the presence of the finite-thickness layer [10]. However, this situation is modified by the presence of the material gradient in the metal layer. It is seen from Fig.7.11 that the material gradient in the aluminum layer does not change the start value of the mode but only the mode shape. Plus gradient coefficients

Fig. 7.11 Phase velocity c of Type 1 transverse surface waves in PZT-4 substrate with a gradient aluminum layer. H = h/λ , cl = 3251 m/s, cs = 2597 m/s, cBG = 2258 m/s, and c(H = ∞) = 2568 m/s.

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make the mode shift to the lower right corner while minus gradient coefficients the top left corner. Furthermore, the minus gradient coefficients modify the wholly anomalous dispersion curve in the case of m = 0 (i.e., a nongraded aluminum layer) into partly anomalous dispersion with partly normal dispersion, where the phase velocity c firstly increases to a certain value lower than the bulk-shear-wave velocity in the substrate 2597 m/s and then decreases gradually to the asymptotic value 2568 m/s.

7.4.2 Functionally graded substrate Different from the structure considered in the last subsection, we consider in this subsection a functionally graded piezoelectric material (FGPM), taken to be of class 6mm (or ∞ m), occupying the half-space x > 0, as shown in Fig.7.12. The polar axis of the substrate is oriented along the z direction of Cartesian coordinates (x, y, z). It is assumed that the physical properties of the FGPM change gradually along the x direction only. Let a metal layer of uniform thickness h be deposited perfectly on the FGPM substrate, resulting in a surface at x = −h free of external forces. Without loss of generality, it is assumed that the waves propagate in the positive direction of the y-axis, such that the nonzero field quantities representing the motion are only functions of the coordinates (x, y) and time t.

Fig. 7.12 A graded piezoelectric half-space covered by a metal/dielectric layer of uniform thickness.

Let w and ϕ denote separately the mechanical displacement and the electrical potential function of the FGPM substrate. The coupled field equations read [27] c44,x w,x + c44 ∇2 w + e15,x ϕ,x + e15 ∇2 ϕ = ρ w, ¨ e15,x w,x + e15 ∇2 w − ε11,x ϕ,x − ε11 ∇2 ϕ = 0

(7.52)

where ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 , and c44 , e15 , ε11 and ρ are the elastic, piezoelectric, dielectric constants and mass density of the substrate which are all functions of x (i.e., the substrate is functionally graded). For simplicity, all the material parameters of the substrate are taken to have the same exponential function variation, as [27] c44 (x) = c044 eα x ,

e15 (x) = e015 eα x ,

0 αx e , ε11 (x) = ε11

ρ (x) = ρ 0 eα x

(7.53)

where α is the coefficient characterizing the profile of the material gradient and the superscript 0 is used to denote the values of material parameters at x= 0. Substitution of

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

249

Eq.(7.53) into Eq.(7.52) yields c044 (α w,x + ∇2 w) + e015 (αϕ,x + ∇2 ϕ ) = ρ 0 w, ¨ 0 (αϕ + ∇2 ϕ ) − e0 (α w + ∇2 w) = 0 ε11 ,x ,x 15

(7.54)

For the metal layer of finite thickness, let w0 denote its mechanical displacement in the z direction. The governing field equation is [10] ∇2 w0 − (1/c2l )w¨ 0 = 0

(7.55)

where cl = (c044 /ρ 0 )1/2 is the bulk-shear-wave velocity, with c044 and ρ 0 representing the shear modulus and mass density of the metal layer, respectively. 1 0zx = 0 at x = −h. ° 2 w = w0 , 0zx = zx , The boundary and continuity conditions are: ° 0 3 w and ϕ → 0 as x → +∞. ϕ = 0 at x = 0. ° Built upon our previous work [28], we consider the following transverse waves satis3 fying attenuation condition °: ¾ w(x, y,t) = A1 erx exp[ik(y − ct)] (x > 0) (7.56) 0 )A1 erx ] exp[ik(y − ct)] ϕ (x, y,t) = [A2 esx + (e015 /ε11 0

0

w0 = [A3 e−b kx + A4 eb kx ] exp[ik(y − ct)] (−h 6 x 6 0)

(7.57)

where A1 , A2 , A3√and A4 are arbitrary constants, k(= 2π/λ ) is the wavenumber, λ is the wavelength, i = −1, and c is the phase velocity. The attenuation coefficients r and s of the substrate are given by r = −[α /2 + (α 2 /4 + b2 k2 )1/2 ],

s = −[α /2 + (α 2 /4 + k2 )1/2 ]

(7.58)

0 )/ρ 0 ]1/2 is the bulk-shear-wave velocwhere b = (1 − c2 /c2s )1/2 , and cs = [(c044 + e015 2 /ε11 ity in the substrate. The assumption that α < 0 is needed for the derivation of Eq.(7.56), which directly leads to the existence condition of transverse surface waves in the FGPM substrate, i.e., c < cs [33]. The decay parameter b0 in Eq.(7.57) is defined as q q b0 = 1 − c2 /c2l or b0 = i c2 /c2l − 1 (7.59)

where cl is the bulk-shear-wave velocity in the metal layer. Substitution of Eqs.(7.56) and (7.57) and the corresponding stress components into 1 and ° 2 yields four linear, homogethe remaining boundary and continuity conditions ° neous algebraic equations for coefficients A1 , A2 , A3 and A4 . For nontrivial solutions of these coefficients, the determinant of the coefficient matrix of the linear equations has to vanish. This leads to the following dispersion relation of the transverse waves: 0

0

c¯44 r + c044 b0 k(1 − e2b kh )/(1 + e2b kh ) − se215 /ε11 = 0

(7.60)

0 is the equivalent shear modulus of the FGPM substrate. where c¯44 = c044 + e015 2 /ε11

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In Eq.(7.60), b0 can not only take real values but also imaginary values, depending on whether the surface wave velocity c is smaller or greater than the bulk-shear-wave velocity in the metal layer, cl . Therefore, it is convenient to classify the present problem into three physical situations for which dispersion curves of different types may be found [10]: Type 1: cl > cs > cBG , b0 always real Type 2: cs > cBG > cl , b0 always imaginary Type 3: cs > cl > cBG , b0 real or imaginary Here, cBG is the velocity of the B-G wave on the surface of a piezoelectric substrate coated with an infinitely thin layer of conducting material. Here, we only discuss the effects of material gradient on the dispersion behavior of the Type 1 wave. For those who are interested in the effects of material gradient on the other two types of wave, please refer to our recent work [37]. For convenience, we introduce two dimensionless quantities m = α h and H = h/λ . Using Eqs.(7.58) and (7.59), we can then rewrite Eq.(7.60) separately for real b0 and imaginary b0 , as q q c¯44 M + e215 /ε11 N − c044 2πH 1 − c2 /c2l tanh(2πH 1 − c2 /c2l ) = 0 (7.61a) q q c¯44 M + e215 /ε11 N + c044 2πH c2 /c2l − 1 tan(2πH c2 /c2l − 1) = 0 (7.61b) p p where M = −m/2 − m2 /4 + (1 − c2 /c2s )4π2 H 2 and N = m/2 + m2 /4 + 4π2 H 2 . The Type 1 situation, shown in Fig.7.13, permits only one mode of wave propagation, which does not exist in the absence of piezoelectricity, whether the piezoelectric substrate is graded or not. For comparison, the dispersion curve for the case when the piezoelectric substrate is not graded (i.e., α = 0 or, equivalently, m = 0) is included too, which

Fig. 7.13 Phase velocity c of Type 1 transverse surface waves in the graded PZT-4 substrate carrying an aluminum layer for selected values of gradient coefficient. H = h/λ , cl = 3251 m/s, cs = 2597 m/s, cBG = 2258 m/s, and c(H = ∞) = 2568 m/s.

7 One Type of Transverse Surface Waves in Piezoelectric Layered Solids

251

shows excellent agreement with the results obtained by Curtis and Redwood [10]. The totally anomalous dispersion in the case of m = 0 (i.e., a pure piezoelectric substrate) is altered by the presence of material gradient in the substrate. Not only is the starting value of the fundamental mode but also its cutoff frequency changed as the substrate becomes graded. For small values of α , the wholly anomalous dispersion when α = 0 is modified into partly normal dispersion and partly anomalous dispersion. On the other hand, if the gradient coefficient is sufficiently large, the mode is changed into totally normal dispersion. For simplicity, the unrealistic assumption Eq.(7.53) on the gradient properties of the material parameters in the piezoelectric substrate is selected in the current study. It presents the advantages of mathematical simplicity and familiarity, but the inconvenience of describing a somewhat unrealistic inhomogeneity, because it either blows up or vanishes as x → ∞. These problems can be overcome by considering that it occurs sufficiently far away from the interface, and by focusing on the near-the-surface localization of the wave. For those who are interested in material gradient of arbitrary functions can get an asymptotic solution by using the procedure introduced in the last subsection in which the WKB technique is presented.

7.5 Summary As shown in this chapter, one type of transverse surface wave can exist in piezoelectric layered structures under specific physical condition under which no Love wave can exist. First, closed-form dispersion relations are presented for a piezoelectric layer and a metal/dielectric substrate. Then a hard metal interlayer is introduced to remove the undesired mode in the former simple structure, and general closed-form dispersion relations are obtained for a piezoelectric layer and a metal/dielectric substrate with multiple metal interlayers. Furthermore, the effects of initial stresses on the transverse surface wave are shown for a piezoelectric substrate and a metal/dielectric layer. Finally, the effects of material gradient in either piezoelectric substrate or metal/dielectric layer on the transverse surface wave are separately presented. This type of transverse surface wave is closely related to a bulk shear wave, though little reported and used at present may have certain advantages over the traditional surface waves utilized for electro-acoustic devices due to its unique features like high performance and simple particle motion and so on.

Acknowledgments The author (Z.H. Qian) would like to thank the support by the Program for New Century Excellent Talents in Universities (No. NCET-12-0625), the NUAA Fundamental Research Funds (No. NE2013101), and the National Natural Science Foundation of China (No. 11232007). One of the authors (F. Jin) gratefully acknowledges the support by the

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National Natural Science Foundation of China (No. 10972171) and the Program for New Century Excellent Talents in Universities (No. NCET-08-0429).

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[21] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a 6mm piezoelectric material carrying a prestressed metal layer of finite thickness. Appl. Phys. Lett., 2009, 94, 1-3. [22] Suresh S, Mortensen A. Fundamentals of Functionally Graded Materials— Processing and Thermomechanical Behavior of Graded Metals and Metal- Ceramic Composites. Cambridge University Press, 1998. [23] Liu J, Wang Z K, Zhang L. Love waves in layered structures with graded materials. Acta Mech. Solida Sin., 2004, 25, 165-170. [24] Li X Y, Wang Z K, Huang S H. Love waves in functionally graded piezoelectric materials. Int. J. Solids Struct., 2004, 41, 7309-7328. [25] Liu J, Wang Z K. The propagation behavior of Love waves in a functionally graded layered piezoelectric structure. Smart Mater. Struct., 2005, 14, 137-146. [26] Liu J, Cao X S, Wang Z K. Propagation of Love waves in a smart functionally graded piezoelectric composite structure. Smart Mater. Struct., 2007, 16, 13-24. [27] Du J, Jin X, Wang J, et al. Love wave propagation in functionally graded piezoelectric material layer. Ultrasonics, 2007, 46, 13-22. [28] Qian Z H, Jin F, Wang Z K, et al. Transverse surface waves on a piezoelectric material carrying a functionally graded layer of finite thickness. Int. J. Eng. Sci., 2007, 45, 455-466. [29] Qian Z H, Jin F, Lu T J, et al. Effect of initial stress on Love waves in a piezoelectric structure carrying a functionally graded material layer. Ultrasonics, 2010, 50, 84-90. [30] Qian Z H, Jin F, Kishimoto K, et al. Propagation behavior of Love waves in a functionally graded half-space with initial stress. Int. J. Solids Struct., 2009, 46, 1354-1361. [31] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a layered structure with a functionally graded piezoelectric substrate and a hard dielectric layer. Ultrasonics, 2009, 49, 293-297. [32] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in an FGM layered structure. Acta Mech., 2009, 207, 183-193. [33] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in functionally graded piezoelectric materials with exponential variation. Smart Mater. Struct., 2008, 17, 065005. [34] Qian Z H, Hirose S, Kishimoto K. Modulation of Bleustein-Gulyaev waves in a functionally graded piezoelectric substrate by a finite-thickness metal waveguide layer. JSME Trans. J. Solid Mech. Mater. Eng., 2009, 3, 1182-1192. [35] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a piezoelectric material carrying a gradient metal layer of finite thickness. Int. J. Eng. Sci., 2009, 47, 1049-1054. [36] Qian Z H, Jin F, Hirose S, et al. Effects of material gradient on transverse surface waves in piezoelectric coupled solid media. Appl. Phys. Lett., 2009, 95, 073501. [37] Qian Z H, Jin F, Lu T J, et al. Transverse surface waves in a functionally graded piezoelectric substrate coated with a finite-thickness metal waveguide layer. Appl. Phys. Lett., 2009, 94, 023501. [38] Achenbach J D. Wave Propagation in Elastic Solids. Oxford: North-Holland Publishing Company, 1973. [39] Ishimaru A. Electromagnetic Wave Propagation, Radiation and Scattering. New Jersey: Prentice-Hall, 1991. [40] Dlale F, Erdogan F. On the mechanical modeling of the interfacial region in bonded halfplanes. ASME J. Appl. Mech., 1988, 55, 317-324.

Chapter 8 Theoretical Investigation of Force-frequency and Electroelastic Effects of Thickness Mode Langasite Resonators

HaiFeng Zhang Department of Engineering Technology, University of North Texas, Denton, TX 76207, USA

Abstract Piezoelectric resonators are critical devices for the modern electronics industry. For many applications, these resonators are influenced by biasing fields, with mechanical and electrical biasing fields among the most important ones. The reason that the character of a resonator changes under a biasing field is primarily due to the inherent nonlinear material properties of the piezoelectric material. Among commonly used piezoelectric materials for resonators, langasite single crystals are very new and promising. They combine many of the advantages of quartz and lithium niobate—having high electromechanical coupling and good frequency-temperature characteristics. Thus, langasite is a potential material to substitute for current applications dominated by quartz. In this chapter, the influence of mechanical and electrical biasing fields on resonators made of langasite is investigated systematically. The perturbation theory of electroelasticity for large deformations and strong fields is first presented. Based on this theory, the force-frequency effect and electroelastic effect of langasite resonators are discussed analytically and numerically. The results of this research are expected to be fundamentally important to the piezoelectric resonator research community. Keywords langasite resonators, force-frequency effect, electroelastic effect

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8.1 Introduction Piezoelectric materials, or more generally, electroelastic materials, produce a voltage in response to an applied force. Similarly, a change in dimensions can be induced by the application of a voltage to these materials. There are many materials that exhibit this kind of behavior including quartz (SiO2 ), gallium phosphate (GaPO4 ), barium titanate (BaTiO3 ), lithium niobate (LiNbO3 ), lead zirconate titanate (PZT), aluminum nitride (AlN), zinc oxide (ZnO), polyvinylidene fluoride (PVDF), and so on. They are widely used in modern sensors, actuators, ultrasonic transducers, ultrasonic probes, piezoelectric motors, as well as frequency control and timing products (resonators and filter). Depending upon its material properties, each kind of piezoelectric materials has its specific applications. For example, quartz crystals are currently the primary material for piezoelectric resonators. A piezoelectric resonator is actually a vibrating crystal that operates at a desired resonant frequency. They have numerous applications in modern communications and aviation, as well as for atomic clock, televisions, disk drives, radios· · · and many other applications. It is hard to imagine the status of the modern electronic industry without a crystal resonator. Quartz has combined unique behavior of high Q, good frequency-temperature behavior, and low aging effects. However, low electromechanical coupling hinders its application in filter design, because filter design normally needs wider bandwidth. Recently, new piezoelectric materials such as langasite have been investigated for their potential to improve the performance of resonators made from quartz. Langasite single crystal belongs to the gallo-germanate family, which includes La3 Ga5 SiO14 (LGS), La3 Ga5.5 Nb0.5 O14 (LGN), and La3 Ga5.5 Ta0.5 O14 (LGT). These materials have superior behavior such as high Q, stable frequency-temperature behavior, no phase transition to a high temperature region and high electromechanical coupling coefficient. Thus, langasite is a potential material to substitute for current applications dominated by quartz. Military needs are a primary drive of modern piezoelectric resonator technology. Modern military systems require that crystal resonators are stable over a wide range of parameters (time, temperature, acceleration, electrical field, radiation, etc.), have low noise, require low power, are small in size, have fast warm-up, and have low life-cycle cost. Among these requirements, the research on the influence of environmental effects (biasing fields) on the frequency stability of piezoelectric resonator is a very important subject. A resonator’s frequency shift under biasing fields can be explained by the theory of infinitesimal fields superposed on finite biasing fields [1], in which the inherent nonlinear material properties of the crystal contribute to the frequency shift. Among the numerous biasing fields, acceleration and electrical fields are two critical ones. Acceleration will induce the force-frequency effect that describes the shift in resonant frequency, a resonator experiences due to the application of a mechanical load. A clear understanding of this effect is essential for many device designs such as pressure sensors. The electrical field-frequency effect or electroelastic effect, refers to the resonator frequency shift that occurs when it is subject to a DC electrical field. A deep understanding of this effect is crucial for voltage sensor design. In order to model these two effects, first-order perturbation integral theory was established thirty years ago [2]. It may be used

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to estimate the effect of biasing fields on BAW and SAW piezoelectric resonators under the assumption that the biasing fields only cause linear deformation and the frequency of interests are simple without degeneracy. Methods to quantify the frequency shift of a crystal resonator with arbitrary cut angle caused by mechanical force and electrical field can be theoretical or numerical. Theoretical methods are most useful, but are limited to simple geometry and often require tedious calculations; numerical methods are not as accurate as theoretical estimations but can be used for the frequency shift estimations of resonators with more complex geometry and boundary conditions. In this chapter, theoretical and numerical estimations of resonator frequency shifts by both mechanical force and electrical fields are investigated for arbitrary crystal cuts, Experimental measurements of force-frequency effect and electroelastic effect of thickness mode langasite resonators are presented and the applications of the measurement results on sensors and material constants determination of langasite single crystals are also discussed. Since nonlinear material constants include third-order elastic, piezoelectric, dielectric and electrostrictive constants, they all contribute to the frequency shift caused by biasing fields. Thus, a quantitative analysis that describes the different contribution from each group of nonlinear material constants is necessary. This analysis supplies invaluable reference information for experimental determination of third-order material constants by resonator methods. In this chapter, sensitivity analysis is carried out for both the forcefrequency effect and the electrical field-frequency effect, using both theoretical and numerical methods.

8.2 Perturbation integral for resonator frequency shift calculations The resonant frequency of a resonator depends on its geometry, material constants and boundary conditions. When a langasite resonator is subjected to an external mechanical load, the geometry changes slightly. In this case, the material constants may be characterized as effective constants that will change with the external mechanical load. Thus, a resonant frequency will shift with the applied load. The shifted value may be estimated accurately by the perturbation integral theory. The expression to estimate the first-order perturbation of a specific mode can be found from [2] ∆ωM = ω − ωM Z

= 2ωM Z

× where

V

1 M M M M M ρ0 (uM 1 u1 + u2 u2 + u3 u3 )dV

M M M M M (cˆK α Lγ uM γ ,K uα ,L + 2eˆKLγ φK uγ ,L − εˆKL φ,K φ,L )dV

(8.1)

258

HaiFeng Zhang 0 δ +c 0 0 cˆK α Lγ = TKL αγ K α LN ωγ ,N + cKNLγ ωα ,N + cK α Lγ AB SAB + kAK α Lγ EA

(8.2)

0 + b0 0 eˆKLγ = eKLM ωγ0,M − kKLγ AB SAB AKLγ EA 0 +χ 0 εˆKL = b0KLAB SAB KLA EA

(8.3) (8.4)

with b0AKLγ = bABCD + ε0 δAB δCD − ε0 (δAC δBD + δAD δBC ) In Eqs.(8.1)-(8.4), ωM is the unperturbed angular frequency, ω is the perturbed angular frequency, and ∆ωM is the angular frequency shift. Here, cˆK α Lγ , eˆKLγ , and εˆKL are the effective elastic, piezoelectric, and dielectric constants respectively; uM γ is a specific mode shape function in the unperturbed condition and φ M is the electrical potential for this 0 , S0 , and E 0 are the initial stress, strain and electrical field respecspecific mode. TKL AB A tively with ωγ ,N defining the displacement gradient. cK α Lγ and eKLM are the second-order elastic and piezoelectric constants. cK α Lγ AB , kAK α Lγ , and χKLA are the third-order elastic, piezoelectric, and dielectric constants respectively, and bKLAB are the electrostrictive constants. ε0 is the permittivity of free space. It is worth noting that the definition of the energy density used here differs from that of Sorokin et al. [3]. The resulting differences in material constants are discussed in Ref.[4]. The displacement gradient is the summation of strain and the rigid body rotation tensor. It may be written as

ωγ ,N = Sγ N + Ωγ N Physically, it can be easily verified that a rotation of the position of a crystal resonator will not cause a resonant frequency shift. Theoretically, it has been proven in Ref.[5] that an arbitrary pure homogeneous infinitesimal rigid body rotation has no influence on frequency shift. Thus, in the case considered here, the displacement gradient may be obtained directly from the static strain solution. The frequency shift, as given by Eq.(8.1), is a function of the mode, the natural frequency without perturbation, and the effective material constants. From Eqs.(8.2)-(8.4), the effective materials constants are functions of the initial stress and strain and the electrical field.Thus, estimates of the frequency shift of a resonator under an external mechanical load require the mode shape and stress, strain, and electrical field distribution for the resonator. Then, the static solution for the electrical field, initial stress, and strain are required. These solutions may be obtained analytically or numerically. For a resonator with a simple geometry, the perturbation integral Eq.(8.1) may be simplified greatly. However, for a resonator with a relatively complex geometry, the perturbation integral can be carried out only numerically. In such cases, the finite element solution combined with numerical integration may be done to estimate the resonator frequency shift accurately. The initial stress, strain and field can be solved by the finite element method (FEM) for the element stress, strain and field. The volume integral can be treated as the summation of each element volume. The perturbation integral for the finite element approach may then be expressed as ∆ωM = ∆ω1 + ∆ω2 − ∆ω3

(8.5)

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where ELN

∆ω1 =

0 0 (N)δαγ + 2cK α LN Sγ0N (N) + cK α Lγ AB SAB (N) + kAK α Lγ EA0 (N) ∑ (TKL

N=1

+2cK α LN Ωγ0N (N))uγM,K (N)uM α ,L (N)V (N) ELN

∆ ω2 = 2

0 (N) ∑ (eKLM (Sγ0M (N) + Ωγ0M (N)) − kKLγ AB SAB

N=1

+b0AKLγ EA0 (N))φKM (N)uM γ ,L (N)V (N) ELN

∆ ω3 =

0 M M (N) + χKLA EA0 (N))φ,K (N)φ,L (N)V (N) ∑ (b0KLAB SAB

N=1

0 (N), S0 (N), E 0 (N), and V (N) are the element Here, N is the element number and TKL γN A stress, strain, field and volume. The rigid body rotation tensor Ωγ0N (N) is calculated from the nodal displacements. The analytical solutions to be derived in Sections 8.3 and 8.4 are used for the mode shape uk (N) and field φ (N).

8.3 Force-frequency effect of thickness mode langasite resonators In this section, the frequency shift is analyzed theoretically and numerically for thin, circular langasite plates subjected to a pair of diametrical forces. In addition, the sensitivity of the force-frequency effect is analyzed with respect to the nonlinear material constants. The results are anticipated to be valuable for experimental measurements of nonlinear material constants as well as for device design.

8.3.1 Background Piezoelectric resonators based on langasite and its isomorphs [6-8] are of current interest in the piezoelectric device community because of the combination of moderately strong piezoelectric coupling, low acoustic loss, zero temperature coefficient of frequency, and reduced nonlinear effects occurring in simple orientations. One of these nonlinear effects is the force-frequency effect, which describes the resonant frequency shift that occurs when the resonator is subjected to an external mechanical load (here, an applied diametrically oriented force pair). Depending upon the design of the resonator mounting structure, this effect can cause problems with resonators used in timing devices when external accelerations are experienced. One approach to counteract the frequency shift caused by acceleration is to actively adjust the frequency with another nonlinear effect, namely the electroelastic effect [4,9,10]. Alternatively, the force-frequency effect can also be used

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to produce a force sensor [11,12]. The first examination of the force-frequency effect in langasite was published by Boy et al. in 2001 [13]. Their work on langasite included both theoretical and experimental components, drawing significant attention to the “major discrepancy” found for the magnitude of the effect in langasite. The experimental measurements of Boy et al. also included langatate (LGT), subsequently measured by Kim and Ballato along with langanite (LGN) [14]. Neither work included calculations for LGT or LGN as the required nonlinear material constants have not yet been measured. Independent calculations on the force-frequency effect in langasite were reported in Ref.[15] based upon an assumed isotropic biasing stress. The isotropic stress assumption was consistent with previous analyses of quartz resonators [16], and the results for quartz were shown to be consistent with the earlier work. The implication was that the results for langasite are a similarly good approximation. Unfortunately, these prior analyses ignored the influence of linear and nonlinear piezoelectric constants. For langasite single crystals that have moderately high piezoelectric coupling, those assumptions may be invalid. Consequently, in this section we improve our calculations of the force-frequency effect in langasite resonators by taking into account the full set of material nonlinearities and by determining more accurately the biasing stress and strain fields. The complete set of linear and nonlinear material constants may be necessary for estimations of the force-frequency effect for langasite resonators. The force-frequency effect is caused by intrinsic nonlinear material properties of the single crystal. The nonlinear material properties are characterized by the third-order material constants including the third-order elastic, piezoelectric, dielectric and electrostrictive constants. A detailed description of this phenomenon requires the theory of infinitesimal fields superposed on finite biasing fields [17], which describes the influence of a biasing effect (mechanical stress, electrical field) on the natural frequency of piezoelectric resonators. For most (but not all) cases of interest, the shifted value of the natural frequency can be estimated by the first-order perturbation integral [2], as discussed in Section 8.2. To calculate the frequency shift of resonators subjected to a static mechanical load, one must first obtain the static solution for the strain and electrical field. However, because of the anisotropic material properties of the single crystal, it is difficult to obtain such solutions analytically. Previous research [13,15,16] has included several different approaches to both the isotropic and anisotropic solutions. The isotropic case provides a convenient initial estimate since the static solutions can easily be obtained in an analytical form that then enables a simple solution for the overall effect. Nevertheless, the true anisotropic solution differs sufficiently from the isotropic solution such that one must solve the fully anisotropic problem. An alternative method for this case is to use the finite element method (FEM). FEM is now commonly used in the basic analyses of quartz resonators [18-22]. A previous FEM result for frequency shift estimates of quartz resonators can be found in Ref.[12], for which only the third-order elastic constants are considered. For quartz resonators with low piezoelectric coupling coefficient, this assumption may be accurate enough. However, it is unclear for langasite resonators with high piezoelectric coupling, whether other nonlinear material constants play an important role in this response. Given the difficulty of obtaining an analytical solution that includes all of the

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anisotropy and nonlinearity, in this section, FEM is used to estimate accurately the forcefrequency effect of langasite resonators. The static anisotropic solution for a diametrical biasing stress is obtained using COMSOL 3.2. Combined with the analytical mode shape solutions, the force-frequency effect for a langasite resonator with arbitrary material orientations is obtained. All the third-order material constants including the third-order elastic, piezoelectric, dielectric and electrostrictive constants are used in the calculations. The results are compared both to an isotropic analytical solution and the limited experimental results reported in the literature. Finally, the influence from each group of nonlinear material constants is examined quantitatively by isolating the contributions of different sets of nonlinear material constants to the force-frequency effect for resonators with arbitrary material orientations. This sensitivity analyses is similar in objective but greatly expanded beyond that presented in Fig.5 of Ref.[13] where the authors are simply attempting to explain the discrepancy in the magnitude of the theoretical and experimental results.

8.3.2 Perturbation theory The Perturbation theory for the resonator calculation is introduced in Section 8.2. The theory leads to the perturbation integral Eqs.(8.1) and (8.5) with respect to the theoretical and finite element forms, respectively. These two expressions are now used in the following frequency shift estimation for the thickness mode langasite resonator.

8.3.3 Unperturbed mode Consider the free vibrations of a thin plate cut from a single crystal with arbitrary symmetry, as shown schematically in Fig.8.1. The plate is fully coated with thin, massless electrodes on both sides. The boundary conditions are assumed traction free. Here, we approximate the exact mode as that for an infinite plate for simplicity. Later, we restrict our integration area to a region near the center of the plate in order to represent more closely the response expected from a trapped energy mode [23]. Such an approximation

Fig. 8.1 A doubly rotated langasite plate with electrodes on both sides.

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HaiFeng Zhang

is reasonable since errors in the mode shape result in higher-order behavior in the perturbation integral. In this case, the governing equations may be written as c2 jk2 uk ,22 + e22 j φ,22 = ρ u¨ j

(8.6)

e2k2 uk ,22 − ε22 φ,22 = 0

(8.7)

The boundary conditions are written as T2 j = c2 jk2 uk ,2 + e22 j φ,2 = 0

φ = 0,

(8.8)

at x2 = ±h

(8.9)

The general solution for this case can be summarized as 3

3

i=1

i=1

u j (x2 ,t) = ∑ Dij sin λi x2 e−iω t = ∑ ki Bij sin λi x2 e−iω t 3

µ

φ (x2 ,t) = ∑ K i=1

i e11m

ε11

( j = 1, 2, 3)

(8.10)

¶

Bim sin λi x2 + P1 x2 + P2

e−iω t

(8.11)

where P1 = −

1 3 i e11m i ∑ K ε11 Bm sin λi h, h i=1

P2 = 0

Thus, the ith mode shape may be defined as uij (x2 ) = Dij sin λi x2

(8.12)

The amplitudes Dij are obtained by solving the eigenvalue problem (c02 jk2 − cδ jk )Dk = 0 where c02 jk2 = c2 jk2 +

e22 j e22k ε22

Here, (uij (x2 ))M = Dij sin λi x2 is the ith normalized mode shape and K i is the normals 2 ρω 2 with ω as the angular frequency. Substituting ized weighting coefficient, λi = ci Eqs.(8.10) and (8.11) into boundary conditions, Eqs.(8.8) and (8.9) gives the transcendental equation that is used to determine the resonant frequencies. This equation is p p p Det{Bim [c02 jk2 ω h ci /ρ cos(ω h ci /ρ ) − (e22 j e22m /ε22 ) sin ω h ci /ρ ]} = 0 (8.13) Equation (8.13) is solved numerically for the frequencies of interest. Alternatively, the resonant frequency can be approximated by the antiresonant frequency as [24]

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Theoretical Investigation of Force-frequency and Electroelastic Effects

ω ≈ ω1 =

(2q − 1) 4h

r

cj ρ

( j = 1, 2, 3; q = 1, 2, 3, · · · )

263

(8.14)

Equation (8.14) is often an accurate approximation for any overtone for materials with low piezoelectric coupling coefficient. For materials with high piezoelectric coupling coµ ¶2 1 efficient, the approximation is accurate for high overtones (q > 4) where the 2q − 1 reduction in effective coupling as a function of increasing harmonic results in a low effective piezoelectric coupling coefficient. Langasite has moderately high piezoelectric coupling. Therefore, Eq.(8.14) is expected to give accurate results only for high overtones.

8.3.4 Diametrical force 8.3.4.1 Analytical solution Consider a doubly rotated langasite plate fully coated with electrodes on both sides as shown in Fig.8.1. The material orientation is shown in Fig.8.2 which follows IEEE standard 176-1987 [24]. A pair of equal forces F is applied in the diametrical direction shown in Fig.8.3. The three-dimensional linear equations of piezoelectricity can be written as [25] Si j = Si jkl Tkl − dki j Ek

(8.15)

Di = dikl Tkl + εik Ek

(8.16)

T ji, j = 0

(8.17)

Di,i = 0

(8.18)

where the strain and the electric field are written as Skl =

Fig. 8.2 Plate orientation of a doubly rotated langasite resonator (Y Xwl Φ /Θ ).

uk,l + ul,k 2

(8.19)

Fig. 8.3 A pair of diametrical forces applied on the LGS resonator.

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HaiFeng Zhang

Ek = −φ,k

(8.20)

As an approximation [16], the following stresses (at the center of an isotropic plate subjected to a pair of diametrical forces) are used throughout the thin plate: T11 = −

6F , π2hD

T33 =

2F , π2hD

T13 = 0

(8.21)

Lee et al. [26] has verified that the use of Eqs.(8.21) as the uniform stress solution, from which the strain solution is obtained from anisotropic constitutive equation leads to a force-frequency effect that closely matches experiments. Based on this approach, the stress distribution for the resonator subjected to a pair of diametrical forces is obtained as 2 F (1 + 2 cos 2ψ ) π 2hD 2 F T33 = − (1 − 2 cos 2ψ ) π 2hD 4 F T13 = − sin 2ψ π 2hD T11 = −

(8.22) (8.23) (8.24)

Then from the anisotropic strain and stress in Eq.(8.15), the strain components can be obtained. 8.3.4.2 Finite element solution ◦

◦

A finite element model for a doubly rotated langasite resonator (Y Xwl Φ = 20 /Θ = 30 ) is also constructed using FEMLAB 3.2 (COMSOL 3.2). The model includes 2224 Lagrangian quadratic elements with the number of boundary elements as 1640, such that there are 3 layers of elements. The sample radius is 6.5 mm, the thickness is 0.65 mm, and a pair of diametrical forces is applied along the X-axis. The radial edge of the plate has zero charge. The convergence of the numerical solution is verified by refining the element size and by comparison with analytical solutions when available. In the analytical approach, because of the isotropic material properties assumption, the stress distribution is uniform. The strain is then calculated by an anisotropic stress-strain constitutive equation, and the electrical charge and field are assumed zero. In the finite element approach, the plate is fully anisotropic and the influence of electrical charge and field are included in the analysis. Therefore, the FEM solution gives a more realistic result. Figure 8.4 shows an example finite element solution for stress T12 . It can be seen that the solution is different from the isotropic solution, particularly away from the center of the plate, which illustrates a fundamental limitation of the isotropic assumption and the advantage of the FEM approach.

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Fig. 8.4 Static solution for shear stress T12 determined by the finite element method for an LGS resonator (Y Xwl Φ = 20◦ /Θ = 30◦ ), the thickness is 0.65 mm and the radius is 6.5 mm.

8.3.5 Results For a resonator with arbitrary material orientation, the force-frequency effect for a specific mode is dependent on all material constants including second-and third-order constants, after transforming them to the new coordinate. Secondly, the unperturbed natural frequency and mode shape for the specific mode are determined. Thirdly, the static stress, strain, and electrical field for the resonator under biasing diametrical forces are obtained. Finally, Eqs.(8.1) and (8.5), respectively, are used for the calculation of the force-frequency effect analytically and numerically. Example results are presented in this section. 8.3.5.1 Force-frequency effect The force-frequency effect is often characterized by the force sensitivity coefficient, which can be found in Ref.[27]. It is defined as K f (φ , θ , ψ ) =

∆ f 2hD ∆ f T hickness Diameter = f0 F f0 (2h/n) f0 Force (Acoustic velocity)/2

(8.25)

where ∆ f is the natural frequency shift, h is the plate thickness, D is the plate diameter, f0 is the unperturbed natural frequency, F is the diametrical force, and n is the harmonic overtone order of the fundamental frequency. This coefficient is now being examined for a variety of plate cuts. Figure 8.5 shows the force-frequency effect of a Y-cut LGS resonator resulting from a pair of applied diametrical forces. The approximate analytical results are obtained under the assumption that the thin plate is isotropic. Therefore, only the second-order and the third-order elastic constants are included in the perturbation integral calculations. FEM was also used to calculate the static strain solution for the anisotropic plate. All nonlinear material constants are included in the perturbation integral calculations. Figure 8.5 also

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Fig. 8.5 The comparison between FEM result and theoretical result with different integral radius for the fast shear mode. The plot with markers is the FEM result with different integral radius, the line without markers is the theoretical result.

shows the results obtained using FEM to determine the influence of the integral volume with respect to the approximate analytical solution. Thus, R1 defines the integral radius as a fraction of the plate radius. It is clear that the FEM results are closer to the approximate analytical result when the integral volume is restricted to the region near the center of the thin plate. This is because the FEM result assumes the analytical solution of the infinite plate, which may not apply near the edge of the plate. When the integration volume is restricted to the center region, the real mode shape is closer to the approximate analytical solution. Therefore, it appears that when R1 (integral radius, shown in Fig.8.6) = 0.35R, the FEM result fits the analytical result well. Thus, this integration radius will be used for the other results shown here. A more detailed analysis involving the exact trapped energy mode shapes is the subject of future work.

Fig. 8.6 The sketch of integral radius.

Figure 8.7 shows the force-frequency effect of the three thickness mode langasite resonator with respect to azimuth angle. The integral radius R1 = 0.35 R is used for the FEM estimation, the resulting force-frequency effect is changing smoothly with respect to the azimuth angle and it is symmetric with respect to Ψ = 90◦ as expected. The FEM results

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267

Fig. 8.7 The force-frequency effect of Y-cut thickness mode LGS resonators as a function of azimuth angle.

fit the analytical result well. Figures 8.8-8.10 show the force-frequency effect for the doubly rotated LGS resonators (Y Xwl Φ /Θ , Φ = 0◦ , 20◦ , 40◦ , 60◦ , Θ = −90◦ to 90◦ ) when diametrical forces are applied along the transformed X-axis. Because the interest here is on the behavior with respect to the third-order material constants, only the contribution to the force-frequency effect from the third-order material constants is presented such that contributions from second-order material constants are excluded. The analytical results are shown as solid lines while the numerical results are denoted with squares. The FEM results and the approximate analytical results agree well for most cases, while there are certain orientations which have relatively large differences (e.g., the orientation: Y Xwl

Fig. 8.8 Force-frequency effect of doubly rotated LGS resonator (Y Xwl Φ /Θ ) for mode A. The solid lines correspond to analytical solution, while the symbols denote the FEM solution.

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HaiFeng Zhang

Φ = 60◦ /Θ = −90◦ to 90◦ ). For modes B and C, there are considerable jumps of the force-frequency effect at the cut Y Xwl Φ = 0◦ /Θ = ±43◦ . This result is due to a degeneracy that occurs at this cut. In this case, the first-order perturbation integral does not apply, but the second-order perturbation integral [28] may be used. The intersections with the X-axis also indicate a possible stress compensation cut which can be used in the case which requires a minimum sensitivity to diametrical forces.

Fig. 8.9 Force-frequency effect of doubly rotated LGS resonator (Y Xwl Φ /Θ ) for mode B. The solid lines correspond to analytical solution, while the symbols denote the FEM solution.

Fig. 8.10 Force-frequency effect of doubly rotated LGS resonator (Y Xwl Φ /Θ ) for mode C. The solid lines correspond to analytical solution, while the symbols denote the FEM solution.

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8.3.5.2 Sensitivity to nonlinear material constants As stated in Eq.(8.1), the natural frequency shift is related to all nonlinear material constants including the third-order elastic, piezoelectric, dielectric and electrostrictive constants. However, the contribution to the frequency shift from each group of nonlinear material constants depends on the cut angle. Therefore, to investigate the different contributions from different material constants quantitatively, we plot the contribution from each group of nonlinear material constants as a function of cut angle. Because nonlinear material constants are the major focus area in this chapter, the contribution of the second-order material constants is not shown in all sensitivity plots. Therefore, it is clear from the figures shown subsequently that the total contribution from all nonlinear material constants does not match the complete curve, since it includes second-order effects as well. Figure 8.11 shows the force-frequency effect for cuts Y Xwl Φ = 0◦ /Θ = −90◦ to ◦ 90 . The results for Mode A (longitudinal mode) and B (fast shear) show that the primary contributions to the force-frequency effect come from the third-order elastic constants without any contribution from the third-order piezoelectric, electrostrictive and the thirdorder dielectric constants. For Mode C, it is observed that the cuts Y Xwl Φ = 0◦ /Θ = 0◦ to 30◦ have small contributions from the third-order piezoelectric constants. Figure 8.12 shows the results for cuts Y Xwl Φ = 20◦ /Θ = −90◦ to 90◦ . Compared with the case shown in Fig.8.11, we can see that except for the major contribution from the third-order elastic constants, the third-order piezoelectric constants play a more important role for a specific cut range (Y Xwl Φ = 0◦ /Θ = −30◦ to 30◦ for Mode A and C, Y Xwl Φ = 20◦ /Θ = −60◦ to 45◦ for Mode B). Therefore, if we want to determine the third-order elastic constants of LGS using applied diametrical load for LGS resonators, these cut ranges should be avoided. In Fig.8.13, results for Mode C for the cut range YXwl Φ =40◦ / Θ = −30◦ to 30◦ shows that the influence of the third-order piezoelectric constants on the force-frequency effect is larger relative to other cut ranges. Therefore, this cut range may be good for determination of the third-order piezoelectric constants using an applied diametrical load by resonator methods. Figure 8.14 shows the results for the cuts Y Xwl Φ = 60◦ /Θ = −90◦ to 90◦ , we do not observe considerable contributions to forcefrequency effect from the third-order piezoelectric constants for Modes A and B. For Mode C, the cut range Y Xwl Φ = 60◦ /Θ = −40◦ to 0◦ has little contribution from the third-order piezoelectric constants. The sensitivity analyses for resonators with a variety of cuts range has been carried out. Next, we will show our comparison with previous reported experimental results. 8.3.5.3 Comparison with previous experimental results We compared our theoretical analysis result to the experimental result by Boy et al. [13]. The sample is a Y-cut 9.98 MHz 3rd overtone langasite resonator. The thickness is 0.41 mm, the external diameter is 13.2 mm, the electrodes are 6 mm, the radius of the curvature is 500/500 mm. Figure 8.15 shows the comparison of force-frequency effect for this

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Fig. 8.11 The plots for the sensitivity of all groups of nonlinear material constants to force-frequency effect (Φ = 0◦ ). TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (thirdorder dielectric constants), ES (electrostrictive constants).

langasite resonator. The approximate analytical result fits the experimental results very well. It is worth noting that approximate analytical calculation is restricted to plano-plano resonator model.

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Fig. 8.12 The plots for the sensitivity of all group of nonlinear material constants to force-frequency effect (Φ = 20◦ ). TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

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Fig. 8.13 The plots for the sensitivity of all group of nonlinear material constants to force-frequency effect (Φ = 40◦ ). TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

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Fig. 8.14 The plots for the sensitivity of all group of nonlinear material constants to force-frequency effect (Φ = 60◦ ). TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

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Fig. 8.15 The comparison of the theoretical result and the experimental data [13] of a Y-cut 3rd overtone 9.98 MHz langasite resonator.

8.3.6 Conclusions In this section, the force-frequency effect and the related sensitivity analysis of thickness mode langasite resonator were examined both analytically and numerically. It was shown that the similarity of FEM results and theoretical results depends on the integral volume. The integral region close to the center of the resonator gives closer agreement between the FEM and analytical result. In addition, possible stress compensation cuts were found and discussed. The sensitivity analysis shows that the electrostrictive constants and the third-order dielectric constants contribute little to the force-frequency effect for the majority of cuts examined. The major contributions to the force-frequency effect come from the third-order elastic constants. The contribution from the third-order piezoelectric constants is small in most cases. However, there do exist some cut ranges where the contributions from the third-order piezoelectric constants are large enough that they cannot be ignored. Determination of the third-order elastic constants using an applied diametrical load on a resonator is proven feasible for cut ranges where the contribution from the third-order elastic constants dominate. The comparison between our theoretical calculation result to previous experimental data for a Y-cut 3rd overtone langasite resonator shows good agreement.

8.4 Electroelastic effect of thickness mode langasite resonators In this section, the resonant frequency shift of a thickness mode langasite resonator is analyzed with respect to a DC electric field applied in the thickness direction. The vibration

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modes of a thin langasite plate fully coated with an electrode are analyzed. The analysis is based on the theory for small fields superposed on a bias in electroelastic bodies and the first-order perturbation integral theory. The electroelastic effect of the resonator is analyzed by both analytical and finite element methods. The complete set of nonlinear elastic, piezoelectric, dielectric permeability and electrostrictive constants of langasite is used in the theoretical and numerical analysis. The sensitivity of electroelastic effect to nonlinear material constants is analyzed.

8.4.1 Introduction The electroelastic effect of piezoelectric resonators is defined as the resonant frequency shift that occurs with respect to an applied DC electric field. This phenomenon can be explained by the nonlinear theory of piezoelectricity. This interesting effect has been applied to frequency-temperature compensation [29] and electrostatic voltage sensors [30-32]. A recent attempt has also been made to use the electroelastic effect to reduce the acceleration sensitivity of quartz resonators [33]. Finally, the measurement of electroelastic effect can be used to determine the third-order piezoelectric constants [34]. While most of the applications mentioned above use quartz as the resonator material, La3 Ga5 SiO14 single crystals are of recent interest. Langasite belongs to point group 32, such that it has the same symmetry as quartz. It has good temperature behavior and piezoelectric coupling factor, low acoustic loss, and high Q factor [7]. Resonators made from this material are expected to perform better than those made from quartz [8]. The frequency shift of langasite resonators caused by temperature was investigated by Fritze et al. [6] and Mateescu et al. [35]. The force-frequency effect was studied by Kim et al. [14] and Kosinski et al. [15]. Research associated with the electroelastic effect for langasite resonators has not yet been reported. The sensitivity of the electroelastic effect to nonlinear material constants is critical for accurate determination of nonlinear material constants by resonator methods. Related work for quartz resonators was reported by Brendel [36] for one specific cut, while the analysis for langasite resonators has not been conducted. In this section, the frequency shift of a langasite resonator with arbitrary orientation under DC electric field is discussed based on concepts regarding small fields superposed on finite biasing fields in a thermoelectroelastic body [37] and perturbation theory [2]. The complete set of third-order material constants including third-order elastic, piezoelectric, dielectric, and electrostrictive constants is needed for this calculation. The contributions of nonlinear material constants to the electroelastic effect of langasite resonators are analyzed for several crystal cuts. By comparing the contribution of each group of nonlinear material constants to the electroelastic effect, specific cuts of interest are identified with respect to the nonlinear material constants. In Section 8.2, the first-order perturbation integral theory was introduced. Using this theory, the general solution of the dynamic response of an infinite, thin piezoelectric plate without DC biasing electric field is ob-

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tained in Section 8.4.2. The static stress, strain and electric field distribution of a doubly rotated langasite plate under a biasing static electric field in the thickness direction are obtained analytically and numerically in Section 8.4.3. In Section 8.4.4, the electroelastic effect and sensitivity analysis are obtained by both analytically and by finite element method. Finally, conclusions are made in Section 8.4.5.

8.4.2 Unperturbed modes The unperturbed modes are the same as the ones discussed in Section 8.3.2.

8.4.3 Biasing electric field When a DC electric field is applied in the thickness direction, the biasing electric field gives rise to static strain, stress, and electric field solutions. For resonators with simple geometry, such as the plano-plano configuration considered here, the solution may be obtained by analytical methods. However, for resonators with relatively complex geometry, such as plano-convex or double bevel configurations, the solution may be obtained by finite element methods. It is worth noting that the analytical solutions ignore edge effects that can be obtained by finite element method. 8.4.3.1 Analytical solution Consider a doubly rotated langasite plate fully coated with electrodes on both sides as shown in Fig.8.1. A static voltage ±V /2 is applied on the upper and lower surfaces of the plate. The material orientation is shown in Fig.8.2 which follows the IEEE standard 176-1987. The linear constitutive equations of piezoelectricity can be written [25] as Si0j = si jkl Tkl0 − dki j Ek0

(8.26)

D0i

(8.27)

= dikl Tkl0 + εik Ek0 0 =0 T ji, j D0i,i = 0

(8.28) (8.29)

where the strain and the electric field are written as 0 = (ω + ω )/2 Skl k,l l,k

Ek0

= −ϕ,k0

(8.30) (8.31)

In Eqs.(8.30) and (8.31), ωk is the displacement component and ϕ 0 is the electric potential. The boundary conditions are written as

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V ϕ 0 = ± , x2 = ±h 2 0 T2 j = 0, x2 = ±h

277

(8.32) (8.33)

Consider a possible solution for an infinite, thin plate as: Ti0j = 0, Si j = Ki j = const., ϕ,k0 = Fk = const. in Ω . Obviously, the boundary conditions as given by Eqs.(8.32) and (8.33) are satisfied, as is the equations of motion Eqs.(8.28) and (8.29). The static solutions are obtained as Si0j = dki j Ek0 in Ω Ti0j

= 0 in Ω V Ek0 = − in Ω 2h

(8.34) (8.35) (8.36)

The analytical solution is obtained. Next, we will discuss the way to use FEM to calculate the static solution. 8.4.3.2 Finite element solution A finite element model for a doubly rotated (Y Xwl Φ = 20◦ /Θ = 30◦ ) langasite resonator is constructed using FEMLAB 3.2. The model includes 2224 lagrangian quadratic elements with the number of boundary elements as 1640, such that there are 3 layers of elements. The sample radius is 6.5 mm, the thickness is 0.65 mm, and a 1000 V DC voltage is applied along the plate thickness direction. A voltage of this magnitude is appropriate for the example results calculated here. Such a voltage is simple to implement experimentally and results in a natural frequency shift (in the range of 0.07-1.5 Hz/V) that can be measured easily. The radial edge of the plate has zero charge. The convergence of the numerical solution is verified by refining the element size and by comparison with

Fig. 8.16 Static solution for shear strain S12 determined by the finite element method. The sample is an LGS resonator (Y Xwl Φ = 20◦ /Θ = 30◦ ) with the thickness = 0.65 mm and the radius = 6.5 mm.

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the analytical solutions. In the analytical solution, the strain, stress and field are assumed to be uniform based on the thin plate theory. An example finite element result, shown in Fig.8.16, shows the shear strain S12 of the sample above. The solution is basically uniform (solid gray throughout most of the plate) except for some edge effects (darker gray near the edges).

8.4.4 Results After the unperturbed natural frequency, mode shape, static stress, strain, and field are obtained, all material constants are transformed to the new coordinate. Then Eqs.(8.1) and (8.5) respectively may be used for the calculation of the electroelastic effect analytically and numerically. Example results are presented in this section. 8.4.4.1 Electroelastic effect The results for the electroelastic effect are shown in Fig.8.17. The electroelastic effect is df represented by , where, d f is the natural frequency shift, f0 is the natural frequency f0 E without any biasing electric field, and E is the electric field. For each mode (Mode A (longitudinal), Mode B (fast shear), Mode C (slow shear)), the electroelastic effect is plotted

Fig. 8.17 Electroelastic effect of LGS thickness mode resonator (Y Xwl Φ /Θ ).

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for four groups of resonator orientations: Y Xwl Φ = 0◦ to 60◦ /Θ = 0◦ , Y Xwl Φ = 0◦ to 60◦ /Θ = 20◦ , Y Xwl Φ = 0◦ to 60◦ /Θ = 40◦ , and Y Xwl Φ = 0◦ to 60◦ /Θ = 60◦ , respectively. The solid line is the analytical result, while the squares denote the finite element results. As expected, the electroelastic effect changes smoothly with cut angle. The comparison between the finite element result and the analytical result shows the consistency of these two methods. A slight difference between these two methods is observed that can be explained by the zero stress assumption for the analytical stress solution. Edge effects also contribute to this difference. The intersection of the curve with the X-axis shows a possible cut which is insensitive to electric field. This crossing point is not observed for Mode A or Mode C. Such a cut exists only for cut Y Xwl Φ = 50◦ /Θ = 20◦ in Mode B. It is worth noting that the electroelastic effect of Mode C is much larger in magnitude than the other two modes. Thus, Mode C of cut Y Xwl Φ = 30◦ /Θ = 0◦ may be an ideal mode for voltage sensor applications. 8.4.4.2 Sensitivity to nonlinear material constants Results showing the electroelastic effect for selected langasite resonators using both analytical and numerical solutions are now presented. Because the interest here is on the behavior with respect to third-order material constants, only the contribution to the electroelastic effect from third-order material constants is presented such that contributions from second-order material constants are excluded. Figures 8.18-8.21 show the analyti-

Fig. 8.18 Contribution to electroelastic effect from nonlinear material constants for Y Xwl Φ = 0◦ to 60◦ /Θ = 0◦ . The notation used includes TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

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cal and numerical results of the electroelastic effect for Modes A, B, and C. Cut angles of Θ = 0◦ , 20◦ , 40◦ , 60◦ are shown in Fig.8.18-8.21 respectively for Φ = 0◦ to 60◦ . The analytical results are shown as solid lines while the numerical results are denoted with a square. The first observation of the results is that the numerical results fit all analytical results very well. Thus, it may be concluded that the assumptions used in the analysis are sufficient for the resonators considered here. Specific observations of the results for each cut may also be made. Figure 8.18 shows the electroelastic effect for the Y Xwl

Fig. 8.19 Contribution to electroelastic effect from nonlinear material constants for Y Xwl Φ = 0◦ to 60◦ /Θ = 20◦ . The notation used includes TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

Φ = 0◦ to 60◦ /Θ = 0◦ . The results for Mode A (longitudinal mode) show that the primary contributions to the frequency shift come from third-order elastic and piezoelectric effects with little contribution from electrostrictive constants and no effective contribution from the third-order dielectric constants. For Mode B, it may be observed that the cuts Y Xwl Φ = 25◦ to 35◦ /Θ = 0◦ have zero contribution from the electrostrictive constants, such that this cut range could be termed the third-order elastic-piezoelectric cut (TOE-TOP cut). Thus, these cuts are ideal for experimental measurements to determine third-order elastic and piezoelectric constants without the need to consider the influence of electrostrictive constants. For Mode C, the same cut range also exists. Figure 8.19 shows the results for cuts Y Xwl Φ = 0◦ to 60◦ /Θ = 20◦ . Compared with the Y Xwl Φ = 0◦ to 60◦ /Θ = 0◦ cuts, these cuts have a larger contribution from electrostrictive constants. For Mode A, cut Y Xwl Φ = 25◦ /Θ = 20◦ may be designated as a third-order

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elastic-piezoelectric cut (TOE-TOP cut). It should also be noted that for Mode C, cut Y Xwl Φ = 48◦ /Θ = 20◦ (the crossing point) is a third-order piezoelectric constants cut (TOP cut), because the electroelastic response is the same as that obtained by considering effects from third-order piezoelectric constants only. Special cuts for other values of Θ are also expected and require further investigation. In Fig.8.20, we notice in Mode C, the influence of the electrostrictive constants to electroelastic effect is the largest for the cut range Y Xwl Φ = 0◦ to 22◦ /Θ = 40◦ . It even surpasses the contribution from the third-

Fig. 8.20 Contribution to electroelastic effect from nonlinear material constants for Y Xwl Φ = 0◦ to 60◦ /Θ = 40◦ . The notation used includes TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

order elastic and piezoelectric constants. Therefore, this cut range may be appropriate for determination of electrostrictive constants if third-order elastic and piezoelectric constants are known. In Fig.8.21, Y Xwl Φ = 25◦ /Θ = 60◦ of Mode C is the “third-order elastic constants cut (TOE cut)”. We note that for cut Y Xwl Φ = 35◦ /Θ = 60◦ , the third-order piezoelectric constants have zero contribution to the electroelastic effect which means that this cut can be used to determine electrostrictive and third-order elastic constants. This cut may be denoted as an “electrostrictive-third-order elastic constants cut”.

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Fig. 8.21 Contribution to electroelastic effect from nonlinear material constants for Y Xwl Φ = 0◦ to 60◦ /Θ = 60◦ . The notation used includes TOE (third-order elastic constants), TOP (third-order piezoelectric constants), TOD (third-order dielectric constants), ES (electrostrictive constants).

8.4.5 Conclusions In this section, analytical and finite element solution for determining the electroelastic effect and the sensitivity analysis of thickness mode langasite resonator are obtained. The FEM results were shown to fit the analytical results very well for the range of cuts examined. Thus, the applicability of the assumptions necessary for the analytical solution was verified for plano-plano resonator configurations. The results show the cuts that may be used for the case when the frequency stability is required when the resonator is exposed to external field. There also exist possible cuts for voltage sensor applications. The contribution to the electroelastic effect from each group of constants depends on the cut angle and mode. Overall, it was observed that the third-order dielectric constants contribute little to the electroelastic effect. The major contribution usually comes from third-order elastic and piezoelectric constants. Also, there exist special cuts that are dominated by a specific group of constants such as a third-order elastic constants cut or a third-order piezoelectric constants cut. A pure electrostrictive constants cut was not observed for the cases examined. The contribution from electrostrictive constants is generally small although some cuts did have a significant contribution that cannot be ignored. The results from this anal-

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ysis provide valuable insight into the sensitivity of the electroelastic effect to the nonlinear material behavior.

References [1] Tiersten H F. On the accurate description of piezoelectric resonators subject to biasing deformations. Int. J. Eng. Sci., 1995, 33, 2239-2259. [2] Tiersten H F. Perturbation theory for linear electroelastic equations for small fields superposed on a bias. J. Acoust Soc. Am., 1978, 64, 832-837. [3] Sorokin B P, Turchin P P, Burkov S I, et al. Influence of static electric field, mechanical pressure and temperature on the propagation of acoustic waves in La3 Ga5 SiO14 piezoelectric single crystals. Proceedings of IEEE International Frequency Control Symposium, 1996, 161169. [4] Zhang H, Turner J A, Yang J, et al. Electroelastic effect of thickness mode langasite resonators. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2007, 2120-2128. [5] Tiersten H F, Sinha B K. Temperature dependence of the resonant frequency of electrode doubly rotated quartz thickness-mode resonators. J. Acoust. Soc. Am., 1979, 50, 8038-8051. [6] Fritze H M, Schulz H S, Seh H, et al. High temperature operation and stability of langasite resonators. Materials Research Society Symposium Proceedings, 2005, 835, 157-162. [7] Naumenko N. Optimal cuts of langasite, La3 Ga5 SiO14 for SAW devices. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2001, 48, 530-537. [8] Smythe R C, Helmbold R C, Hague G E. Langasite, langanite, and langatate bulk-wave Y-cut resonators. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2000, 47, 355-360. [9] Rosati V J, Filler R L. Reduction of the effects of vibration on sc-cut quartz crystal oscillators. Proceedings of IEEE International Frequency Control Symposium, 1981, 117-121. [10] Rosati V J. Suppression of vibration-induced phase noise in crystal oscillators: an update. Proceedings of IEEE International Frequency Control Symposium, 1987, 409-412. [11] Barthod C. New force sensor based on a double ended tuning fork. Proceedings of IEEE International Frequency Control Symposium, 2000, 74-78. [12] Clayton L D, Eernisse E P. Quartz resonator frequency shifts computed using the finite element method. Int. J. Numer. Meth. Eng., 1993, 36, 385-401. [13] Boy J J, Besson R J, Bigler E, et al. Theoretical and experimental studies of the forcefrequency effect in BAW LGS and LGT resonators. Proceedings of IEEE International Frequency Control Symposium, 2001, 223-226. [14] Kim Y, Ballato A. Force-frequency effect of Y-cut langanite and Y-cut langatate. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2003, 50, 1678-1682. [15] Kosinski J A, Pastore R A, Yang X, et al. Stress-induced frequency shifts in langasite thickness-mode resonators. Proceedings of IEEE International Frequency Control Symposium, 2003, 716-722. [16] Sinha B K. Stress-induced frequency shifts in thickness-mode resonators. Proceedings of IEEE Ultrasonics Symposium, 1980, 813-818. [17] Yang J. An Introduction to the Theory of Piezoelectricity. Advances in Mechanics and Mathematics, Band 9. Springer-Verlag, GmbH, 2005. [18] Gehin C, Samper S, Teisseyre Y. Mounting characterization of a piezoelectric resonator using FEM. Proceedings of IEEE International Frequency Control Symposium, 1997, 630-633.

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[19] Cowdrey D R, Willis J R. Finite element calculation relevant to AT-cut quartz resonators. Proceedings of IEEE International Frequency Control Symposium, 1973, 7-10. [20] Yong Y K, Patel M S. The impact of finite element analysis on the design of quartz resonators. Proceedings of IEEE International Frequency Control Symposium, 2006, 9-23. [21] Patel M S, Yong Y K, Tanaka M, et al. Drive level dependency in quartz resonators. Proceedings of IEEE International Frequency Control Symposium, 2005, 793-801. [22] Dulmet B. Finite element analysis of activity-dips in BAW resonators and sensors. Proceedings of IEEE International Frequency Control Symposium, 2002, 179-190. [23] Stevens D S, Tiersten H F. An analysis of sc-cut quartz trapped energy resonators with rectangular electrodes.Proceedings of IEEE International Frequency Control Symposium, 1981, 205-212. [24] IEEE Standard on Piezoelectricity. IEEE, New York, 1987. [25] Tiersten H F. Linear Piezoelectric Plate Vibrations. New York: Plenum, 1969. [26] Lee P C Y, Wu K M. In-plane accelerations and forces on frequency changes in doubly-rotated quartz plates. J. Acoust. Soc. Am., 1984, 75, 1105-1117. [27] Ratajski J M. Force-frequency coefficient of singly rotated vibrating quartz crystals. IBM J. Res. Dev., 1968, 12, 92-99. [28] Kosinski J A, Pastore R A, Yang J, et al. Perturbation theory for degenerate acoustic eigenmodes. Proceedings of IEEE International Frequency Control Symposium, 2003, 734-741. [29] Chen Q, Zhang T, Wang Q M. Frequency-temperature compensation of piezoelectric resonators by electric dc bias field. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2005, 52, 1627-1631. [30] Ishido M, Ojima K, Kikuta N. Electrostatic voltage sensor and displacement sensor, using saw directional coupler. Proceedings of IEEE International Ultrasonics Symposium, 1991, 341-344. [31] Ishido M, Zhu X. Electrostatic voltage sensor using saw oscillator with waveguide. Proceedings of IEEE International Ultrasonics Symposium, 1987, 383-387. [32] Bauerschmidt P, Lerch R. Optical voltage sensor based on a quartz resonator. Proceedings of IEEE International Ultrasonics Symposium, 1996, 383-387. [33] Yong Y K, Patel M S. Application of a DC-bias to reduce acceleration sensitivity in quartz resonators. Int. J. Appl. Electrom. Mech., 2005, 22, 69-82. [34] Hruska K, Janik L. Change in elastic coefficient and module of alfa quartz in an electric field. Czech J. Appl. Phys., 1968, 18, 112-116. [35] Mateescu I, Zelenka J, Nosek J, et al. Frequency-temperature characteristics of the langasite resonators.Proceedings of IEEE International Ultrasonics Symposium, 2001, 263-267. [36] Brendel R. Material nonlinear piezoelectric coefficients for quartz. J. Appl. Phys., 1983, 54, 5339-5346. [37] Yang J. Equations for small fields superposed on finite biasing fields in a thermoelectroelastic body. IEEE Trans. Ultrason., Ferroelectr., Freq. Contr., 2003, 50, 187-192.

Chapter 9 Wave Propagation in a Piezoelectric Plate with Surface Effect

WeiQiu Chen Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

Abstract There is an increasing academic interest in the mechanical behavior of bodies of small size because some size-dependent phenomena have been observed experimentally or through molecular dynamics simulation. Continuum mechanics has demonstrated very powerful to explain these particular phenomena in a relatively simple theoretical framework. One such continuum theory is to treat the surface of the body as a material surface, which is endowed with different material properties than the bulk material. In this chapter, we will follow the same idea to consider surface effect in a piezoelectric plate and study its influence on the wave propagation characteristic (i.e., dispersion relation). A novel method is developed to establish the surface piezoelectricity theory using the state-space formulations, based on which an approximate transfer relation between state vectors at different positions can be easily obtained. The effect of surface piezoelectricity is then taken into account by establishing the effective boundary conditions for the plate of bulk material. Both the Love type and Rayleigh-Lamb type waves are considered and numerical results are presented. Keywords piezoelectric plate, surface effect, state-space formulations, effective boundary conditions, wave propagation, dispersion

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9.1 Introduction With the development of modern technology, the size of structures or components gets smaller and smaller. Many micro- and even nano-sized multifunctional structures and devices have been widely used in science and engineering, and are known as the microelectro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS). In a usual macro-structure, there is no need to consider the material heterogeneity regarding the surface and bulk regions, which can be attributed to the different environmental constraints imposed on the atoms at different positions. In MEMS or NEMS, the aspect surface-to-volume ratio is much larger than that of a macro-structure, and the surface may play an important role in determining the mechanical behavior of these tiny systems [1,2]. The surface/interface effects have already been noticed and studied for more than one hundred years since Gibbs’ pioneering contribution, with a massive body of literature that can be found on the subject. Of particular relevance to the present work are studies using continuum mechanics methods. Gurtin and Murdoch [3] developed a rigorous mathematical framework for the continuum theory of a deformable material surface. It is shown that the conventional boundary conditions in the classical elasticity are replaced by a set of two-dimensional (2D) differential equations governing the surface deformation [4]. Based on this continuum theory (hereafter referred to as the GM theory), Murdoch [5] studied the propagation of Love and Rayleigh waves in elastic bodies with material boundaries, revealing that the wave propagation behavior is sensitive to the relative values of the residual stress, elastic moduli, and density of the surface. Gurtin and Murdoch [6] showed that the reflection of plane harmonic waves by the free surface of an elastic halfspace with material boundary becomes significantly different from the classical theory at very high frequencies. Murdoch [7] also investigated the propagation of interfacial Stoneley waves along a material interface between two half-spaces, showing that such waves exist in almost all cases, in contrast to the classical elasticity theory which predicts that no Love type waves can propagate along the interface, and that the existence of Rayleigh type waves requires similar acoustical properties of the two half-spaces. It is interesting to point out that, through a particular procedure following the development of plate theories, Mindlin [8] was able to derive the approximate or effective boundary conditions on the plane boundary of a plate covered with a very thin layer. Tiersten [9] presented a comprehensive study of surface waves guided by thin films and showed that Mindlin’s treatment is accurate enough when the wave length is much larger than the thickness of the film. It has been shown that the Mindlin-Tiersten model (hereafter abbreviated as the MT model) is exactly the same as the linearized GM theory [4,5,7] when the residual surface tension is absent and when the surface parameters are properly defined. The effective boundary conditions can also be derived by assuming that the surface or interface is a material layer of a small thickness h and using an asymptotic analysis or Taylor’s series expansion. Theoretically, such a method allows the effective boundary conditions accurate up to an arbitrary order, O(hn ), here n is an integer. Rokhlin and Wang [10] considered the problem of wave propagation along an isotropic interface between two solids, for which the asymptotic expressions of reflection and transmission

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coefficients were used, and presented the effective boundary conditions for the first order, O(h). B¨ovik [11] considered a more general curved surface embedded in-between two media (either solid or fluid), and also presented the first-order effective boundary conditions. Niklasson et al. [12] formulated the second-order effective boundary conditions to replace an anisotropic coating for a two-dimensional problem. Niklasson et al. [13] extended their analysis to the case of a thin anisotropic layer lying between two isotropic layers. Recently, Ting [14] derived the O(hn ) effective boundary conditions for an anisotropic elastic layer attached perfectly or imperfectly to a body; his derivation was also for 2D problems and the Stroh formalism was employed. The O(h) and O(h2 ) effective boundary conditions for a thin piezoelectric layer have also been obtained by Johansson and Niklasson [15] for 1D and 2D problems. Their work was later extended to the establishment of a 2D theory of an arbitrary order for an elastic plate attached with a thin piezoelectric layer [16]. In this chapter, we will present a theory of surface piezoelectricity that can be truncated up to an arbitrary order, say O(hn ), for the plane surface of a 3D piezoelectric body. The surface is modeled as a thin piezoelectric layer that may be endowed with different material properties than the bulk material. The state-space formalism is employed, which enables an easy derivation of a transfer relation between the state vectors at the top and bottom surfaces of the layer. The transfer matrix, when expressed in terms of power series, naturally provides an asymptotic truncation scheme for the effective boundary conditions, which governs the motion of the piezoelectric material surface. The proposed theory can be used conveniently to study the surface effect in micro/nano-sized piezoelectric structures or devices.

9.2 Isotropic elastic material surface In this section, we first consider the surface model of an isotropic elastic material as a starting point. The governing equations of a three-dimensional isotropic elastic medium are = ρ u¨i 1 εi j = (ui, j + u j,i ) 2 i j = λ εkk δi j + 2 µεi j ji, j

(9.1) (9.2) (9.3)

where i j , εi j and ui are the stress tensor, strain tensor, and displacement vector, respectively; ρ is the density, and λ and µ are Lam´e constants of an isotropic material; δi j is the Kronecker delta. i or j following the comma indicates differentiation with respect to xi or x j in a Cartesian coordinate system, and the over dot implies differentiation with respect to time. The convention of summation over repeated indices is also employed. In this chapter, we will model the surface of a body as a material layer, with possibly different material properties than the bulk material. For simplicity, we consider a plane

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material surface so that the layer has two parallel plane surfaces (i.e., the top and bottom surfaces) with a uniform thickness h, as demonstrated in Fig.9.1. Instead of treating the material layer as a different material phase directly, we will present the so-called effective boundary conditions at x3 = 0, which can take account of the effect of that layer in an approximate sense, and greatly simplify the succeeding mathematical analysis.

Fig. 9.1 A surface material layer attached to the bulk material.

For illustration, here we further confine ourselves to the two-dimensional case (i.e., plane strain). In this case, the displacement component u2 vanishes, while u1 and u3 are functions of x1 and x3 only. For comparison, we first follow the method of B¨ovik [11] (also see Niklasson et al. [12,13]) to derive the effective boundary conditions, which are exact up to the order O(h). Thus, we assume s j3 (x1 , −h) =

s j3 (x1 , 0) −

∂

s (x , 0) j3 1

∂ x3

h + O(h2 )

( j = 1, 3)

(9.4)

where the superscript s denotes the surface layer. We consider the following boundary conditions at the top surface x3 = −h and the continuity conditions at the interface x3 = 0: s (x , −h) = −p j j3 1 s (x , 0) = u (x , 0) (x , 0), u j 1 j3 1 j 1

s (x , 0) = j3 1

(9.5) (9.6)

where j = 1, 3 and the quantities without superscript s are affiliated with the bulk material. Here p j are the prescribed surface tractions at x3 = −h. From Eq.(9.1), we get for the considered plane-strain problem s j3,3

= ρ s u¨sj −

s j1,1

( j = 1, 3)

(9.7)

When j = 3, we obtain

∂

s (x , 0) 33 1

∂ x3

= ρ s u¨3 (x1 , 0) −

∂

13 (x1 , 0)

∂ x1

(9.8)

where the continuity conditions (9.6) have been noticed. From the constitutive equations, and making use of the geometric relations (9.2), we can get

∂ 2 us3 ∂ s11 ∂ 2 us = (λ s + 2µ s ) 21 + λ s ∂ x1 ∂ x1 ∂ x3 ∂ x1

(9.9)

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

∂ us3 1 = s ∂ x3 λ + 2µ s

s − 33

∂ us1 λs λ s + 2µ s ∂ x1

289

(9.10)

The above two equations further give rise to

∂ s33 ∂ s11 4(λ s + µ s )µ s ∂ 2 us1 λs + = ∂ x1 λ s + 2µ s ∂ x12 λ s + 2µ s ∂ x1

(9.11)

Thus, in view of Eqs.(9.11) and (9.6), we obtain from Eq.(9.7) when j = 1

∂

s (x , 0) 13 1

∂ x3

= ρ s u¨1 (x1 , 0) −

4(λ s + µ s )µ s ∂ 2 u1 (x1 , 0) λs ∂ − 2 s s s s λ + 2µ λ + 2µ ∂ x1

33 (x1 , 0)

∂ x1

(9.12)

Then, in terms of Eqs.(9.8) and (9.12), we derive from Eq.(9.4) µ ¶ λs ∂ 33 (x1 , 0) 4(λ s + µ s )µ s ∂ 2 u1 (x1 , 0) s (x , 0) + ρ u ¨ (x , 0) + h − 13 1 1 1 λ s + 2µ s ∂ x1 λ s + 2µ s ∂ x12 = −p1 + O(h2 )

(9.13) µ

¶ ∂ 13 (x1 , 0) s − ρ u¨3 (x1 , 0) h = −p3 + O(h2 ) 33 (x1 , 0) + ∂ x1

(9.14)

Equations (9.13) and (9.14) represent the effective boundary conditions at x3 = 0, in which the effect of surface layer has been taken into consideration by the inclusion of parameters with superscript s, in an approximate sense that they are accurate up to the order of O(h). Equations (9.13) and (9.14) can also be directly obtained from Eq.(9.10) of Niklasson et al. [12] using a proper degenerate analysis. The O(h2 ) effective boundary conditions have also been derived by Niklasson et al. [12] using the same argument, but involving a more tedious mathematical manipulation. Now we will present a different yet simpler strategy to derive the same things. To this end, we can rewrite Eqs.(9.1) - (9.3) for the plane-strain problem as

∂V = AV ∂ x3 where V = [u1 , u3 , pose, and

T 13 , 33 ]

(9.15)

is the state vector, here the superscript T denotes the trans-

 0

−

∂ ∂ x1

1 µ

 0

   λ ∂ 1  − 0 0  λ + 2 µ ∂ x λ + 2µ 1  A= 2 2 ∂ 4( λ + µ ) µ ∂ λ ∂  0 0 − ρ 2 − 2  ∂t λ + 2µ ∂ x1 λ + 2µ ∂ x1   2 ∂ ∂ ρ 2 − 0 0 ∂t ∂ x1

           

(9.16)

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Equation (9.15) is known as the state equation of the problem [17-23]. It is seen that the matrix A contains partial differential operators ∂ /∂ x1 and ∂ /∂ t. To solve a practical boundary-value problem based on such a state equation, the next step is to eliminate these operators by either directly assuming the variations of state variables, or by using integral transforms, with respect to x1 and/or t, as shown in Refs. [24-29], to name a few. Here, without such a step, we write directly the solution to Eq.(9.15) as V(x3 ) = exp(Ax3 )V(0)

(9.17)

where the operators ∂ /∂ x1 and ∂ /∂ t in the operational matrix A have been seen as parameters so that Eq.(9.17) should be understood in an operational interpretation [30]. Setting x3 = −h in Eq.(9.17), gives (9.18)

V(−h) = exp(−Ah)V(0)

which relates the state vector at x3 = −h to that at x3 = 0 through the transfer matrix exp(−Ah). According to the definition, the matrix exponential can be expressed in terms of the following power series: 1 (−1)n n n exp(−Ah) = I − Ah + A2 h2 + · · · + A h + O(hn+1 ) 2 n!

(9.19)

We may obtain from Eqs.(9.18) and (9.19) V(−h) = (I − Ah)V(0) + O(h2 )

(9.20a)

or     u1 (x1 , −h)  u1 (x1 , 0)            u3 (x1 , −h) u3 (x1 , 0) =   13 (x1 , 0)  13 (x1 , −h)          33 (x1 , −h) 33 (x1 , 0) 

∂ − ∂ x1

1 µ



0 0       λ ∂ 1   − 0 0   λ + 2µ ∂ x1 λ + 2µ   − h 2 2  ∂ 4(λ + µ )µ ∂ λ ∂  ρ  − 0 0 −  ∂ t2 λ + 2µ ∂ x12 λ + 2µ ∂ x1      ∂2 ∂ 0 ρ 2 − 0 ∂t ∂ x1   u1 (x1 , 0)       u3 (x1 , 0) × + O(h2 ) (9.20b) (x , 0)   13 1     33 (x1 , 0) The third and fourth equations in Eq.(9.20) are exactly the same as Eqs.(9.13) and (9.14) once incorporating the boundary conditions at x3 = −h.

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

291

If we keep the O(h2 ) terms in the series expansion (9.19), we can get the O(h2 ) effective boundary conditions, which can be shown to be the same as those presented in Niklasson et al. [12] for an isotropic material layer. In fact, we can derive the effective boundary conditions accurate up to an arbitrary order as simple as that presented above. This will be shown in the next section for a plane material surface with coupling between the elastic and electric fields.

9.3 Surface piezoelectricity We now turn to develop a theory (for a plane boundary) that accounts for the surface effect in a piezoelectric body without residual stress and pre-existing electric displacement. The basic equations for piezoelectricity may be grouped into three catalogues: the divergence, the gradient and the constitutive equations [31]. The divergence equations include Eq.(9.1) for the mechanical equilibrium and the following Gaussian equation for the electric equilibrium: Di,i = 0 (9.21) where Di is the electric displacement vector. Equations (9.1) and (9.21) indicate that the body force and free electric charge are assumed to be absent. In addition to Eq.(9.2), which gives the mechanical strain-displacement relations, the gradient equations for the electric field are given by Ei = −φ,i (9.22) where Ei and φ are the electric field vector and electric potential, respectively. Since coupling exists between the elastic and electric fields in a piezoelectric material, the constitutive relations become ij

= ci jkl εkl − eki j Ek ,

Di = eikl εkl + κik Ek

(9.23)

where ci jkl , eki j and κi j are the elastic, piezoelectric and dielectric constants, respectively. The state equation for a general piezoelectric material then can be derived from Eqs.(9.1), (9.2), (9.21) - (9.23) as follows [32]: #½ ¾ ½ ¾ " ½ ¾ D11 C−1 ∂ u u u 33 = =A (9.24) T ∂ x3 3 3 3 D21 + K −D11 where u = [u1 , u2 , u3 , φ ]T is the generalized displacement vector, and i = [ 1i , are the generalized stress vectors; and     c1i1 j c1i2 j c1i3 j e j1i ∂2  c2i1 j c2i2 j c2i3 j e j2i    ρ ∂ t 2 I3×3 03×1  Ci j =   c3i1 j c3i2 j c3i3 j e j3i  , K = 01×3 0 ei1 j ei2 j ei3 j −κi j

T 2i , 3i , Di ]

(9.25)

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WeiQiu Chen

  ∂    I   ∂ x1 D11 = −C−1 [C C ] , 31 32 33   ∂ I    ∂ x2   ∂  · ¸· ¸   I ∂ ∂ Q11 Q12 ∂ x1 D21 = − I I , Q21 Q22  ∂ x1 ∂ x2  ∂ I  ∂ x2 Qi j = Ci j − Ci3 C−1 C 3 j 33

(9.26)

By treating the partial operators ∂ /∂ x1 , ∂ /∂ x2 and ∂ /∂ t as usual parameters, the solution to the state equation (9.24) may be formally written as ½ ¾ ½ ¾ u(x3 ) u(0) = exp(Ax3 ) (9.27) 3 (x3 ) 3 (0) which should be understood in an operational sense, as mentioned before. Setting x3 = −h in Eq.(9.27) leads to the following transfer relation between the state vectors at the bottom and top surfaces of the layer: ½ ¾ ½ ¾ u(−h) u(0) = exp(−Ah) (9.28) 3 (−h) 3 (0) Now making use of the power series expression for the matrix exponential, we obtain ½ ¾ · ¸½ ¾ 1 (−1)n n n u(−h) u(0) = I − Ah + A2 h2 + · · · + A h + O(hn+1 ) (9.29) 2 n! 3 (−h) 3 (0) For simplicity, we assume that the upper surface x3 = −h of the material boundary layer is free from tractions as well as surface electric charge, and we also ignore the electric field in the free space. Thus, we have the following boundary conditions: 3 (−h) = 0

(9.30)

With this, we can derive the following relations, which are accurate up to the order of O(hn ) at x3 = 0 from the last four equations in Eq.(9.29) as 1 1 3 (0) − [A21 u(0) + A22 3 (0)]h +

+···+

(−1)n n [A21 u(0) + An22 n!

1 2 [A u(0) + A222 2 21

3 (0)]h

n

where An21 and An22 are the sub-matrices of An such that · n ¸ A11 An12 An = An21 An22

=0

3 (0)]h

2

(9.31)

(9.32)

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

293

With this notation, An21 6= (A121 )n , i.e., the superscript n of the former is not an index (power), but it indicates a sub-matrix of An , the nth power of A. By taking into consideration the concrete form of the operational matrix A as well as the following continuity conditions at x3 = 0 between the surface layer and the bulk material: s us (0) = u(0) (9.33) 3 (0) = 3 (0), where as before, the superscript s has been used to indicate the material surface, the O(h) effective boundary conditions can be obtained as s s s T 3 (0) − [(D21 + K )u(0) − (D11 ) 3 (0)]h

=0

(9.34)

The O(h2 ) effective boundary conditions can be obtained as s s s T 3 (0) − [(D21 + K )u(0) − (D11 ) 3 (0)]h

1 + {[(Ds21 + Ks )Ds11 − (Ds11 )T (Ds21 + Ks )]u(0) 2 +[(Ds21 + Ks )(Cs11 )−1 + ((Ds11 )T )2 ] 3 (0)}h2 = 0

(9.35)

The other higher-order effective boundary conditions may also be obtained in a similar manner. Note that these effective boundary conditions are just the governing equations for the surface piezoelectricity for a plane piezoelectric surface in absence of residual stress and electric displacement. They will be employed in the following section to investigate the surface effect on the wave propagation in a piezoelectric plate.

9.4 Waves in a piezoelectric plate with surface effect In the following, we consider a thin piezoelectric plate which is polarized in the thickness direction (the x3 direction), for which the constitutive relations read as µ ¶ ∂ u1 ∂ u2 ∂ u3 ∂φ ∂ u1 ∂ u3 ∂φ + c12 + c13 + e31 , + + e15 , 11 = c11 13 = c44 ∂ x1 ∂ x2 ∂ x3 ∂ x3 ∂ x3 ∂ x1 ∂ x1 µ ¶ ∂ u1 ∂ u2 ∂ u3 ∂φ ∂ u2 ∂ u3 ∂φ + c11 + c13 + e31 , + + e15 , 22 = c12 23 = c44 ∂ x1 ∂ x2 ∂ x3 ∂ x3 ∂ x3 ∂ x2 ∂ x2 µ ¶ ∂ u1 ∂ u2 ∂ u3 ∂φ ∂ u1 ∂ u2 + c13 + c33 + e33 , + 33 = c13 12 = c66 ∂ x1 ∂ x2 ∂ x3 ∂ x3 ∂ x2 ∂ x1 (9.36) µ D1 = e15 D3 = e31

∂ u1 ∂ u3 + ∂ x3 ∂ x1

¶

∂φ − κ11 , ∂ x1

µ D2 = e15

∂ u1 ∂ u2 ∂ u3 ∂φ + e31 + e33 − κ33 ∂ x1 ∂ x2 ∂ x3 ∂ x3

∂ u2 ∂ u3 + ∂ x3 ∂ x2

¶ − κ11

∂φ , ∂ x2

(9.37)

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where we have c66 = (c11 − c12 )/2 for poled ceramics, which exhibits the hexagonal material symmetry (i.e., transverse isotropy). In the above equations, we have made use of the gradient relations (9.2) and (9.22). The corresponding operational matrix A can be obtained as A = [Ai j ](i, j = 1, 2), with    1  ∂ e15 ∂ 0 0 − − 0 0 0   ∂ x1 c44 ∂ x1    c44     ∂ e15 ∂      1 0 0 − −    0 0 0  ∂ x2 c44 ∂ x2     c 44 A11 =   , A12 =  ,    ∂ ∂ κ33 e33   −g1    −g1 0 0 0    0 ∂ x1 ∂ x2 α α      e33 c33   ∂ ∂ 0 0 − −g2 −g2 0 0 α α ∂ x1 ∂ x2   2 2 2 2 ∂ ∂ ∂ ∂ (cˆ11 + c66 ) 0 0   ρ ∂ t 2 + cˆ11 ∂ x2 − c66 ∂ x2 ∂ x1 ∂ x2 1 2     ∂2 ∂2 ∂2 ∂2  (cˆ11 + c66 ) ρ 2 − c66 2 + cˆ11 2 0 0    ∂ x1 ∂ x2 ∂t A21 =  ∂ x1 ∂ x2 ,   2   ∂  0 0 ρ 2 0    ∂t 

0

0



0

κˆ 11 Λ

∂ ∂ 0 0 −g1 −g2  ∂ x ∂ x1  1    ∂ ∂    0 0 −g1 −g2   ∂ x2 ∂ x2   A22 =     ∂ ∂  −  − 0 0   ∂ x ∂ x 1 2    e15 ∂  e15 ∂ − − 0 0 c44 ∂ x1 c44 ∂ x2 where Λ = ∂ 2 /∂ x12 + ∂ 2 /∂ x22 is the planar Laplacian, and

κˆ 11 = κ11 + e215 /c44 , g1 = (c13 κ33 + e31 e33 )/α ,

cˆ11 = c13 g1 + e31 g2 − c11 ,

g2 = (c13 e33 − e31 c33 )/α ,

α = c33 κ33 + e233

Thus, the O(h) effective boundary conditions are obtained as µ ¶ 2 ∂2 ∂2 ∂ 2 u2 (0) s ∂ s s h (0) − ρ + c ˆ h − c h u1 (0) − (cˆs11 + cs66 )h 13 11 66 2 2 2 ∂t ∂ x1 ∂ x2 ∂ x1 ∂ x2 ∂ 33 (0) ∂ D3 (0) +gs1 h + gs2 h = 0, ∂ x1 ∂ x1 µ ¶ 2 ∂ 2 u1 (0) ∂2 ∂2 s s s ∂ s s (0) − ( c ˆ + c )h ρ h + c ˆ h u2 (0) − − c h 23 11 11 66 66 ∂ x1 ∂ x2 ∂ t2 ∂ x12 ∂ x22

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

∂ D3 (0) = 0, ∂ x2 ∂ x2 2 ∂ 13 (0) ∂ 23 (0) s ∂ u3 (0) + h+ h = 0, 33 (0) − ρ h 2 ∂t ∂ x1 ∂ x2 es ∂ 13 (0) es15 ∂ 23 (0) s hΛφ (0) + 15 h D3 (0) − κˆ 11 + s h =0 cs44 ∂ x1 c44 ∂ x2 +gs1 h

∂

33 (0)

+ gs2 h

295

(9.38)

In the following, we will confine ourselves to the O(h) effective boundary conditions (or surface piezoelectricity theory) only, and the higher-order ones are omitted for brevity.

9.4.1 Decomposition To facilitate the analysis, we introduce two displacement functions Ψi and two stress functions Γi (i = 1, 2) through

∂Ψ1 ∂Ψ2 ∂Ψ1 ∂Ψ2 − , u2 = − ∂ x2 ∂ x1 ∂ x1 ∂ x2 ∂Γ1 ∂Γ2 ∂Γ1 ∂Γ2 − , − 13 = − 23 = ∂ x2 ∂ x1 ∂ x1 ∂ x2

u1 = −

(9.39) (9.40)

by which, the expressions for 13 and 23 in Eq.(9.36) for the bulk material give rise to · ¸ · ¸ ∂ ∂Ψ1 ∂ ∂Ψ2 Γ1 − c44 Γ2 + c44 u3 + e15 φ − c44 − − = 0, ∂ x2 ∂ x3 ∂ x1 ∂ x3 · ¸ · ¸ ∂ ∂Ψ1 ∂ ∂Ψ2 Γ1 − c44 − Γ2 + c44 u3 + e15 φ − c44 =0 (9.41) ∂ x1 ∂ x3 ∂ x2 ∂ x3 According to Ding et al. [33], we can obtain from the above equations

∂Ψ1 =0 ∂ x3 ∂Ψ2 Γ2 + c44 u3 + e15 φ − c44 =0 ∂ x3 Γ1 − c44

(9.42) (9.43)

Substitution from Eqs.(9.36), (9.37) and (9.40) into the first two of Eq.(9.1), the equations of motion, yields ∂A ∂B ∂A ∂B − − = 0, − =0 (9.44) ∂ x2 ∂ x1 ∂ x1 ∂ x2 where

∂Γ1 ∂ 2Ψ1 −ρ , ∂ x3 ∂ t2 ∂ u3 ∂φ ∂Γ2 ∂ 2Ψ2 B = c11 ΛΨ2 − c13 − e31 + −ρ ∂ x3 ∂ x3 ∂ x3 ∂ t2 A = c66 ΛΨ1 +

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Similarly, Eq.(9.44) implies

∂Γ1 ∂ 2Ψ1 −ρ =0 ∂ x3 ∂ t2 ∂ 2Ψ2 ∂ u3 ∂φ ∂Γ2 c11 ΛΨ2 − c13 − e31 + −ρ =0 ∂ x3 ∂ x3 ∂ x3 ∂ t2 c66 ΛΨ1 +

(9.45) (9.46)

Then, we can derive from the fifth of Eq.(9.36), the third of Eq.(9.37), the third of Eq.(9.1), and Eq.(9.21): c33

∂ u3 ∂φ ∂ u3 ∂φ + e33 = 33 + c13 ΛΨ2 , e33 − κ33 = D3 + e31 ΛΨ2 ∂ x3 ∂ x3 ∂ x3 ∂ x3 ∂ 2 u3 ∂ 33 ∂ D3 ∂Ψ2 = ρ 2 + ΛΓ2 , = e15 Λ − e15 Λu3 + κ11 Λφ ∂ x3 ∂t ∂ x3 ∂ x3

(9.47) (9.48)

In such a way, we have transformed the original governing equations for the bulk piezoelectric material into the ones that are expressed by two displacement functions Ψi (i = 1, 2), two stress functions Γi (i = 1, 2), the displacement component u3 , the normal stress component 33 , the electric potential φ and the electric displacement component D3 . We can view Eqs.(9.42), (9.43) and (9.47) as the modified constitutive relations, and Eqs.(9.45), (9.46) and (9.48) as the modified equilibrium equations. The decomposition technique has been widely employed in the analysis of plate and shell structures made of transversely isotropic (or spherically isotropic) materials, see Refs. [25-29,31,33,34] for examples. By making use of the modified constitutive relations, the modified equilibrium equations can be rewritten in terms of Ψ1 , Ψ2 , u3 and φ as

∂ 2Ψ1 ∂ 2Ψ1 − ρ =0 ∂ t2 ∂ x32

(9.49)

∂ 2Ψ2 ∂ u3 ∂φ ∂ 2Ψ2 − (c13 + c44 ) − (e31 + e15 ) −ρ =0 2 ∂ x3 ∂ x3 ∂ t2 ∂ x3

(9.50)

∂ 2 u3 ∂ 2φ ∂Ψ2 ∂ 2 u3 + e Λ φ + e − (c + c )Λ − ρ =0 33 13 44 15 ∂ x3 ∂ t2 ∂ x32 ∂ x32

(9.51)

∂ 2 u3 ∂ 2φ ∂Ψ2 − κ11 Λφ − κ33 2 − (e31 + e15 )Λ =0 2 ∂ x3 ∂ x3 ∂ x3

(9.52)

c66 ΛΨ1 + c44 c11 ΛΨ2 + c44 c44 Λu3 + c33

e15 Λu3 + e33

It can be seen that Ψ1 is decoupled from the other three unknowns. The decomposition also can be applied to the effective boundary conditions. We may obtain from the first two of Eq.(9.38) · ¸ 2 ∂ s s ∂ Ψ2 (0) s s cˆ hΛΨ2 (0) + ρ h − Γ2 (0) + g1 h 33 (0) + g2 hD3 (0) ∂ x1 11 ∂ t2 · ¸ 2 ∂ s ∂ Ψ1 (0) s + ρ h − c66 hΛΨ1 (0) − Γ1 (0) = 0, ∂ x2 ∂ t2

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

297

· ¸ ∂ ∂ 2Ψ2 (0) s s Γ (0) + g h (0) + g hD (0) cˆs11 hΛΨ2 (0) + ρ s h − 2 33 3 1 2 ∂ x2 ∂ t2 · ¸ ∂ ∂ 2Ψ1 (0) − ρ sh − cs66 hΛΨ1 (0) − Γ1 (0) = 0 ∂ x1 ∂ t2 which, along with Eqs.(9.42) and (9.43), yield

∂ 2Ψ1 (0) ∂Ψ1 (0) − cs66 hΛΨ1 (0) − c44 =0 ∂ t2 ∂ x3 ∂ 2Ψ2 (0) cˆs11 hΛΨ2 (0) + ρ s h + c44 u3 (0) + e15 φ (0) ∂ t2 ∂Ψ2 (0) −c44 + gs1 h 33 (0) + gs2 hD3 (0) = 0 ∂ x3 ρ sh

(9.53)

(9.54)

In view of Eq.(9.40), the last two equations in Eq.(9.38) can also be rewritten as

∂ 2 u3 (0) − hΛΓ2 = 0, ∂ t2 es s hΛφ (0) − 15 D3 (0) − κˆ 11 hΛΓ2 = 0 cs44 33 (0) − ρ

s

h

(9.55)

Noticing Eqs.(9.43) and (9.47), we can further transform Eqs.(9.54) and (9.55) to

∂ 2Ψ2 (0) + c44 u3 (0) + e15 φ (0) ∂ t2 ∂Ψ2 (0) ∂ u3 (0) ∂ φ (0) −c44 + (gs1 c33 + gs2 e33 )h + (gs1 e33 − gs2 κ33 )h =0 ∂ x3 ∂ x3 ∂ x3

(cˆs11 − gs1 c13 − gs2 e31 )hΛΨ2 (0) + ρ s h

∂ u3 (0) ∂ φ (0) ∂ 2 u3 (0) + e33 − c13 ΛΨ2 (0) − ρ s h ∂ x3 ∂ x3 ∂ t2 ∂Ψ2 (0) +c44 hΛu3 (0) + e15 hΛφ (0) − c44 hΛ = 0, ∂ x3 ∂ u3 (0) ∂ φ (0) s e33 hΛφ (0) − κ33 − e31 ΛΨ2 (0) − κˆ 11 ∂ x3 ∂ x3 es es15 es15 ∂Ψ2 (0) + 15 c hΛu (0) + e hΛ φ (0) − c44 hΛ =0 44 3 15 s s s c44 c44 c44 ∂ x3

(9.56)

c33

(9.57)

It is seen from Eqs.(9.53), (9.56) and (9.57) that Ψ1 is again decoupled from the other three unknowns in the equations governing the piezoelectric surface.

9.4.2 Two independent classes of motion Consider a thin piezoelectric plate of thickness H with identical material boundaries at

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the top and bottom. The origin of the coordinates now is placed at the center of the plate, see Fig.9.2.

Fig. 9.2 A piezoelectric plate with identical top and bottom material surfaces.

The equations governing the bulk material are given by Eqs.(9.49) - (9.52), while those governing the surface materials are given by, in the new coordinate system,

∂ 2Ψ1 (−H) ∂Ψ1 (−H) − cs66 hΛΨ1 (−H) − c44 =0 ∂ t2 ∂ x3 ∂ 2Ψ1 (H) ∂Ψ1 (H) −ρ s h + cs66 hΛΨ1 (H) − c44 =0 ∂ t2 ∂ x3

ρ sh

(9.58) (9.59)

∂ 2Ψ2 (−H) ∂ t2 ∂Ψ2 (−H) ∂ u3 (−H) +c44 u3 (−H) + e15 φ (−H) − c44 + (gs1 c33 + gs2 e33 )h ∂ x3 ∂ x3 ∂ φ (−H) s s +(g1 e33 − g2 κ33 )h = 0, ∂ x3 ∂ u3 (−H) ∂ φ (−H) ∂ 2 u3 (−H) c33 + e33 − c13 ΛΨ2 (−H) − ρ s h (9.60) ∂ x3 ∂ x3 ∂ t2 ∂Ψ2 (−H) +c44 hΛu3 (−H) + e15 hΛφ (−H) − c44 hΛ = 0, ∂ x3 ∂ u3 (−H) ∂ φ (−H) s hΛφ (−H) e33 − κ33 − e31 ΛΨ2 (−H) − κˆ 11 ∂ x3 ∂ x3 es es es ∂Ψ2 (−H) + 15 c44 hΛu3 (−H) + 15 e15 hΛφ (−H) − 15 c44 hΛ =0 s s c44 c44 cs44 ∂ x3 (cˆs11 − gs1 c13 − gs2 e31 )hΛΨ2 (−H) + ρ s h

−(cˆs11 − gs1 c13 − gs2 e31 )hΛΨ2 (H) − ρ s h +c44 u3 (H) + e15 φ (H) − c44

∂ 2Ψ2 (H) ∂ t2

∂Ψ2 (H) ∂ u3 (H) − (gs1 c33 + gs2 e33 )h ∂ x3 ∂ x3

∂ φ (H) = 0, ∂ x3 ∂ u3 (H) ∂ φ (H) ∂ 2 u3 (H) c33 + e33 − c13 ΛΨ2 (H) + ρ s h ∂ x3 ∂ x3 ∂ t2 −(gs1 e33 − gs2 κ33 )h

(9.61)

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

−c44 hΛu3 (H) − e15 hΛφ (H) + c44 hΛ

299

∂Ψ2 (H) = 0, ∂ x3

∂ u3 (H) ∂ φ (H) s − κ33 − e31 ΛΨ2 (H) + κˆ 11 hΛφ (H) ∂ x3 ∂ x3 es es es ∂Ψ2 (H) − 15 c44 hΛu3 (H) − 15 e15 hΛφ (H) + 15 c44 hΛ =0 s s c44 c44 cs44 ∂ x3

e33

where we have noticed that the derivation of the equations governing the material boundary at x3 = H requires the series expansion with respect to a positive quantity h, which is in contrast to the minus one (−h) in Eq.(9.19). As mentioned before, either in the governing equations for the bulk material or in the equations governing the material boundary, the unknown function Ψ1 is decoupled from the other three unknowns Ψ2 , u3 , and φ . Thus, we may call the motion represented by Eq.(9.49) and boundary conditions (9.58) and (9.59) the first class, with an unknown function Ψ1 . This actually corresponds to an isochoric deformation since we have ∆ = ε11 + ε22 + ε33 = 0. It is interesting to note that, the first class of motion only involves an in-plane mechanical deformation (u1 , u2 6= 0 and u3 = 0), without coupling with the electric field (φ = 0). The second class of motion is represented by Eqs.(9.50) - (9.52) and the boundary conditions (9.60) and (9.61), with three mutually coupled unknowns Ψ2 , u3 and φ . This kind of motion is characterized by the absence of the x3 -component of the rotation vector. Generally, it has both in-plane and out-of-plane deformations and there is a coupling between the electric and elastic fields.

9.4.3 Love type or SH waves The horizontally polarized shear waves (SH) are represented by u1 = u3 = 0,

u2 = u2 (x1 , x3 ,t)

(9.62)

This kind of motion belongs to the first class, and can be obtained by taking

Ψ1 = Ψ1 (x1 , x3 ,t)

(9.63)

which, according to Eq.(9.49), satisfies the following governing equation: c66

∂ 2Ψ1 ∂ 2Ψ1 ∂ 2Ψ1 + c44 −ρ =0 2 2 ∂ t2 ∂ x1 ∂ x3

(9.64)

and the boundary conditions at the material surfaces

ρ sh

∂ 2Ψ1 (−H) ∂ 2Ψ1 (−H) ∂Ψ1 (−H) s − c44 − c h =0 66 ∂ t2 ∂ x3 ∂ x12

−ρ s h

∂ 2Ψ1 (H) ∂ 2Ψ1 (H) ∂Ψ1 (H) s + c h − c44 =0 66 2 2 ∂t ∂ x3 ∂ x1

(9.65) (9.66)

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The wave solution of Eq.(9.64) can be sought as

Ψ1 = f (x3 ) exp[i(ξ x1 − ω t)]

(9.67)

where ξ is the wavenumber along the x1 direction, ω is the circular frequency, and f (x3 ) represents the mode distribution along the plate thickness. Substituting Eq.(9.67) into Eq.(9.64) yields c44 f 00 + (ρω 2 − c66 ξ 2 ) f = 0 (9.68) where the prime denotes differentiation with respect to x3 . The solution of Eq.(9.68) is f (x3 ) = E sin(β x3 ) + F cos(β x3 )

(9.69)

where E and F are undetermined constants, and c44 β 2 = ρω 2 − c66 ξ 2 > 0

(9.70)

The substitution from Eqs.(9.67) and (9.69) into the boundary conditions gives (ξ 2 cs66 h − ω 2 ρ s h)[−E sin(β H) + F cos(β H)] − c44 β [E cos(β H) + F sin(β H)] = 0, −(ξ 2 cs66 h − ω 2 ρ s h)[E sin(β H) + F cos(β H)] − c44 β [E cos(β H) − F sin(β H)] = 0 (9.71) Subtraction and addition manipulation leads to 2(ξ 2 cs66 h − ω 2 ρ s h)F cos(β H) − 2c44 β F sin(β H) = 0, 2(ξ 2 cs66 h − ω 2 ρ s h)E sin(β H) + 2c44 β E cos(β H) = 0

(9.72)

We can immediately find the following two possible solutions:

ω 2 ρ s h − ξ 2 cs66 h c44 β 2 s ξ c66 h − ω 2 ρ s h (ii) E = 0, F = 6 0, tan(β H) = c44 β (i) E 6= 0, F = 0, cot(β H) =

(9.73) (9.74)

The first solution represents the antisymmetric SH modes and the second corresponds to the symmetric SH modes. These two types of modes are separate. It can be seen that, when the material constants vanish in the surface layer or the thickness of the surface layer tends to zero, we can recover the classical result of SH waves in an elastic plate [35], i.e., (i) E 6= 0, F = 0, cos(β H) = 0 (ii) E = 0, F 6= 0, sin(β H) = 0

nπ 2 nπ or β H = 2

or β H =

(n = 1, 3, 5, · · · )

(9.75)

(n = 0, 2, 4, · · · )

(9.76)

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

301

If the surface layer has exactly the same material properties as the bulk material, we obtain from Eqs.(9.73) and (9.74), (i) E 6= 0, F = 0,

cot(β H) = β h

(9.77)

(ii) E = 0, F 6= 0,

tan(β H) = −β h

(9.78)

where Eq.(9.70) has been noticed. It may be seen that the approximation introduced in the effective boundary conditions are only applicable for lower modes when the surface layer is thin. In fact, Eqs.(9.77) and (9.78) can be obtained from Eqs.(9.73) and (9.74) by considering a plate with thickness H + h, and noticing that cos[β (H + h)] = cos(β H) cos(β h) − sin(β H) sin(β h) = cos(β H) − β h sin(β H), sin[β (H + h)] = sin(β H) cos(β h) + cos(β H) sin(β h) = sin(β H) + β h cos(β H) where the Taylor’s expansion approximations cos(β h) → 1 and sin(β h) → β h have been utilized, indicating that β h should be small. The characteristic equation of dispersion of SH waves in the piezoelectric plate with surface effect are given in Eqs.(9.73) and (9.74), namely

ω 2 ρ s h − ξ 2 cs66 h c44 β 2 s ξ c66 h − ω 2 ρ s h Symmetric SH modes: tan(β H) = c44 β Antisymmetric SH modes: cot(β H) =

(9.79) (9.80)

where c44 β 2 = ρω 2 − c66 ξ 2 > 0. If ρω 2 − c66 ξ 2 < 0, we take c44 β 2 = c66 ξ 2 − ρω 2 , and the solution of Eq.(9.68) becomes f (x3 ) = E sinh(β x3 ) + F cosh(β x3 ) (9.81) The corresponding characteristic equations are given by

ω 2 ρ s h − ξ 2 cs66 h c44 β 2 s ω ρ h − ξ 2 cs66 h Symmetric SH modes: tanh(β H) = c44 β Antisymmetric SH modes: coth(β H) =

(9.82) (9.83)

It is noted here that, the above dispersion characteristic equations for SH waves in a piezoelectric plate are exactly the same as those for SH waves in the corresponding elastic plate. We first check the validity of the surface model derived above by considering a homogeneous piezoelectric plate without surface effect, of which the material properties are

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1 given in Table 9.1 [36]. The total thickness of the plate is denoted by 2H ° , and artificial surface layers of thickness h are assumed to exist near the top and bottom boundaries. In such a case, the surface layer has exactly the same properties as the intermediate bulk layer, which has a thickness of 2(H − h). The corresponding characteristic equations are described by Eqs.(9.77) and (9.78), which for the current problem may be rewritten as

Antisymmetric SH modes: cot[β (H − h)] = β h

(9.84)

Symmetric SH modes: tan[β (H − h)] = −β h

(9.85)

It is easy to observe that β = 0 is a solution of Eq.(9.85), which corresponds to the lowest symmetric mode. Table 9.1 Bulk material properties of PZT-4. Elastic/(109 N/m2 )

c11 = 139,

c12 = 77.8,

Piezoelectric/(C/m2 )

e15 = 12.7,

e31 = −5.2,

Dielectric/(10−11 F/m)

κ11 = 646.4,

Density/(kg/m3 )

ρ = 7500

c13 = 74.3,

c33 = 115,

c44 = 25.6

e33 = 15.1

κ33 = 562.2

Figures 9.3 and 9.4 depict the first three branches of the frequency spectrum for the antisymmetric and symmetric wave modes respectively, where the dimensionless frequency p and wavenumber are defined as Ω = ω H ρ /c44 and χ = ξ H. As we can see, the introduction of an artificial surface layer, which is treated approximately using the effec-

Fig. 9.3 Frequency spectrum of antisymmetric SH wave modes in a PZT-4 plate without surface effect. solid line, h/H = 0; dotted line, h/H = 0.1; circle, h/H = 0.2. Only when the surface effect is absent (i.e., the surface layer has the same properties as the bulk material), we use 2H to indicate the total thickness of the plate. Otherwise, 2H is the thickness of the bulk layer as indicated in Fig.9.2. 1 °

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

303

tive boundary conditions, will lead to some numerical deviation from the actual solution (h = 0), except for the lowest symmetric mode. Such deviation becomes more prominent for higher frequencies. In particular, when the nondimensional wavenumber χ exceeds approximately 4.57, a new branch emerges when h/H = 0.2, which is solely due to the introduction of the artificial surface layer and its approximate treatment. A close check indicates that this new artificial branch, for which ρω 2 − c66 ξ 2 < 0, corresponds to the Love type wave that may exist in a layered half-space [37].

Fig. 9.4 Frequency spectrum of symmetric SH wave modes in a PZT-4 plate without surface effect. solid line, h/H = 0; dotted line, h/H = 0.1; circle, h/H = 0.2.

We now consider a PZT-4 plate with surface effect. To study the effect of material boundaries on the wave propagation behavior, we introduce the following dimensionless quantities: cs ρs h rρ = , rc = 66 , rh = (9.86) ρ c44 H which define the ratios of density, elastic modulus, and thickness between the surface and bulk materials, respectively. Figure 9.5 depicts the frequency spectra when the density ratio rρ = 5.0 and the modulus ratio rc = 10.0 while the thickness ratio rh = 0 and 0.01. In this case, the surface layer is much stiffer than the bulk material. We observe that the presence of surface layers can either lower or raise the frequency, depending on the magnitude of wavenumber. Specifically, when the wavenumber is small, the frequency is lowered, while it is raised after the wavenumber arrives at a certain critical value, where the frequency is unchanged whether the surface layers are taken into account or not. These critical wavenumbers can be easily determined from the condition ω 2 ρ s = ξ 2 cs66 , according to Eqs.(9.79) and (9.80). It is worth mentioning that the difference between the results of a plate with and without additional surface layers is more significant for stiff surface layers than for soft surface

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layers. The results of the latter case with rρ = rc = 0.1 are shown in Fig.9.6, from which the difference between h/H = 0 and h/H = 0.01 can be hardly distinguished.

Fig. 9.5 SH wave frequency spectrum of a PZT-4 plate with stiff surface layer (rρ = 5.0, rc = 10.0). solid line, h/H = 0; dotted line, h/H = 0.01.

Fig. 9.6 SH wave frequency spectrum of a PZT-4 plate with soft surface layer (rρ = rc = 0.1). solid line, h/H = 0; dotted line, h/H = 0.01.

9.4.4 Rayleigh-Lamb type waves The Rayleigh-Lamb type waves in a piezoelectric plate are confined to the x1 -x3 plane, bearing the form of

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

u1 = u1 (x1 , x3 ),

u2 = 0,

u3 = u3 (x1 , x3 ),

φ = φ (x1 , x3 )

305

(9.87)

These belong to the second class of motion, and can be obtained by taking

Ψ2 = Ψ2 (x1 , x3 ,t),

u3 = u3 (x1 , x3 ,t),

φ = φ (x1 , x3 ,t)

(9.88)

which satisfy, according to Eqs.(9.50) - (9.52), the following governing equations: c11 c44

∂ 2Ψ2 ∂ 2Ψ2 ∂ 2Ψ2 ∂ u3 ∂φ + c − (c + c ) ρ − (e + e ) − = 0, 44 13 44 31 15 ∂ x3 ∂ x3 ∂ t2 ∂ x12 ∂ x32

∂ 2 u3 ∂ 2 u3 ∂ 2 u3 ∂ 2φ ∂ 2φ ∂ 3Ψ2 + c33 2 + e15 2 + e33 2 − (c13 + c44 ) 2 − ρ 2 = 0, (9.89) 2 ∂t ∂ x1 ∂ x3 ∂ x1 ∂ x3 ∂ x1 ∂ x3 e15

∂ 2 u3 ∂ 2 u3 ∂ 2φ ∂ 2φ ∂ 3Ψ2 + e − κ − κ − (e + e ) =0 33 11 33 31 15 ∂ x12 ∂ x32 ∂ x12 ∂ x32 ∂ x12 ∂ x3

along with the following boundary conditions: 2 ∂ 2Ψ2 (−H) s ∂ Ψ2 (−H) h + ρ ∂ t2 ∂ x12 ∂Ψ2 (−H) ∂ u3 (−H) +c44 u3 (−H) + e15 φ (−H) − c44 + (gs1 c33 + gs2 e33 )h ∂ x3 ∂ x3 ∂ φ (−H) +(gs1 e33 − gs2 κ33 )h = 0, ∂ x3 2 ∂ u3 (−H) ∂ φ (−H) ∂ 2Ψ2 (−H) s ∂ u3 (−H) c33 − ρ h (9.90) + e33 − c13 ∂ x3 ∂ x3 ∂ t2 ∂ x12

(cˆs11 − gs1 c13 − gs2 e31 )h

+c44 h e33

∂ 2 u3 (−H) ∂ 2 φ (−H) ∂ 3Ψ2 (−H) + e h − c h = 0, 44 15 ∂ x12 ∂ x12 ∂ x12 ∂ x3

2 ∂ u3 (−H) ∂ φ (−H) ∂ 2Ψ2 (−H) s ∂ φ (−H) − κˆ 11 h − κ33 − e31 2 ∂ x3 ∂ x3 ∂ x1 ∂ x12

+

es15 ∂ 2 u3 (−H) es15 ∂ 2 φ (−H) es15 ∂ 3Ψ2 (−H) c h + e h − c h =0 44 44 15 cs44 cs44 cs44 ∂ x12 ∂ x12 ∂ x12 ∂ x3

2 ∂ 2Ψ2 (H) s ∂ Ψ2 (H) h − ρ ∂ t2 ∂ x12 ∂Ψ2 (H) ∂ u3 (H) +c44 u3 (H) + e15 φ (H) − c44 − (gs1 c33 + gs2 e33 )h ∂ x3 ∂ x3 ∂ φ (H) −(gs1 e33 − gs2 κ33 )h = 0, ∂ x3 ∂ u3 (H) ∂ φ (H) ∂ 2Ψ2 (H) ∂ 2 u3 (H) c33 + e33 − c13 + ρ sh 2 ∂ x3 ∂ x3 ∂ t2 ∂ x1

−(cˆs11 − gs1 c13 − gs2 e31 )h

(9.91)

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WeiQiu Chen

−c44 h e33

∂ 2 u3 (H) ∂ 2 φ (H) ∂ 3Ψ2 (H) − e15 h + c44 h = 0, 2 2 ∂ x1 ∂ x1 ∂ x12 ∂ x3

2 ∂ u3 (H) ∂ φ (H) ∂ 2Ψ2 (H) s ∂ φ (H) ˆ h − κ33 − e31 + κ 11 ∂ x3 ∂ x3 ∂ x12 ∂ x12

−

es15 ∂ 2 u3 (H) es15 ∂ 2 φ (H) es15 ∂Ψ2 (H) c − e h + s c44 h 2 =0 44 15 s s 2 2 c44 c44 c44 ∂ x1 ∂ x1 ∂ x1 ∂ x3

The plane-harmonic wave solution of Eq.(9.89) may be sought in the form of

Ψ2 = g1 (x3 ) exp[i(ξ x1 − ω t)], u3 = g2 (x3 ) exp[i(ξ x1 − ω t)],

(9.92)

φ = g3 (x3 ) exp[i(ξ x1 − ω t)] where gi are unknown mode distribution functions along the plate thickness. Substituting into Eq.(9.89) yields −c11 ξ 2 g1 + c44 g001 − (c13 + c44 )g02 − (e31 + e15 )g03 + ρω 2 g1 = 0, −c44 ξ 2 g2 + c33 g002 − e15 ξ 2 g3 + e33 g003 + (c13 + c44 )ξ 2 g01 + ρω 2 g2 = 0, −e15

ξ 2g

00 2 2 0 00 2 + e33 g2 + κ11 ξ g3 − κ33 g3 + (e31 + e15 )ξ g1

(9.93)

=0

One group of solutions to Eq.(9.93) may be assumed as g1 = As cosh(λ x3 ),

g2 = Bs sinh(λ x3 ),

g3 = Cs sinh(λ x3 )

(9.94)

where As , Bs and Cs are constants to be determined. This kind of solutions corresponds to the symmetric wave modes (with respect to the x1 -axis). Substitution of Eq.(9.94) into Eq.(9.93) leads to (−c11 ξ 2 + c44 λ 2 + ρω 2 )As − (c13 + c44 )λ Bs − (e31 + e15 )λ Cs = 0, (c13 + c44 )ξ 2 λ As + (−c44 ξ 2 + c33 λ 2 + ρω 2 )Bs + (−e15 ξ 2 + e33 λ 2 )Cs = 0, (9.95) (e31 + e15 )ξ 2 λ As + (−e15 ξ 2 + e33 λ 2 )Bs + (κ11 ξ 2 − κ33 λ 2 )Cs = 0 For nontrivial solutions, we must have ¯ ¯ ¯ −c11 ξ 2 + c44 λ 2 + ρω 2 −(c13 + c44 )λ −(e31 + e15 )λ ¯¯ ¯ ¯ ¯ (c13 + c44 )ξ 2 λ −c44 ξ 2 + c33 λ 2 + ρω 2 −e15 ξ 2 + e33 λ 2 ¯ = 0 ¯ ¯ ¯ ¯ (e31 + e15 )ξ 2 λ −e15 ξ 2 + e33 λ 2 κ11 ξ 2 − κ33 λ 2 ¯

(9.96)

which is the characteristic equation, a cubic algebraic equation of λ 2 , from which three couples of roots ±λi (i = 1, 2, 3) can be obtained. Another group of solutions to Eq.(9.93) takes the following form: g1 = Aa sinh(λ x3 ),

g2 = Ba cosh(λ x3 ),

g3 = Ca cosh(λ x3 )

(9.97)

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

307

where Aa , Ba and Ca are constants to be determined. The waves represented by Eq.(9.97) are antisymmetric with respect to the x1 -axis. It can be shown that the characteristic equation for the antisymmetric wave modes is exactly the same as Eq.(9.96). Thus, the complete solution to Eq.(9.93) is 3

3

i=1 3

i=1 3

g1 = ∑ Asi cosh(λi x3 ) + ∑ Aai sinh(λi x3 ), g2 = ∑ pi Asi sinh(λi x3 ) + ∑ pi Aai cosh(λi x3 ), i=1 3

i=1 3

i=1

i=1

(9.98)

g3 = ∑ qi Asi sinh(λi x3 ) + ∑ qi Aai cosh(λi x3 ) where Asi and Aai (i = 1, 2, 3) are undetermined constants, and pi and qi can be determined from any two independent equations of the following set: (c13 + c44 )λi pi + (e31 + e15 )λi qi = −c11 ξ 2 + c44 λi2 + ρω 2 , (−c44 ξ 2 + c33 λi2 + ρω 2 )pi + (−e15 ξ 2 + e33 λi2 )qi = −(c13 + c44 )ξ 2 λi , (9.99) (−e15 ξ 2 + e33 λi2 )pi + (κ11 ξ 2 − κ33 λi2 )qi = −(e31 + e15 )ξ 2 λi This set is derived by substituting λi , which is solved from the characteristic equation, back into Eq.(9.95) and assuming As or Aa equal 1 since pi and qi are the ratios between the constants. Now substituting Eqs.(9.92) and (9.98) into the boundary conditions (9.90) and (9.91), we get two sets of linear algebraic equations about Asi and Aai , respectively:    As1  Ks As2 = 0 (9.100)   As3    Aa1  Ka Aa2 = 0 (9.101)   Aa3 where Kisj and Kiaj , the elements of matrices Ks and Ka are given by K1is = {(cˆs11 − gs1 c13 − gs2 e31 )hξ 2 + ρ s hω 2 −[(gs1 c33 + gs2 e33 )hpi + (gs1 e33 − gs2 κ33 )hqi ]λi } cosh(λi H) +(c44 pi + e15 qi − c44 λi ) sinh(λi H), K2is

= [(c33 pi + e33 qi )λi + c13 ξ 2 ] cosh(λi H) 2

s

2

2

(9.102) 2

+(c44 hξ pi − ρ hω pi + e15 hξ qi − c44 hξ λi ) sinh(λi H), K3is

= [(e33 pi − κ33 qi )λi + e31 ξ 2 ] cosh(λi H) µ s ¶ e es15 es15 s 2 ˆ c hp − κ hq + e hq − c h λ + 15 i 44 44 i ξ sinh(λi H) 15 i 11 i cs44 cs44 cs44

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WeiQiu Chen

K1ia = {(cˆs11 − gs1 c13 − gs2 e31 )hξ 2 + ρ s hω 2 −[(gs1 c33 + gs2 e33 )hpi + (gs1 e33 − gs2 κ33 )hqi ]λi } sinh(λi H) +(c44 pi + e15 qi − c44 λi ) cosh(λi H), K2ia

= [(c33 pi + e33 qi )λi + c13 ξ 2 ] sinh(λi H)

(9.103)

+(c44 hξ 2 pi − ρ s hω 2 pi + e15 hξ 2 qi − c44 hξ 2 λi ) cosh(λi H), K3ia = [(e33 pi − κ33 qi )λi + e31 ξ 2 ] sinh(λi H) µ s ¶ e15 es15 es15 s ˆ + s c44 hpi − κ11 hqi + s e15 hqi − s c44 hλi ξ 2 cosh(λi H) c44 c44 c44 For nontrivial solutions, the determinant of coefficient matrix in Eq.(9.100) or (9.101) should vanish, giving rise to the following characteristic equations: |Ks | = 0

(9.104)

|Ka | = 0

(9.105)

The first is the dispersion equation for the symmetric Rayleigh-Lamb waves in the piezoelectric plate with surface effect, while the second is for the antisymmetric RayleighLamb waves. Figure 9.7 is the frequency spectrum for the Rayleigh-Lamb wave in a piezoelectric plate with or without the surface layer. All parameters are the same as those defined before except the modulus ratio rc , which now reads as rc = csi j /ci j = esi j /ei j = κisj /κi j

(9.106)

It is assumed to be the same for all material constants except the density.

Fig. 9.7 Rayleigh-Lamb wave frequency spectrum of a PZT-4 plate with stiff surface layer (rρ = 5.0, rc = 10.0). solid line, h/H = 0; dotted line, h/H = 0.01.

9 Wave Propagation in a Piezoelectric Plate with Surface Effect

309

The effect of stiff surface layer on the Rayleigh-Lamb wave propagation characteristics can be clearly seen from Fig.9.7, which also shows the intersection of the third antisymmetric branch and the second symmetric branch, as indicated by the rectangle. Like SH waves, the effect of soft surface layer on Rayleigh-Lamb waves can be neglected, and the results are not given here for the sake of brevity.

9.5 Summary This chapter presents an alternative approach that can be readily used to derive the governing equations of surface piezoelectricity. In contrast to the GM theory, we treat the surface of a piezoelectric body as a thin layer of small thickness. We start form the threedimensional theory of piezoelectricity, which is formulated in a mixed form in the state space, and use the series expression for the transfer matrix to derive the governing differential equations of various orders for a piezoelectric plane boundary. A transversely polarized piezoelectric plate with surface effect is considered. Two independent classes of motion are identified. While the first class is related to the elastic properties of the plate only, the second exhibits coupling between the elastic and electric fields. For either class, both symmetric and antisymmetric waves exist in the plate. Numerical results show that the surface effect plays an important role in the wave propagation. In particular, when the surface material is much stiffer (and faster) than the bulk material, then new wave modes may appear at a large wavenumber. Finally, it is worth pointing out that, when the residual stress is absent, the GM theory includes both the effect of inertia and effect of elasticity of the surface. For example, for isotropic elastic surface in plane-strain state, the GM theory takes the following form [4,6]: ˜ 13 (x1 , 0) = ρ

s u¨ (x , 0) − (λ ˜ s + 2µ˜ s ) ∂ 1 1

˜ 33 (x1 , 0) = ρ

2u

1 (x1 , 0) ∂ x12

s u¨ (x , 0) 3 1

(9.107) (9.108)

where λ˜ s and µ˜ s are surface Lam´e moduli, and ρ˜ s is the surface mass density [3]. As pointed out by Gurtin and Murdoch [4,5,7] and also mentioned earlier in this chapter, these relations are found identical with the MT model of the effective boundary conditions for a thin layer deposited on a body, provided that

ρ˜ s = ρ s h,

µ˜ s = µ s h,

λ˜ s = 2λ s µ s h/(λ s + 2µ s )

(9.109)

Thus ρ˜ s and µ˜ s are merely the scaled versions of their bulk counterparts. With Eq.(9.109), Eqs.(9.107) and (9.108) can be rewritten in Mindlin’s form: 13 (x1 , 0) = ρ

s hu¨

1 (x1 , 0) −

4µ s h(λ s + µ s ) ∂ 2 u1 (x1 , 0) λ s + 2µ s ∂ x12

(9.110)

310

WeiQiu Chen 33 (x1 , 0) = ρ

s hu¨

3 (x1 , 0)

(9.111)

By comparison, we can find that the first term in the parentheses in Eq.(9.13) or (9.14) is omitted above. From the expression, we can see that such terms may become significant for a nano-sized structure for which the thickness of the surface may be not small enough compared with the bulk size. They also become comparably large if the stresses in the bulk vary acutely in the plane parallel with the surface. In the present theory, we may identify these as the thickness effect, in addition to the inertia effect and elasticity effect in the GM theory or MT model as mentioned above. For the problem of wave propagation in a layered elastic half-space, B¨ovik [38] has actually shown that the thickness effect becomes very important at a large wavenumber, especially when the deposited layer is faster than the substrate.

Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos. 11090333 and 11272281). The author also acknowledges Mr. Bin Wu for pointing out a sign error in a formula in an earlier version of the manuscript.

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